Contents

Introduction 1

1 The stable module category. Phantom maps. 6 1.1 Self-injective algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 The stable module category . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Phantom maps in the stable module category . . . . . . . . . . . . . 12 1.4 Some basic properties of phantom maps . . . . . . . . . . . . . . . . 15

2 Phantom maps out of countable-dimensional modules 19 2.1 Milnor exact sequence. lim^{1} expression . . . . . . . . . . . . . . . . . 19 2.2 Examples of phantom maps . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Universal phantom maps. An application. . . . . . . . . . . . . . . . 27

3 An overview of purity 29 3.1 Basic definitions and properties . . . . . . . . . . . . . . . . . . . . . 29 3.2 Long pure-exact sequences . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 Pure extensions. The functor Pext. . . . . . . . . . . . . . . . . . . . 42 3.4 Pext^{∗}, lim^{∗}, and cardinality . . . . . . . . . . . . . . . . . . . . . . . 46 3.5 Flat modules and purity . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.6 Endofinite modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.7 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.8 Projectivity and purity . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4 Phantom maps, purity and

the structure of the stable module category 59 4.1 Phantom maps as degree one pure extensions . . . . . . . . . . . . . 59 4.2 Universal phantom maps . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3 Pure-Heller operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4 Composites of phantom maps . . . . . . . . . . . . . . . . . . . . . . 66 4.5 Pure-global dimension of group algebras . . . . . . . . . . . . . . . . 68 4.6 Vanishing multifold phantom maps. . . . . . . . . . . . . . . . . . . . 69 4.7 Bounding the phantom index . . . . . . . . . . . . . . . . . . . . . . 73

5 The Extended Milnor Sequence 84 5.1 The phantom spectral sequence . . . . . . . . . . . . . . . . . . . . . 84 5.2 The transgression d_{2}^{01} . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.3 Lifting direct systems from the stable module category . . . . . . . . 91 5.4 An even longer Milnor sequence . . . . . . . . . . . . . . . . . . . . . 93

6 Purity, phantom maps, and Rickard’s idempotent modules 96 6.1 Rickard’s idempotent modules . . . . . . . . . . . . . . . . . . . . . . 96 6.2 Pure-injective Rickard’s module E . . . . . . . . . . . . . . . . . . . . 101 6.3 The pure-injective Klein modules . . . . . . . . . . . . . . . . . . . . 106 6.4 Pure-projective dimension of Rickard’s module F . . . . . . . . . . . 111 6.5 Modules of Omega type: flat cohomology . . . . . . . . . . . . . . . . 115 6.6 Modules of Omega type: all maps phantom . . . . . . . . . . . . . . . 122

A Contracting homotopy for the standard pure resolution 127

B Artinian x-torsion k[x]-modules 133

C Lifting tubes from the stable category 137

Bibliography 144

Introduction

The study of infinite-dimensional modules is becoming a major area of research in the representation theory of finite groups and finite-dimensional algebras. Much attention is devoted to the stable module category StMod(Λ), where Λ is a finite-dimensional self-injective algebra (for example the group algebra kG of a finite group G over a field k). This category encodes much of the homological information about the algebra Λ. We present the category StMod(Λ) in Chapter 1, along with its attributes such as the Heller operator Ω and the exact triangles.

It is often desirable in mathematics to study an infinite-dimensional concept by means of its finite-dimensional building pieces. We take on this issue for the morphisms in the category StMod(Λ). Specifically, we study phantom maps.

Definition . A map M→N in StMod(Λ) is phantom if its restriction to every finitedimensional submodule of M is zero, or equivalently if for every map A→M in StMod(Λ), with A a finite-dimensional module, the composite A→M→N is zero.

Phantom maps capture the indeterminacy in the knowledge of a map M→N in StMod(Λ) from the knowledge of a consistent system of (restriction) maps from the finite-dimensional submodules of M to N. The space of morphisms M → N in the stable module category StMod(Λ) is denoted \underline{Hom}_{Λ}(M, N), and the subspace of phantom maps is denoted \underline{Ph}_{Λ}(M, N).

In Chapter 2, we investigate phantom that emanate from countable-dimensional modules. This restriction allows for certain interesting results, such as the Milnor short exact sequence, and a situation where all maps are phantom but not necessarily zero. Specifically, the following appears as Theorem 2.2.1 and Corollary 2.2.2, and is from Gnacadja [31, Theorem 9 and Corollary 10].

Theorem . Let G be a finite group and k be a field. Let α, β ∈ H^{m}(G, k) with m ≥ 1, and t ∈ {0, …, m − 1}. If (α, β) is a regular sequence, then all maps F(α) → Ω^{t}F(β) are phantom. If in addition H^{m−t−1}(G, k) ≠ 0 then there are nonzero maps F(α) → Ω^{t}F(β). In particular, if α and β are homogeneous elements of H^{∗}(G, k) of positive degrees such that (α, β) is a regular sequence, then all maps F(α) → Ω^{−1} F(β) are phantom and there are nonzero ones.

In Theorem 2.2.3 (Gnacadja [31, Theorem 11]), we take this result down to a more concrete level with G the Klein four-group V_{4} = ℤ/2 × ℤ/2. The modules F which appear above are certain of Rickard’s idempotent modules. (Chapter 6 includes an introduction to these very important modules.)

It quickly became evident that phantom maps were related to purity theoretic properties. We provide a survey of the notion of purity in Chapter 3. We start with the basic notions of pure homological algebra. In particular, we introduce the pure extensions functor Pext^{∗}. We discuss the interpretation of the functor Pext^{∗} in terms of the derived functors of the functor \underleftarrow{lim}. Then we discuss more specialized topics: flat modules in pure-exact sequences, endofinite modules, and functorial preservation of purity theoretic properties. We end with an investigation of how ordinary projectivity interacts with purity (Section 3.8). We obtain in Theorem 3.8.5 that over a finitedimensional self-injective algebra, every short pure-exact sequence can be trimmed down to one where all modules are projective-free.

Most of the material in Chapter 4 deals with composites of phantom maps. We show how the pure-global dimension of the algebra Λ controls their vanishing. We introduce the notion of phantom dimension of a finite-dimensional self-injective algebra in a very natural way. Then we define the phantom index ph.index(G) of a finite group G as the supremum of the phantom dimensions of the group algebras kG for various fields k. Expanding a technique suggested by Jon Carlson, we develop a method for bounding the phantom index. Specifically, we have the following result as Corollary 4.7.6.

Theorem . Let G be a finite 2-group which is not elementary abelian. Let H_{1}, …, H_{r} be index two subgroups in G with corresponding degree one cohomology classes ξ_{1}, … ξ_{r} ∈ H^{1}(G, 𝔽_{2}) composing to zero in H^{r}(G, 𝔽_{2}). Then

ph.index(G) ≤ ∑_{i=1}^{i=r} ph.index(H_{i}).

We apply this result to the groups C_{2^{m},2^{n}} = ℤ_{2^{m}} × ℤ_{2^{n}}, D_{2^{n}}, Q_{2^{n}}, and SD_{2^{n}}. This certainly does not exhaust the list of groups for which computations can be carried out. A striking result is that the quaternion groups Q_{2^{n}} have phantom index either two or three (Theorem 4.7.11). This is also the case for the Klein four-group V_{4} = ℤ/2 × ℤ/2 (Theorem 4.6.12). That in these two examples the phantom index is not precisely determined is part of what probably remains the most crucial question in this thesis’s topic. The following question is posed a number of times throughout Chapter 4, in general or case-specific forms, from various perspertives.

Question . Do there exist nonzero twofold phantom maps ?

In Chapter 5, we see another interesting interaction between purity and phantom maps. Most of the material is included in our preprint [15] with Benson. We study the phantom spectral sequence

E_{2}^{pq} = \begin{matrix}lim \\ ←------- \end{matrix}_{α}^{p}Ext_{Λ}^{q}(M_{α}, N) ⟹ Ext_{Λ}^{n}(M, N) = H^{n},

where {M_{α}}_{α} is a direct system of finite-dimensional Λ-modules with direct limit \begin{matrix}lim \\ -------→ \end{matrix}M_{α} = M. The fundamental sequence for this spectral sequence is an extension of the Milnor short exact sequence of Chapter 2. The following result appears as Theorem 5.1.1.

Theorem . Let Λ be a finite-dimensional self-injective algebra over a field k. Let {M_{α}}_{α} be a direct system of finite-dimensional Λ-modules with direct limit M = \begin{matrix}lim \\ -------→ \end{matrix}_{α}M_{α}. Let N be a Λ-module. The following is an exact sequence:

0 ⟶ \underline{Ph}_{Λ}(M, N) ⟶ \underline{Hom}_{Λ}(M, N) ⟶ \begin{matrix}lim \\ ←------- \end{matrix}_{α}\underline{Hom}_{Λ}(M_{α}, N) ⟶^{d_{2}^{01}}

⟶^{d_{2}^{01}} Pext_{Λ}^{2}(M, ΩN) ⟶ \underline{Ph}_{Λ}^{2}(M, Ω^{−1}N) ⟶ 0 .

We further investigate the transgression d_{2}^{01}. The insight we get allows us to show that certain pathologies do occur. The following statement condenses Theorem 5.3.1 and Corollary 5.3.2. Theorem . Let Λ be a finite-dimensional self-injective algebra over a field k. Suppose a module M has pure-projective dimension at least two and has no nonzero phantom self-maps. Then there exists a direct system of maps {M_{α} → Y }_{α} in StMod(Λ) which does not lift to Mod(Λ). Also there exists a degree two non-split pure extension of M which is split as an ordinary degree two extension of M. In other words, the natural (forgetful) map Pext_{Λ}^{2}(M, −) → Ext_{Λ}^{2}(M, −) can be non-injective.

Finally, in Chapter 6, we investigate properties of Rickard’s idempotent modules in relation to purity and phantom maps. In the first section, we introduce these very important modules, along with the necessary background on varieties of modules. Two sections deal with pure-injectivity and one with the pure-projective dimension. In particular, we see in Section 6.3 that the indecomposable pure-injective Klein modules (modules over the Klein four-group V_{4} = ℤ/2 × ℤ/2) fall in four families. The modules in three of these families are or are duals of Rickard’s modules. The last two sections deal with the so-called Omega type modules. They are generalizations of certain of Rickard’s idempotent modules, and share some of their properties from which purity theoretic results were derived.

The appendices A, B, and C provide the details for certain topics used in the thesis and which are of interest in their own right. The material is not new. But the result we elaborate on in Appendix A for example is widely accepted although the necessary calculations seem nowhere to be found.

I would like to thank my advisor Dave Benson for providing me with guidance and assistance throughout my research and the preparation of this thesis. Many thanks also to Jon Carlson for a number a helpful discussions and suggestions.

Chapter 1

The stable module category.

Phantom maps.

1.1 Self-injective algebras

We assume in this chapter that Λ is a finite-dimensional algebra over a field k. We assume moreover that Λ is self-injective, i.e., that the Λ-module Λ is injective. The case of most importance to us is when Λ is the group algebra kG over k of a finite group G. The algebra kG is indeed a finite-dimensional self-injective algebra. This follows from the fact that it is a Frobenius algebra; see for example [7, Proposition 1.6.2].

The fact that the ring Λ is self-injective implies that a Λ-module is projective if and only if it is injective. This result is very well known but its origin seems to be much less. The proof is quite straightforward if the ring Λ a finite-dimensional algebra over a field k and one is interested only in finite-dimensional modules; see for example [7, Proposition 1.6.2]. Carlson [16, Theorem 11.2] provides a proof that applies to all modules when Λ is the group algebra kG of a finite group G. But the result in its greatest generality is due to Faith and Walker [29, Theorem 5.3] and Faith [28, Theorems A and 5].

The stable module category we are about to introduce is an appropriate categorical setting for homological algebra over a finite-dimensional self-injective algebra.

1.2 The stable module category

Proposition-Definition 1.2.1. We say that a module morphism f : M → N is projective if it satisfies the following equivalent conditions:

(i) The map f : M → N factors through the projective cover P_{N} → N of N.

(ii) The map f : M → N factors through every epimorphism P → N with P a projective module.

(iii) The map f : M → N factors through every epimorphism Y → N.

(iv) The map f : M → N factors through the injective envelope M → I_{M} of M.

(v) The map f : M → N factors through every monomorphism M → I with I an injective module.

(vi) The map f : M → N factors through every monomorphism M → X. If Λ = kG, the group algebra of a finite group G, then we have the trace, or transfer map

Tr_{G} : Hom_{k}(M, N) → Hom_{kG}(M, N), u ↦ Tr_{G}u = ∑_{g∈G} gu

where gu is defined by (gu)(x) = g((u(g^{−1}x))) for x ∈ M. The conditions above are equivalent to the next one. (vii) The map f : M → N is the trace of some map u ∈ Hom_{k}(M, N).

The projective maps M → N form a vector space denoted PHom_{Λ}(M, N). The collection PHom_{Λ} of all projective maps is a two-sided ideal in the category Mod(Λ).

Definition 1.2.2. The stable module category StMod(Λ) is the quotient of the category Mod(Λ) by the ideal PHom_{Λ} of projective maps. Specifically:

• The objects of the category StMod(Λ) are the objects of the category Mod(Λ), i.e., all Λ-modules.

• The space of morphisms M → N is

\underline{Hom}_{Λ}(M, N) = \frac{Hom_{Λ}(M, N)}{PHom_{Λ}(M, N)}.

• The composition of morphisms in StMod(Λ) is naturally induced from the composition of morphisms in Mod(Λ) via the quotient functor Mod(Λ) → StMod(Λ).

The stable module category stmod(Λ) of finite-dimensional Λ-modules is the quotient of the category mod(Λ) by the ideal of projective maps in mod(Λ). It is also the full subcategory of StMod(Λ) of finite-dimensional modules.

Under the quotient functor Mod(Λ) → StMod(Λ) (resp. mod(Λ) → stmod(Λ)), the modules that map to zero are the projective modules, and the morphisms that map to zero are the projective maps. Two modules M and N are stably isomorphic, i.e., are isomorphic in StMod(Λ), if and only if there are projective modules P and Q such that M ⊕ Q ≅ N ⊕ P.

Unlike the category Mod(Λ), the category StMod(Λ) is not abelian: it is additive but does not have kernels and cokernels. However, it does have a very useful substitute for short exact sequences, namely the exact triangles, also called distinguished triangles, or just triangles. To put it more formally, the category StMod(Λ) is triangulated with translation functor Ω^{−1}, where Ω is the Heller operator. We are merely going to introduce the functors Ω and Ω^{−1}. Treatments of triangulated categories and of the stable module category abound in the literature. See for instance Weibel [56, Section 10.2] and Carlson [16, Section 5 and Theorem 11.4].

Theorem 1.2.3 (Schanuel’s Lemma). Consider, over an arbitrary ring Λ, two short exact sequences

0 → K′ → P′ → M → 0 and 0 → K′′ → P′′ → M → 0

abutting at the same module M, with both modules P′ and P′′ projective. Then

K′ ⊕ P′′ ≅ K′′ ⊕ P′,

i.e., when this makes sense, the modules K′ and K′′ are stably isomorphic.

See [7, Lemma 1.5.3]. With this, we can now define the Heller operator.

Theorem-Definition 1.2.4. The Heller operator is the functor Ω : StMod(Λ) → StMod(Λ) such that Ω(M) is the stable isomorphism class consisting of the modules that are kernels of epimorphisms P → M with P a projective module.

Every Λ-module has a projective cover. It is often convenient to regard Ω(M) as the kernel of “the” projective cover of M.

Every Λ-module also has an injective hull, and the cokernel of the injective hull of M is denoted Ω^{−1}(M).

It is a consequence of the self-injectivity of the algebra Λ that the modules M, Ω^{−1}Ω(M) and ΩΩ^{−1}(M) are stably isomorphic.

We now discuss the notions of projective part and projective-free part.

Every Λ-module has maximal projective submodules/summands. This follows from Bass’s Theorem P [4]. We will denote by Proj(M), and call projective part of M any chosen maximal projective submodule/summand of a module M. A chosen complement of Proj(M) in M will be denoted Ω^{0}(M) and called a projective-free part or a projective-free core of M. We have

M = Proj(M) ⊕ Ω^{0}(M),

and the module Ω^{0}(M) is projective-free, i.e., it has no nonzero projective submodules/summands.

Given two decompositions of M,

M = X⊕Y and M = X′⊕Y′,

with the modules X and X′ projective and the modules Y and Y′ projective-free, every automorphism M→M of M induces isomorphisms X ⟶^{≅} X′ and Y ⟶^{≅} Y′. (Also, the induced map X → Y′ is zero, but the induced map Y → X′ need not be zero.) So projective part and projective-free core are essentially unique.

If X and Y are a projective submodule and a projective-free submodule of M, and M = X⊕Y, then we can take Proj(M) = X and Ω^{0}(M) = Y.

Every projective submodule module of M is contained (as a direct summand) in a projective part of M.

If X and Z are submodules of M with X projective and X∩Z = 0, then there is a submodule Y in M with

Y ⊇ Z and M = X⊕Y.

This is immediate from the commutative triangle below.

\xymatrix{X⊕Z \ar[r]^{⊆} \ar[d]_{_{pr_{X}}} & M \ar@{-->}[dl]^{_{Pr_{X}}} \\ X}

So if X is a projective part of M, and Z is a (necessarily projective-free) submodule of M with X∩Z = 0, then there is a projective-free core Y in M with

Y ⊇ Z and M = X⊕Y.

Two modules are stably isomorphic if and only if they have isomorphic projectivefree cores.

Our next topic is a discussion of what exact triangles in the category StMod(Λ) are. A sequence of Λ-homomorphisms

X ⟶^{f} Y ⟶^{g}Z ⟶^{h} Ω^{−1}X

is an exact triangle in StMod(Λ) if there exists a short exact sequence

0 ⟶ X′ ⟶^{f′} Y′ ⟶^{g′} Z′ ⟶ 0

such that the maps f′ : X′⟶Y′ and g′ : Y′⟶Z′ are stably equivalent to the maps f : X⟶Y and g : Y ⟶Z respectively (in particular, the modules X′, Y′, Z′ are stably isomorphic to the modules X, Y, Z respectively), and the map h : Z⟶Ω^{−1}X represents the degree one extension

0 ⟶ X′ ⟶^{f′} Y′ ⟶^{g′} Z′ ⟶ 0

in the isomorphisms

\underline{Hom}_{Λ}(Z, Ω^{−1}X) ≅ Ext_{Λ}^{1}(Z, X) ≅ Ext_{Λ}^{1}(Z′, X′).

A short exact sequence

0 ⟶ X ⟶^{f} Y ⟶^{g}Z ⟶ 0

gives rise to the exact triangle

X ⟶^{f} Y ⟶^{g}Z ⟶^{h} Ω^{−1}X

where the map h : Z⟶Ω^{−1}X arises in the following commutative diagram.

[figure: 0; X; f; Y; g; Z; h; 0; 0; X; (inj.); Ω^{−1}X; 0]

Since an exact triangle is completely determined by its first two maps (actually by any of its maps), we often omit the third map.

The properties of exact triangles are discussed in detail in Weibel [56, Section 10.2] and Carlson [16, Section 5 and Theorem 11.4].

We close this section with two useful lifting properties. They are immediate consequences of Proposition-Definition 1.2.1, Conditions (iii) and (vi) respectively. Proposition 1.2.5.

• Given a module epimorphism v : X → M and a module morphism f : L → M, if there is a commutative triangle in StMod(Λ) as on the LHS below

\xymatrix{ & L \ar@{-->}[dl]_{g} \ar[d]^{f} &  & L \ar@{.>}[dl]_{h} \ar[d]^{f} \\ X \ar[r]^{v} & M & X \ar[r]^{v} & M}

(i.e. there is a module morphism g : L → X such that the difference f − vg : L → M is a projective map), then there is a commutative triangle in Mod(Λ) as on the RHS above.

• Given a module monomorphism u : M → Y and a module morphism f : M → N, if there is a commutative triangle in StMod(Λ) as on the LHS below

\xymatrix{M \ar[r]^{u} \ar[d]_{f} & Y \ar@{-->}[dl]^{g} & M \ar[r]^{u} \ar[d]_{f} & Y \ar@{.>}[dl]^{h} \\ N &  & N}

(i.e. there is a module morphism g : Y → N such that the difference f − gu : L → M is a projective map), then there is a commutative triangle in Mod(Λ) as on the RHS above.

1.3 Phantom maps in the stable module category

Definition 1.3.1. A map M→N in Mod(Λ) is said to be phantom if its restriction to every finitedimensional submodule of M is projective, or equivalently if for every map A→M in Mod(Λ), with A a finitely generated module, the composite A→M→N is projective. A map M→N in StMod(Λ) is said to be phantom if its restriction to every finitedimensional submodule of M is zero, or equivalently if for every map A→M in StMod(Λ), with A a finitely generated module, the composite A→M→N is zero. A phantom map is said to be trivial if it is projective when regarded in Mod(Λ) and is zero when regarded in StMod(Λ).

Note that a map in Mod(Λ) is phantom if and only if its image in StMod(Λ) is phantom. Phantom maps M→N are the obstructions to the determination of a map M→N in StMod(Λ) from the knowledge of its restrictions to the finite-dimensional submodules of M. More generally, if {M_{α}}_{α} is a direct system of finite-dimensional modules with direct limit M, phantom maps M→N are the obstructions to the unique determination of a map M→N in StMod(Λ) from the knowledge of a direct system of maps M_{α} → N in StMod(Λ) which lifts to a direct system of maps in Mod(Λ). Closely related to this is the problem of “liftability” of direct systems from StMod(Λ) to Mod(Λ). We will deal with this problem in Section 5.3. Of course phantom maps M→N are of interest only if M is not finitely generated.

There is an analogous notion in homotopy theory (whence we borrowed the name). Let X = ⋃_{n=1}^{∞} X^{n} and Y be CW-complexes, with X^{n} the n-dimensional skeleton of X. A (continuous) map X→Y is said to be phantom if its restriction to each skeleton X^{n} is nullhomotopic. Alex Heller was one of the first to consider phantom maps and the terminology is attributed to him. It is believed that he felt the name “phantom” was appropriate for a notion defined in terms of skeleta. And indeed, what we have are really phantoms, because these are objects that we do not see in finite dimensions but that may well be present.

Phantom maps have been extensively studied from various perspectives by several people over the past few years. McGibbon [46] wrote a good survey on phantom maps from the classical homotopy theory perpective. Phantom maps in a stable homotopy category are studied in Christensen and Strickland [20] and Christensen’s Ph.D. thesis [19]. There are a number of contributions from Neeman, for instance [48]. Beligiannis [5, 6] also investigated phantom maps in his work on homological algebra inside a triangulated category. These recent works (with the exception of McGibbon’s expository article) were driven by the search for a high-level categorical context to encompass the various situations where phantom maps occur. There are, however, a lot of questions in specific situations. Our focus will be on phantom maps in the stable module category.

We denote by Ph_{Λ}(M, N) and \underline{Ph}_{Λ}(M, N) the spaces of phantom maps in Mod(Λ) and StMod(Λ) respectively. Of course,

Ph_{Λ}(M, N) ⊇ PHom_{Λ}(M, N),

and we have

\underline{Ph}_{Λ}(M, N) = \frac{Ph_{Λ}(M, N)}{PHom_{Λ}(M, N)}.

It is plain that

Ph_{Λ}(M, N) = \begin{matrix}lim \\ ←------- \end{matrix}_{A f.d. ⊆M}PHom_{Λ}(A, N)

and

\underline{Ph}_{Λ}(M, N) = Ker (\underline{Hom}_{Λ}(M, N) → ∏_{A f.d. ⊆M} \underline{Hom}_{Λ}(A, N)) .

In fact, we have

\underline{Ph}_{Λ}(M, N) = Ker (\underline{Hom}_{Λ}(M, N) → \begin{matrix}lim \\ ←------- \end{matrix}_{A f.d. ⊆M} \underline{Hom}_{Λ}(A, N)) .

Put in another way, we have the exact sequence

0 ⟶ \underline{Ph}_{Λ}(M, N) ⟶ \underline{Hom}_{Λ}(M, N) ⟶ \begin{matrix}lim \\ ←------- \end{matrix}_{A f.d. ⊆M} \underline{Hom}_{Λ}(A, N).

We will have more to say about exact sequences like this one; see Proposition 2.1.1 and Theorem 5.1.1.

1.4 Some basic properties of phantom maps

The following result was suggested to us by the referee of our article [31].

Proposition 1.4.1. Let f : M → N be a phantom map in Mod(Λ). If M is projective-free then Im f ⊆ Rad N. If N is projective-free then Ker f ⊇ Soc M.

Proof. First assume that the map f is projective. So f factors through the injective hull M ↪ I_{M} of M and through the projective cover P_{N} ↠ N of N. If M is projective-free then Im(M ↪ I_{M}) ⊆ Rad(I_{M}), and so Im f ⊆ Rad N. If N is projective-free then Ker(P_{N} ↠ N) ⊇ Soc(P_{N}), and so Ker f ⊇ Soc M. Now suppose the map f is phantom. If M is projective-free then for every finite-dimensional submodule A ⊆ M, Im(f_{|A}) ⊆ Rad N. So Im f ⊆ Rad N. If N is projectivefree then for every finite-dimensional submodule A ⊆ M, Ker(f_{|A}) ⊇ Soc A. So Ker f ⊇ Soc M.

We now discuss how phantom maps behave with respect to restriction, induction, and co-induction. First, a brief reminder about these notions.

Let R be a subring of a ring S.

If M is an S-module, we have the restricted module M↓_{R}^{S} which is M regarded as an R-module.

If M is an R-module, we have the induced module M ↑_{R}^{S}= S⊗_{R}M. It is an S-module via s·(σ ⊗ m) = (sσ) ⊗ m .

If M is an R-module, we have the co-induced module M⇈_{R}^{S}= Hom_{R}(S, M). It is an S-module via (s·f)(σ) = f(σs) .

These three notions are also defined for maps, and we will use the notations Res_{S,R}, Ind_{R,S}, and CoInd_{R,S}.

For an R-module M, we have the canonical R-monomorphism

M ↪ M↑^{S}↓_{R}= (S⊗_{R}M)↓_{R}, m ↦ 1⊗m.

For an S-module M, we have the canonical S-monomorphism

M ↪ M↓_{R}⇈^{S}= Hom_{R}(S, M↓_{R}), m ↦ (σ ↦ σm).

If as an R-module, S is projective, then if an S-module P is projective, the restricted R-module P ↓_{R}^{S} is projective. So if a map f in Mod(S) factors through a projective module, then the restricted map Res_{S,R}(f) in Mod(R) factors through a projective module.

If an R-module P is projective, then the induced S-module P↑_{R}^{S} is projective. So if a map f in Mod(R) factors through a projective module, then the induced map Ind_{R,S}(f) in Mod(S) factors through a projective module.

If as an R-module I is injective, then the co-induced S-module I⇈_{R}^{S} is injective. So if a map f in Mod(R) factors through an injective module, then the co-induced map CoInd_{R,S}(f) in Mod(S) factors through an injective module.

If as an R-module S is free, then induction and co-induction coincide.

The fundamental rules governing the mutual behaviour of these notions are the Nakayama relations on the left hand side below

Hom_{R}(M, N↓_{R}^{S}) ≅ Hom_{S}(M↑_{R}^{S}, N)  Ext_{R}^{∗}(M, N↓_{R}^{S}) ≅ Ext_{S}^{∗}(M↑_{R}^{S}, N)
Hom_{R}(M↓_{R}^{S}, N) ≅ Hom_{S}(M, N⇈_{R}^{S})  Ext_{R}^{∗}(M↓_{R}^{S}, N) ≅ Ext_{S}^{∗}(M, N⇈_{R}^{S})

and the Eckmann-Shapiro relations on the right hand side above. They hold provided that S is projective as an R-module. See [7, Proposition 2.8.3 and Corollary 2.8.4].

Now let Λ and Γ be finite-dimensional self-injective k-algebras, with Γ a subalgebra of Λ. By the discussion above, restriction, induction, and co-induction of modules and maps are well-defined at the level of stable module categories.

Note that if H is a subgroup of a finite group G, then the group algebra kG is free as a module over its subalgebra kH. Indeed we have

kG = ⨁_{g∈H\G} k[Hg],

where the expression g ∈ H\G means that g runs over a set of representatives of right cosets of H in G. So the theorem below does apply to the situations most relevant to us.

Theorem 1.4.2. Let Λ and Γ be finite-dimensional self-injective k-algebras, with Γ a subalgebra of Λ and Λ free as a module over Γ.

• If a map f in StMod(Λ) is phantom, then the restricted map Res_{Λ,Γ}(f) in StMod(Γ) is phantom.

• If a map f in StMod(Γ) is phantom, then the induced map Ind_{Γ,Λ}(f) in StMod(Λ) is phantom.

Proof. We prove the first statement. The second one is proven similarly. Let f : M → N be a phantom map in Mod(Λ). Let A be a finite-dimensional Γ-module and u : A → M ↓_{Γ} be a Γ-homomorphism. Let v : A↑^{Λ}→ M be the Λ-homomorphism corresponding to u via the Nakayama isomorphism

Hom_{Γ}(A, M↓_{Γ}) ≅ Hom_{Λ}(A↑^{Λ}, M).

The composite Λ-map

A↑^{Λ}⟶^{v}M ⟶^{f} N

is projective. So the composite Γ-map

A↑^{Λ}↓_{Γ}⟶^{Res_{Λ,Γ}(v)} M↓_{Γ}⟶^{Res_{Λ,Γ}(f)} N↓_{Γ}

is projective. Then the composite Γ-map

A ↪ A↑^{Λ}↓_{Γ}⟶^{Res_{Λ,Γ}(v)} M↓_{Γ}⟶^{Res_{Λ,Γ}(f)} N↓_{Γ}

is also projective. Now the composite

A ↪ A↑^{Λ}↓_{Γ}⟶^{Res_{Λ,Γ}(v)} M↓_{Γ}

is just the map u : A → M↓_{Γ}. So we have got that the composite

A ⟶^{u} M↓_{Γ}⟶^{Res_{Λ,Γ}(f)} N↓_{Γ}

is projective.

We can now formulate the Nakayama-Eckmann-Shapiro relations for phantom maps.

Theorem 1.4.3. Let Λ and Γ be finite-dimensional self-injective algebras over a field k, with Γ a subalgebra of Λ and Λ free as a Γ-module. The Nakayama-Eckmann-Shapiro isomorphisms restrict to isomorphisms

Ph_{Γ}(M, N↓_{Γ}^{Λ}) ≅ Ph_{Λ}(M↑_{Γ}^{Λ}, N)  \underline{Ph}_{Γ}(M, N↓_{Γ}^{Λ}) ≅ \underline{Ph}_{Λ}(M↑_{Γ}^{Λ}, N)
Ph_{Γ}(M↓_{Γ}^{Λ}, N) ≅ Ph_{Λ}(M, N⇈_{Γ}^{Λ})  \underline{Ph}_{Γ}(M↓_{Γ}^{Λ}, N) ≅ \underline{Ph}_{Λ}(M, N⇈_{Γ}^{Λ})

Phantom maps out of

countable-dimensional modules

2.1 Milnor exact sequence. lim^{1} expression

Proposition 2.1.1. Let (M_{i})_{i≥1} be a direct sequence of finite-dimensional Λ-modules with direct limit M = \begin{matrix}lim \\ -------→ \end{matrix}_{i≥1}M_{i}. Let N be a Λ-module. Then we have the Milnor short exact sequence

0 → \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}\underline{Hom}_{Λ}(M_{i}, ΩN) → \underline{Hom}_{Λ}(M, N) → \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}\underline{Hom}_{Λ}(M_{i}, N) → 0,

and we have an isomorphism

\underline{Ph}_{Λ}(M, N) ≅ \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}\underline{Hom}_{Λ}(M_{i}, ΩN).

Proof. Let φ_{i} : M_{i} → M_{i+1} and φ̄_{i} : M_{i} → M, i ≥ 1, be the maps the direct sequence (M_{i})_{i≥1} and its direct limit M are equipped with. Define:

δ : ⨁ M_{i} → ⨁ M_{i}, x = (x_{i})_{i≥1} ↦ δ(x) = (x_{1}, x_{2} − φ_{1}(x_{1}), x_{3} − φ_{2}(x_{2}), …) and

π : ⨁ M_{i} → M, y = (y_{i})_{i≥1} ↦ π(y) = ∑ φ̄_{i}(y_{i}).

Then by the definition of \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}, we have the short exact sequence

0 ⟶ ⨁ M_{i} ⟶^{δ} ⨁ M_{i} ⟶^{π} M ⟶ 0.

Applying the functor \underline{Hom}_{Λ}(−, N) to it, we get the exact sequence

∏_{i≥1} \underline{Hom}_{Λ}(M_{i}, ΩN) ⟶^{δ^{∗}} ∏_{i≥1} \underline{Hom}_{Λ}(M_{i}, ΩN) ⟶^{ϕ^{∗}} \underline{Hom}_{Λ}(M, N) ⟶^{π^{∗}}

⟶^{π^{∗}} ∏_{i≥1} \underline{Hom}_{Λ}(M_{i}, N) ⟶^{δ^{∗}} ∏_{i≥1} \underline{Hom}_{Λ}(M_{i}, N),

where the map ϕ : M → ⨁ Ω^{−1}M_{i} is such that we have the exact triangle

⨁ M_{i} ⟶^{δ} ⨁ M_{i}. ⟶^{π} M ⟶^{ϕ} ⨁ Ω^{−1}M_{i}

We now use the explicit description of the functor \begin{matrix}lim \\ ←------- \end{matrix}^{1} for inverse systems indexed by the positive integers. The description is presented in Jensen [40, Chapter 2]; he attributes it to Eilenberg. In our situation, we obtain:

Coker (∏_{i≥1} \underline{Hom}_{Λ}(M_{i}, ΩN) ⟶^{δ^{∗}} ∏_{i≥1} \underline{Hom}_{Λ}(M_{i}, ΩN)) = \begin{matrix}lim \\ ←------- \end{matrix}^{1}\underline{Hom}_{Λ}(M_{i}, ΩN)

and

Ker (∏_{i≥1} \underline{Hom}_{Λ}(M_{i}, N) ⟶^{δ^{∗}} ∏_{i≥1} \underline{Hom}_{Λ}(M_{i}, ΩN)) = \begin{matrix}lim \\ ←------- \end{matrix}\underline{Hom}_{Λ}(M_{i}, N).

Whence the Milnor exact sequence.

This method for finding the Milnor exact sequence is well known in Homotopy Theory. It is used for instance in [38, Proposition 2.2.4].

The Mittag-Leffler condition will be instrumental in this chapter, especially when in the next section we use the \begin{matrix}lim \\ ←------- \end{matrix}^{1} expression of Proposition 2.1.1 to show that there indeed are nonzero phantom maps. We collect the essential facts about this condition in the following proposition.

Proposition 2.1.2. For an inverse sequence (X_{m})_{m≥1} of vector spaces, consider the following statements:

(i) The inverse sequence (X_{m})_{m≥1} satisfies the Mittag-Leffler condition, i.e. for every n, the decreasing sequence ( Im(X_{m} → X_{n}) )_{m≥n} is eventually constant.

(ii) \begin{matrix}lim \\ ←------- \end{matrix}_{m}^{1}X_{m} = 0.

(iii) \begin{matrix}lim \\ ←------- \end{matrix}_{m}^{1}X_{m} is countably generated.

We have (i) ⇒ (ii). If the vector spaces X_{m} are countably generated, conditions (i), (ii) and (iii) are equivalent.

The Mittag-Leffler condition was introduced by Grothendieck in [34, Chapitre 0, Section 13]. Weaker properties (flasqueness and weak flasqueness) are treated in Jensen [40, Chapter 1]. The implication (i) ⇒ (ii) follows from Theorem 1.8 from this latter reference. The equivalence of conditions (i), (ii) and (iii) when all spaces X_{m} are countably generated is a result of Emmanouil [25, Proposition A.3]. He obtains it by adapting a result of Gray [33] for ℤ-modules, arguing with k-dimension instead of with cardinality.

We close this Section with another \begin{matrix}lim \\ ←------- \end{matrix}^{1} expression for the space of phantom maps.

Proposition 2.1.3. If the direct sequence (M_{i})_{i≥1} is an ascending tower of finitedimensional modules, then we have an isomorphism

\underline{Ph}_{Λ}(M, N) ≅ \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}Hom_{Λ}(M_{i}, ΩN).

We proved this in [31, Theorem 2]. It is this isomorphism that we will generalize when the restriction to phantom maps out of countably generated modules is discarded (see Corollary 4.1.4). The assumption that the maps M_{i} → M_{i+1} be injective will turn out to be unncessary.

The surjective map Ph_{Λ}(M, N) ↠ \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}Hom_{Λ}(M_{i}, ΩN) which gives rise to the isomorphism \underline{Ph}_{Λ}(M, N) →^{≅} \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}Hom_{Λ}(M_{i}, ΩN) is a connecting homomorphism occuring in an application of the Snake Lemma. It can be explicitly described as follows. Let P_{N} → N be the projective cover of N. Let f ∈ Ph_{Λ}(M, N). Let, for each i ≥ 1, g_{i} : M_{i} → P_{N} such that f_{|M_{i}} is the composite M_{i} →^{g_{i}} P_{N} → N. Set h_{i} := g_{i} − g_{i+1}_{|M_{i}}. Then

(P_{N} → N) ∘ h_{i} = f_{i} − f_{i+1}_{|M_{i}} = 0,

so h_{i} ∈ Hom_{Λ}(M_{i}, ΩN). The map Ph_{Λ}(M, N) ↠ \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}Hom_{Λ}(M_{i}, ΩN) sends f to the class of (h_{i})_{i≥1}.

2.2 Examples of phantom maps

This is taken from Section 3 of our article [31].

To every homogeneous cohomology element α ∈ H^{m}(G, k), with m ≥ 1, is associated a module F(α) as follows. We have natural isomorphisms

H^{m}(G, k) ≅ \underline{Hom}_{kG}(Ω^{m}k, k) ≅ \underline{Hom}_{kG}(k, Ω^{−m}k) ≅ \underline{Hom}_{kG}(Ω^{−m}k, Ω^{−2m}k) ≅ · · · .

Then we have the direct sequence

k ⟶^{α} Ω^{−m}k ⟶^{α} Ω^{−2m}k ⟶^{α} · · · .

The module F(α) is defined to be its homotopy colimit in StMod(kG). This means that F(α) fits in the exact triangle

⨁_{i≥0} Ω^{−im}k ⟶^{1−α} ⨁_{i≥0} Ω^{−im}k ⟶ F(α),

where the map ⨁_{i≥0} Ω^{−im}k ⟶^{1−α} ⨁_{i≥0} Ω^{−im}k is built from the maps (Id, −α) : Ω^{−im}k ⟶ Ω^{−im}k ⊕ Ω^{−(i+1)m}k. The direct sequence lifts to the category Mod(kG), and each lift gives rise to a colimit (direct limit) in Mod(kG) which is stably isomorphic to F(α). The construction and many relevant questions are studied by Rickard in [53]. We discuss Rickard’s modules in a greater generality in Chapter 6. We just mention here that F(α) is periodic of period dividing m, and that F(α) ≅ F(α^{n}) for every n ≥ 1.

Theorem 2.2.1. Let α, β ∈ H^{m}(G, k) with m ≥ 1, and t ∈ {0, …, m − 1}. If (α, β) is a regular sequence, then all maps F(α) → Ω^{t}F(β) are phantom. If in addition H^{m−t−1}(G, k) ≠ 0 then there are nonprojective maps F(α) → Ω^{t}F(β).

As H^{0}(G, k) = k ≠ 0, we have:

Corollary 2.2.2. Let α, β be homogeneous elements of H^{∗}(G, k) of positive degrees. If (α, β) is a regular sequence, then all maps F(α) → Ω^{−1} F(β) are phantom and there are nonprojective ones.

Proof (Theorem 2.2.1). We argue in the context of the category StMod(kG). Let f_{r} : Ω^{−rm}k → Ω^{t}F(β), r ≥ 0, be a sequence of maps consistent with the direct system

k ⟶^{α} Ω^{−m}k ⟶^{α} Ω^{−2m}k ⟶^{α} ⋯ ⟶^{α} Ω^{−rm}k ⟶^{α} Ω^{−(r+1)m}k ⟶^{α} · · · .

Fix r, i ≥ 0. For s large enough, there is a commutative diagram as follows, where u represents a cohomology class in H^{∗}(G, k).

\xymatrix{ & Ω^{t−sm}k \ar[d] \\ Ω^{−rm}k \ar@{-->}[ur]^{u} \ar[r]_{f_{r}} & Ω^{t}F(β)}

Then we have a commutative diagram as on the LHS below, and we may choose j large enough so as to have a commutative diagram as on the RHS below, with v representing a cohomology class in H^{∗}(G, k).

\xymatrix{Ω^{−rm}k \ar[r]^{u} \ar[d]_{α^{i}} \ar[rr]^{f_{r}} & Ω^{t−sm}k \ar[r] & Ω^{t}F(β) & Ω^{−rm}k \ar[r]^{u} \ar[d]_{α^{i}} & Ω^{t−sm}k \ar[d]^{β^{j}} \\ Ω^{−(r+i)m}k \ar[urr]_{f_{r+i}} &  &  & Ω^{−(r+i)m}k \ar@{-->}[r]^{v} & Ω^{t−(s+j)m}k}

The latter diagram translates into the equality β^{j}u = vα^{i} in the cohomology ring H^{∗}(G, k). But (α^{i}, β^{j}) is a regular sequence. So u is divisible by α^{i} in H^{∗}(G, k). This holds for every i ≥ 1, so u = 0. It follows that f_{r} = 0. Now let c := t + 1. We assume H^{m−c}(G, k) ≠ 0 and show that the inverse system

\underline{Hom}_{kG}(k, Ω^{c}F(β)) ⟵^{α^{∗}} \underline{Hom}_{kG}(Ω^{−m}k, Ω^{c}F(β)) ⟵^{α^{∗}} \underline{Hom}_{kG}(Ω^{−2m}k, Ω^{c}F(β)) ⟵^{α^{∗}} ⋯

fails the Mittag-Leffler condition. It will follow that its \begin{matrix}lim \\ ←------- \end{matrix}^{1} is non vanishing, hence that \underline{Ph}_{kG}(F(α), Ω^{t}F(β)) ≠ 0. (See Section 2.1.)

Let γ ∈ H^{m−c}(G, k) and i ≥ t. Let f_{i} : k ⟶^{α^{i}} Ω^{−im}k →^{γ} Ω^{c−(i+1)m}k → Ω^{c}F(β).

f_{i} is in the image of \underline{Hom}_{kG}(k, Ω^{c}F(β)) ⟵^{(α^{∗})^{i}} \underline{Hom}_{kG}(Ω^{−im}k, Ω^{c}F(β)). Assume, for contradiction, that we have a commutative diagram as on the LHS below.

\xymatrix{k \ar[r]^{f_{i}} \ar[d]_{α^{i+1}} & Ω^{c}F(β) & k \ar[r]^{γα^{i}} \ar[d]_{α^{i+1}} & Ω^{c−(i+1)m}k \ar[d]^{β^{j}} \\ Ω^{−(i+1)m}k \ar@{-->}[ur]_{h} &  & Ω^{−(i+1)m}k \ar@{-->}[r]^{v} & Ω^{c−(i+j+1)m}k}

Then there is a commutative diagram as on the RHS above. This translates into the equality β^{j}γα^{i} = vα^{i+1} in the ring H^{∗}(G, k). Since (α, β) is a regular sequence, we get β^{j}γ = vα, and α divides γ. But degγ = m − c < m = degα. This forces γ = 0. But γ could have been chosen nonzero. So there is no map h as assumed, whence f_{i} is not in the image of \underline{Hom}_{kG}(k, Ω^{c}F(β)) ⟵^{(α^{∗})^{i+1}} \underline{Hom}_{kG}(Ω^{−(i+1)m}k, Ω^{c}F(β)).

In the following particular case we actually describe the projective maps [and hence the nonprojective (phantom) maps].

Set G = ⟨g, h : g^{2} = 1 = h^{2}, gh = hg⟩, the Klein four group. Assume k has characteristic 2. Let u := g − 1, v := h − 1 ∈ kG. u and v are in Ωk, and we have two elements α and β of H^{1}(G, k) = Hom_{kG}(Ωk, k) defined by

α = v^{∗} : \begin{cases}u ↦ 0 \\ v ↦ 1 \end{cases} , β = u^{∗} : \begin{cases}u ↦ 1 \\ v ↦ 0 \end{cases} .

H^{∗}(G, k) is the polynomial ring k[α, β] with α and β in degree 1. We will use diagrammatic representations of modules. See Carlson [16, Section 7] for more on the use of diagrams to produce examples of modules over group algebras. Refer to Benson and Carlson [9] for a thorough study of the topic. Below are diagrams for kG, Ωk, and k.

[figure: 1; u; v; uv = vu; kG; Ωk; k; u; v; uv = vu; 1; action of u; action of v]

One can establish that Ω^{−n}k has the following diagrammatic representation.

[figure: a_{0}; a_{1}; a_{2}; a_{3}; a_{4}; a_{5}; a_{2n−2}; a_{2n−1}; a_{2n}]

Moreover the map Ω^{−(n+1)}α : Ω^{−n}k ⟶^{α} Ω^{−(n+1)}k can be represented by the inclusion Ω^{−n}k ⊂ Ω^{−(n+1)}k consistent with the above labelled diagram. We then get for F(α) the following diagram.

[figure: a_{0}; a_{1}; a_{2}; a_{3}; a_{4}; a_{5}; a_{2s−2}; a_{2s−1}; a_{2s}; a_{2s+1}; a_{2s+2}]

Likewise F(β) is represented by the following diagram

[figure: b_{0}; b_{1}; b_{2}; b_{3}; b_{4}; b_{5}; b_{2t−2}; b_{2t−1}; b_{2t}; b_{2t+1}; b_{2t+2}]

Theorem 2.2.3. Let f ∈ Hom_{kG}(F(α), F(β)). Then f is phantom, and Imf ⊆ ⨁_{t=0}^{∞} kb_{2t}. Let f(a_{1+2s}) = ∑_{t=0}^{∞} λ_{st}b_{2t} for each s ≥ 0, and σ_{m} := ∑_{s+t=m} λ_{st} for each m ≥ 0. f is projective if and only if σ_{m} = 0 for m large enough.

Proof. The module F(β) has period 1, so f is phantom by Corollary 2.2.2. Then Imf ⊆ Rad F(β) = ⨁_{t=0}^{∞} kb_{2t} by Proposition 1.4.1. The injective hull of F(α) is j : F(α) ↪ J where J is the free kG-module with kG-basis {c_{s}}_{s≥0} and j is given by j(a_{2s+1}) = vc_{s} + uc_{s+1} [and j(a_{2s}) = uvc_{s} = vuc_{s}], s ≥ 0.

[figure: a_{0}; a_{1}; a_{2}; a_{3}; a_{4}; a_{5}; a_{6}; c_{0}; c_{1}; c_{2}; c_{3}; F(α); J; j]

Factoring f through j amounts to solving the system of equations

vx_{s} + ux_{s+1} = ∑_{t=0}^{∞} λ_{st}b_{2t} , s ≥ 0 ,

for (x_{s})_{s≥0} in F(β). As both u and v annihilate ⨁_{t=0}^{∞} kb_{2t}, the system is solvable in F(β) if and only if it is in ⨁_{t=0}^{∞} kb_{1+2t}. Assume this latter condition holds. So we have a solution (x_{s})_{s≥0} with

x_{s} = ∑_{t=0}^{∞} μ_{st}b_{2t+1}.

Then λ_{s,0} = μ_{s,0} and λ_{st} = μ_{st} + μ_{s+1,t−1} for t ≥ 1, whence μ_{0,t} = ∑_{l=0}^{t} λ_{l,t−l} = σ_{t}. So σ_{t} = 0 for t large enough. For the converse set μ_{st} := ∑_{l=0}^{t} λ_{s+l,t−l} for s, t ≥ 0. Then μ_{0,t} = σ_{t} and for s ≥ 1, μ_{st} = μ_{s−1,t+1} + λ_{s−1,t+1}. If we assume that σ_{t} = 0 for t large enough then it follows by induction that for every s ≥ 0, μ_{st} = 0 for t large enough. Set x_{s} := ∑_{t=0}^{∞} μ_{st}b_{2t+1}. (x_{s})_{s≥0} is a solution as desired. Note that the fact that f is phantom can also be proved similarly, that is by showing that for every n ≥ 0 the system of equations

vx_{s} + ux_{s+1} = ∑_{t=0}^{∞} λ_{st}b_{2t} , 0 ≤ s ≤ n

27

is solvable for (x_{0}, x_{1}, …, x_{n}, x_{n+1}) in F(β).

We now say a word on the existence of regular sequences in H^{∗}(G, k). Let H(G, k) denote H^{∗}(G, k) if p = 2, and the subring ⨁_{n=0}^{∞} H^{2n}(G, k) of H^{∗}(G, k) if p > 2. H(G, k) is a commutative graded subring with identity of H^{∗}(G, k). The following proposition is a particular case of Evens [26, Theorem 10.3.1], which in turn is an adaptationgeneralization to group cohomology of a theorem of J. Duflot [24] in equivariant cohomology.

Proposition 2.2.4. Assume k has characteristic p. The depth of H^{∗}(G, k) regarded as a module over H(G, k) is at least equal to the p-rank of the center of a Sylow p-subgroup of G.

Hence if a Sylow p-subgroup of G has a central subgroup isomorphic to ℤ/p × ℤ/p, then H^{∗}(G, k) has regular sequences (α, β) of homogeneous positive degree elements.

We will present a more general situation where all maps between two modules are phantom (Section 6.6) after we deal with Rickard and so-called Omega type modules more extensively.

2.3 Universal phantom maps. An application.

The universal phantom map already appeared in the proof of Proposition 2.1.1. It is the map ϕ : M → ⨁ Ω^{−1}M_{i} such that the following is an exact triangle.

⨁ M_{i} ⟶^{δ} ⨁ M_{i} ⟶^{π} M ⟶^{ϕ} ⨁ Ω^{−1}M_{i}

We recall that this means that we have a commutative diagram

[figure: 0; ⨁ M_{i}; δ; ⨁ M_{i}; π; M; ϕ; 0; 0; _{⨁ M_{i}}; _{⨁ I_{M_{i}}}; ⨁ Ω^{−1}M_{i}; 0]

where the map ⨁ M_{i} → ⨁ I_{M_{i}} is built up in the natural way from with the injective hulls M_{i} → I_{M_{i}}.

Proposition 2.3.1. The map ϕ : M → ⨁ Ω^{−1}M_{i} is a weakly universal phantom map out of M, in the sense that every phantom map out of M factors (non uniquely) in StMod(Λ) through ϕ.

Proof. If f : M → N is a phantom map, then we have a commutative diagram in StMod(Λ) as below by the axioms of a triangulated category.

\xymatrix{⨁_{i=1}^{∞} M_{i} \ar[r]^{π} \ar@{-->}[dr]_{zero} & M \ar[r]^{ϕ} \ar[d]^{f} & ⨁_{i=1}^{∞} Ω^{−1}M_{i} \ar@{-->}[dl]^{f̃} \\  & N}

Corollary 2.3.2. Let M, M′, M′′ be Λ-modules with M countably generated. If f : M → M′ and f′ : M′ → M′′ are phantom maps then the composite f′f is zero.

Proof. We make use of the notations of Proposition 2.3.1 and its proof, and argue in StMod(Λ). We have f = f̃ϕ so f′f factors through f′f̃. This latter map is phantom out of a direct sum of finitely generated modules, and hence is zero.

The construction of the universal map ϕ and its use to prove Corollary 2.3.2 have their analogues in the theory of spectra. They appear in Ohkawa [49, Proposition 11], Neeman [48, Remark 4.3 and Corollary 4.4], and Christensen and Strickland [20, Lemma 4.4 and Corollary 4.7].

The composition of phantom maps is one of the features that require a more careful investigation when the restriction to maps out of countably generated modules is abandoned. We will have more to say about this in Chapter 4 (Sections 4.4, 4.6, and 4.7).

An overview of purity

The most comprehensive references we know of on purity are the books of Jensen and Lenzing [41] and Prest [50]. Both actually survey model theory. The former book especially seems appropriate for those unaware of this branch of mathematics. The notion of purity is quite old and to a certain extent less known to algebraists than to logicians. But it has gained much interest over the past few years because it relates in a quite natural way to phantom maps in topological and in algebraic settings. Also pure-homological properties of modules are receiving increasing attention because of the growing interest in infinite-dimensional representations.

This chapter is mostly expository. We survey the notions and collect the results on purity that we are going to use in subsequent chapters. There will be almost no proofs. Unless otherwise stated, the most appropriate reference will be Jensen and Lenzing [41]. Only the last section (Section 3.8) contains new results. There we investigate how projectivity and purity co-exist when the ring is a finite-dimensional self-injective algebra over a field.

3.1 Basic definitions and properties

In this section, we give the most basic definitions and properties on purity relevant to our work.

Let Λ denote an arbitrary ring. Modules are assumed to be left Λ-modules unless otherwise specified.

We first recall the notion of finite presentation. It is very much present in purity theory and is a condition stronger than finite generation when the ring is not Nœtherian.

Definition 3.1.1. A presentation of a module M is an exact sequence

R ⟶ G ⟶ M ⟶ 0.

(The choice of notation G and R are meant to signify “generators” and “relators.”) A finite presentation of the module M is a presentation as above where the modules G and R are finitely generated. The module M is said to be finitely presented if it has a finite presentation.

Of course, if the ring Λ is Nœtherian then finite presentation and finite generation are synonymous.

It is well known, and proven in various manners in the literature, that every module is the direct limit of some direct system of finitely presented modules.

We will also need the notion of duality.

The dual of a left module M is D(M) = Hom_{ℤ}(M, ℚ/ℤ); it is a right Λ-module. If Λ is a finite-dimensional algebra over a field k, we also have the k-dual M^{∗} = Hom_{Λ}(M, k).

Theorem-Definition 3.1.2. A short exact sequence of modules

0 ⟶ A ⟶ B ⟶ C ⟶ 0

is said to be pure-exact if it satisfies the following equivalent conditions:

(i) For every finitely presented module M, the sequence has the lifting property

\xymatrix{ &  &  & M \ar[d] \ar@{-->}[dl] \\ 0 \ar[r] & A \ar[r] & B \ar[r] & C \ar[r] & 0,}

i.e. the Hom_{Λ}(M, −)-induced sequence

0 ⟶ Hom_{Λ}(M, A) ⟶ Hom_{Λ}(M, B) ⟶ Hom_{Λ}(M, C) ⟶ 0

is exact.

(ii) For every finitely presented right module N, the (N⊗_{Λ}−)-induced sequence

0 ⟶ N ⊗_{Λ} A ⟶ N ⊗_{Λ} B ⟶ N ⊗_{Λ} C ⟶ 0

is exact.

(iii) For every right module N (not assumed to be finitely presented), the (N⊗_{Λ}−)- induced sequence

0 ⟶ N ⊗_{Λ} A ⟶ N ⊗_{Λ} B ⟶ N ⊗_{Λ} C ⟶ 0

is exact.

(iv) The sequence is the direct limit of a direct system of short split-exact sequences.

(v) If a system of linear equations in A with coefficients in Λ admits a solution in B, then it admits a solution in A. (The module A is regarded as a submodule of B, and no assumptions are made on the cardinalities of the set of equations and of the set of unknowns.)

(vi) The dual sequence

0 ⟶ D(C) ⟶ D(B) ⟶ D(A) ⟶ 0

is split-exact.

If Λ is a finite-dimensional algebra over a field k, then the above conditions are equivalent to:

(vii) The k-dual sequence

0 ⟶ C^{∗} ⟶ B^{∗} ⟶ A^{∗} ⟶ 0

is split-exact.

There are obvious notions of pure monomorphism and pure epimorphism, pure submodule and pure quotient.

Remark 3.1.3. Let A be a pure submodule of a module B. Then for every left ideal I of Λ, we have

A ∩ IB = IA.

More generally, if C is a submodule of B, then

A ∩ IC ⊆ IA , A ∩ IC = IA ∩ IC .

This follows from Condition (v) in Theorem-Definition 3.1.2.

Remark 3.1.4. Condition (iv) in Theorem-Definition 3.1.2 can be made more explicit as follows. Suppose

0 ⟶ A ⟶ B ⟶ C ⟶ 0

is a short pure-exact sequence, and let {C_{α}}_{α} be a direct system of finitely presented modules with direct limit \begin{matrix}lim \\ -------→ \end{matrix}C_{α} = C. Then the direct system \{C_{α}\}_{α} lifts to a direct system of split-exact sequences

0 ⟶ A ⟶ B_{α} ⟶ C_{α} ⟶ 0

whose direct limit is the pure-exact sequence

0 ⟶ A ⟶ B ⟶ C ⟶ 0.

All maps of the direct system {A}_{α} are Id_{A}. If the maps of the direct system {C_{α}}_{α} are injective, then the maps of the direct system {B_{α}}_{α} are also injective. The construction of such a direct system is presented in full details in Dauns [23, proof of Theorem 18-3.3].

Theorem-Definition 3.1.5. A module M is said to be pure-projective if it satisfies the following equivalent conditions:

(i) The module M has the lifting property with respect to pure-exact sequences:

\xymatrix{ &  &  & M \ar[d] \ar@{-->}[dl] \\ 0 \ar[r] & A \ar[r] & B \ar[r] & C \ar[r] & 0;}

i.e. given any short pure-exact sequence

0 ⟶ A ⟶ B ⟶ C ⟶ 0,

the Hom_{Λ}(M, −)-induced sequence

0 ⟶ Hom_{Λ}(M, A) ⟶ Hom_{Λ}(M, B) ⟶ Hom_{Λ}(M, C) ⟶ 0

is exact.

(ii) Every pure-exact sequence

0 ⟶ A ⟶ B ⟶ M ⟶ 0

splits.

(iii) The module M is a summand of a (possibly infinite) direct sum of finitely presented modules.

If Λ is a finite-dimensional algebra over a field k, then the above conditions are equivalent to:

(iv) (Crawley, Jønsson, and Warfield; see Anderson and Fuller [2, Corollary 26.6])

The module M is a (possibly infinite) direct sum of finitely-dimensional modules. Theorem-Definition 3.1.6. A module M is said to be pure-injective if it satisfies the following equivalent conditions:

(i) The module M has the extension property with respect to pure-exact sequences:

\xymatrix{0 \ar[r] & A \ar[r] \ar[d] & B \ar[r] \ar@{-->}[dl] & C \ar[r] & 0 \\  & M}

i.e. given any short pure-exact sequence

0 ⟶ A ⟶ B ⟶ C ⟶ 0,

the Hom_{Λ}(−, M)-induced sequence

0 ⟶ Hom_{Λ}(C, M) ⟶ Hom_{Λ}(B, M) ⟶ Hom_{Λ}(A, M) ⟶ 0 is exact.

(ii) Every pure-exact sequence

0 ⟶ M ⟶ B ⟶ C ⟶ 0

splits. (iii) The module M is a summand of the dual D(N) = Hom_{ℤ}(N, ℚ/ℤ) of some right

Λ-module N. If Λ is a finite-dimensional algebra over a field k, then the above conditions are equivalent to: (iv) The module M is a summand of the k-dual N^{∗} = Hom_{Λ}(N, k) of some right

Λ-module N.

(v) The module M is a summand of a (possibly infinite) direct product of finitedimensional modules.

Remark 3.1.7. Let Λ be a finite-dimensional algebra over a field k. Then the dual N^{∗} = Hom_{Λ}(N, k) of a right Λ-module M is a pure-injective left Λ-module. If {M_{λ}}_{λ} is an inverse system of finite-dimensional modules, then the inverse limit M = \begin{matrix}lim \\ ←------- \end{matrix}M_{λ} is a pure-injective module. Indeed, we have M = N^{∗}, where N is the direct limit of the direct system {M_{λ}^{∗}}_{λ}.

Obviously, (infinite) direct sums of pure-projective modules are pure-projective, and finite direct sums and (infinite) direct products of pure-injective modules are pure-injective.

By a result of Kiełpiński and Simson [42, Theorem 2.1], unless all Λ-modules are both pure-projective and pure-injective [i.e. the ring Λ has (left) pure-global dimension zero (see the end of Section 3.3)], there are modules that are pure-projective but not pure-injective, and there are modules that are pure-injective but not pureprojective.

There is a property equivalent to pure-injectivity, namely algebraic compactness. Definition 3.1.8. A module M is said to be algebraically compact provided that given any system of linear equations in M with coefficients in Λ, if every finite subsystem admits a solution, then the system itself admits a solution. Theorem 3.1.9 (Stenström, Warfield, ...). For a module J, the following are equivalent:

(i) The module J is pure-injective.

(ii) The module J is algebraically compact.

(iii) Given modules M, N, and maps f : M ⟶ J and g : M ⟶ N, if for every finitely generated submodule A of M, the restriction f_{|A} : A ⟶ J factors through the restriction g_{|A} : A ⟶ N, then f : M ⟶ J factors through g : M ⟶ N.

The equivalence of conditions (i) and (ii) was proved independently by several authors. It seems that the credit for most general statements and complete proofs should go to Stenström [54, Proposition 9.2] and Warfield [55, Theorem 2]. Following the proof in the latter reference are some notes on the history of the result. See also the books of Dauns [23, Theorem 18-4.4] and Jensen and Lenzing [41, Theorem 7.1]. The latter reference uses material from model theory. The proof of the equivalence of Conditions (ii) and (iii) is a matter of verifying that (iii) says in terms of maps what (ii) says in terms of equations (Definition 3.1.8).

We would not be complete on pure-injectivity without mentioning the Gruson– Jensen’s condition. See Jensen and Lenzing [41, Corollary 7.34].

Theorem 3.1.10 ( Gruson and Jensen ). For a module M, the following are equivalent:

(i) The module M is pure-injective.

(ii) For every index set I, the components summation map M^{(I)} = ⨁_{I} M ⟶ M extends to a map M^{I} = ∏_{I} M ⟶ M.

(ii) For every infinite index set I, the diagonal embedding M ⟶ M^{I}/M^{(I)} splits.

Another characterization of pure-injectivity using the double dual.

Proposition 3.1.11. Let M be a module. It embeds in its double dual via the map

i_{M} : M ⟶ D^{2}(M), m ⟼ (i_{m} : D(M) ⟶ ℚ/ℤ, f ↦ i_{m}(f) = f(m)),

and this is a pure monomorphism. (Moreover the module D^{2}(M) is pure-injective.) The same holds with −^{∗} in place of D(−) when the ring Λ is a finite-dimensional algebra over a field k.

See [14, Theorem 2.4.2], or [41, Proposition 7.8] for a more general formulation We now state a characterization of pure monomorphisms which suggests a relation

to phantom maps and perhaps a generalization thereof. Proposition 3.1.12. Let N, N′ be modules, with N a submodule of N′. The following are equivalent:

(i) N is a pure submodule of N′.

(ii) Given a finitely generated module A, a finitely presented module B, and homomorphisms f : A ⟶ N and g : A ⟶ B, if there exists a homomorphism h′ : B ⟶ N′ such that f = h′g, then there exists a homomorphism h : B ⟶ N such that f = hg.

This characterization appears in Lazard [45, Lemme 2.2] and Jensen and Lenzing [41, Proposition 7.16] with the modules A and B free finitely generated. Here too, we are in essence saying in terms of morphisms what Condition (v) of Theorem-Definition 3.1.2 says in terms of equations.

3.2 Long pure-exact sequences

Definition 3.2.1. A long exact sequence

⋯ ⟶ M_{i+1} ⟶ M_{i} ⟶ M_{i−1} ⟶ ⋯

is said to be pure-exact if the short exact sequences

0 ⟶ K_{i+1} ⟶ M_{i} ⟶ K_{i} ⟶ 0,

where K_{i} = Ker(M_{i−1} ⟶ M_{i−2}) = Im(M_{i} ⟶ M_{i−1}), are pure-exact.

There are several equivalent conditions that can be derived from the characterizations of short pure-exact sequences (Theorem-Definition 3.1.2). We first need to introduce a terminology.

Definition 3.2.2. A totally split exact sequence is a (long exact) sequence which is the Yoneda splice of short split-exact sequences.

Theorem 3.2.3. For a long exact sequence

⋯ ⟶ M_{i+1} ⟶ M_{i} ⟶ M_{i−1} ⟶ ⋯

the following conditions are equivalent:

(i) The sequence is pure-exact.

(ii) For every right Λ-module N, the long sequence

⋯ ⟶ N ⊗_{Λ} M_{i+1} ⟶ N ⊗_{Λ} M_{i} ⟶ N ⊗_{Λ} M_{i−1} ⟶ ⋯

is an exact sequence of modules over the center Z(Λ) of Λ (in particular of abelian groups).

(iii) The sequence is the direct limit of a direct system of totally split exact sequences.

We now discuss a way to find totally split sequences. Recall the notion of contracting homotopy.

Definition 3.2.4. A contracting homotopy of a complex

⋯ ⟶^{d_{i+2}} M_{i+1} ⟶^{d_{i+1}} M_{i} ⟶^{d_{i}} M_{i−1} ⟶^{d_{i−1}} ⋯

is a sequence of abelian group homomorphisms s_{i} : M_{i} ⟶ M_{i+1} satisfying the identity

s_{i−1}d_{i} + d_{i+1}s_{i} = Id_{M_{i}}.

It is plain that if a complex has a contracting homotopy, then it is acyclic, i.e. it is an exact sequence.

Proposition 3.2.5. The following conditions for a sequence

A ⟶^{α} B ⟶^{β} C

of modules and module homomorphisms are equivalent:

(i) The sequence 0 ⟶ A ⟶^{α} B ⟶^{β} C ⟶ 0 is split-exact.

(ii) We have βα = 0 (i.e. the sequence is a complex) and there are module homomorphisms α′ : B ⟶ A and β′ : C ⟶ B such that αα′ + β′β = Id_{B}.

Corollary 3.2.6. If a complex has a contracting homotopy consisting of module homomorphisms, then it is a totally split exact sequence.

We now apply all this to a canonical construction. Given a direct system of modules {M_{α}}_{α}, there is what we will call the standard pure resolution of the direct limit M = \begin{matrix}lim \\ -------→ \end{matrix}M_{α}. It is a long pure-exact sequence abutting at M and whose terms are certain direct sums of copies of the modules M_{α}. The construction of this resolution follows. We will use multicomponent indices. A hat over a component of a multicomponent index will mean that the component is skipped.

Given a (t + 1)-tuple (α_{0}, α_{1}, …, α_{t}) of indices, we let M_{α_{0}α_{1}...α_{t}} denote a copy of the module M_{α_{0}}. Then for any integer n ≥ −1 and index α, we set

R_{n;α} = ⨁_{α_{0}≤α_{1}≤···≤α_{n}≤α} M_{α_{0}α_{1}...α_{n}α}.

In particular,

R_{−1;α} = M_{α} and R_{0;α} = ⨁_{α_{0}≤α} M_{α_{0}α} = ⨁_{α_{0}≤α} M_{α_{0}}.

We define maps

j_{n,i;α_{0}α_{1}...α_{n}α} : M_{α_{0}α_{1}...α_{n}α} ⟶ M_{α_{0}α_{1}...α̂_{i}...α_{n}α}

for n ≥ 0, 0 ≤ i ≤ n, as follows:

\begin{aligned}1 ≤ i ≤ n & : & j_{n,i;α_{0}α_{1}...α_{n}α} & : & M_{α_{0}α_{1}...α_{n}α} & ⟶^{Id_{Mα_{0}}} & M_{α_{0}α_{1}...α̂_{i}...α_{n}α} \\ i = 0, n ≥ 1 & : & j_{n,0;α_{0}α_{1}...α_{n}α} & : & M_{α_{0}α_{1}...α_{n}α} & ⟶^{f_{α_{0}α_{1}}} & M_{α_{1}...α_{n}α} \\ i = 0, n = 0 & : & j_{0,0;α_{0}α} & : & M_{α_{0}α} & ⟶^{f_{α_{0}α}} & M_{α} \end{aligned}

Then we define maps

∂_{n;α_{0}α_{1}...α_{n}α} : M_{α_{0}α_{1}...α_{n}α} ⟶ ⨁_{0≤i≤n} M_{α_{0}α_{1}...α̂_{i}...α_{n}α} ↪ R_{n−1;α}

by

∂_{n;α_{0}α_{1}...α_{n}α} = ⨁_{0≤i≤n}(−1)^{i}j_{n,i;α_{0}α_{1}...α_{n}α}.

In a more explit manner, we have for n ≥ 1:

∂_{n;α_{0}α_{1}...α_{n}α} = (M_{α_{0}α_{1}...α_{n}α} ⟶^{f_{α_{0}α_{1}}} M_{α_{1}...α_{n}α}) +

+ ∑_{1≤i≤n}(−1)^{n} (M_{α_{0}α_{1}...α_{n}α} ⟶^{Id_{Mα_{0}}} M_{α_{0}α_{1}...α̂_{i}...α_{n}α});

and for n = 0:

∂_{0;α_{0}α} = (M_{α_{0}α} ⟶^{f_{α_{0}α}} M_{α}) .

The reader should think of the map ∂_{n;α_{0}α_{1}...α_{n}α} as performing alternative sums. Finally we define the maps

∂_{n;α} : R_{n;α} ⟶ R_{n−1;α}

by

∂_{n;α} = ⨁_{α_{0}≤α_{1}≤···≤α_{n}≤α} ∂_{n;α_{0}α_{1}...α_{n}α}.

On another hand, we define

s_{−1;α} : M_{α} ⟶^{Id_{Mα}} M_{αα} ↪ M_{0;α};

and, for n ≥ 0,

s_{n;α_{0}α_{1}...α_{n}α} : M_{α_{0}α_{1}...α_{n}α} ⟶^{(−1)^{n−1}Id_{Mα_{0}}} M_{α_{0}α_{1}...α_{n}αα} ↪ R_{n+1;α}.

And we then define the maps

s_{n;α} : R_{n;α} ⟶ R_{n+1;α}

by

s_{n;α} = ⨁_{α_{0}≤α_{1}≤···≤α_{n}≤α} s_{n;α_{0}α_{1}...α_{n}α}.

Proposition 3.2.7. The sequence

⋯ ⟶^{∂_{n+2;α}} R_{n+1;α} ⟶^{∂_{n+1;α}} R_{n;α} ⟶^{∂_{n;α}} ⋯ ⟶^{∂_{2;α}} R_{1;α} ⟶^{∂_{1;α}} R_{0;α} ⟶^{∂_{0;α}} M_{α} ⟶ 0

is totally split, for it is a complex admitting the sequence of module homomorphisms s_{n;α} : R_{n;α} ⟶ R_{n+1;α}, n ≥ −1, as a contracting homotopy.

The proof of this statement is a series of tedious calculations where the real difficulty is to carefully keep track of multicomponent indices. The long but rigorous definitions of the maps provide an appropriate frame of reference for the verifications. The complete proof is in Appendix A (Proposition A.1).

Note that the long sequences form a direct system indexed by {α} in an obvious way. We set

R_{n} = ⨁_{α_{0}≤α_{1}≤···≤α_{n}} M_{α_{0}α_{1}...α_{n}} = \begin{matrix}lim \\ -------→ \end{matrix}_{α}R_{n;α}

and

∂_{n} = \begin{matrix}lim \\ -------→ \end{matrix}_{α}∂_{n;α}.

Then we have:

Proposition-Definition 3.2.8. The sequence

⋯ ⟶^{∂_{n+2}} R_{n+1} ⟶^{∂_{n+1}} R_{n} ⟶^{∂_{n}} R_{n−1} ⟶^{∂_{n−1}} ⋯ ⟶^{∂_{2}} R_{1} ⟶^{∂_{1}} R_{0} ⟶^{∂_{0}} M ⟶ 0

is pure-exact. It is the standard pure resolution of M associated to the direct system {M_{α}}_{α}.

Note that if the modules M_{α} are pure-projective, then so are the modules R_{n}.

3.3 Pure extensions. The functor Pext.

Definition 3.3.1. A pure-projective resolution of a module M is a pure-exact sequence

⋯ ⟶ R_{2} ⟶ R_{1} ⟶ R_{0} ⟶ M ⟶ 0

in which the modules R_{n} are pure-projective. A pure-injective resolution of a module N is a pure-exact sequence

0 ⟶ N ⟶ S^{0} ⟶ S^{1} ⟶ S^{2} ⟶ ⋯

in which the modules S^{n} are pure-injective.

We mentioned after Definition 3.1.1 that every module is the direct limit of a direct system of finitely presented modules. Combining this with Proposition-Definition 3.2.8, we obtain that every module has a (standard) pure-projective resolution. Proposition 3.1.11 implies that every module has a pure-injective resolution. And in fact, it is a result of various people including Warfield [55, Proposition 6] (see also [41, Proposition 7.6]) that any module has a pure-injective envelope. [It seems that the conditions for the existence of pure-projective covers have not been (successfully) studied. However, see Jensen and Lenzing [41, Theorem B.14].]

Standard homological algebra techniques show that the Comparison Theorem holds, namely that maps between modules lift to chain maps between resolutions, and that two chain maps lifting the same module homomorphism are chain homotopic. Moreover, by an inductive argument based on the classical degree shifting technique, one can show that the cohomology of the complex obtained by applying the functor Hom_{Λ}(−, N) to a pure-projective resolution of M and the cohomology of the complex obtained by applying the functor Hom_{Λ}(M, −) to a pure-injective resolution of N are naturally isomorphic. We thus have all the conditions for a good definition of the pure extensions functor. Theorem-Definition 3.3.2. Let M and N be modules. Consider a pure-projective resolution

⋯ ⟶ R_{2} ⟶ R_{1} ⟶ R_{0} ⟶ M ⟶ 0

of M, and a pure-injective resolution

0 ⟶ N ⟶ S^{0} ⟶ S^{1} ⟶ S^{2} ⟶ ⋯

of N. There are natural (degree-wise) isomorphisms between Pext_{Λ}^{∗}(M, N) and the cohomologies of the complexes

0 ⟶ Hom_{Λ}(M, S^{0}) ⟶ Hom_{Λ}(M, S^{1}) ⟶ Hom_{Λ}(M, S^{2}) ⟶ ⋯

and

0 ⟶ Hom_{Λ}(R^{0}, N) ⟶ Hom_{Λ}(R^{1}, N) ⟶ Hom_{Λ}(R^{2}, N) ⟶ · · · .

Pext_{Λ}^{∗} is a bifunctor on Λ-modules.

The above Theorem-Definition does not really present Pext_{Λ}^{∗}(M, N) as (equivalence classes of) pure extensions. But we do have this interpretation. Definition 3.3.3. Let M and N be modules, and n be a positive integer. A degree n pure extension of M by N is a degree n extension of M by N which is a pure-exact sequence, i.e. a pure-exact sequence

0 ⟶ N ⟶ X_{i−1} ⟶ X_{i−2} ⟶ ⋯ ⟶ X_{1} ⟶ X_{0} ⟶ M ⟶ 0.

A degree 0 pure extension of M by N is just a module homomorphism M ⟶ N.

The equivalence of degree n pure extensions is defined in the same way as is the equivalence of degree n extensions. We would like however to stress a point that one can overlook if not careful enough. When thinking of the equivalence of degree n pure extensions in terms of transformation of the degree one factors of an n-fold Yoneda product, all n-factors at every stage of the transformation must be degree one pure extensions. The set of equivalence classes of degree n pure extensions of M by N, equipped with the Baer sum, is an abelian group.

Theorem 3.3.4. The abelian group of equivalence classes of degree n pure extensions of M by N is isomorphic to Pext_{Λ}^{n}(M, N). In the diagram below (where the top row is a pure-projective resolution of M),

[figure: ⋯; ∂_{n+2}; R_{n+1}; ∂_{n+1}; R_{n}; ∂_{n}; ξ̂; R_{n−1}; ∂_{n−1}; ⋯; ∂_{1}; R_{0}; ∂_{0}; M; 0; ξ :; 0; N; X_{n−1}; ⋯; X_{0}; M; 0]

the morphism ξ̂: R_{n} ⟶ N represents the pure extension ξ in

Pext_{Λ}^{n}(M, N) = Ker (Hom_{Λ}(∂_{n+1}, N) : Hom_{Λ}(R_{n}, N) ⟶ Hom_{Λ}(R_{n+1}, N)) ; Im (Hom_{Λ}(∂_{n}, N) : Hom_{Λ}(R_{n−1}, N) ⟶ Hom_{Λ}(R_{n}, N)).

There is the Yoneda composition

Pext_{Λ}^{i}(L, M) × Pext_{Λ}^{j}(M, N) ⟶ Pext_{Λ}^{i+j}(L, N)

of pure extensions. We would like to point out an implication of the Yoneda composition maps

Hom_{Λ}(L, M) × Pext_{Λ}^{j}(M, N) ⟶ Pext_{Λ}^{j}(L, N)

and

Pext_{Λ}^{i}(L, M) × Hom_{Λ}(M, N) ⟶ Pext_{Λ}^{i}(L, N).

Theorem 3.3.5. The push-out and the pull-back of pure extensions along morphisms are pure extensions.

There are obvious notions of pure-projective dimension (p.proj.dim_{Λ}) and pure-injective dimension (p.inj.dim_{Λ}) of a Λ-module, and of pure-global dimension (p.gl.dim) of a ring. Regarding this, we have the following result of Kiełpiński and Simson [42, Proposition 1.6].

Theorem 3.3.6. The pure-global dimension of the ring Λ equals the supremum of the pure-projective (resp. pure-injective) dimensions of the modules with finite pureprojective (resp. pure-injective) dimension.

We close this section by mentioning the natural forgetful transformation

Pext_{Λ}^{∗} ⟶ Ext_{Λ}^{∗}

which basically takes a pure extension and regards it as an ordinary extension. The transformation

Pext_{Λ}^{1} ⟶ Ext_{Λ}^{1}

is a subfunctor of the identity transformation on Ext_{Λ}^{1}. Close to nothing was known about the kernel for the transformation

Pext_{Λ}^{n} ⟶ Ext_{Λ}^{n}

for n ≥ 2. We will show later (Corollary 5.3.2) that the transformation

Pext_{Λ}^{2} ⟶ Ext_{Λ}^{2}

can be noninjective.

The reader might wonder about a functor Ptor which would relate to the functor Tor in the same way as the functor Pext relates to the functor Ext. The fact is that, by Condition (ii) of Theorem-Definition 3.1.2, we have Ptor_{Λ}^{i} = 0 for i ≥ 1.

3.4 Pext^{∗}, lim^{∗}, and cardinality

Let \{M_{α}\}_{α} be a direct system with direct limit M = \begin{matrix}lim \\ -------→ \end{matrix}M_{α}. Consider the associated standard pure resolution

⋯ ⟶^{∂_{n+2}} R_{n+1} ⟶^{∂_{n+1}} R_{n} ⟶^{∂_{n}} R_{n−1} ⟶^{∂_{n−1}} ⋯ ⟶^{∂_{2}} R_{1} ⟶^{∂_{1}} R_{0} ⟶^{∂_{0}} M ⟶ 0.

(See Proposition-Definition 3.2.8.) Let N be a module. Applying Hom_{Λ}(−, N) to the above resolution, we obtain the following complex:

0 ⟶ Hom_{Λ}(M, N) ⟶^{δ^{0}} ∏_{α_{0}} Hom_{Λ}(M_{α_{0}}, N) ⟶^{δ^{1}} ∏_{α_{0}≤α_{1}} Hom_{Λ}(M_{α_{0}α_{1}α_{2}}, N) ⟶^{δ^{2}}

⟶^{δ^{2}} ∏_{α_{0}≤α_{1}≤α_{2}} Hom_{Λ}(M_{α_{0}α_{1}}, N) ⟶^{δ^{3}} · · · .

It is the standard complex for the purpose of calculating the derived functors of \begin{matrix}lim \\ ←------- \end{matrix}_{α} on the inverse system {Hom_{Λ}(M_{α}, N)}_{α}; see Jensen [40, Chapter 1]. Therefore we have: Theorem 3.4.1. Let \{M_{α}\}_{α} be a direct system with direct limit M = \begin{matrix}lim \\ -------→ \end{matrix}M_{α}. If all modules M_{α} are finitely presented, or more generally pure-projective, then we have isomorphisms

Pext_{Λ}^{n}(M, N) ≅ \begin{matrix}lim \\ ←------- \end{matrix}_{α}^{n}Hom_{Λ}(M_{α}, N)

for all integers n ≥ 0. Theorem 3.4.2. If a directed poset A has cardinality ≤ ℵ_{t}, then the functor \begin{matrix}lim \\ ←------- \end{matrix}_{A}^{n} vanishes for n ≥ t + 2.

The result appears in Jensen [40, Theorem 3.1]. He credits the result to Goblot [32] (with a more complicated proof), and mentions special cases due to Jensen [39] and Mitchell [47]. Combining it with Theorem 3.4.1, we obtain: Corollary 3.4.3. If a module is ℵ_{t}-presented, then it has pure-projective dimension at most t + 1.

Note that in order to obtain this Corollary from the above Theorem, one needs to use the fact that if a module M is ℵ_{t}-presented, then it is the direct limit of some direct system of finitely presented modules indexed by a directed poset of cardinality ≤ ℵ_{t}.

The result is immediate for countably-presented modules.

Let {M_{i}}_{i≥1} be a direct sequence of finitely presented modules, with structural maps φ_{i} : M_{i} ⟶ M_{i+1}. Let M be the direct limit, with structural maps φ̄_{i} : M_{i} ⟶ M. Then define the maps

δ : ⨁ M_{i} ⟶ ⨁ M_{i}, x = (x_{i})_{i≥1} ↦ δ(x) = (x_{1}, x_{2} − φ_{1}(x_{1}), x_{3} − φ_{2}(x_{2}), …)

and

π : ⨁ M_{i} ⟶ M, y = (y_{i})_{i≥1} ↦ π(y) = ∑ φ̄_{i}(y_{i}).

The sequence

0 ⟶ ⨁ M_{i} ⟶^{δ} ⨁ M_{i} ⟶^{π} M ⟶ 0

is pure-exact. And so the module M has pure-projective dimension one or is pureprojective.

The following is a Theorem of Gruson and Jensen [35, Section 2], [36, Proposition 10.5].

Theorem 3.4.4. If a ring has cardinality at most ℵ_{t}, then it has pure-global dimension at most t + 1.

3.5 Flat modules and purity

We start by noticing the following:

Proposition 3.5.1. For a module M, the following are equivalent:

(i) The module M is flat, i.e. Tor_{1}^{Λ}(−, M) = 0. (ii) Every short exact sequence

0 ⟶ Y ⟶ X ⟶ M ⟶ 0

abutting at M is pure-exact. On another hand, it follows from the long exact sequence for the functor Tor

that if the last two modules in a short exact sequence are flat, then so is the first. Therefore: Proposition 3.5.2. A long exact sequence

⋯ → M_{i+1} → M_{i} → M_{i−1} → ⋯ → M_{1} → M_{0} → 0

in which all terms are flat modules is pure-exact.

Now we have the following: Proposition 3.5.3. The following conditions for a module M are equivalent:

(i) The module M is pure-projective and flat. (ii) The module M is projective.

The implication (ii) ⇒ (i) is obvious and the reverse implication follows from Proposition 3.5.1.

Therefore we have: Proposition 3.5.4. Every projective resolution of a flat module M is a pureprojective resolution of the module M. In particular, we have

Pext_{Λ}^{∗}(M, −) = Ext_{Λ}^{∗}(M, −),

and the pure-projective dimension of M equals its projective dimension.

3.6 Endofinite modules

Let M be a left Λ-module. The evaluation map

End_{Λ}(M) × M ⟶ M, (f, x) ⟼ f(x)

equips M with a structure of left module over the ring End_{Λ}(M).

We will sometimes refer to the End_{Λ}(M)-submodules of M as the endosubmodules of M, and to the End_{Λ}(M)-endomorphisms of M as the endoselfmorphisms of M.

If the ring Λ is commutative, then every endo-submodule is a submodule, and every endo-selfmorphism an endomorphism. It does not have to be the case otherwise. The subgroups of finite definition (defined below) are examples of endo-submodules.

Let X and M be a right and a left Λ-modules. The abelian group X⊗_{R}M becomes an End_{Λ}(M)-module if we let f ∈ End_{Λ}(M) act as Id_{X} ⊗f. And then for every x ∈ X, we have the End_{Λ}(M)-module homomorphism

τ_{M,X,x} : M → X⊗_{R}M, m ↦ x⊗m.

Proposition-Definition 3.6.1. A subgroup of finite definition of M is an endosubmodule of M which is the kernel the map τ_{M,X,x} for some finitely presented right module X and some x ∈ X.

The subgroups of finite definition were introduced by Gruson and Lenzing in [35]. See Jensen and Lenzing [41, Chapter 6] for other characterizations and their properties. One characterization is in terms of finite linear systems in finitely many variables, and justifies the terminology.

Definition 3.6.2. The endolength of a Λ-module M is the length of M regarded as a End_{Λ}(M)-module. The module M is said to be endofinite if it has finite endolength.

One can find some basic facts about endofinite modules in Crawley-Boevey [22, Section 4] and [21, Section 1.3]. A certain amount of work is needed to prove the following:

Theorem 3.6.3. Every endofinite module is pure-injective.

We actually have more than this: Every endofinite module M is Σ-pure-injective, i.e. every (infinite) direct sum of copies of M is pure-injective. The two really fundamental results are in the following Theorem.

Theorem 3.6.4.

• A module is endofinite if and only if it satisfies both the ascending and the descending chain conditions on subgroups of finite definition.

• A module is Σ-pure-injective if and only if it satisfies the descending chain condition on subgroups of finite definition.

This theorem is part of Theorem 3.1 of Crawley-Boevey [22], where the statement is followed by a listing of several references. The first part of Theorem 3.6.4 is restated as Proposition 4.1 and proved in the same article. The second statement is proved in Jensen and Lenzing [41, Theorem 8.1], and is attributed to Gruson and Lenzing on one hand, and Zimmermann on another hand.

We would now like to present an example of how to obtain an endofinite module. Next will be an example of how to obtain a Σ-pure-injective module.

The following example is taken from Crawley-Boevey [21] (first example in Section 1.3).

Example 3.6.5. Let B be an algebra over a field L. Let A be a K-subalgebra of B, where K is a subfield of L. Let M be a B-module. The L-linear endomorphisms of M are endo-selfmorphisms of the restricted A-module M ↓_{A}^{B}. So every endosubmodule of M↓_{A}^{B} is an L-vector subspace of M. So if M is finite-dimensional as a B-module (i.e. as a L-vector space), then the restricted A-module M↓_{A}^{B} is endofinite.

Now this may well happen with the module M infinite-dimensional as a K-vector space. An instance of this is if the field extension K ⊂ L has infinite degree (for instance is transcendental) and B = L ⊗_{K} A. A good illustration of this special situation is with A the Kronecker algebra, the path K-algebra of the Kronecker quiver

K = \xymatrix{∘ \ar[r] \ar[r] & ∘} ,

and M the regular K(X)-Kronecker module

Φ = \xymatrix{K(X) \ar[r]^{1} \ar[r]_{X} & K(X)} .

The next example is taken from Crawley-Boevey [21, Section 2.5, Exemple 4].

Example 3.6.6. Let M be a Λ-module, where Λ is an algebra over a field k. Suppose that M has an endomorphism τ which is locally nilpotent, i.e. we have

M = ⋃_{n=1}^{∞} Ker(τ^{n}).

We have a structure of Λ-k[X]-bimodule on M with X acting as τ. Note that the k[X]-module structure on M is restricted from the End_{Λ}(M)-module structure. Every one-dimensional k-subspace of Ker τ is obviously a simple k[X]-submodule of M. So Ker τ is a semisimple k[X]-submodule of M. In addition, since τ is locally nilpotent, Ker τ is an essential k[X]-submodule of M (i.e. every nonzero submodule has nonzero intersection with it). We conclude that Soc_{k[X]}(M) = Ker τ.

Suppose Ker τ is finite-dimensional. Then, by Proposition B.3 (Appendix B), the k[X]-module M is Artinian. It follows that the End_{Λ}(M)-module M is also Artinian.

Thus in particular, M satisfies the descending chain condition on subgroups of finite definition. We summarize what we have got in the following Proposition.

Proposition 3.6.7. Let M be a Λ-module, where Λ is an algebra over a field k. If the module M has an endomorphism τ which is locally nilpotent and has finitedimensional kernel, then it is Σ-pure-injective. Also, the kernels Ker(τ^{n}) are finitedimensional, and the module M is countable-dimensional.

The following result of Krause [44, Proposition 8.3 (2)] is an instance of Example 3.6.6 above. It is also presented in Crawley-Boevey [21, Section 2.5, Exemple 5]. The idea is also used in the work of Benson and Gnacadja [14, Theorem 5.3.1]. We refer the reader to Appendix C and Section 6.2 for the terminology and facts about tubes.

Proposition 3.6.8. Suppose a module M is the direct limit of a tube (M_{i}, σ_{i}, τ_{i})_{i≥1} in mod(Λ), where Λ is an algebra over a field k. Then the endomorphism τ = \begin{matrix}lim \\ -------→ \end{matrix} τ_{i} : M ⟶ M of M is locally nilpotent and has finite-dimensional kernel. In particular, the module M is Σ-pure-injective.

To see this, one just needs to observe that, with the appropriate identifications via monomorphisms, we have Ker τ^{i} = M_{i} and M = ⋃ M_{i}. See Theorem 6.2.1 for an application of this result.

3.7 Functoriality

We discuss how various purity properties are preserved by functors.

Proposition 3.7.1. Let Γ and Λ be two rings, and let Φ : Mod(Γ) ⟶ Mod(Λ) be a covariant functor which is additive and preserves direct limits. Then Φ preserves pure-exact sequences.

Proof. The fact that Φ is additive implies that it preserves split-exact sequences. By Theorem-Definition 3.1.2, the additional fact that Φ preserves direct limits implies that it preserves pure-exact sequences.

The oldest reference we were able to find this result in is Lazard [45, Corollaire 2.5].

Proposition 3.7.2. Let Γ and Λ be two rings, and let Φ : Mod(Γ) ⟶ Mod(Λ) be a covariant functor which is additive, right-exact, takes Γ to Λ, and preserves direct sums. Then Φ preserves pure-projective modules.

Proof. The fact that Φ is additive, right-exact and takes Γ to Λ implies that it preserves finitely presented modules. The fact that it is additive implies that it preserves direct summands. By Theorem-Definition 3.1.5, the additional fact that Φ preserves direct sums implies that it preserves pure-projective modules.

Proposition 3.7.3. Let Γ and Λ be two rings, and let Φ : Mod(Γ) ⟶ Mod(Λ) be a covariant functor which is additive and preserves direct sums and direct products. Then Φ preserves pure-injective modules.

Proof. Let M be a Γ-module and I be an arbitrary index set. Consider the summation map π : M^{(I)} = ⨁_{I} M ⟶ M. Since the functor Φ is covariant additive and preserves direct sums, we have that Φ(π) is the summation map π : Φ(M)^{(I)} = ⨁_{I} Φ(M) ⟶ Φ(M). Suppose that M is pure-injective. Then the map π extends to M^{I} = ∏_{I} M. Since the functor Φ preserves direct products, we have that the map Φ(π) extends to Φ(M)^{I} = ∏_{I} Φ(M). We conclude that the module Φ(M) is pure-injective by the Gruson–Jensen’s characterization of pure-injective (Theorem 3.1.10, Condition (ii)).

3.8 Projectivity and purity

We start with two results that are well-known and widely used whenever finitedimensional self-injective algebras are present. Theorem 3.8.1. Let Λ be a finite-dimensional self-injective algebra over a field k. The following are equivalent for a Λ-module M:

• The module M is projective.

• The module M is injective.

• SocM = (SocΛ)M. Theorem 3.8.2. Let Λ be a finite-dimensional self-injective algebra over a field k. The following are equivalent for a Λ-module M:

• The module M is projective-free.

• (SocΛ)M = 0.

Now we use them to find out how projective and projective-free modules can occur in pure-exact sequences. Theorem 3.8.3. Let Λ be a finite-dimensional self-injective algebra over a field k. Let

0 ⟶ L ⟶ M ⟶ N ⟶ 0

be a short pure-exact sequence of Λ-modules. If M is projective, then both L and N are projective. If both L and N are projective-free, then M is projective-free. Proof. We combine Remark 3.1.3 with Theorem 3.8.1, and with Theorem 3.8.2, to obtain the first and the second statements respectively. Since L is a pure submodule of M, we have L ∩ (Soc Λ)M = (Soc Λ)L. Suppose M is projective. Then Soc M = (Soc Λ)M. So

Soc L = L ∩ Soc M = L ∩ (Soc Λ)M = (Soc Λ)L.

Thus L is projective, whence injective, and the sequence splits. With L projective-free, we obtain 0 = (Soc Λ)L = L ∩ (Soc Λ)M. With N projectivefree, we obtain (Soc Λ)N = 0, whence (Soc Λ)M ⊆ L. So if both L and N are projective-free, then (Soc Λ)M = 0, M is projective-free.

The first statement of Theorem 3.8.3 is actually not new and holds in a greater generality. We have the following result of Fieldhouse [30, Proposition 10.2]. The notion of perfect ring has several interesting characterizations, including those of Bass [4, Theorem P]. A finite-dimensional self-injective algebra is perfect. Theorem 3.8.4. The following are equivalent for a ring Λ.

• The ring Λ is perfect (in the sense of Bass).

• Pure submodules of flat Λ-modules are direct summands.

• Every short pure-exact sequence of Λ-modules with a flat module in the middle splits.

Theorem 3.8.3 can be generalized as follows. Theorem 3.8.5. Let Λ be a finite-dimensional self-injective algebra over a field k. Let

0 ⟶ L ⟶ M ⟶ N ⟶ 0

be a pure-exact sequence of Λ-modules. Given projective-free cores Ω^{0}(L) of L and Ω^{0}(N) of N, a projective-free core Ω^{0}(M) of M can be chosen so that the sequence above restricts to a pure-exact sequence of projective cores

0 ⟶ Ω^{0}(L) ⟶ Ω^{0}(M) ⟶ Ω^{0}(N) ⟶ 0.

Also,

Proj(M) ≅ Proj(L) ⊕ Proj(N).

Proof. We denote by π the pure epimorphism M → N, and regard L as a pure submodule of M. Consider a decomposition N = X_{N}⊕Y_{N} of N, with X_{N} a projective part and Y_{N} a projective-free core. Since pull-back preserves short pure-exact sequences (Theorem 3.3.5), we have the short pure-exact sequence

0 ⟶ L ⟶ π^{−1}(Y_{N}) ⟶^{π} Y_{N} ⟶ 0.

Let L = X_{L} ⊕ Y_{L} be a decomposition of L with X_{L} a projective part and Y_{L} a projective-free core. There is a submodule Y in π^{−1}(Y_{N}) such that Y ⊇ Y_{L} and π^{−1}(Y_{N}) = X_{L} ⊕ Y. (See the discussion of projective parts and projective-free cores in Section 1.2.) Since X_{L} ∩ Y = 0 and Y_{L} ⊆ Y, we have L ∩ Y = (X_{L} ⊕ Y_{L}) ∩ Y = Y_{L}. So the pure-exact sequence

0 ⟶ L ⟶ π^{−1}(Y_{N}) ⟶^{π} Y_{N} ⟶ 0

is the direct sum of the trivial sequence

0 ⟶ X_{L} ⟶^{=} X_{L} ⟶ 0 ⟶ 0

and the pure-exact sequence

0 ⟶ Y_{L} ⟶^{⊆} Y ⟶^{π_{|Y}} Y_{N} ⟶ 0.

By Theorem 3.8.3, the module Y is projective-free. In order to prove that Y is actually a projective-free core of M , we show that the quotient module M/Y is projective. We have

M/π^{−1}(Y_{N}) ≅ N/Y_{N} = (X_{N}⊕Y_{N})/Y_{N} ≅ X_{N} ,

so the quotient M/π^{−1}(Y_{N}) is projective. We also have

π^{−1}(Y_{N})/Y = (X_{L}⊕Y)/Y ≅ X_{L} ,

so the quotient π^{−1}(Y_{N})/Y is projective. Now that the module M/Y is projective follows from the fact that it fits in the exact sequence

0 ⟶ π^{−1}(Y_{N})/Y ⟶ M/Y ⟶ M/π^{−1}(Y_{N}) ⟶ 0.

Corollary 3.8.6. Let X be a module over a finite-dimensional self-injective k-algebra Λ (k a field). Suppose X is projective-free. Then any pure-projective (respectively pure-injective) resolution of X can be reduced to a pure-projective (respectively pureinjective) resolution of X where all terms are projective-free modules, by means of applying Ω^{0}. In particular, in a pure-projective (respectively pure-injective) resolution of X of minimal length, no terms are projective modules. The module X has a pureprojective (respectively pure-injective) resolution of minimal length in which all terms are projective-free modules.

This corollary tells us in particular that we cannot investigate the natural transformation Pext_{Λ}^{∗} ⟶ Ext_{Λ}^{∗} by working with resolutions applicable to both homological algebra theories (pure and ordinary) involved.

We close this section with a result on how varieties and complexities of modules in a pure-exact sequence are related. We will briefly introduce in Section 6.1 the notion of varieties for modules over group algebras. We however elect to state the result here because it is a direct application of Theorem 3.8.3 and the introduction we provide in Section 6.1 is not sufficient for the unaware reader (for the purpose of proving Theorem 3.8.7 below). So we really direct this result to the reader familiar with the material of Benson-Carlson-Rickard’s paper [12].

Theorem 3.8.7. Let G be a finite group and k be a field. Let

0 ⟶ L ⟶ M ⟶ N ⟶ 0

be a pure-exact sequence of kG-modules. We have

V_{G}(M) = V_{G}(L) ∪ V_{G}(N)

and

c_{G}(M) = max(c_{G}(L), c_{G}(N)).

More generally, suppose

⋯ ⟶ M_{i−1} ⟶ M_{i} ⟶ M_{i+1} ⟶ M_{i+2} ⟶ ⋯

is a long pure-exact sequence, and let K_{i} = Ker(M_{i} → M_{i+1}) = Im(M_{i−1} → M_{i}). Then

⋃_{i even ∈ ℤ}V_{G}(M_{i}) = ⋃_{i∈ℤ}V_{G}(K_{i}) = ⋃_{i odd ∈ ℤ}V_{G}(M_{i}).

Proof. We use the notations and Lemma 10.3 of Benson-Carlson-Rickard [12]. We have to show that V_{G}(L)∪V_{G}(N) ⊆ V_{G}(M). Let V ∈ V_{G}(k). Then the exact sequence

0 ⟶ κ(V) ⊗ L ⟶ κ(V) ⊗ M ⟶ κ(V) ⊗ N ⟶ 0

is pure-exact, by Proposition 3.7.1. Suppose V ∉ V_{G}(M). Then κ(V) ⊗ M is projective. So, by Theorem 3.8.3, both κ(V) ⊗ L and κ(V) ⊗ N are projective, whence V ∉ V_{G}(L) ∪ V_{G}(N).

The result was proved in Benson and Gnacadja [14, Theorem 4.3.4] by utilizing the fact that the variety of a kG-module M is covered by the images of the varieties of M regarded as module over the elementary abelian p-subgroups of G.

Phantom maps, purity and

the structure of the stable module

category

In all of this chapter, the ring Λ is a self-injective finite-dimensional algebra over a field k.

4.1 Phantom maps as degree one pure extensions

Definition 4.1.1. We say that an exact triangle

N → X → M → Ω^{−1}N

is pure-exact if the degree one extension of M by N it represents is a pure extension.

All characterizations of pure-exact sequences in Mod(Λ) carry over to pure-exact triangles in StMod(Λ). We just mention the ones that will be explicitly used in this chapter. Theorem 4.1.2. For an exact triangle

N → X → M → Ω^{−1}N,

the following are equivalent:

(i) The triangle is pure-exact.

(ii) For every finite-dimensional module A, the triangle has the lifting property

\xymatrix{ &  & A \ar[d] \ar@{-->}[dl] \\ N \ar[r] & X \ar[r] & M \ar[r] & Ω^{−1}N}

in StMod(Λ).

(iii) For every finite-dimensional module A, the \underline{Hom}_{Λ}(A, −)-induced sequence

0 ⟶ \underline{Hom}_{Λ}(A, N) ⟶ \underline{Hom}_{Λ}(A, X) ⟶ \underline{Hom}_{Λ}(A, M) ⟶ 0

is exact.

(iv) For every finite-dimensional module A, the \underline{Hom}_{Λ}(A, −)-induced map

\underline{Hom}_{Λ}(A, M) ⟶ \underline{Hom}_{Λ}(A, Ω^{−1}N)

is zero.

Only the implication (ii) ⇒ (i) requires some attention. It easily follows from the first of the two statements in Proposition 1.2.5.

The following fact should now be obvious to the reader.

Proposition 4.1.3. The Heller operator Ω and its inverse Ω^{−1} preserve pure-exact triangles, as well as pure-projective and pure-injective modules, and phantom maps.

Condition (iv) of Theorem 4.1.2 just says that the map M → Ω^{−1}N is phantom. We have thus obtained:

Corollary 4.1.4. The isomorphism

Ext_{Λ}^{1}(M, N) ⟶^{≅} \underline{Hom}_{Λ}(M, Ω^{−1}N),

which to every degree one extension of M by N assigns the map M → Ω^{−1}N which completes it to the right into an exact triangle, restricts to the isomorphism

Pext_{Λ}^{1}(M, N) ⟶^{≅} \underline{Ph}_{Λ}(M, Ω^{−1}N).

If \{M_{α}\}_{α} is a direct system of finite-dimensional modules with direct limit \begin{matrix}lim \\ -------→ \end{matrix}M_{α} = M, then

\underline{Ph}_{Λ}(M, N) ≅ \begin{matrix}lim \\ ←------- \end{matrix}\underline{Hom}_{Λ}(M_{α}, ΩN).

The following result is now immediate:

Theorem 4.1.5.

• A module M has no outbound phantom maps if and only if it is pure-projective, i.e. is a direct sum of finite-dimensional modules.

• A module M has no inbound phantom maps if and only if it is pure-injective, i.e. is a direct summand of a direct product of finite-dimensional modules.

• The category StMod(Λ) is phantomless if and only if the algebra Λ has pureglobal dimension zero.

Note that the fact that a pure-injective module receives no nontrivial phantom maps also follows from Condition (iii) of Theorem 3.1.9 characterizing pure-injectivity.

4.2 Universal phantom maps

Theorem 4.2.1. Let p : M′ → M be a pure epimorphism and j : N → N′ be a pure monomorphism, and let f : M → N be a map. If M is finite-dimensional, then the following are equivalent:

(i) The map f : M ⟶ N is projective.

(ii) The composite map M′ ⟶^{p}M ⟶^{f} N is projective. (iii) The composite map M ⟶^{f} N ⟶^{j}N′ is projective. Thus the following are equivalent: (i’) The map f : M ⟶ N is phantom. (ii’) The composite map M′ ⟶^{p}M ⟶^{f} N is phantom. (iii’) The composite map M ⟶^{f} N ⟶^{j}N′ is phantom. If M′ is pure-projective and N′ is pure-injective then the following are equivalent: (i”) The map f : M ⟶ N is phantom. (ii”) The composite map M′ ⟶^{p}M ⟶^{f} N is projective. (iii”) The composite map M ⟶^{f} N ⟶^{j}N′ is projective. Proof. We have the implication (ii) ⇒ (i) because the epimorphism M′ → M splits. The implication (iii) ⇒ (i) follows from Condition (ii) of Proposition 3.1.12 characterizing pure monomorphisms.

Let M, N be modules. Let R→M be a pure epimorphism with R a pure-projective module, and N→S be pure monomorphism with S a pure-injective module. (We know that such morphisms exist; see Section 3.3.) Complete these maps to get exact triangles in StMod(Λ):

R ⟶ M ⟶^{ϕ_{M}} X Y ⟶^{ϕ^{N}} N ⟶ S.

The maps ϕ_{M} and ϕ^{N} are phantom by conditions (ii”) and (iii”) of Theorem 4.2.1. We moreover have the following which is easy to check.

Theorem 4.2.2.

• The map ϕ_{M} : M ⟶ X is a weakly universal phantom map out of M, in the sense that it is phantom and every phantom map out M factors (non uniquely) through it in StMod(Λ).

• The map ϕ^{N} : Y ⟶ N is a weakly universal phantom map into N, in the sense that it is phantom and every phantom map into N factors (non uniquely) through it in StMod(Λ).

We now want to show that the target of the universal outbound phantom map and the source of the universal inbound phantom map are, to a certain extent, unique. Toward this goal, we are going to present the purity version of Schanuel’s lemma (Theorem 1.2.3) and introduce the pure-Heller operators Θ̃ and Θ^{−1}.

4.3 Pure-Heller operators

Theorem 4.3.1 (Schanuel’s Lemma, purity version). Consider, over an arbitrary ring Λ, two short pure-exact sequences

0 → K′ → R′ → M → 0 and 0 → K′′ → R′′ → M → 0

abutting at the same module M, with both modules R′ and R′′ pure-projective. Then

K′ ⊕ R′′ ≅ K′′ ⊕ R′.

This is proved in the same way the ordinary Schanuel’s Lemma (Theorem 1.2.3) is proved; see [7, Lemma 1.5.3].

Definition 4.3.2. If M is a module over an arbitrary ring Λ, we let Θ̃(M) denote any module K which fits in a short pure-exact sequence

0 → K → R → M → 0

with R a pure-projective module, i.e., which is the kernel of some pure epimorphism onto M from a pure-projective module.

By Schanuel’s Lemma, purity version (Theorem 4.3.1), the modules Θ̃(M) are determined “up to pure-projective summands.”

On the other hand, since we have genuine pure-injective envelopes (Warfield [55, Proposition 6], ...), we can set the following definition.

Definition 4.3.3. If M is a module over an arbitrary ring Λ, we let Θ^{−1}(M) denote the cokernel of “the” pure-injective envelope of M.

The definition of Θ^{−1}(M) is “up to non-unique isomorphism”. When the minimality conditions required for the operator Θ^{−1} are not necessary, we will use the notation Θ̃^{−1}. We have Θ̃^{−1}(M) = Θ^{−1}(M) ⊕ (pure − injective).

We call Θ̃, Θ^{−1} and Θ̃^{−1} the pure Heller operators.

We can now state:

Proposition 4.3.4.

• The weakly universal phantom map out of a module M is the map

ϕ_{M} : M ⟶ Ω^{−1}Θ̃M

in the exact triangle

Θ̃M ⟶ R ⟶ M ⟶ Ω^{−1}Θ̃M

where the map R ⟶ M is a pure epimorphism and the module R is pureprojective.

• The weakly universal phantom map into a module N is the map

ϕ^{N} : ΩΘ^{−1}N ⟶ N

in the exact triangle

ΩΘ^{−1}N ⟶ N ⟶ S(N) ⟶ Θ^{−1}N

where the map N ⟶ S(N) is the pure-injective envelope of N.

We mention that the universal outbound phantom map presented here does generalize the one of Proposition 2.3.1.

We now note the following:

Proposition 4.3.5. The Heller operator Ω and its inverse Ω^{−1} commute with the pure-Heller operators Θ̃, Θ^{−1}, Θ̃^{−1}.

Proof. The various commutativity relations follow readily from Proposition 4.1.3. For instance, to show that the operators Ω and Θ̃ commute, we consider a pure-exact triangle

Θ̃M ⟶ R ⟶ M

with the module R pure-projective, and note that we have the pure-exact triangle

ΩΘ̃M ⟶ ΩR ⟶ ΩM

with the module ΩR pure-projective, so that the modules ΩΘ̃M and Θ̃ΩM are stably isomorphic.

Since the genuine Heller operator Ω is a self-equivalence of the category StMod(Λ), it is worth trying to understand the composite operator Θ^{−1}Θ̃ = Θ̃Θ^{−1}. It seems that one should not expect this to be the identity. We do have two distinguished maps

ψ_{M} : M ⟶ Θ^{−1}Θ̃M and ψ^{M} : Θ̃Θ^{−1}M ⟶ M

arising from the following commutative diagrams with pure-exact rows.

[figure: 0; Θ̃M; pure − proj.; M; ψ_{M}; 0; 0; Θ̃M; pure − inj.; Θ^{−1}Θ̃M; 0]

[figure: 0; Θ̃Θ^{−1}M; ψ^{M}; pure − proj.; Θ^{−1}M; 0; 0; M; pure − inj.; Θ^{−1}M; 0]

Question 4.3.6. What are the properties of the maps ψ_{M} : M ⟶ Θ^{−1}Θ̃M and ψ^{M} : Θ̃Θ^{−1}M ⟶ M, and their composite ψ_{M}ψ^{M} : Θ̃Θ^{−1}M ⟶ Θ^{−1}Θ̃M ?

4.4 Composites of phantom maps

Definition 4.4.1. An n-fold phantom map is a map M → N which has a factorization

M = M_{0} → M_{1} → ⋯ → M_{n−1} → M_{n} = N

where each of the n maps M_{i} → M_{i+1} (0 ≤ i ≤ n − 1) is phantom. We denote \underline{Ph}_{Λ}^{n}(M, N) the space of n-fold phantom maps M → N.

Theorem 4.4.2. The natural isomorphism

\underline{Hom}_{Λ}(M, N) ⟶^{≅} Ext_{Λ}^{n}(M, Ω^{n}N)

restricts to an isomorphism

\underline{Ph}_{Λ}^{n}(M, N) ⟶^{≅} Im (Pext_{Λ}^{n}(M, Ω^{n}N) ⟶ Ext_{Λ}^{n}(M, Ω^{n}N)) .

For n = 1, we already have the isomorphism

Pext_{Λ}^{1}(M, ΩN) ⟶^{≅} \underline{Ph}_{Λ}(M, N)

by Theorem 4.1.4. For n ≥ 2, the map Pext_{Λ}^{n}(X, Y) ⟶ Ext_{Λ}^{n}(X, Y) need not be injective, so the surjection Pext_{Λ}^{n}(M, Ω^{n}N) ↠ \underline{Ph}_{Λ}^{n}(M, N) need not be an isomorphism (see Corollary 5.3.2).

Proof (Theorem 4.4.2). Given a n-fold phantom map M → N, we construct a degree n extension (i.e. a representative of the class of extensions) which corresponds to it under the isomorphism

\underline{Hom}_{Λ}(M, N) ≅ Ext_{Λ}^{n}(M, Ω^{n}N)

and is a pure extension. So consider a map

f : M = M_{0} ⟶^{f_{0}} M_{1} ⟶^{f_{1}} ⋯ ⟶^{f_{n−2}} M_{n−1} ⟶^{f_{n−1}} M_{n} = N

with all the maps f_{i} : M_{i} → M_{i+1} phantom. Then for each i = 0, …, n − 1, we have, by Theorem 4.1.4, a pure-exact triangle

ΩM_{i+1} → X_{i} → M_{i} →^{f_{i}} M_{i+1}.

Then the map f_{i} : M_{i} → M_{i+1} corresponds, under the isomorphism

\underline{Hom}_{Λ}(M_{i}, M_{i+1}) ≅ Ext_{Λ}^{1}(Ω^{i}M_{i}, Ω^{i+1}M_{i+1}),

to the pure-exact triangle

Ω^{i+1}M_{i+1} → Ω^{i}X_{i} → Ω^{i}M_{i}.

These pure-exact triangles compose to yield the sequence

Ω^{n}N → Ω^{n−1}X_{n−1} → ⋯ → ΩX_{1} → X_{0} → M

which represents the map f : M → N and a degree n pure extension of M by Ω^{n}N.

There are weakly universal n-fold phantom maps. They are obtained by composing the relevant universal (1-fold) phantom maps. We obtain:

Theorem 4.4.3.

• Given a module M, the natural map

ϕ_{M}^{n} : M ⟶ Ω^{−n}Θ̃^{n}M

is a weakly universal n-fold phantom map out of M.

• Given a module N, the natural map

ϕ_{n}^{N} : Ω^{n}Θ^{−n}N ⟶ N

is a weakly universal n-fold phantom map into N.

It is worth noticing the natural maps

Pext_{Λ}^{n}(M, N) ⟶ Pext_{Λ}^{n−i}(M, Ω^{−i}N)

where i = 0, …, n − 1. By Corollary 5.3.2, the map

Pext_{Λ}^{2}(M, N) ⟶ Pext_{Λ}^{1}(M, Ω^{−1}N)

need not be injective.

4.5 Pure-global dimension of group algebras

The group algebras are the finite-dimensional self-injective algebras we are the most interested in. We are merely going to restate Theorem 4.5.3 of Benson and Gnacadja [14]. It is derived from various results of Auslander, Baer, Brune, Lenzing.

Theorem 4.5.1. Let k be a field of characteristic p and cardinality ℵ_{t}, where t ∈ {0, 1, …; ∞}.

• If p = 0 or G has (possibly trivial) cyclic Sylow p-subgroups then kG has pureglobal dimension zero.

• If p ≠ 2 and G has noncyclic Sylow p-subgroups then the pure-global dimension of kG is t + 1.

• If p = 2 and G has Sylow 2-subgroups which are not cyclic, dihedral, semidihedral or generalized quaternion, then the pure-global dimension of kG is t + 1.

• If p = 2 and G has Sylow 2-subgroups isomorphic to ℤ/2 × ℤ/2 then the pureglobal dimension of kG is equal to one if k is countable and two if k is uncountable.

• In the remaining cases in characteristic two, namely when the Sylow 2-subgroup is dihedral of order greater than four, semidihedral or generalized quaternion, then the pure-global dimension of kG is equal to one if k is countable, two if k has cardinality ℵ_{1}, and is somewhere between 2 and t + 1 if k has cardinality ℵ_{t} with t > 1.

4.6 Vanishing multifold phantom maps.

The following is immediate from Theorem 4.4.2.

Theorem 4.6.1.

• If the module M has pure-projective dimension n, then there are no nontrivial (n + 1)-fold phantom maps out of M; \underline{Ph}_{Λ}^{n+1}(M, −) = 0.

• If the module N has pure-injective dimension n, then there are no nontrivial (n + 1)-fold phantom maps into N; \underline{Ph}_{Λ}^{n+1}(−, N) = 0.

• If the algebra Λ has pure-global dimension n, then there are no nontrivial (n+1)- fold phantom maps in the category StMod(Λ); \underline{Ph}_{Λ}^{n+1} = 0.

Combining this with Theorem 3.4.4, we get:

Theorem 4.6.2. If the algebra Λ has cardinality at most ℵ_{t}, then there are no nontrivial (t + 2)-fold phantom maps in the category StMod(Λ); \underline{Ph}_{Λ}^{t+2} = 0.

Probably the most important open problem in this subject is the following question. Question 4.6.3. Do there exist nonvanishing two-fold phantom maps (over a finitedimensional self-injective algebra of pure-global dimension at least two) ? Definition 4.6.4.

• The outbound phantom dimension of a module M is out.ph.dim_{Λ}(M) = min\{n ≥ 1 : \underline{Ph}_{Λ}^{n}(M, −) = 0\}.

• The inbound phantom dimension of a module N is in.ph.dim_{Λ}(N) = min\{n ≥ 1 : \underline{Ph}_{Λ}^{n}(−, N) = 0\}.

• The phantom dimension of the algebra Λ is ph.dim(Λ) = min\{n ≥ 1 : \underline{Ph}_{Λ}^{n}(−, −) = 0\}.

Obviously we have: Proposition 4.6.5. The phantom dimension of the algebra Λ is equal to the minimum of the outbound phantom dimensions of Λ-modules, and to the minimum of the inbound phantom dimensions of Λ-modules.

We can rephrase Theorem 4.6.1 as follows. Theorem 4.6.6.

out.ph.dim_{Λ}(M) < p.proj.dim_{Λ}(M)
in.ph.dim_{Λ}(N) < p.inj.dim_{Λ}(N)
ph.dim(Λ) < p.gl.dim(Λ)

We also introduce the following terminology:

Definition 4.6.7.

• The phantom index ph.index(G) of a finite group G is the supremum of the phantom dimensions of the group algebras kG as k ranges over all fields.

• The purity index p.index(G) of a finite group G is the supremum of the pureglobal dimensions of the group algebras kG as k ranges over all fields.

We obviously have:

Proposition 4.6.8. If the finite group G has finite phantom index, then

ph.index(G) < p.index(G).

If k has characteristic zero or a prime number not dividing the order of G, then the group algebra kG has pure-global dimension zero, and hence its phantom dimension is one. So the indices of Definition 4.6.7 are also obtained by taking the respective suprema as k ranges over all fields whose characteristic divides the order of G. In particular, if G is a p-group, where p is a prime positive integer, then it is enough to take the suprema as k ranges over all fields with characteristic p.

There are no evidence that the purity and phantom indices ought to be finite. However, as we will see later (Section 4.7, especially Theorem 4.7.7), certain finite 2-groups have finite phantom index.

We note the following, an easy consequence of Theorem 4.5.1 combined with Theorem 4.1.5.

Theorem 4.6.9. The following conditions for a finite group G are equivalent:

(i) The group G has purity index zero.

(ii) The group G has phantom index one.

(iii) All the Sylow subgroups of G are cyclic.

There is a group theoretic characterization of finite groups with cyclic Sylow subgroups. Theorem 4.6.10 below is taken from Hall’s book [37, Theorem 9.4.3].

For m, n, r positive integers, we let 𝔐_{m,n,r} denote the metacyclic group ⟨a⟩ ⋊ ⟨b⟩ with a and b of order m and n respectively, and b^{−1}ab = a^{r}. Theorem 4.6.10. The following conditions for a finite group G are equivalent: (iii) All the Sylow subgroups of G are cyclic.

(iv) G ≅ 𝔐_{m,n,r} for some positive integers m, n, r such that m, n, r−1 are pairwise

relatively prime. Question 4.6.3 can be restated as:

Question 4.6.11. Can we have ph.dim(Λ) > 2 (where Λ is a finite-dimensional selfinjective algebra with p.gl.dim(Λ) ≥ 2) ?

More plainly, this question is asking whether the notions of phantom dimension have any relevance.

There is a notion of phantom maps in the derived category of a ring. It follows from an example of Neeman [48, end part of the article] (which he credits to Keller) that in that setting, the answer to the Questions 4.6.3, 4.6.11 is yes. The example takes advantage of the special way flat modules behave with respect to purity (see Section 3.5). We cannot do this here: every flat Λ-module is projective.

We would like to formulate the question for a specific case.

Let G be the Klein four group ℤ/2 × ℤ/2, and let the field k have characteristic two. By Theorem 4.5.1, p.gl.dim(kG) equals one or two according to whether k is countable or uncountable respectively. On another hand, by the results of Section 2.2 (and also by Theorem 4.5.1) we have ph.dim(kG) ≥ 2 regardless of the cardinality of k. So we have:

Theorem 4.6.12. Let G be the Klein four group ℤ/2 × ℤ/2, and let the field k have characteristic two. The phantom dimension of the group algebra kG satisfies:

• If k is countable, then ph.index(kG) = 2.

• If k is uncountable, then either ph.index(kG) = 2 or ph.index(kG) = 3.

In particular, the Klein four group has purity index two and phantom index either two or three.

Now we ask:

Question 4.6.13. Let G be the Klein four group ℤ/2 × ℤ/2. Is the phantom index of G equal to two or three ? Can we have ph.dim(kG) = 3 (where k is an uncountable field of characteristic two) ?

4.7 Bounding the phantom index

By Theorem 4.6.2, if the algebra Λ has cardinality at most ℵ_{t}, then ph.dim(Λ) ≤ t+2. So if G is a finite group and the field k has cardinality ℵ_{t} or less, then ph.dim(kG) ≤ t + 2. We develop a method which, for certain groups G, gives an upper bound for ph.dim(kG) which does not depend on the cardinality of the field k, and hence an upper bound for ph.index(G). It is based on the knowledge of ph.dim(kH) for certain appropriately chosen subgroups H of G. We already have such an upper bound for the Klein four group (Theorem 4.6.12). We draw upon this fact in some of the actual applications of the technique. The method was first suggested by Carlson, and used by Benson and Gnacadja in [14, Section 4.6]. Here we adopt a more systematic approach and obtain concrete results for many more groups.

First we need some preparation with group cohomology. The notation k^{+} below represents the underlying abelian group of the field k.

Proposition 4.7.1. Let G be a finite group and k a field. Let A be the augmentation ideal of kG, i.e. the ideal of dimension |G| − 1 with k-basis {g − 1_{G} : g ∈ G \ {1_{G}}}. We have:

Hom_{kG}(A, k) ≅ H^{1}(G, k) ≅ Hom_{Group}(G, k^{+}).

Two elements η ∈ Hom_{kG}(A, k) and κ ∈ Hom_{Group}(G, k^{+}) correspond to each other if they are related by the equation

η(g − 1_{G}) = κ(g).

For the first isomorphism, one just needs to note that the augmentation ideal is a realization of the Heller translate Ωk of k. The second isomorphism is obtained by analysing cohomology with the bar resolution; see for example Evens [26, Section 2.3].

Let p be a prime positive integer. The isomorphism H^{1}(G, 𝔽_{p}) ≅ Hom_{Group}(G, 𝔽_{p}^{+}) from Proposition 4.7.1 realizes a one-to-one correspondence between the nonzero elements of H^{1}(G, 𝔽_{p}) and the normal subgroups of G of index p. This is because the assignment to a nonzero (hence surjective) group morphism G → 𝔽_{p}^{+} of its kernel is a one-to-one correspondence between the nonzero elements of Hom_{Group}(G, 𝔽_{p}^{+}) and the normal subgroups of G of index p. A cohomology class ζ ∈ H^{1}(G, 𝔽_{p}) and an index p subgroup H of G correspond to each other if H = Ker(κ), where κ is the element of Hom_{Group}(G, 𝔽_{p}^{+}) corresponding to ζ.

Suppose now that we have an arbitrary field k of characteristic p. We regard the cohomology H^{∗}(G, 𝔽_{p}) as the 𝔽_{p}-subspace 1⊗H^{∗}(G, 𝔽_{p}) of H^{∗}(G, k) = k⊗_{𝔽_{p}}H^{∗}(G, 𝔽_{p}). Then to every index p normal subgroup H of G corresponds a cohomology class in H^{1}(G, k) lying in H^{1}(G, 𝔽_{p}).

We now concentrate on the case p = 2. Proposition 4.7.1 shows how to match an index two (necessarily normal) subgroup H of G with the corresponding element of H^{1}(G, 𝔽_{2}) regarded as an element of Hom_{𝔽_{2}G}(A, 𝔽_{2}). We are going to discuss how to match an index two subgroup H of G with the corresponding element of H^{1}(G, 𝔽_{2}) regarded as a degree one extension, i.e., an element of Ext_{𝔽_{2}G}^{1}(𝔽_{2}, 𝔽_{2}).

Let H be a subgroup of G of index two. We denote by ξ̃ the unique (nonzero) element of Hom_{Group}(G, 𝔽_{2}^{+}) with kernel H, and by e and a the elements H and G \ H of the quotient group G/H. We have the 𝔽_{2}G-maps

τ : 𝔽_{2} → 𝔽_{2}[G/H], 1 ↦ e + a and ε : 𝔽_{2}[G/H] → 𝔽_{2}, e, a ↦ 1.

They fit in the following short exact sequence over 𝔽_{2}G

ξ : 0 → 𝔽_{2} →^{τ} 𝔽_{2}[G/H] →^{ε} 𝔽_{2} → 0.

This sequence does not split because the action of the group G is not trivial on the kG-module 𝔽_{2}[G/H], whereas it is on the other two modules in the sequence. But H acts trivially on 𝔽_{2}[G/H], so the restriction ξ↓_{H}^{G} splits. We conclude that ξ is the extension in Ext_{𝔽_{2}G}^{1}(𝔽_{2}, 𝔽_{2}) = H^{1}(G, 𝔽_{2}) corresponding to the group homomorphism ξ̃.

Now let k be an arbitrary field of characteristic 2. We have the group morphism ξ̃ : G → k^{+} with kernel H and image the subfield 𝔽_{2} of k. The corresponding extension in Ext_{kG}^{1}(k, k) = H^{1}(G, k) is

ξ : 0 → k →^{τ} k[G/H] →^{ε} k → 0,

obtained by tensoring with k over 𝔽_{2} what we had above over the algebra 𝔽_{2}G. We will say that ξ is the element of Ext_{kG}^{1}(k, k) = H^{1}(G, k) corresponding to the subgroup H. Note that k[G/H] = k↓_{H}↑^{G}.

Let M be a kG-module. By tensoring ξ and M over k, we obtain the short exact sequence

ξ ⊗ M : 0 → M ⟶^{τ⊗M} M↓_{H}↑^{G}⟶^{ε⊗M} M → 0.

This element of Ext_{kG}^{1}(M, M) is the cup product of ξ ∈ H^{1}(G, k) and Id_{M} ∈ Ext_{kG}^{0}(M, M).

Let N be a kG-module. The functor Êxt_{kG}^{∗}(−, N) applied to the short exact sequence ξ ⊗ M gives rise to the long exact sequence

⋯ ⟶^{(τ⊗M)^{∗}} Êxt_{kG}^{n−1}(M, N) ⟶Êxt_{kG}^{n}(M, N) ⟶^{(ξ⊗M)^{∗}} Êxt_{kG}^{n}(M ↓_{H}↑^{G}, N) ⟶^{(τ⊗M)^{∗}}

⟶^{(τ⊗M)^{∗}} Êxt_{kG}^{n}(M, N) ⟶Êxt_{kG}^{n+1}(M, N) ⟶^{(ξ⊗M)^{∗}} · · · .

The connecting map Êxt_{kG}^{n}(M, N) ⟶ Êxt_{kG}^{n+1}(M, N) is the cup product multiplication by the element ξ ⊗ M = ξId_{M} of Ext_{kG}^{1}(M, M), i.e by the element ξ of H^{1}(G, k).

For the other two maps, we have the following commutative diagrams. (See [7, Section 2.8, Exercise] for the first diagram.)

\xymatrix{Êxt_{kG}^{n}(M, N) \ar[r]^{(ξ⊗M)^{∗}} \ar@{-->}[dr]_{_{Res_{G,H}}} & Êxt_{kG}^{n}(M↓_{H}↑^{G}, N) \ar[d]_{≅}^{Eckmann−Shapiro} \\  & Ext_{kH}^{n}(M↓_{H}, N↓_{H})}

\xymatrix{Êxt_{kG}^{n}(M↓_{H}↑^{G}, N) \ar[r]^{(τ⊗M)^{∗}} & Êxt_{kG}^{n}(M, N) \\ Ext_{kH}^{n}(M↓_{H}, N↓_{H}) \ar[u]_{≅}^{Eckmann−Shapiro} \ar[ur]_{_{Tr_{G,H}}}}

We have thus obtained:

Lemma 4.7.2. Let G be a finite group and H be an index two subgroup of G. Let k be a field of characteristic two. Let ξ be the degree one cohomology class in H^{1}(G, k) corresponding to the subgroup H. For every kG-modules M and N, we have the long exact sequence

⋯ ⟶^{Tr_{G,H}} Êxt_{kG}^{n−1}(M, N) ⟶^{ξ·} Êxt_{kG}^{n}(M, N) ⟶^{Res_{G,H}} Êxt_{kH}^{n}(M↓_{H}, N↓_{H}) ⟶^{Tr_{G,H}}

⟶^{Tr_{G,H}} Êxt_{kG}^{n}(M, N) ⟶^{ξ·} Êxt_{kG}^{n+1}(M, N) ⟶^{Res_{G,H}} · · · .

We now have all the necessary background to prove the following theorem providing an inductive procedure to bound the phantom dimension.

Theorem 4.7.3. Let k be a field of characteristic two and G be a finite group. Suppose that G has index two subgroups H_{1}, …, H_{r} with corresponding degree one cohomology classes ξ_{1}, … ξ_{r} ∈ H^{1}(G, k) composing to zero in H^{r}(G, k). Then

ph.dim(kG) ≤ ∑_{i=1}^{i=r} ph.dim(kH_{i}).

Proof. Set d_{i} = ph.dim(kH_{i}) and n = ∑_{i=1}^{i=r} d_{i}. Let f : M → N be a n-fold phantom map in StMod(kG). We can write f = f_{1}⋯f_{r} where for each i, f_{i} : M_{i} → M_{i−1} is a d_{i}-fold phantom map (M_{r} = M and M_{0} = N). We regard f as an element of Êxt_{kG}^{r}(Ω^{−r}M, N) and f_{i} as an element of Ext_{kG}^{1}(Ω^{−i}M_{i}, Ω^{−i+1}M_{i−1}). The restricted map Res_{G,H_{i}}(f_{i}) is a d_{i}-fold phantom map in the category StMod(kH_{i}) by Theorem 1.4.2. So Res_{G,H_{i}}(f_{i}) = 0. By the exact sequence

\underline{Hom}_{kG}(Ω^{−i}M_{i}, Ω^{−i+1}M_{i−1}) ⟶^{ξ_{i}·} Ext_{kG}^{1}(Ω^{−i}M_{i}, Ω^{−i+1}M_{i−1}) ⟶^{Res_{G,H_{i}}}

⟶^{Res_{G,H_{i}}} Ext_{kH_{i}}^{1}(Ω^{−i}M_{i}↓_{H_{i}}, Ω^{−i+1}M_{i−1}↓_{H_{i}})

of Lemma 4.7.2, we have f_{i} = ξ_{i}g_{i} for some g_{i} ∈ Êxt_{kG}^{0}(Ω^{−i}M_{i}, Ω^{−i+1}M_{i−1}) = \underline{Hom}_{kG}(Ω^{−i}M_{i}, Ω^{−i+1}M_{i−1}). Then

f = f_{1}⋯f_{r}ξ_{1}g_{1}⋯ξ_{r}g_{r}.

We are regarding the collection Êxt_{kG}^{∗} as a graded algebra over the cohomology ring H^{∗}(G, k). (For details on this, refer for instance to Section 6, especially Proposition 6.8 of Carlson’s book [16].) So we have

f = (−1)^{ℓ}(ξ_{1}⋯ξ_{r})(g_{1}⋯g_{r})

(where ℓ = ∑_{1≤j<i≤r} deg(ξ_{i}) deg(g_{j}) = 0 ). So f = 0.

The hypothesis of Theorem 4.7.3 cannot hold if the finite group G is elementary abelian, i.e. G ≅ ℤ/2 × ⋯ × ℤ/2. Indeed in that case the cohomology ring H^{∗}(G, k) is the polynomial ring in rank_{2}(G) variables, an integral domain. A theorem of Serre asserts that the condition holds for all other finite 2-groups. We formulate it below in a way sufficient for our needs. Refer to Evens’s book [26, Corollary 6.4.2 and Theorem 6.4.1] for more general statements and proofs. See also Benson’s book [8, Section 4.7]. Theorem 4.7.4 (Serre). Let G be a finite 2-group which is not elementary abelian. Then there are degree one cohomology classes ξ_{1}, … ξ_{r} ∈ H^{1}(G, 𝔽_{2}) composing to zero in H^{r}(G, 𝔽_{2}).

Under the hypothesis and in the notations of Serre’s Theorem above, there are index two subgroups H_{1}, …, H_{r} in G such that for every field k of characteristic two, the corresponding elements of H^{1}(G, k) compose to zero. Such elements can be obtained by taking elements in H^{1}(G, 𝔽_{2}) satisfying the required condition and regarding them as elements of H^{1}(G, k) = k⊗_{𝔽_{2}}H^{1}(G, 𝔽_{2}) lying is the 𝔽_{2}-subspace 1⊗H^{1}(G, 𝔽_{2}).

We can now state the following more specialized version of Theorem 4.7.3. Theorem 4.7.5. Let G be a finite 2-group which is not elementary abelian. Let H_{1}, …, H_{r} be index two subgroups in G with corresponding degree one cohomology classes ξ_{1}, … ξ_{r} ∈ H^{1}(G, 𝔽_{2}) composing to zero in H^{r}(G, 𝔽_{2}). Then for any field k (of characteristic two), we have

ph.dim(kG) ≤ ∑_{i=1}^{i=r} ph.dim(kH_{i}).

Corollary 4.7.6. Let G be a finite 2-group which is not elementary abelian. Let H_{1}, …, H_{r} be index two subgroups in G with corresponding degree one cohomology classes ξ_{1}, … ξ_{r} ∈ H^{1}(G, 𝔽_{2}) composing to zero in H^{r}(G, 𝔽_{2}). Then

ph.index(G) ≤ ∑_{i=1}^{i=r} ph.index(H_{i}).

Suppose G is a finite 2-group of 2-rank at most two and order larger than four. Then any subgroup of G of order larger than four is not elementary abelian. Moreover the subgroups of order four of G have finite phantom index: the phantom index of the cyclic group is one (Theorem 4.6.9) and that of the Klein four group ℤ/2×ℤ/2 is either two or three (Theorem 4.6.12). Therefore, by applying Corollary 4.7.6 repeatedly, we obtain a finite upper bound for ph.index(G). We have thus obtained.

Theorem 4.7.7. If a finite 2-group has 2-rank at most two, then it has finite phantom index.

We are going to apply Corollary 4.7.6 to the following 2-groups:

• The abelian groups C_{2^{n−1},2} = ℤ/2^{n−1} × ℤ/2 with n ≥ 2. (Theorem 4.7.8) More generaly, the abelian groups C_{2^{m},2^{n}} = ℤ/2^{m} × ℤ/2^{n} with m, n ≥ 1. (Theorem 4.7.9)

• The dihedral groups D_{2^{n}} with n ≥ 2. (Theorem 4.7.10)

• The quaternion groups Q_{2^{n}} with n ≥ 3. (Theorem 4.7.11)

• The semidihedral groups SD_{2^{n}} with n ≥ 4. (Theorem 4.7.12)

The facts about the index two subgroups of D_{2^{n}}, Q_{2^{n}}, and SD_{2^{n}} that we use can be found in Aschbacher’s book [3, Chapter 8, Exercise 3].

Theorem 4.7.8. Let C_{2^{n−1},2} be the abelian group ℤ/2^{n−1} ×ℤ/2 with n ≥ 2. We have

ph.index(C_{2^{n−1},2}) ≤ 3·2^{n−2}.

Proof. We set d_{n} = ph.index(C_{2^{n−1},2}). For n ≥ 3, the mod 2 cohomology ring of C_{2^{n−1},2} is

H^{∗}(C_{2^{n−1},2}, 𝔽_{2}) = 𝔽_{2}[α, β, γ]/(α^{2})

with deg(α) = deg(β) = 1 and deg(γ) = 2. See for instance in Benson’s book [7] Proposition 3.5.5 and the Remark following Theorem 3.5.6. The degree one cohomology class α corresponds to the (naturally embedded) copy of C_{2^{n−2},2} in C_{2^{n−1},2}. Since α^{2} = 0, we have d_{n} ≤ 2d_{n−1} for n ≥ 3. It follows that d_{n} ≤ 2^{n−2}d_{2} for n ≥ 3. By Theorem 4.6.12, d_{2} ≤ 3. So d_{n} ≤ 3·2^{n−2}. Theorem 4.7.9. Let C_{2^{m},2^{n}} be the abelian group ℤ/2^{m} × ℤ/2^{n} with m, n ≥ 1. We have

ph.index(C_{2^{m},2^{n}}) ≤ 3·2^{m+n−2}.

Proof. We set d_{m,n} = ph.index(C_{2^{m},2^{n}}). For m, n ≥ 2, the mod 2 cohomology ring of C_{2^{m},2^{n}} is

H^{∗}(C_{2^{m},2^{n}}, 𝔽_{2}) = 𝔽_{2}[α, β, γ, δ]/(α^{2}, β^{2})

with deg(α) = deg(β) = 1 and deg(γ) = deg(δ) = 2. See again Proposition 3.5.5 and the Remark following Theorem 3.5.6 in Benson’s book [7]. The degree one cohomology classes α and β correspond to the (naturally embedded) copies of C_{2^{m−1},2^{n}} and C_{2^{m},2^{n−1}} in C_{2^{m},2^{n}}. If m = 1, n ≥ 2, we have the group C_{2,2^{n}} with mod 2 cohomology ring

H^{∗}(C_{2,2^{n}}, 𝔽_{2}) = 𝔽_{2}[α, β, δ]/(β^{2})

where deg(α) = deg(β) = 1 and deg(δ) = 2, and the degree one cohomology class β corresponds to the naturally embedded subgroup C_{2,2^{n−1}} in C_{2,2^{n}}. In either case, the relation β^{2} = 0 implies that we have d_{m,n} ≤ 2d_{m,n−1} for n ≥ 2. It follows that d_{m,n} ≤ 2^{n−1}d_{m,1} for n ≥ 2. By Theorem 4.7.8, d_{m,1} ≤ 3·2^{m−1}. So d_{n,m} ≤ 3·2^{m+n−2}.

Theorem 4.7.10. Consider the dihedral group D_{2^{n}} with n ≥ 2. We have

ph.index(D_{2^{n}}) ≤ 3·2^{n−2}.

Proof. We set d_{n} = ph.index(D_{2^{n}}). For n ≥ 3, the mod 2 cohomology ring of D_{2^{n}} is

H^{∗}(D_{2^{n}}, 𝔽_{2}) = 𝔽_{2}[α, β, γ]/(αβ)

with deg(α) = deg(β) = 1 and deg(γ) = 2. See for instance Theorem 2.7 in Chapter IV of Adem and Milgram’s book [1]. See also Carlson [16, Theorem 7.8]. The group D_{2^{n}} has three index two subgroups: one is isomorphic to the cyclic group ℤ/2^{n−1}, and the other two are isomorphic to D_{2^{n−1}}. The degree one cohomology classes α and β correspond to the two copies of D_{2^{n−1}} in D_{2^{n}}. Since αβ = 0, we have d_{n} ≤ 2d_{n−1} for n ≥ 3 as in the proof of Theorem 4.7.8. The conclusion follows. Theorem 4.7.11. For each n ≥ 3, the quaternion group Q_{2^{n}} has phantom index either two or three. Proof. We set d_{n} = ph.index(Q_{2^{n}}). For n ≥ 4, the mod 2 cohomology ring of Q_{2^{n}} is

H^{∗}(Q_{2^{n}}, 𝔽_{2}) = 𝔽_{2}[α, β, γ]/(αβ, α^{3} + β^{3})

with deg(α) = deg(β) = 1 and deg(γ) = 4. See for instance Lemma 2.11 in Chapter IV of Adem and Milgram’s book [1]. The group Q_{2^{n}} has three index two subgroups: one is isomorphic to the cyclic group ℤ/2^{n−1}, and the other two are isomorphic to Q_{2^{n−1}}. The degree one cohomology classes α and β correspond to the two subgroups Q_{2^{n−1}}. Since αβ = 0, we again have d_{n} ≤ 2d_{n−1} for n ≥ 3, and we could conclude as in Theorems 4.7.8 and 4.7.10. But here, we proceed differently in order to obtain the very small bound given by the Theorem. The degree one cohomology class ε = α + β corresponds to the cyclic group ℤ/2^{n−1}. And we have ε^{3} = 0. So we obtain

d_{n} ≤ 3·ph.index(ℤ/2^{n−1}) = 3.

We now treat the case n = 3. The mod 2 cohomology ring of Q_{8} is

H^{∗}(Q_{8}, 𝔽_{2}) = 𝔽_{2}[α, β, γ]/(α^{2} + αβ + β^{2}, α^{2}β + αβ^{2})

with deg(α) = deg(β) = 1 and deg(γ) = 4. See Exemple 1.9 in Chapter V or Lemma 2.10 in Chapter IV of Adem and Milgram’s book [1]. The degree one cohomology classes α, β, α + β correspond to the three copies of the cyclic group ℤ/4 in Q_{8}. And we have αβ(α + β) = 0. So d_{n} ≤ 3·ph.index(ℤ/4) = 3.

Theorem 4.7.12. Consider the semidihedral group SD_{2^{n}} with n ≥ 4. We have

ph.index(SD_{2^{n}}) ≤ 3(1 + 2^{n−3}).

Proof. For n ≥ 4, the mod 2 cohomology ring of SD_{2^{n}} is

H^{∗}(SD_{2^{n}}, 𝔽_{2}) = 𝔽_{2}[α, β, γ, δ]/(α^{3}, αβ, αγ, γ^{2} + δβ^{2})

with deg(α) = deg(β) = 1, deg(γ) = 3, and deg(δ) = 4. See Section 15, especially Theorem 15.5, of Benson and Carlson [9]. The cohomology was originaly calculated by Evens and Priddy [27]. The group SD_{2^{n}} has three index two subgroups. They are isomorphic to ℤ/2^{n−1}, D_{2^{n−1}}, and SD_{2^{n−1}}. The degree one cohomology classes α, β, and ε = α + β correspond respectively to the subgroups D_{2^{n−1}}, Q_{2^{n−1}}, and ℤ/2^{n−1}. There are at least three relations we can use, namely

α^{3} = 0 , αβ = 0 , α^{2}ε = 0.

It turns out that the second one gives the best bound. It yields the inequality

ph.index(SD_{2^{n}}) ≤ ph.index(D_{2^{n−1}}) + ph.index(Q_{2^{n−1}}) ≤ 3·2^{n−3} + 3 = 3(1 + 2^{n−3}).

For the sole reason that the phantom index for the quaternion groups is so constrained, we think that it is appropriate to ask the question that was already posed for the Klein four group (Question 4.6.13).

Question 4.7.13. Let G be the quaternion group Q_{8}, or more generally a quaternion group Q_{2^{n}} with n ≥ 3. Is the phantom index of G equal to two or three ? Can we have ph.dim(kG) = 3 (where k is an uncountable field of characteristic two) ?

The Extended Milnor Sequence

We consider the phantom spectral sequence

E_{2}^{pq} = \begin{matrix}lim \\ ←------- \end{matrix}_{α}^{p}Ext_{Λ}^{q}(M_{α}, N) ⟹ Ext_{Λ}^{n}(M, N) = H^{n}.

This spectral sequence has been studied by Christensen [19] in relation to purity and phantom maps in the derived category of a ring. The result about the phantom filtration is from his work. The five term fundamental exact sequence for this spectral sequence turns out to be an extension of the Milnor sequence established under certain countability hypotheses in [31, Theorem 2] and Proposition 2.1.1. We quite explicitly describe and study the transgression d_{2}^{01}. All this is used to prove that there can exist a direct system of maps in the stable module category which does not lift to the module category. Finally, we present an even longer extension of the Milnor sequence, and an application thereof. Most of the material in this chapter is the subject of our joint article [15] with Benson.

In all of this chapter, the ring Λ is a self-injective finite-dimensional algebra.

5.1 The phantom spectral sequence

Let {M_{α}}_{α} be a direct system of finite-dimensional Λ-modules. We consider the associated standard pure-projective resolution

⋯ ⟶^{∂_{n+2}} R_{n+1} ⟶^{∂_{n+1}} R_{n} ⟶^{∂_{n}} R_{n−1} ⟶^{∂_{n−1}} ⋯ ⟶^{∂_{2}} R_{1} ⟶^{∂_{1}} R_{0} ⟶^{∂_{0}} M ⟶ 0

of the direct limit M = \begin{matrix}lim \\ -------→ \end{matrix}M_{α} (see Proposition-Definition 3.2.8). The resolution is the Yoneda splice of the short pure-exact sequences

0 ⟶ K_{p+1} ⟶ R_{p} ⟶ K_{p} ⟶ 0 (K_{0} = M).

Let N be a module.

We have the spectral sequence

E_{2}^{pq} = \begin{matrix}lim \\ ←------- \end{matrix}_{α}^{p}Ext_{Λ}^{q}(M_{α}, N) ⟹ Ext_{Λ}^{n}(M, N) = H^{n}.

In Christensen [19, Section 9.1], this spectral sequence is constructed as the one associated to the exact couple

\xymatrix{D_{1} \ar[rr]^{i_{1}} &  & D_{1} \ar[dl]^{j_{1}} \\  & E_{1} \ar[ul]^{k_{1}}}

where

D_{1}^{pq} = Ext_{Λ}^{q}(K_{p}, N) and E_{1}^{pq} = Ext_{Λ}^{q}(R_{p}, N) ,

and the maps

i_{1}^{pq} : D_{1}^{pq} ⟶ D_{1}^{p−1,q+1}, j_{1}^{pq} : D_{1}^{pq} ⟶ E_{1}^{p,q}, and k_{1}^{pq} : E_{1}^{pq} ⟶ D_{1}^{p+1,q}

are Ext_{Λ}^{q}(−, N)-induced by the universal phantom map K_{p−1} ⟶ Ω^{−1}K_{p}, the pure epimorphism R_{p} ⟶ K_{p}, and the pure monomorphism K_{p+1} ⟶ R_{p} respectively.

Let

Ext_{Λ}^{n}(M, N) = P^{0}Ext_{Λ}^{n}(M, N) ⊇ P^{1}Ext_{Λ}^{n}(M, N) ⊇ ⊇ P^{2}Ext_{Λ}^{n}(M, N) ⊇ ⋯ ⊇ P^{n−1}Ext_{Λ}^{n}(M, N) ⊇ P^{n}Ext_{Λ}^{n}(M, N) be the resulting filtration of Ext_{Λ}^{n}(M, N). With n = p + q, we have

P^{p}Ext_{Λ}^{n}(M, N) = Im(D_{1}^{pq} ⟶^{i_{1}^{pq}} D_{1}^{p−1,q+1} ⟶^{i_{1}^{p−1,q+1}} ⋯ ⟶^{i^{2,n−2_{1}}} D_{1}^{1,n−1} ⟶^{i_{1}^{1,n−1}} D_{1}^{0,n}).

So in the commutative square (where q ≥ 1)

\xymatrix{D_{1}^{pq} = Ext_{Λ}^{q}(K_{p}, N) \ar[dr]^{≅} \ar[rrr]^{i_{1}^{1,n−1}i^{2,n−2_{1}}···i_{1}^{p−1,q+1}i_{1}^{pq}} &  &  & Ext_{Λ}^{n}(M, N) = D_{1}^{0,n} \ar[dl]_{≅} \\  & \underline{Hom}_{Λ}(Ω^{−p}K_{p}, Ω^{−n}N) \ar[r] & \underline{Hom}_{Λ}(M, Ω^{−n}N)}

the bottom map is \underline{Hom}_{Λ}(−, Ω^{−n}N)-induced by the universal p-fold phantom map M ⟶ Ω^{−p}K_{p}.

So for n ≥ 1, the resulting filtration of \underline{Hom}_{Λ}(M, Ω^{−n}N) is

\underline{Hom}_{Λ}(M, Ω^{−n}N) = \underline{Ph}_{Λ}^{0}(M, Ω^{−n}N) ⊇ \underline{Ph}_{Λ}(M, Ω^{−n}N) ⊇ ⊇ \underline{Ph}_{Λ}^{2}(M, Ω^{−n}N) ⊇ ⋯ ⊇ \underline{Ph}_{Λ}^{n−1}(M, Ω^{−n}N) ⊇ \underline{Ph}_{Λ}^{n}(M, Ω^{−n}N) .

In Jensen [40, Théorème 4.2], the spectral sequence is constructed from a bicomplex. With this approach, we are able to describe the transgression d_{2}^{01} quite explicitly (see Proposition 5.2.1).

Let the following be an injective resolution of N.

0 ⟶ N ⟶^{∂^{−1}} J^{0} ⟶^{∂^{0}} J^{1} ⟶^{∂^{1}} J^{2} ⟶^{∂^{2}} J^{3} ⟶^{∂^{3}} ⋯

Then the bicomplex is

E_{0}^{pq} = Hom_{Λ}(R_{p}, J^{q})

where the horizontal and vertical differentials are

d_{h}^{pq} = Hom_{Λ}(∂_{p+1}, J^{q}) : E_{0}^{pq} ⟶ E_{0}^{p+1,q}

and

d_{v}^{pq} = (−1)^{p}Hom_{Λ}(R_{p}, ∂^{q}) : E_{0}^{pq} ⟶ E_{0}^{p,q+1}

respectively. (We use the Cartan-Eilenberg sign convention.)

With the five-term exact sequence with ΩN in place of N, we obtain: Theorem 5.1.1. Let Λ be a finite-dimensional self-injective algebra over a field k. Let {M_{α}}_{α} be a direct system of finite-dimensional Λ-modules with direct limit M = \begin{matrix}lim \\ -------→ \end{matrix}_{α}M_{α}. Let N be a Λ-module. The following is an exact sequence:

0 ⟶ \underline{Ph}_{Λ}(M, N) ⟶ \underline{Hom}_{Λ}(M, N) ⟶ \begin{matrix}lim \\ ←------- \end{matrix}_{α}\underline{Hom}_{Λ}(M_{α}, N) ⟶^{d_{2}^{01}}

⟶^{d_{2}^{01}} Pext_{Λ}^{2}(M, ΩN) ⟶ \underline{Ph}_{Λ}^{2}(M, Ω^{−1}N) ⟶ 0 .

This sequence generalizes the Milnor exact sequence of [31, Theorem 2] and Proposition 2.1.1 obtained with the assumption that M is countable-dimensional. We refer to it as the Extended Milnor Sequence.

The sequence shows that lifting filtered systems with terminal object {{M_{α}}_{α}; N} from the stable module category StMod(Λ) to the module category Mod(Λ) depends upon the vanishing of the transgression d_{2}^{01} or, equivalently, the faithfulness of the natural (forgetful) functor Pext_{Λ}^{2} → Ext_{Λ}^{2}. This in turn depends upon the existence of nonvanishing two fold phantom maps.

Of course, if \underline{Ph}_{Λ}^{2}(M, Ω^{−1}N) = 0 and Pext_{Λ}^{2}(M, ΩN) ≠ 0, then there are systems {{M_{α}}_{α}; N} in StMod(Λ) which do not lift to Mod(Λ). However, we still do not whether the transformation Pext_{Λ}^{2} → Ext_{Λ}^{2} can vanish while Pext_{Λ}^{2} is nonzero. In Section 5.3, we show how to construct direct systems {{M_{α}}_{α}; N} in StMod(Λ) which do not lift to Mod(Λ) as long as the module M has pure-projective dimension at least two and has no phantom self-maps. But first, we need to better understand the transgression d_{2}^{01}. We investigate it in the next section.

The term \begin{matrix}lim \\ ←------- \end{matrix}_{α}\underline{Hom}_{Λ}(M_{α}, N) in the sequence does not really depend on the choice of a direct system {M_{α}}_{α} with direct limit M. It actually is the space

Ind \underline{Hom}_{Λ}(M, N) of morphisms M → N in the Ind completion of the category StMod(Λ). We can now restate Theorem 5.1.1 in the following more intrinsic manner. Theorem 5.1.2. Let Λ be a finite-dimensional self-injective algebra over a field k. Let M and N be Λ-modules. The following is an exact sequence:

0 ⟶ \underline{Ph}_{Λ}(M, N) ⟶ \underline{Hom}_{Λ}(M, N) ⟶ Ind \underline{Hom}_{Λ}(M, N) ⟶^{d_{2}^{01}}

⟶^{d_{2}^{01}} Pext_{Λ}^{2}(M, ΩN) ⟶ \underline{Ph}_{Λ}^{2}(M, Ω^{−1}N) ⟶ 0 .

Some people, including myself, Benson, and Christensen, have considered investigating the category obtained by quotienting out the ideal of phantom maps. The subject is still more than embryonic. Here we merely rewrite the Extended Milnor Sequence from that perspective. The notation \underline{\underline{Hom}}_{Λ}(−, −) is for the space of morphisms is this new category. Proposition 5.1.3. The following is an exact sequence:

0 → \underline{\underline{Hom}}_{Λ}(M, N) → Ind \underline{Hom}_{Λ}(M, N) ⟶^{d_{2}^{01}} Pext_{Λ}^{2}(M, ΩN) → \underline{Ph}_{Λ}^{2}(M, Ω^{−1}N) → 0.

5.2 The transgression d_{2}^{01}

Let \{f_{α}\}_{α} be an element of \begin{matrix}lim \\ ←------- \end{matrix}_{α}\underline{Hom}_{Λ}(M_{α}, N). It can be represented by a module homomorphism f : R_{0} = _{⨁α}M_{α} ⟶ N. The map R_{1} ⟶^{∂_{1}} R_{0} ⟶^{f} N is projective. Let π_{N} : P_{N} ⟶ N be the projective cover of N. We have the commutative diagram

\xymatrix{ & R_{2} \ar[r]^{∂_{2}} \ar@{.>}[d]_{h} & R_{1} \ar[r]^{∂_{1}} \ar@{-->}[d]_{g} & R_{0} \ar[d]_{f} \\ 0 \ar[r] & ΩN \ar[r]^{⊆} & P_{N} \ar[r]^{−π_{N}} & N \ar[r] & 0.}

It shows that d_{h}^{01}(f) = d_{v}^{10}(g) and d_{h}^{10}(g) = h; here f and h should be regarded as maps R_{0} ⟶ N ⊆ J^{1} and R_{2} ⟶ ΩN ⊆ J^{0} = P_{N} respectively. So we have obtained:

Proposition 5.2.1. The module homomorphism h = {h_{αβγ}}_{α≤β≤γ} above represents the element d_{2}^{01}(\{f_{α}\}_{α}) of Pext_{Λ}^{2}(M, ΩN) ≅ \begin{matrix}lim \\ ←------- \end{matrix}_{α}^{2}Hom_{Λ}(M_{α}, ΩN).

We now present a construction that we will use to get an interesting commutative square involving the transgression d_{2}^{01} (Theorem 5.2.2).

We have the following commutative diagram with exact rows and columns.

[figure: 0; 0; 0; 0; K_{2}; j; P; π; Ω^{−1}K_{2}; η; 0; 0; R_{1}; _{\binom{∂_{1}}{j̃}}; R_{0} ⊕ P; (λ μ); Y; ζ; 0; 0; K_{1}; R_{0}; M; 0; 0; 0; 0]

In this diagram, the map j : K_{2} ⟶ P is the injective hull of K_{2} and the map j̃: R_{1} ⟶ P extends it.

By applying the functor \underline{Hom}_{Λ}(−, N) to the distinguished triangle

R_{1} ⟶ R_{0} ⟶ Y ⟶ Ω^{−1}R_{1}

we get that the space \begin{matrix}lim \\ ←------- \end{matrix}_{α}\underline{Hom}_{Λ}(M_{α}, N) is the cokernel of the map

\underline{Hom}_{Λ}(Ω^{−1}R_{1}, N) ⟶ \underline{Hom}_{Λ}(Y, N).

The space Pext_{Λ}^{2}(M, ΩN) is the cokernel of the map

Hom_{Λ}(R_{1}, ΩN) ⟶ Hom_{Λ}(K_{2}, ΩN),

and hence also of the map

\underline{Hom}_{Λ}(R_{1}, ΩN) ⟶ \underline{Hom}_{Λ}(K_{2}, ΩN).

We note that in the notation of Proposition 5.2.1, the morphism h factors as

R_{2} ⟶ K_{2} ⟶^{h′} ΩN

and the morphism h′ is a representative in Hom_{Λ}(K_{2}, ΩN) of d_{2}^{01}({f_{α}}_{α}).

Theorem 5.2.2. The following square is commutative:

\xymatrix{\underline{Hom}_{Λ}(Y, N) \ar[r] \ar[d] & \underline{Hom}_{Λ}(K_{2}, ΩN) \ar[d] \\ \underleftarrow{lim}_{α}\underline{Hom}_{Λ}(M_{α}, N) \ar[r]^{d_{2}^{01}} & Pext_{Λ}^{2}(M, ΩN)}

Proof. Let u be a morphism Y ⟶ N. Then we have the commutative diagram on the left-hand side below.

\xymatrix{K_{2} \ar[r]^{j} \ar@{.>}[dd]_{v} & P \ar[r]^{π} \ar@{-->}[dd]_{w_{1}} & Ω^{−1}K_{2} \ar[d]^{η} & K_{2} \ar[r] \ar[dd]_{v} & R_{1} \ar[r]^{∂_{1}} \ar[dd]_{w_{2}} & R_{0} \ar[d]^{λ} \\  &  & Y \ar[d]^{u} &  &  & Y \ar[d]^{u} \\ ΩN \ar[r]^{⊆} & P_{N} \ar[r]^{π_{N}} & N & ΩN \ar[r]^{⊆} & P_{N} \ar[r]^{−π_{N}} & N}

The morphism v : K_{2} ⟶ ΩN represents the image of the morphism u under the map

\underline{Hom}_{Λ}(Y, N) ⟶ \underline{Hom}_{Λ}(K_{2}, ΩN) ⟶ Pext_{Λ}^{2}(M, ΩN).

Let w_{2} : R_{1} \underrightarrow{j̃} P ⟶^{w_{1}} P_{N}. Using the 3 × 3 commutative diagram constructed earlier in this section, we get the one on the right-hand side above. It shows that the morphism v : K_{2} ⟶ ΩN also represents the image of the morphism u under the map

\underline{Hom}_{Λ}(Y, N) ⟶ \begin{matrix}lim \\ ←------- \end{matrix}_{α}\underline{Hom}_{Λ}(M_{α}, N) ⟶^{d_{2}^{01}} Pext_{Λ}^{2}(M, ΩN).

5.3 Lifting direct systems from the stable module

category

We are now ready to show how to find a system in StMod(Λ) which does not lift to Mod(Λ). The notations are from the previous sections.

Theorem 5.3.1. Suppose the module M has pure-projective dimension at least two and has no nonzero phantom self-maps. Then there exists a direct system of maps {M_{α} → Y }_{α} in StMod(Λ) which does not lift to Mod(Λ).

Proof. Such a system has to be an element of \begin{matrix}lim \\ ←------- \end{matrix}_{α}\underline{Hom}_{Λ}(M_{α}, Y) whose image via the transgression d_{2}^{01} is a nonzero element of Pext_{Λ}^{2}(M, ΩY). Consider the commutative diagram below,

\xymatrix{\underline{Hom}_{Λ}(Y, Y) \ar[r] \ar[d] & \underline{Hom}_{Λ}(K_{2}, ΩY) \ar[d] & \underline{Hom}_{Λ}(K_{2}, K_{2}) \ar[l] \ar[d] \\ \underleftarrow{lim}_{α}\underline{Hom}_{Λ}(M_{α}, Y) \ar[r]^{d_{2}^{01}} & Pext_{Λ}^{2}(M, ΩY) & Pext_{Λ}^{2}(M, K_{2}) \ar[l]}

and take the image of Id_{Y} in \begin{matrix}lim \\ ←------- \end{matrix}_{α}\underline{Hom}_{Λ}(M_{α}, N). Its image via the transgression d_{2}^{01} is also obtained by mapping Id_{K_{2}} from the upper right-hand corner. The image of Id_{K_{2}} in Pext_{Λ}^{2}(M, K_{2}) is the canonical degree two pure extension of M by K_{2}; it is non-split (as a pure extension) because the module M has pure-projective dimension at least two. So we obtain that the image of Id_{K_{2}} in Pext_{Λ}^{2}(M, ΩY) is nonzero by argueing that the map Pext_{Λ}^{2}(M, K_{2}) → Pext_{Λ}^{2}(M, ΩY) is injective. The sequence

0 ⟶ Ω^{−1}K_{2} ⟶^{η} Y ⟶^{ζ} M ⟶ 0

in the 3×3 diagram of Section 5.2 is pure-exact; the map M ⟶ Ω^{−2}K_{2} completing it to the right into an exact triangle is phantom – specifically, it is the universal two-fold phantom map out of M. Applying the functor Pext_{Λ}^{∗}(M, −) thereupon, we get the exact sequence

Pext_{Λ}^{1}(M, ΩM) → Pext_{Λ}^{2}(M, K_{2}) → Pext_{Λ}^{2}(M, ΩY).

Now Pext_{Λ}^{1}(M, ΩM) ≅ \underline{Ph}_{Λ}(M, M) = 0.

Corollary 5.3.2. Suppose the module M has pure-projective dimension at least two and has no nonzero phantom self-maps. Then there exists a degree two non-split pure extension of M which is split as an ordinary degree two extension of M. In other words, the natural (forgetful) map Pext_{Λ}^{2}(M, −) → Ext_{Λ}^{2}(M, −) can be non-injective.

Proof. From the Extended Milnor Sequence (Theorem 5.1.1), we have the exact sequence

\underline{Hom}_{Λ}(M, Y) ⟶ \begin{matrix}lim \\ ←------- \end{matrix}_{α}\underline{Hom}_{Λ}(M_{α}, Y) ⟶^{d_{2}^{01}} Pext_{Λ}^{2}(M, ΩY) ⟶ Ext_{Λ}^{2}(M, ΩY).

The first map is not surjective, whence the second map is nonzero, and the third map is not injective.

If a module M has pure-projective dimension at least two and is pure-injective, then it satisfies the hypothesis of Theorem 5.3.1. Our joint paper [14] with Benson provides examples of such modules. By Theorems 5.1.2 and 5.4.6 of this paper, (see also Theorems 6.3.2 and 6.4.2 in this thesis) we can take for Λ the group algebra kV_{4}, where V_{4} is the Klein four group ℤ/2 × ℤ/2 and k is an uncountable field of characteristic 2, and for M Rickard’s module F(U), where U is the collection of all proper closed homogeneous subvarieties of the cohomology variety V_{E}(k) = 𝔸^{2}(k). The next chapter discusses the pure homological properties of Rickard’s idempotent modules in more details.

5.4 An even longer Milnor sequence

The Extended Milnor Sequence of Theorem 5.1.1 is a direct application of the wellknown five term fundamental sequence

\xymatrix{0 \ar[r] & E_{2}^{10} \ar[r] & H^{1} \ar[r] & E_{2}^{01} \ar[r]^{d_{2}^{01}} & E_{2}^{20} \ar[rr] \ar@{->>}[dr] &  & H^{2} \\  &  &  &  &  & F^{2}H^{2} \ar@{^{(}->}[ur]}

associated to any first quadrant cohomology spectral sequence

E_{2}^{pq} ⟹ H^{n}.

Observe that we also have the following exact sequences

0 ⟶ F^{2}H^{2} ⟶ F^{1}H^{2} ⟶ E_{∞}^{11} ⟶ 0,

0 ⟶ E_{∞}^{11} ⟶ E_{2}^{11} ⟶^{d_{2}^{11}} E_{2}^{30},

E_{2}^{11} ⟶^{d_{2}^{11}} E_{2}^{30} ⟶ E_{3}^{30} ⟶ 0;

and we have E_{∞}^{11} = E_{3}^{11}.

So we have the following eight term exact sequence

[figure: 0; E_{2}^{10}; H^{1}; E_{2}^{01}; d_{2}^{01}; E_{2}^{20}; F^{1}H^{2}; F^{2}H^{2}]

[figure: F^{1}H^{2}; E_{2}^{11}; d_{2}^{11}; E_{2}^{30}; E_{3}^{30}; 0; E_{3}^{11}]

The reason why this sequence is less popular might be that when dealing with a spectral sequence, one is usually interested in knowing the space H^{n}, not a filtered or graded version of it. But in our case, the (phantom) filtration is what we are interested in.

Let’s specify some of the terms in the eight term sequence for the phantom spectral sequence

E_{2}^{pq} = \begin{matrix}lim \\ ←------- \end{matrix}_{α}^{p}Ext_{Λ}^{q}(M_{α}, N) ⟹ Ext_{Λ}^{n}(M, N) = H^{n}.

E_{2}^{10} = \begin{matrix}lim \\ ←------- \end{matrix}^{1}Hom_{Λ}(M_{α}, N) ≅ Pext_{Λ}^{1}(M, N) ≅ \underline{Ph}_{Λ}(M, Ω^{−1}N)
H^{1}  = Ext_{Λ}^{1}(M, N) ≅ \underline{Hom}_{Λ}(M, Ω^{−1}N)
E_{2}^{01} = \begin{matrix}lim \\ ←------- \end{matrix}Ext_{Λ}^{1}(M_{α}, N) ≅ \begin{matrix}lim \\ ←------- \end{matrix}\underline{Hom}_{Λ}(M_{α}, Ω^{−1}N)
E_{2}^{20} = \begin{matrix}lim \\ ←------- \end{matrix}^{2}Hom_{Λ}(M_{α}, N) ≅ Pext_{Λ}^{2}(M, N)
F^{2}H^{2} = P^{2}Ext_{Λ}^{2}(M, N) ≅ \underline{Ph}_{Λ}^{2}(M, Ω^{−2}N)
F^{1}H^{2} = P^{1}Ext_{Λ}^{2}(M, N) ≅ \underline{Ph}_{Λ}(M, Ω^{−2}N) ≅ Pext_{Λ}^{1}(M, Ω^{−1}N)
≅ \begin{matrix}lim \\ ←------- \end{matrix}^{1}Hom_{Λ}(M_{α}, Ω^{−1}N)
E_{2}^{11} = \begin{matrix}lim \\ ←------- \end{matrix}^{1}Ext_{Λ}^{1}(M_{α}, N) ≅ \begin{matrix}lim \\ ←------- \end{matrix}^{1}\underline{Hom}_{Λ}(M_{α}, Ω^{−1}N)
E_{2}^{30} = \begin{matrix}lim \\ ←------- \end{matrix}^{3}Hom_{Λ}(M_{α}, N) ≅ Pext_{Λ}^{3}(M, N)
H^{2}  = Ext_{Λ}^{2}(M, N) ≅ \underline{Hom}_{Λ}(M, Ω^{−2}N)

We would like to note in particular the following result.

Theorem 5.4.1. Let Λ be a finite-dimensional self-injective algebra over a field k. Let {M_{α}}_{α} be a direct system of finite-dimensional Λ-modules, with direct limit M = \begin{matrix}lim \\ -------→ \end{matrix}_{α}M_{α}. Let N be a Λ-module. There is an exact sequence

0 ⟶ \underline{Ph}_{Λ}^{2}(M, N) ⟶ \begin{matrix}lim \\ ←------- \end{matrix}^{1}Hom_{Λ}(M_{α}, ΩN) ⟶ \begin{matrix}lim \\ ←------- \end{matrix}^{1}\underline{Hom}_{Λ}(M_{α}, ΩN).

Now we put two final remarks.

Remark 5.4.2. We have

E_{n+1}^{n,0} = E_{∞}^{n,0} ≅ P^{n}Ext_{Λ}^{n}(M, N) ≅ \underline{Hom}_{Λ}(M, Ω^{−n}N).

So we have the exact sequence

E_{n}^{0,n−1} ⟶^{d_{n}^{0,n−1}} E_{n}^{n,0} ⟶ \underline{Hom}_{Λ}^{n}(M, Ω^{−n}N)

where the map d_{n}^{0,n−1} : E_{n}^{0,n−1} ⟶ E_{n}^{n,0} is the transgression.

Remark 5.4.3. From the inverse system of exact sequences

0 ⟶ PHom_{Λ}(M_{α}, N) ⟶ Hom_{Λ}(M_{α}, N) ⟶ \underline{Hom}_{Λ}(M_{α}, N) ⟶ 0,

we obtain for every i ≥ 0 the exact sequence

\begin{matrix}lim \\ ←------- \end{matrix}^{i}PHom_{Λ}(M_{α}, N) ⟶ \begin{matrix}lim \\ ←------- \end{matrix}^{i}Hom_{Λ}(M_{α}, N) ⟶ \begin{matrix}lim \\ ←------- \end{matrix}^{i}\underline{Hom}_{Λ}(M_{α}, N).

From the inverse system of exact sequences

0 ⟶ Hom_{Λ}(M_{α}, N) ⟶ Hom_{Λ}(M_{α}, J_{N}) ⟶ PHom_{Λ}(M_{α}, Ω^{−1}N) ⟶ 0,

we obtain for every i ≥ 1 the isomorphisms

\begin{matrix}lim \\ ←------- \end{matrix}^{i}PHom_{Λ}(M_{α}, Ω^{−1}N) ⟶^{≅} \begin{matrix}lim \\ ←------- \end{matrix}^{i+1}Hom_{Λ}(M_{α}, N) ≅ Pext_{Λ}^{i+1}(M, N).

So we have for every i ≥ 1 the exact sequence

Pext_{Λ}^{i+1}(M, N) ⟶ Pext_{Λ}^{i}(M, Ω^{−1}N) ⟶ \begin{matrix}lim \\ ←------- \end{matrix}^{i}\underline{Hom}_{Λ}(M_{α}, Ω^{−1}N).

In case i = 1, this sequence is the portion

E_{2}^{20} ⟶ F^{1}H^{2} ⟶ E_{2}^{11}

of the eight term exact sequence.

Purity, phantom maps, and

Rickard’s idempotent modules

In this chapter, we discuss some pure homological properties of Rickard’s idempotent modules established in Benson and Gnacadja [14, Sections 5.1, 5.3, 5.4]. The idemptotent modules E and F were introduced by Rickard in [53]. They have since then been used in virtually every work dealing with infinite-dimensional modular representations of finite groups.

Throughout this chapter, G denotes a finite group and k a field of prime characteristic.

6.1 Rickard’s idempotent modules

We are going to summarize the fundamental facts about these modules (at least those we immediately need) in Theorem 6.1.2 below. We need start with some preparation with terminology, notations, and certain basic facts.

Definition 6.1.1. A thick subcategory of stmod(kG) is one which is full, triangulated, and closed under taking direct summands. A tensor-ideal subcategory of stmod(kG) is a thick subcategory C such that if M is in C and N is in stmod(kG), then M⊗_{k}N is in C. If C is a subcategory of StMod(kG), an object N in StMod(kG) is said to be C-local if for every object M in C, \underline{Hom}_{kG}(M, N) = 0. Associated to a tensor-ideal subcategory C of stmod(kG) are two important full subcategories of StMod(kG): the category C^{→} whose objects are the direct limits of direct systems of objects in C, and the category C^{⊥} of C-local objects.

Theorem 6.1.2 (Rickard). Let C be a tensor-ideal subcategory of the stable category stmod(kG) of finite-dimensional modules. There exists an essentially unique exact triangle

T_{C} : E_{C} ⟶^{ε} k ⟶^{η} F_{C} ⟶^{δ} Ω^{−1}E_{C}

in StMod(kG) where the module E_{C} is in C^{→}, i.e. is stably isomorphic to a direct limit of modules in C, and the module F_{C} is in C^{⊥}, i.e. is C-local. The map ε : E_{C} ⟶ k is the universal map among maps from objects of C^{→} to k. The map η : k ⟶ F_{C} is the universal map among maps from k to objects of C^{⊥}. The modules E_{C} and F_{C} are idempotent, i.e. there are stable isomorphisms E_{C} ⊗_{k} E_{C} ≅ E_{C} and F_{C} ⊗_{k} F_{C} ≅ F_{C}. Also the module E_{C} ⊗_{k} F_{C} is projective.

The modules E and F are typically infinite-dimensional.

The tensor-ideal subcategories of stmod(kG) most dealt with in modular representation theory are the ones associated to (collections of) subvarieties of the cohomology variety V_{G}(k). See Theorem 6.1.5 below.

The cohomology variety V_{G}(k) is the homogeneous prime ideal spectrum of the commutative ring H^{∗}(G, k)/\sqrt{0}, where \sqrt{0} is the nilradical. Equivalently, it is the homogeneous prime ideal spectrum of the commutative ring H^{#}(G, k) defined as follows:

H^{#}(G, k) = \begin{cases}H^{∗}(G, k) = ⨁_{i=0}^{∞} H^{i}(G, k) & if p = 2, \\ H^{even}(G, k) = ⨁_{i=0}^{∞} H^{2i}(G, k) & if p > 2. \end{cases}

The cohomology variety V_{G}(k) has an affine space structure. This variety is actually the variety of the trivial kG-module k. The variety V_{G}(M) of an arbitrary finitedimensional kG-module M is defined as follows.

Let I_{G}(M) be the annihilator of the H^{∗}(G, k)-module Ext_{kG}^{∗}(M, M). The variety V_{G}(M) is the homogeneous prime ideal spectrum of the commutative ring H^{∗}(G, k)/\sqrt{I_{G}(M)}. It is a closed homogeneous subvariety of V_{G}(k).

The study of varieties in relation to cohomology was initiated by Quillen in [51, 52]. The subject has evolved thanks to major inputs from several people, including (in alphabetical order) but not limited to Alperin, Avrunin, Benson, Carlson, Chouinard, Evens, Quillen, Scott, Venkov. The reader is invited to consult the comprehensive expositions of Benson [8, Chapter 5] and Evens [26, Chapter 8] when reference is needed. Benson, Carlson and Rickard attempted a definition of varieties for infinitedimensional modules in [11]. They successfully did so in [12]. The notation for the variety of (not assumed to be finite-dimensional) module M is V_{G}(M). The variety V_{G}(M) is actually a set of closed homogeneous irreducible nonzero subvarieties of the cohomology variety V_{G}(k). If M is finite-dimensional then the relationship between the varieties V_{G}(M) and V_{G}(M) is that V_{G}(M) is the set of all closed homogeneous irreducible nonzero subvarieties of V_{G}(M). In particular, the variety V_{G}(k) of the trivial module is the collection of all closed homogeneous irreducible nonzero subvarieties of the cohomology variety V_{G}(k).

There is another notion of variety specific to elementary abelian groups, the notion of rank variety. The rank variety V_{E}^{rk}(M) of a kE-module M, where E is a finite elementary abelian p-group, plays the exact same role as the variety V_{E}(M). Moreover, for an arbitrary finite group G and a kG-module M, the variety V_{G}(M) (or V_{G}(M) when appropriate) is coherently covered by the varieties the modules obtained by restricting the kG-module M to the elementary abelian p-subgroups of G.

We need some preparation in order to introduce the rank variety. Let E = ⟨g_{1}⟩×⋯×⟨g_{r}⟩ ≅ \underbrace{ℤ/p × ⋯ × ℤ/p}_{r} be an elementary abelian p-group of rank r. Let K be a field extension of k.

For every nonzero r-tuple λ = (λ_{1}, …, λ_{r}) in V_{E}(K) = 𝔸^{r}(K), the element u_{λ} = 1+λ_{1}(g_{1}−1)+⋯+λ_{r}(g_{r}−1) of the group algebra KG has multiplicative order p. So we have in KG the subalgebra K⟨u_{λ}⟩, group algebra over K of the group ⟨u_{λ}⟩ ≅ ℤ/p. The algebra K⟨u_{λ}⟩ does not depend directly on the r-tuple λ = (λ_{1}, …, λ_{r}) ∈ V_{E}(K) = 𝔸^{r}(K), but rather on the projective line defined by it. So to every projective line λ ∈ ℙ(V_{E}(K)) = ℙ^{r−1}(K) is associated a so-called shifted K-subalgebra Γ_{λ}.

Let V be a closed homogeneous irreducible nonzero subvariety of V_{E}(k). A generic point of V over K is a point in ℙ(V_{E}(K)), i.e., a projective line in V_{E}(K), whose closure in V_{E}(K) = 𝔸^{r}(K) intersects V_{E}(k) = 𝔸^{r}(k) at V. If the field extension K ⊇ k has transcendence degree at least r −1, then there are generic points over K for all closed homogeneous irreducible nonzero subvarieties of V_{E}(k).

Definition 6.1.3. Let E be an elementary abelian p-group of rank r and k be a field of characteristic p. Let M be a kE-module. Let K be a field extension of k of transcendence degree at least r − 1. The rank variety V_{E}^{rk}(M) of M consists of the closed homogeneous irreducible nonzero subvarieties V of V_{E}(k) with generic points λ over K such that the module (K ⊗_{k} M)↓_{Γ_{λ}}^{KE} is not free.

We now have reasonable background on varieties to return to their relation to thick and tensor-ideal subcategories of the stable category of finite-dimensional modules.

Definition 6.1.4. A set of subvarieties X of V_{G}(k) is be said to be closed under specialization provided that given subvarieties U and U′ in V_{G}(k), if U ∈ X and U′ ⊆ U, then U′ ∈ X.

The following is from [13, Theorem 3.4].

Theorem 6.1.5 (Benson - Carlson - Rickard). Given a tensor-ideal subcategory C of stmod(kG), the set X (C) of the varieties of modules in C is closed under specialization and finite union. Given a set X of closed homogeneous irreducible nonzero subvarieties of V_{G}(k), closed under specialization and finite union, the full subcategory C(X) of stmod(kG), whose objects are all finite-dimensional modules with variety in X, is a tensor-ideal subcategory.

We can now reformulate Rickard’s Theorem 6.1.2 as follows.

Theorem 6.1.6 (Rickard). Let X be a collection of closed homogeneous irreducible nonzero subvarieties of V_{G}(k), closed under specialization and finite union. There exists an essentially unique exact triangle

T (X) : E(X) ⟶^{ε} k ⟶^{η} F(X) ⟶^{δ} Ω^{−1} E(X)

in StMod(kG) where the modules E(X) and F(X) satisfy

V_{G}(E(X)) = X and V_{G}(F(X)) = V_{G}(k) \ X .

If V is a closed homogeneous irreducible nonzero subvarieties of V_{G}(k), the notations E(V), F(V), etc, will stand for E(X_{V}), F(X_{V}), etc, where X_{V} is the set of all closed homogeneous irreducible nonzero subvarieties of V.

If {α_{i}}_{i} is a set of positive degree homogeneous elements of the cohomology ring H^{∗}(G, k), the notations E({α_{i}}_{i}), F({α_{i}}_{i}), etc, will stand for E(V({α_{i}}_{i})), F(V({α_{i}}_{i})), etc, where V({α_{i}}_{i}) is the set of closed homogeneous irreducible nonzero subvarieties of V_{G}(k) defined by the set {α_{i}}_{i}. In particular, there are modules E(α) and F(α) associated to any positive degree homogeneous element of the cohomology ring H^{∗}(G, k). Modules such as F(α) were used in Section 2.2 to provide examples of phantom maps.

6.2 Pure-injective Rickard’s module E

Theorem 6.2.1. Let G be a finite group of p-rank two and let k be an algebraically closed field of characteristic p. Let ζ be a nonzero homogeneous positive degree cohomology class in H^{∗}(G, k). Then the module E(ζ) is pure-injective.

We will prove this Theorem by applying Proposition 3.6.8. We will construct a representative of E(ζ) in Mod(kG) which has a locally nilpotent endomorphism with finite-dimensional kernel. It will follow that the module E(ζ) is Σ-pure-injective.

The terminology we introduce here is inspired by Krause [44, Section 8]. Let (M_{i})_{i≥1} be a sequence of finitely generated modules and σ_{i} : M_{i} ⟶ M_{i+1} and τ_{i+1} : M_{i+1} ⟶ M_{i}, i ≥ 1 be module homomorphisms. Let

σ_{i}^{j} : M_{i} ⟶^{σ_{i}} M_{i+1} ⟶^{σ_{i+1}} ⋯ ⟶^{σ_{i+j−1}} M_{i+j},

τ_{i+j}^{j} : M_{i+j} ⟶^{τ_{i+j}} M_{i+j−1} ⟶^{τ_{i+j−1}} ⋯ ⟶^{τ_{i+1}} M_{i}.

Definition 6.2.2.

• We say that the data (M_{i}, σ_{i}, τ_{i})_{i≥1} form a tube in mod(kG) provided that the sequences

0 ⟶ M_{i} ⟶^{σ_{i}^{j}} M_{i+j} ⟶^{τ_{i+j}^{i}} M_{j} ⟶ 0

are exact, and the squares

\xymatrix{M_{i} \ar[r]^{τ_{i}} \ar[d]_{σ_{i}} & M_{i−1} \ar[d]^{σ_{i−1}} \\ M_{i+1} \ar[r]_{τ_{i+1}} & M_{i}}

are commutative. The direct limit M in Mod(kG) of the direct sequence

M_{1} ⟶^{σ_{1}} M_{2} ⟶^{σ_{2}} M_{3} ⟶^{σ_{3}} ⋯

will be called the direct limit of the tube.

• Regard the modules and maps under consideration as objects and morphisms in stmod(kG). We say that the data (M_{i}, σ_{i}, τ_{i})_{i≥1} form a tube in stmod(kG) provided that we have exact triangles

M_{i} ⟶^{σ_{i}^{j}} M_{i+j} ⟶^{τ_{i+j}^{i}} M_{j} ⟶ Ω^{−1}M_{i}

and (stably) commutative squares

\xymatrix{M_{i} \ar[r]^{τ_{i}} \ar[d]_{σ_{i}} & M_{i−1} \ar[d]^{σ_{i−1}} \\ M_{i+1} \ar[r]_{τ_{i+1}} & M_{i}.}

“The” (homotopy) direct limit M in StMod(kG) of the direct sequence

M_{1} ⟶^{σ_{1}} M_{2} ⟶^{σ_{2}} M_{3} ⟶^{σ_{3}} ⋯

will be called the direct limit of the tube.

We discuss properties of tubes in Appendix C. We show in particular that a tube in stmod(kG) lifts, in the appropriate sense, to a tube in mod(kG) (Theorem C.5). With this fact and Proposition 3.6.8, we are done with the proof of Theorem 6.2.1 if we prove the following.

Theorem 6.2.3. Let G be a finite group of p-rank two and let k be an algebraically closed field of characteristic p. Let V be a closed homogeneous subvariety of dimension one in the cohomology variety V_{G}(k). Then the module E(V) is “the” direct limit in StMod(kG) of a tube in stmod(kG).

We need the following notation. If α is a nonzero homogeneous element of H^{∗}(G, k) of degree m, we let L_{α} denote the kernel of the representative map α̂ : Ω^{m}k → k. These modules play various important roles; see [8, Section 5.9] for more on them.

The following remark summarizes a particular feature of the Octahebral Axiom for triangulated categories, which we will use twice in the proof of Theorem 6.2.3.

Remark 6.2.4. The Octahebral Axiom for triangulated categories asserts that a commutative diagram of four exact triangles

\xymatrix{ &  & C′ \ar[dr] \\  & B \ar[ur] \ar[dr] &  & B′ \ar[ddr] \\  &  & C \ar[ur] \ar[drr] \\ A \ar[uur] \ar[urr] &  &  &  & A′}

can be formed whenever the commutative diagram is given with three exact triangles. On another hand, an exact triangle is determined up to isomorphism by any one of its maps. So a commutative diagram as above is completely determined when two maps not on the same exact triangle are given. In particular, if a commutative diagram as above is given with one arrow not specified, then there exists a unique arrow that fills the gap.

Proof (Theorem 6.2.3). By Quillen’s Dimension Theorem, the cohomology variety V_{G}(k) has dimension two. So the subvariety V has codimension one, and hence it is defined by a single homogeneous cohomology class ζ of positive degree, n say. We have an exact sequence

0 ⟶ L_{ζ} ⟶ Ω^{n}k ⟶^{ζ̂}k ⟶ 0,

from which we derive the following direct sequence of exact triangles in stmod(kG)

[figure: Ω^{−n}L_{ζ}; σ_{1}; k; ζ̂; Ω^{−n}k; ζ̂; Ω^{−n−1}L_{ζ}; Ω^{−2n}L_{ζ^{2}}; σ_{2}; k; ζ̂^{2}; Ω^{−2n}k; ζ̂; Ω^{−2n−1}L_{ζ^{2}}; Ω^{−3n}L_{ζ^{3}}; σ_{3}; k; ζ̂^{3}; Ω^{−3n}k; ζ̂; Ω^{−3n−1}L_{ζ^{3}}; ⋮; ⋮; ⋮; ⋮]

Taking the direct limit in Mod(kG), we obtain the exact triangle

E(V) → k → F(V) → Ω^{−1} E(V).

Now we choose a homogeneous cohomology class η which defines a subvariety of V_{G}(k) intersecting V at a subvariety of dimension zero. We may assume that ζ and η have been chosen to have same degree n. Then η generates the periodicity of L_{ζ^{i}} in the sense of Benson [8, Section 5.10]: tensoring the module L_{ζ^{i}} over k with the short exact sequence

0 ⟶ L_{η} ⟶ Ω^{n}k ⟶^{η̂} k ⟶ 0

yields the stable isomorphism η̂ ⊗ Id_{L_{ζ^{i}}} : Ω^{n}L_{ζ^{i}} ⟶^{≅} L_{ζ^{i}}. So with M_{i} = L_{ζ^{i}}, we have the direct sequence of exact triangles in stmod(kG):

[figure: M_{1}; σ_{1}; k; ζ̂; Ω^{−n}k; ζ̂; Ω^{−1}M_{1}; M_{2}; σ_{2}; k; ζ̂^{2}; Ω^{−2n}k; ζ̂; Ω^{−1}M_{2}; M_{3}; σ_{3}; k; ζ̂^{3}; Ω^{−3n}k; ζ̂; Ω^{−1}M_{3}; ⋮; ⋮; ⋮; ⋮]

Taking the direct limit in Mod(kG), we obtain the exact triangle E(V) → k → F(V) → Ω^{−1} E(V).

We consider in particular the three exact triangles

M_{i} ⟶ k ⟶^{ζ̂^{i}} Ω^{−in}k , M_{i+j} ⟶ k ⟶^{ζ̂^{i+j}} Ω^{−(i+j)n}k , M_{j} ⟶ Ω^{−in}k ⟶^{ζ̂^{j}} Ω^{−(i+j)n}k ,

the latter being obtained by applying Ω^{−in} to the exact triangle

M_{j} ⟶ k ⟶^{ζ̂^{j}} Ω^{−jn}k .

Then by Remark 6.2.4 (Octahedral Axiom), the map σ_{i}^{j} : M_{i} ⟶ M_{i+j} fits in an exact triangle

M_{i} ⟶^{σ_{i}^{j}} M_{i+j} ⟶^{τ_{i+j}^{i}} M_{j},

which fits in the commutative diagram below.

\xymatrix{ &  & M_{j} \ar[dr] \\  & M_{i+j} \ar@{-->}[ur]^{τ_{i+j}^{i}} \ar[dr] &  & Ω^{−in}k \ar[ddr]^{ζ̂^{j}} \\  &  & k \ar[ur]^{ζ̂^{i}} \ar[drr]_{ζ̂^{i+j}} \\ M_{i} \ar[uur]^{σ_{i}^{j}} \ar[urr] &  &  &  & Ω^{−(i+j)n}k}

Remark 6.2.4 also gives the following commutative diagram.

\xymatrix{ &  & M_{i−1} \ar[dr]^{σ_{i−1}} \\  & M_{i} \ar@{-->}[ur]^{τ_{i}} \ar[dr]^{σ_{i}} &  & M_{i} \ar[ddr]^{τ_{i}^{i−1}} \\  &  & M_{i+1} \ar@{-->}[ur]^{τ_{i+1}} \ar[drr]_{τ_{i+1}^{i}} \\ M_{1} \ar[uur]^{σ_{1}^{i−1}} \ar[urr]^{σ_{1}^{i}} &  &  &  & M_{1}}

In particular, we have the commutative squares

\xymatrix{M_{i} \ar[r]^{τ_{i}} \ar[d]_{σ_{i}} & M_{i−1} \ar[d]^{σ_{i−1}} \\ M_{i+1} \ar[r]_{τ_{i+1}} & M_{i}}

So we have a tube (M_{i}, σ_{i}, τ_{i})_{i≥1} in stmod(kG), and the module E(V) is its direct limit.

In our paper [14] with Benson, we ask whether the Rickard’s module E(V) is always pure-injective. Krause shows in [43] that this is true in an appropriate category (Theorem 6.2.5 below).

If C is a thick subcategory of stmod(kG), then the localizing subcategory generated by C in StMod(kG) is the full triangulated subcategory C^{⊕} which is the closure of C under taking (infinite) direct sums and completing exact triangles.

Theorem 6.2.5 (Krause). Let C be a thick subcategory of the stmod(kG). The module E_{C} is endofinite and pure-injective in the localizing subcategory C^{⊕} in StMod(kG) generated by C.

6.3 The pure-injective Klein modules

What we refer to as the Klein modules are simply the modules over the Klein four group V_{4} = ℤ/2 × ℤ/2.

The classification of indecomposable pure-injective Klein modules can be read off from the classification of indecomposable pure-injective Kronecker modules. The latter is provided by Jensen and Lenzing [41, Corollary 8.59]. They attribute the classification to Geigle. We are able to do this because the canonical functors between the category of Kronecker module and the category of Klein modules preserve indecomposability and by Proposition 3.7.3, preserve pure-injectivity. In the following theorem ([14, Theorem 5.1.1]), we combine Rickard’s notation and the terminology of Jensen and Lenzing [41, Chaptery 8], so as to emphasize the correspondence.

Theorem 6.3.1. Let k be a field of characteristic two and G be the Klein four group V_{4} = ℤ/2 × ℤ/2. The indecomposable pure-injective kV_{4}-modules are:

(i) The indecomposable finite-dimensional modules.

(ii) The V -adic modules E(V), where V is a projective line in the cohomology variety V_{G}(k) = 𝔸^{2}(k).

(iii) The Prüfer modules associated the projective lines V in the cohomology variety V_{G}(k) = 𝔸^{2}(k), the k-duals E(V)^{∗}.

(iv) The rational function field F(U), where U is the set of projective lines in the cohomology variety V_{G}(k) = 𝔸^{2}(k).

The correspondences understood in (ii) and (iii) above are just a matter of translating the relevant module constructions from Kronecker modules to Klein modules. The correspondence of (iv) is less trivial. Below is what we are really claiming.

Theorem 6.3.2. Let k be a field of characteristic two and G be the Klein four group V_{4} = ℤ/2 × ℤ/2. Let U be the set of projective lines in the cohomology variety V_{G}(k) = 𝔸^{2}(k). The kG-module F corresponding to the Kronecker module

Φ = \xymatrix{k(X) \ar[r]^{1} \ar[r]_{X} & k(X)}

is a representative in Mod(kG) of Rickard’s module F(U).

Note that we already know from Example 3.6.5 that the module F is pure-injective.

Proof. We will show this by using a strong version of Theorem 6.1.6 with rank varieties. Specifically, we will show that there is a short exact sequence

0 ⟶ k ⟶^{η} F ⟶^{δ}E′ ⟶ 0

and that V_{G}^{rk}(F) = {V_{G}(k)} and V_{G}^{rk}(E′) = U.

Let S be the socle of F, and H (for head) be a k(X)-complement of S in F. Both S and H are copies of the field k(X), and there are two nontrivial elements g_{1} and g_{2} in G such g_{1} − 1 acts as Id_{k(X)} : H ⟶^{=} S and g_{2} − 1 acts as XId_{k(X)} : H ⟶^{X·} S. Let η be the kG-module monomorphism k ⟶ k(X) = S ⊂ F sending 1 to 1. Let δ : F ⟶ E′ be its cokernel.

Let λ = [λ_{1}:λ_{2}] ∈ ℙ^{1}(k), let u_{λ} = 1 + λ_{1}(g_{1} − 1) + λ_{2}(g_{2} − 1) ∈ kG. Then u_{λ} − 1 sends H to S by multiplication by λ_{1} + λ_{2}X; this is an k(X)-isomorphism. Now note that the group ⟨u_{λ}⟩ is just a copy of ℤ/2. The way u_{λ} − 1 acts shows that the Γ_{λ}-module F↓_{Γ_{λ}}^{kG} is free; any k(X)-basis of H is a free Γ_{λ}-basis of F. And since the Γ_{λ}-module k is not free, the Γ_{λ}-module E′↓_{Γ_{λ}}^{kG} is not free. We conclude that the projective lines in 𝔸^{2}(k) = V_{G}(k) (i.e. the elements of U) are in V_{G}^{rk}(E′), and not in V_{G}^{rk}(F).

Let t be an indeterminate. Let λ = [1 : t] ∈ ℙ^{1}(k(t)). Then λ is a generic point for the variety V_{G}(k) over the field k(t). Indeed, the zero polynomial is the only polynomial in two variables over k which is zero on the projective line λ of 𝔸^{2}(k(t)) = V_{G}(k(t)). Let u_{λ} = 1 + (g_{1} − 1) + t(g_{2} − 1) ∈ k(t)G. Also consider the map

ψ : k(t)⊗_{k}S = k(t)⊗_{k}k(X) → k(t), f(t)⊗g(X) → f(t)g(1/t).

The element u_{λ} − 1 of k(t)G acts by sending k(t)⊗_{k}H into k(t)⊗_{k}S by multiplication by μ = 1⊗1 + t⊗X. By Lemma 6.3.3 below, this action is a k(t)-vector space isomorphism of k(t)⊗_{k}H onto Ŝ = Ker ψ. So we have in the Γ_{λ}-module (k(t)⊗_{k}F)↓_{Γ_{λ}}^{k(t)G} the free Γ_{λ}-submodule F̂ = (k(t)⊗_{k}H)⊕Ŝ. Moreover, the monomorphism k(t) ⊗ η : k(t) → k(t) ⊗_{k} S = k(t)⊗_{k}k(X) splits with splitting epimorphism ψ.

We conclude that (k(t) ⊗ F)↓_{Γ_{λ}}^{k(t)G}≅ k(t) ⊕ F̂ and (k(t) ⊗ E′)↓_{Γ_{λ}}^{k(t)G}≅ F̂. In particular, the Γ_{λ}-modules (k(t) ⊗ F)↓_{Γ_{λ}}^{k(t)G} and (k(t) ⊗ E′)↓_{Γ_{λ}}^{k(t)G} are respectively not free and free. We conclude that the variety V_{G}(k) is in V_{G}^{rk}(F) but not in V_{G}^{rk}(E′).

Since V_{G}^{rk}(k) = U ∪{V_{G}(k)}, we have obtained as promised that V_{G}^{rk}(F) = {V_{G}(k)} and V_{G}^{rk}(E′) = U.

Lemma 6.3.3. Let k be a field and x, y be two indeterminates over k. Let μ = x⊗y − 1⊗1 ∈ k(x)⊗_{k}k(y). Consider the ring homomorphism

ψ : k(x)⊗_{k}k(y) → k(x), f(x)⊗g(y) ↦ f(x)g(1/x).

Then multiplication by μ is a k(x)⊗_{k}k(y)-module isomorphism of k(x)⊗_{k}k(y) onto the ideal Ker ψ.

Proof. The ring k(x)⊗_{k}k(y) is an integral domain, being a subring of the fraction field k(x, y). So multiplication by μ is injective. We have ψ(μ) = 1_{k} − 1_{k} = 0, so μ(k(x)⊗_{k}k(y)) ⊆ Ker ψ. Let r = ∑_{i,j≥0}r_{ij}x^{i}⊗y^{j} ∈ k[x]⊗_{k}k[y] ⊂ k(x)⊗_{k}k(y) such that ψ(r) = 0. We have 0 = ψ(r) = ∑_{i,j≥0}r_{ij}x^{i−j}. So setting z_{ij} = x^{i}⊗y^{j} − x^{i−j}⊗1, we have r = ∑_{i,j≥0}r_{ij}z_{ij}. Let z̃_{ij} = ∑_{l=1}^{j} x^{i−l}⊗y^{j−l}. Then μz̃_{ij} = z_{ij}. So setting r̃ = ∑_{i,j≥0}r_{ij}z̃_{ij}, we have r = μr̃ ∈ μ(k(x)⊗_{k}k(y)). So we have obtained that

(Ker ϕ) ∩ (k[x]⊗_{k}k[y]) ⊆ μ(k(x)⊗_{k}k(y)).

The ring k(x)⊗_{k}k(y) is the localization of the ring k[x]⊗_{k}k[y] at the multiplicative set

{f(x)⊗g(y) : 0 ≠ f(x) ∈ k[x] and 0 ≠ g(y) ∈ k[y]}.

It follows that Ker ψ ⊆ μ(k(x)⊗_{k}k(y)). So μ(k(x)⊗_{k}k(y)) = Ker ψ.

The following theorem proposes another way to realize the module F(U). The notations are from Theorem 6.3.2.

Theorem 6.3.4. Let k be a field of characteristic two and a, b ∈ k, not both zero. Let G be the Klein four group V_{4} = ℤ/2 × ℤ/2. Consider the degree one cohomology element

ζ_{a,b} = bu + av ∈ H^{1}(G, k) ⊂ H^{∗}(G, k) = k[u, v].

Let L_{a,b} be the kernel of the representative kG-map

ζ̃_{a,b} : Ωk ⟶ k.

Then L_{a,b} is the Klein module corresponding to the kronecker module

Φ_{a,b} = \xymatrix{k(X) \ar[r]^{a} \ar[r]_{b} & k(X)} .

In particular, the module (L_{1,X}) ↓_{kG}^{k(X)G} is a representative in Mod(kG) of Rickard’s module F(U).

Proof. We choose two nontrivial element g, h ∈ G and a k-basis {u_{∗}, v_{∗}, z} in Ωk such that Ωk is represented by the following diagram

[figure: u_{∗}; v_{∗}; z; g − 1; h − 1]

and so that u, v ∈ H^{1}(G, k) are dual to u_{∗}, v_{∗} respectively with respect to the basis {u_{∗}, v_{∗}, z}. Set w_{a,b} = au_{∗} + bv_{∗}. Then ζ̃_{a,b}(w_{a,b}) = 0, and also ζ̃_{a,b}(z) = 0. So we have L_{a,b} = kw_{a,b}⊕kz as k-vector space. Moreover

(g − 1)w_{a,b} = az and (h − 1)w_{a,b} = bz; .

So L_{a,b} does correspond to Φ_{a,b}.

6.4 Pure-projective dimension of Rickard’s module F

Recall from Proposition 3.5.4 that if a module over an arbitrary ring is flat, then its pure-projective and projective dimensions are equal. We can not take advantage of this in a useful way directly, because over a finite-dimensional self-injective algebra, flat modules are projective. The following theorem of Benson and Gnacadja [14, Corollaries 4.4.4 and 4.4.3] proposes a way around.

Theorem 6.4.1. Let G be a finite p-group and k a field of characteristic p. Let M be a kG-module such that as a module over Ĥ^{∗}(G, k), Ĥ^{∗}(G, M) is nonprojective, flat, and has finite projective dimension. Then the pure-projective dimension of the kG-module M is equal to the projective dimension of the Ĥ^{∗}(G, k)-module Ĥ^{∗}(G, M). Moreover, the module M has a pure-projective resolution of minimal length where the pure-projective modules are stably isomorphic to direct sums of Ω-translates of k.

The following is an application of this theorem in a specific situation.

Theorem 6.4.2. Let E be an elementary abelian p-group of rank r ≥ 2, and k be an algebraically closed field of characteristic p and cardinality ℵ_{t}. Let U be the collection of all proper closed homogeneous irreducible subvarieties of the cohomology variety V_{E}(k) = 𝔸^{r}(k). The module F(U) has pure-projective dimension min(r, t + 1).

The actual thing to prove is:

Theorem 6.4.3. Let E be an elementary abelian p-group of rank r, with r ≥ 2, and k be an algebraically closed field of characteristic p and cardinality ℵ_{t}. Let U be the collection of all proper closed homogeneous irreducible subvarieties of the cohomology variety V_{E}(k) = 𝔸^{r}(k). As a module over the Tate cohomology Ĥ^{∗}(E, k), the Tate cohomology Ĥ^{∗}(E, F(U)) is flat and has projective dimension min(r, t + 1).

We reach this result after a quite elaborate process. Let’s start with a lemma that will be used twice.

Lemma 6.4.4. Let S be a ring. Assume that S = I⊕R where I is a two-sided ideal and R is a subring. Let X be a flat S-module. If Q_{∗} → X is an S-projective resolution of X, then Q_{∗}/IQ_{∗} → X/IX is an R-projective resolution of X/IX. In particular,

proj.dim_{R}(X/IX) ≤ proj.dim_{S}(X).

Suppose in addition that I ⊗_{R} X = 0. Then the R-module X is flat. If P_{∗} → X is an R-projective resolution of X, then S ⊗_{R} P_{∗} → X is an S-projective resolution of X. In particular,

proj.dim_{R}(X) ≤ proj.dim_{S}(X).

Proof. First, note that if Q is a projective S-module, then Q/IQ is a projective Rmodule. And if P is a projective R-module, then S ⊗_{R} P is a projective S-module. Let Q_{∗} → X be an S-projective resolution of X. Since X is S-flat, Q_{∗} → X is an S-pure-projective resolution of X (Proposition 3.5.4). So applying (S/I) ⊗_{S} −, we obtain the R-projective resolution Q_{∗}/IQ_{∗} → X/IX of X. Suppose in addition that I ⊗_{R} X = 0. To see that X is flat as an R-module, note that every R-module can be regarded as an S-module on which I acts as zero, and that under this assumption the tensor product of two R-modules is isomorphic to their tensor product as S-modules. The idea is that the passage from tensoring over R to tensoring over the larger ring S involves no new relations. Let P_{∗} → X is an R-projective resolution of X. This is an R-pure-projective resolution of X since X is R-flat. And we have S ⊗_{R} X ≅ X as S-modules. So applying S ⊗_{R} − to P_{∗} → X, we obtain the S-projective resolution S ⊗_{R} P_{∗} → X of X.

The cohomology ring of the group E when p = 2 is

H^{∗}(E, k) = k[u_{1}, …, u_{r}]

with u_{1}, …, u_{r} in degree one. For p > 2, have

H^{∗}(E, k) = ⋀_{k}(v_{1}, …, v_{r}) ⊗_{k} k[u_{1}, …, u_{r}]

where ⋀_{k}(v_{1}, …, v_{r}) is the exterior k-algebra over the degree one generators v_{1}, …, v_{r}, and the polynomial generators u_{1}, …, u_{r} have degree two (they are the Bockstein of v_{1}, …, v_{r}).

With the same notations, we have

Ĥ^{∗}(E, F(U)) ≅ \begin{cases}k(u_{1}, …, u_{r})_{gr} & if p = 2; \\ ⋀_{k}(v_{1}, …, v_{r}) ⊗_{k} k(u_{1}, …, u_{r})_{gr} & if p > 2. \end{cases}

The notation k(u_{1}, …, u_{r})_{gr} represents the graded ring of fractions in the indeterminates u_{1}, …, u_{r}, namely the subring of the full field of fractions k(u_{1}, …, u_{r}) consisting of the fractions whose denominators are (nonzero) homogeneous polynomials. The Tate cohomology Ĥ^{∗}(E, F(U)) above can be extracted from the work of Carlson, Donovan and Wheeler in [17] and that of Carlson and Wheeler in [18]. But see also Proposition 6.5.15.

The following result of Osofsky is instrumental.

Theorem 6.4.5 (Osofsky). Let k be a field of cardinality ℵ_{t}. The full field of fractions k(x_{1}, …, x_{r}) and the graded ring of fractions k(x_{1}, …, x_{r})_{gr}, as modules over the polynomial ring k[x_{1}, …, x_{r}], are flat and have projective dimension min(r, t+1).

We can now get close to Theorem 6.4.3.

Theorem 6.4.6. As a module over the ordinary cohomology ring H^{∗}(E, k), the Tate cohomology Ĥ^{∗}(E, F(U)) is flat and has projective dimension min(r, t + 1).

Proof. For p = 2, this follows directly from Osofsky’s Theorem 6.4.5. For p > 2, we argue that tensoring with the exterior algebra does not affect the projective dimension.

Specifically, set

\begin{aligned}X & = & Ĥ^{∗}(E, F(U)) ≅ ⋀_{k}(v_{1}, …, v_{r}) ⊗_{k} k(u_{1}, …, u_{r})_{gr}, \\ Y & = & k(u_{1}, …, u_{r})_{gr}, \\ S & = & Ĥ^{∗}(E, k) ≅ ⋀_{k}(v_{1}, …, v_{r}) ⊗_{k} k[u_{1}, …, u_{r}], \\ R & = & k[u_{1}, …, u_{r}], \\ I & = & ⟨v_{1}, …, v_{r}⟩ ⊂ S. \end{aligned}

If P_{∗} → Y is a projective resolution of Y over R, then ⋀_{k}(v_{1}, …, v_{r}) ⊗_{k} Q_{∗} → ⋀_{k}(v_{1}, …, v_{r})⊗_{k} Y = X is a projective resolution of X over ⋀_{k}(v_{1}, …, v_{r})⊗_{k} R = S. So

proj.dim_{S}(X) ≤ proj.dim_{R}(Y).

By Lemma 6.4.4, we have

proj.dim_{R}(X/IX) ≤ proj.dim_{S}(X).

Now we have X/IX ≅ Y as R-modules.

The next stage in the proof of Theorem 6.4.3 is the following result from Benson and Carlson [10] on the ring structure of the Tate cohomology ring of elementary abelian groups.

Theorem 6.4.7 (Benson and Carlson). Let E be an elementary abelian p-group of rank at least two, and k be a field of characteristic p. Then

Ĥ^{∗}(E, k) = H^{−}(E, k) ⊕ H^{∗}(E, k).

The negative Tate cohomology H^{−}(E, k) = ⨁_{i>0} Ĥ^{−i}(E, k) is a square-zero two-sided ideal. The ordinary cohomology H^{∗}(E, k) is a subring. We have

(H^{−}(E, k)·H^{∗}(E, k)) ∩ H^{∗}(E, k) = 0.

To complete the proof of Theorem 6.4.3, observe that since Ĥ^{∗}(E, F(U)) is divisible by the homogeneous elements of H^{∗}(E, k), the fact that

(H^{−}(E, k)·H^{∗}(E, k)) ∩ H^{∗}(E, k) = 0

implies that

H^{−}(E, k) ⊗_{H^{∗}(E,k)} Ĥ^{∗}(E, F(U)) = 0.

Also, the ring Ĥ^{∗}(E, F (U)) is the localization of H^{∗}(E, k) at the multiplicative subset of homogeous elements in the polynomial subring. Localizing Ĥ^{∗}(E, k) at this same multiplicative set again gives Ĥ^{∗}(E, F(U)). So Ĥ^{∗}(E, F(U)) is Ĥ^{∗}(E, k)-flat. Thus we have all the conditions of Lemma 6.4.4, whence the result of Theorem 6.4.3.

The fact that the module F(U) above has flat Tate cohomology is not a singularity, but belongs to a more general pattern which we develop in the next section.

6.5 Modules of Omega type: flat cohomology

The following terminology is from an unpublished work of Benson and Gnacadja.

Definition 6.5.1. We say that a kG-module M is of Omega type if it is the direct limit of a direct system of finite-dimensional modules, each of which is a direct sum of a projective module and Ω-translates of the trivial module k.

The Omega type property is obviously preserved by several module operations, some of which are listed in the following proposition.

Proposition 6.5.2. The tensor product (over the field k) of two modules of Omega type is of Omega type. The Ω-translates of a module of Omega type are of Omega type. The direct limit of a direct system of modules of Omega type is of Omega type.

We would like now to introduce a notion of infinite tensor product.

Let {M_{α}}_{α∈Υ} be a collection of modules indexed by a set Υ, with each module equipped with a distinguished module morphism ε_{α} : k ⟶ M_{α} from the trivial module k. Let P(Υ) be the collection of finite subsets of Υ. The set P(Υ), equipped with the inclusion, is a directed poset. For every σ ∈ P(Υ), let M_{σ} be the tensor product of the modules M_{α} with α ∈ σ. We take the tensor product of the maps ε_{α} : k ⟶ M_{α}, α ∈ σ, to obtain the distinguished map ε_{σ} : k ⟶ M_{σ}. Let σ, τ ∈ P(Υ), with σ ⊂ τ. Then, by tensoring the maps ε_{τ\σ} : k ⟶ M_{τ\σ} and Id_{M_{σ}} : M_{σ} ⟶ M_{σ}, we obtain a map λ_{σ,τ} : M_{σ} ⟶ M_{τ}. The very definition of the maps λ_{∗,∗} shows that if ρ, σ, τ ∈ P(Υ) satisfy ρ ⊂ σ ⊂ τ, then λ_{σ,τ}λ_{ρ,σ} = λ_{ρ,τ}. So we have the direct system {M_{σ}}_{σ∈P(Υ)} with maps λ_{σ,τ}.

Definition 6.5.3. The tensor product of the collection of modules {M_{α}}_{α∈Υ} is

⨂_{α∈Υ} M_{α} = \begin{matrix}lim \\ -------→ \end{matrix}_{σ∈P(Υ)}M_{σ}

where P(Υ) be the directed poset of finite subsets of Υ ordered by the inclusion, and for each σ ∈ P(Υ),

M_{σ} = ⨂_{α∈σ} M_{α}.

Remark 6.5.4. The maps ε_{σ} : k ⟶ M_{σ}, σ ∈ P(Υ), form a direct system of maps. So we have a distinguished map

ε_{Υ} : k ⟶ ⨂_{α∈Υ} M_{α}.

Remark 6.5.5. If the index set Υ is finite, then the tensor product defined here coincides with the usual tensor product.

Proposition 6.5.6. The tensor product of an (infinite) collection of modules of Omega type is of Omega type.

Let V be a homogeneous hypersurface in the cohomology variety V_{G}(k) defined by some nonzero ζ ∈ H^{n}(G, k), n > 0. The module F(V) is of Omega type, being the (homotopy) direct limit of the direct system

k ⟶^{ζ}Ω^{−n}k ⟶^{ζ}Ω^{−2n}k ⟶^{ζ}Ω^{−3n}k ⟶^{ζ}· · · .

Also, the module F(V) comes equipped with a distinguished map ε_{V} : k ⟶ F(V). So we can take the tensor product of an infinite family of such modules.

Theorem 6.5.7. Let V be a collection of homogeneous hypersurfaces and their subvarieties in the cohomology variety V_{G}(k). We have

F(V) = ⨂_{V ∈V} F(V).

In particular, the module F(V) is of Omega type.

Strictly speaking, we should only say that there is a stable isomorphism between the two modules. This theorem is essentially saying that the known fact that F(V ∪W) and F(V)⊗F(W) are stably isomorphic (Rickard [53, Theorem 7.5]) extends to infinite collections of varieties.

We can also provide an expression for the module E(V). But it is complicated and we do not see any immediate interest.

The following lemma is needed for the proof.

Lemma 6.5.8. Let V be a collection of homogeneous subvarieties in the cohomology variety V_{G}(k). Let C(V) be the thick subcategory of stmod(kG) of modules M such that the irreducible components of the variety V_{G}(M) are subvarieties of elements of V. (This notation generalizes the one of Theorem 6.1.5.) Then the subcategory C(V) equals the thick subcategory generated by the thick subcategories C(V), V ∈ V.

Proof. The result is known for finite collections V. This follows by induction from Proposition 7.3 of Rickard [53]. We need to show here that the category C(V) is contained in the thick subcategory generated by the categories C(V), V ∈ V; the reverse inclusion is trivial. But this follows from the result for finite collections of varieties. Indeed, if a finite-dimensional module M is in C(V), then V_{G}(M) ⊆ ⋃_{V ∈W} V for some finite subset W of V.

Proof (Theorem 6.5.7). We have the direct system of short exact sequences

0 ⟶ k ⟶^{ε_{W}} F(W) ⟶^{δ_{W}} Ω^{−1} E(W) ⟶ 0

indexed by the collection P(V) of finite subsets W of V. The direct limit is an exact sequence

0 ⟶ k ⟶^{ε_{V}} F ⟶^{δ_{V}} E′ ⟶ 0

where the the module F is precisely the tensor product ⨂_{V ∈V} F(V). For each W ∈ P(V), the module Ω^{−1} E(W) is in the category C(W)^{⟶}, and hence in the category C(V)^{⟶}. Therefore the module E′ is in the category C(V)^{⟶}. It remains to show that the module F is C(V)-local. For V ∈ V, let P(V, V) be the subset of P(V) consisting of the finite subsets of V containing V. Then P(V, V) is a cofinal directed subposet of P(V). So

F = \begin{matrix}lim \\ -------→ \end{matrix}_{W∈P(V,V)} F(W).

Now for every W ∈ P(V, V), the module F(W) is C(W)-local, and so in particular it is C(V)-local. It follows that the module F is C(V)-local. And since this is true for every V ∈ V, it follows from Lemma 6.5.8 that the module F is C(V)-local.

The following property of Omega type modules is one of the two for which they are of interest to us. Proposition 6.5.9. If a kG-module M is of Omega type, then its Tate cohomology Ĥ^{∗}(G, M) is flat as a module over Ĥ^{∗}(G, k). Proof. The Tate cohomology functor Ĥ^{∗}(G, −) commutes with direct limits. If a module A is a direct sum of a projective module and Ω-translates of the trivial module k, then its cohomology Ĥ^{∗}(G, A) is free as a module over Ĥ^{∗}(G, k). Now the direct limit of a direct system of free modules is flat.

Corollary 6.5.10. Let V be a collection of homogeneous hypersurfaces in the cohomology variety V_{G}(k). The Tate cohomology Ĥ^{∗}(G, F(V)) of Rickard’s module F(V), as a module over the Tate cohomology ring Ĥ^{∗}(G, k), is flat.

Using Theorem 6.5.7, we can provide a description of the Tate cohomology

Ĥ^{∗}(G, F(V)).

First, an easy lemma which will render immediate the case where V consists of a single hypersurface.

Lemma 6.5.11. Let R be a ring with identity. Let α be a central nonnilpotent element of R. Then the direct limit of the direct sequence

R ⟶^{α} R ⟶^{α} R ⟶^{α} ⋯

is the R-module R [\frac{1}{α}], localization of the ring R at the multiplicative set \{α, α^{2}, α^{3}, … \}. Considering the direct sequence indexed by 1, 2, 3, …, the structure map of index i is

R ⟶ R [\frac{1}{α}], r ⟼ \frac{r}{α^{i}} .

Theorem 6.5.12. Let V be a collection of homogeneous hypersurfaces in the cohomology variety V_{G}(k). Consider a collection of nonzero homogenous positive degree cohomology classes in H^{∗}(G, k) defining the elements of V. Then let Φ be the multiplicative subset of the ring H^{∗}(G, k) generated by the collection of cohomology classes. (The set Φ consists of the (finite) products of elements from the collection.) Then the Tate cohomology Ĥ^{∗}(G, F(V)) equals the localization of the Tate cohomology ring Ĥ^{∗}(G, k) at the multiplicative subset Φ.

Proof. First, consider a single hypersurface V in V_{G}(k) defined by the class α ∈ H^{n}(G, k); assume n even if p = char k > 2. The module F(V) is the direct limit of the sequence

k ⟶^{α} Ω^{−n}k ⟶^{α} Ω^{−2n}k ⟶^{α} Ω^{−3n}k ⟶^{α} · · · .

Its Tate cohomology Ĥ^{∗}(G, F(V)) is the direct limit of the direct sequence

Ĥ^{∗}(G, k) ⟶^{α} Ĥ^{∗}(G, Ω^{−n}k) ⟶^{α} Ĥ^{∗}(G, Ω^{−2n}k) ⟶^{α} Ĥ^{∗}(G, Ω^{−3n}k) · · · .

For each i ≥ 1, the Ĥ^{∗}(G, k)-module Ĥ^{∗}(G, Ω^{−in}k) is just a copy of Ĥ^{∗}(G, k) with degree shifted by +in. So by Lemma 6.5.11,

Ĥ^{∗}(G, F(V)) = Ĥ^{∗}(G, k)[α^{−1}].

Now the general situation. The module Ĥ^{∗}(G, F(V)) is the direct limit of the modules Ĥ^{∗}(G, k)[ζ^{−1}], ζ ∈ Φ, and this direct limit is the localization Ĥ^{∗}(G, k)[Φ^{−1}].

We are going to state a version of Theorem 6.5.12 which combines Tate and ordinary cohomologies in a useful way. To this end we first state the following result of Benson and Carlson. It it the combination of Lemma 2.1 and Theorem 3.1 of [10]. We earlier stated and used a specialized version (Theorem 6.4.7).

Theorem 6.5.13 (Benson and Carlson). Let G be finite group of p-rank at least two, and k be a field of characteristic p. If the cohomology ring H^{∗}(G, k) is Cohen-Macaulay, then

Ĥ^{∗}(G, k) = H^{−}(G, k) ⊕ H^{∗}(G, k).

The negative Tate cohomology H^{−}(G, k) = ⨁_{i>0} Ĥ^{−i}(G, k) is a square-zero two-sided ideal. The ordinary cohomology H^{∗}(G, k) is a subring. We have

(H^{−}(G, k)·H^{∗}(G, k)) ∩ H^{∗}(G, k) = 0.

Theorem 6.5.14. Let V be a collection of homogeneous hypersurfaces in the cohomology variety V_{G}(k). Consider a collection of nonzero homogenous positive degree cohomology classes in H^{∗}(G, k) defining the elements of V. Then let Φ be the multiplicative subset of the ring H^{∗}(G, k) generated by the collection of cohomology classes. (The set Φ consists of the (finite) products of elements from the collection.) Suppose that the group G has p-rank at least two, and that the cohomology ring H^{∗}(G, k) is Cohen-Macaulay. Then the Tate cohomology Ĥ^{∗}(G, F(V)) equals the localization of the ordinary cohomology ring Ĥ^{∗}(G, k) at the multiplicative subset Φ.

Proof. The localization annihilates the ideal H^{−}(G, k).

The most common groups with Cohen-Macaulay cohomology are the elementary abelian p-groups, but there are other groups. The following is a particular case of Theorem 6.5.14. We stated and used the result in Section 6.4.

Proposition 6.5.15. Let E be an elementary abelian p-group of rank r, with r ≥ 2, and k be an algebraically closed field of characteristic p and cardinality ℵ_{t}. Let U be the collection of all proper closed homogeneous irreducible subvarieties of the cohomology variety V_{E}(k) = 𝔸^{r}(k). If p = 2, then

Ĥ^{∗}(E, F(U)) = k(u_{1}, …, u_{r})_{gr}

with u_{1}, …, u_{r} in degree one. If p > 2, then

Ĥ^{∗}(E, F(U)) = ⋀_{k}(v_{1}, …, v_{r}) ⊗_{k} k(u_{1}, …, u_{r})_{gr}

where ⋀_{k}(v_{1}, …, v_{r}) is the exterior k-algebra over the degree one generators v_{1}, …, v_{r}, and u_{1}, …, u_{r} have degree two.

Proof. We know the cohomology of the group E. The multiplicative set here consists of the homogeneous positive degree elements in the polynomial subring k[u_{1}, …, u_{r}].

6.6 Modules of Omega type: all maps phantom

We discuss another interesting fact about Omega type modules. Specifically, we show a way to generalize the situation of Theorem 2.2.1 where all maps between two Rickard modules F of Omega type are phantom.

What makes this possible is the following property of Omega type modules and the morphisms between them.

Theorem 6.6.1. Let M be a kG-module of Omega type, displayed as such by a direct system {M_{α}}_{α}. Let N be an arbitrary kG-module. There is an isomorphism

\begin{matrix}lim \\ ←------- \end{matrix}\underline{Hom}_{kG}(M_{α}, N) ≅ Hom_{Ĥ^{∗}(G,k)}(Ĥ^{∗}(G, M), Ĥ^{∗}(G, N)).

In a more intrinsic manner,

Ind \underline{Hom}_{kG}(M, N) ≅ Hom_{Ĥ^{∗}(G,k)}(Ĥ^{∗}(G, M), Ĥ^{∗}(G, N)).

The reader shoud keep in mind that morphisms between cohomology modules are morphisms of graded modules.

Proof. First suppose that M = Ω^{n}k for some integer n. Then Ĥ^{∗}(G, M) = Ĥ^{∗}(G, Ω^{n}k) is a free rank one Ĥ^{∗}(G, k)-module with generator in degree n. So

Hom_{Ĥ^{∗}(G,k)}(Ĥ^{∗}(G, M), Ĥ^{∗}(G, N)) = Hom_{Ĥ^{∗}(G,k)}(Ĥ^{∗}(G, Ω^{n}k), Ĥ^{∗}(G, N))
≅ Hom_{k}(k, Ĥ^{n}(G, N))
≅ Ĥ^{n}(G, N)
≅ \underline{Hom}_{kG}(Ω^{n}k, N)
= \underline{Hom}_{kG}(M, N)

It readily follows from this that we also have the isomorphism

Hom_{Ĥ^{∗}(G,k)}(Ĥ^{∗}(G, M), Ĥ^{∗}(G, N)) ≅ \underline{Hom}_{kG}(M, N)

if the module M is a direct sum of a projective module and finitely many Ω-translates of the trivial module k. Now suppose M is displayed as an Omega type module by the direct system {M_{α}}_{α}. We have

Ĥ^{∗}(G, M) = \begin{matrix}lim \\ -------→ \end{matrix}Ĥ^{∗}(G, M_{α})

and then

Hom_{Ĥ^{∗}(G,k)}(Ĥ^{∗}(G, M), Ĥ^{∗}(G, N)) ≅ \begin{matrix}lim \\ ←------- \end{matrix}Hom_{Ĥ^{∗}(G,k)}(Ĥ^{∗}(G, M_{α}), Ĥ^{∗}(G, N))

≅ \begin{matrix}lim \\ ←------- \end{matrix}\underline{Hom}_{kG}(M_{α}, N).

Theorem 6.6.1 together with the Extended Milnor Sequence (Theorem 5.1.1) give the following.

Theorem 6.6.2. Let M and N be kG-modules, with M of Omega type. Then we have exact sequences

0 ⟶ \underline{Ph}_{kG}(M, N) ⟶ \underline{Hom}_{kG}(M, N) ⟶ Hom_{Ĥ^{∗}(G,k)}(Ĥ^{∗}(G, M), Ĥ^{∗}(G, N))

and

Hom_{Ĥ^{∗}(G,k)}(Ĥ^{∗}(G, M), Ĥ^{∗}(G, N)) ⟶ Pext_{kG}^{2}(M, ΩN) ⟶ \underline{Ph}_{kG}^{2}(M, Ω^{−1}N) ⟶ 0.

Corollary 6.6.3. Let M and N be kG-modules, with M of Omega type. If

Hom_{Ĥ^{∗}(G,k)}(Ĥ^{∗}(G, M), Ĥ^{∗}(G, N)) = 0,

then all maps M ⟶ N are phantom and the canonical map

Pext_{kG}^{2}(M, ΩN) ⟶ \underline{Ph}_{kG}^{2}(M, Ω^{−1}N)

is an isomorphism.

We now want to look for ways to realize the condition of Corollary 6.6.3 (in a manner which encompasses the situation of Theorem 2.2.1).

Lemma 6.6.4. Let R be a commutative ring, and M and N be R-modules. Let Φ be a nonempty subset in R such that for every α ∈ Φ, αM = M (the module M is divisible by α), and such that ⋂_{α∈Φ} αN = 0 (only zero is divisible in N by all elements of Φ). Then Hom_{R}(M, N) = 0.

Proof. Let f ∈ Hom_{R}(M, N). For every α ∈ Φ, f(M) = f(αM) = αf(M). So

Im f = ⋂_{α∈Φ} α Im f ⊆ ⋂_{α∈Φ} αN = 0.

Proposition 6.6.5. Let R be a commutative ring. Let Φ and Ψ be multiplicative subsets in R. Suppose that every couple (α, β) ∈ Φ × Ψ is a regular sequence in R. Suppose also that only zero is divisible in R by all elements of Φ (i.e. ⋂_{α∈Φ} αR = 0). Then

Hom_{R}(R[Φ^{−1}], R[Ψ^{−1}]) = 0,

where R[Φ^{−1}] and R[Ψ^{−1}] are the localizations of the ring R at the multiplicative subsets Φ and Ψ respectively.

Proof. The R-module M = R[Φ^{−1}] is obviously divisible by all elements of Φ. By Lemma 6.6.4, we will be done if we show that only zero is divisible in N = R[Ψ^{−1}] by all elements of Ψ. So let y ∈ ⋂_{α∈Φ} αR[Ψ^{−1}]. Let r ∈ R, β ∈ Ψ such that y = \frac{r}{β}, and for each α ∈ Φ, let r_{α} ∈ R, β_{α} ∈ Ψ such that y = α\frac{r_{α}}{β_{α}}. Then rβ_{α} = αβr_{α} for every α ∈ Φ. Since (α, β) is a regular sequence in R, we have r = r′_{α}α for some r′_{α} ∈ R. Hence r is divisible by α in R. This is true for every α ∈ Φ, so r = 0, whence y = 0.

Theorem 6.6.6. Let E be an elementary abelian p-group of rank r ≥ 2. Let U and V be a collection of hypersurfaces in the cohomological variety V_{E}(k) = 𝔸^{r}(k). Assume that U and V are coprime in the sense that if Φ and Ψ are collections of homogeneous elements of the polynomial subring of H^{∗}(E, k) defining U and V, then for every α ∈ Φ, β ∈ Ψ, α and β are relatively prime. Let t be an arbitrary integer. Then

Hom_{H^{∗}(G,k)}(Ĥ^{∗}(G, F(U)), Ĥ^{∗}(G, Ω^{t} F(V))) = 0.

A fortiori,

Hom_{Ĥ^{∗}(G,k)}(Ĥ^{∗}(G, F(U)), Ĥ^{∗}(G, Ω^{t} F(V))) = 0.

Proof. By Theorem 6.5.14, we have Ĥ^{∗}(G, F(U)) = H^{∗}(G, k)[Φ^{−1}], the localization of H^{∗}(G, k) at the multiplicative set generated by Φ. The Tate cohomology Ĥ^{∗}(G, Ω^{t} F(V)) is H(G, k)[Ψ^{−1}] with degree shifted by −t. With these details, the theorem becomes a direct application of Proposition 6.6.5. Corollary 6.6.7. Let E be an elementary abelian p-group of rank r ≥ 2. Let U and V be collections of hypersurfaces in the cohomological variety V_{E}(k) = 𝔸^{r}(k). Assume that U and V are coprime in the sense that if Φ and Ψ are collections of homogeneous elements of the polynomial subring of H^{∗}(E, k) defining U and V, then for every α ∈ Φ, β ∈ Ψ, α and β are relatively prime. Let t be an arbitrary integer. All maps F(U) ⟶ Ω^{t} F(V) are phantom. The canonical maps

Pext_{kE}^{1}(F(U), Ω^{t} F(V)) ⟶ Ext_{kE}^{1}(F(U), Ω^{t} F(V))

and

Pext_{kE}^{2}(F(U), Ω^{t−2} F(V)) ⟶ \underline{Ph}_{kE}^{2}(F(U), Ω^{t} F(V))

are isomorphisms.

One can probably state the theorem for a larger class of groups, but we limit ourselves to elementary abelian p-groups for simplicity.

This theorem generalizes part of Theorem 2.2.1. But here we do not have any interesting way of ascertaining that there are nonzero (phantom) maps. The latter statement in the theorem is interesting only if there are nonzero two-fold phantom maps. And this remains unknown at this time.

Appendix A

Contracting homotopy for the

standard pure resolution

This appendix provides the details about Proposition 3.2.7 and its proof. As in Section 3.2, we will use multicomponent indices. A hat over a component of a multicomponent index will mean that the component is skipped.

Throughout, we work over an arbitrary ring Λ, and all modules are left Λ-modules. We consider a direct system \{M_{α}\}_{α} of modules with direct limit M = \begin{matrix}lim \\ -------→ \end{matrix}M_{α}.

Given a (t + 1)-tuple (α_{0}, α_{1}, …, α_{t}) of indices, we let M_{α_{0}α_{1}...α_{t}} denote a copy of the module M_{α_{0}}. Then for any integer n ≥ −1 and index α, we set

R_{n;α} = ⨁_{α_{0}≤α_{1}≤···≤α_{n}≤α} M_{α_{0}α_{1}...α_{n}α}.

In particular,

R_{−1;α} = M_{α} and R_{0;α} = ⨁_{α_{0}≤α} M_{α_{0}α} = ⨁_{α_{0}≤α} M_{α_{0}}.

We define maps

j_{n,i;α_{0}α_{1}...α_{n}α} : M_{α_{0}α_{1}...α_{n}α} ⟶ M_{α_{0}α_{1}...α̂_{i}...α_{n}α}

for n ≥ 0, 0 ≤ i ≤ n, as follows:

\begin{aligned}1 ≤ i ≤ n & : & j_{n,i;α_{0}α_{1}...α_{n}α} & : & M_{α_{0}α_{1}...α_{n}α} & ⟶^{Id_{Mα_{0}}} & M_{α_{0}α_{1}...α̂_{i}...α_{n}α} \\ i = 0, n ≥ 1 & : & j_{n,0;α_{0}α_{1}...α_{n}α} & : & M_{α_{0}α_{1}...α_{n}α} & ⟶^{f_{α_{0}α_{1}}} & M_{α_{1}...α_{n}α} \\ i = 0, n = 0 & : & j_{0,0;α_{0}α} & : & M_{α_{0}α} & ⟶^{f_{α_{0}α}} & M_{α} \end{aligned}

Then we define maps

∂_{n;α_{0}α_{1}...α_{n}α} : M_{α_{0}α_{1}...α_{n}α} ⟶ ⨁_{0≤i≤n} M_{α_{0}α_{1}...α̂_{i}...α_{n}α} ↪ R_{n−1;α}

by

∂_{n;α_{0}α_{1}...α_{n}α} = ⨁_{0≤i≤n}(−1)^{i}j_{n,i;α_{0}α_{1}...α_{n}α}.

In a more explit manner, we have for n ≥ 1:

∂_{n;α_{0}α_{1}...α_{n}α} = (M_{α_{0}α_{1}...α_{n}α} ⟶^{f_{α_{0}α_{1}}} M_{α_{1}...α_{n}α}) +

+ ∑_{1≤i≤n}(−1)^{n} (M_{α_{0}α_{1}...α_{n}α} ⟶^{Id_{Mα_{0}}} M_{α_{0}α_{1}...α̂_{i}...α_{n}α});

and for n = 0:

∂_{0;α_{0}α} = (M_{α_{0}α} ⟶^{f_{α_{0}α}} M_{α}) .

The reader should think of the map ∂_{n;α_{0}α_{1}...α_{n}α} as performing alternative sums. Finally we define the maps

∂_{n;α} : R_{n;α} ⟶ R_{n−1;α}

by

∂_{n;α} = ⨁_{α_{0}≤α_{1}≤···≤α_{n}≤α} ∂_{n;α_{0}α_{1}...α_{n}α}.

On another hand, we define

s_{−1;α} : M_{α} ⟶^{Id_{Mα}} M_{αα} ↪ M_{0;α};

and for n ≥ 0,

s_{n;α_{0}α_{1}...α_{n}α} : M_{α_{0}α_{1}...α_{n}α} ⟶^{(−1)^{n−1}Id_{Mα_{0}}} M_{α_{0}α_{1}...α_{n}αα} ↪ R_{n+1;α}.

And we then define the maps

s_{n;α} : R_{n;α} ⟶ R_{n+1;α}

by

s_{n;α} = ⨁_{α_{0}≤α_{1}≤···≤α_{n}≤α} s_{n;α_{0}α_{1}...α_{n}α}.

Proposition A.1. The sequence

⋯ ⟶^{∂_{n+2;α}} R_{n+1;α} ⟶^{∂_{n+1;α}} R_{n;α} ⟶^{∂_{n;α}} R_{n−1;α} ⟶^{∂_{n−1;α}} ⋯ ⟶^{∂_{2;α}} R_{1;α} ⟶^{∂_{1;α}} R_{0;α} ⟶^{∂_{0;α}} M_{α} ⟶ 0

is a complex, i.e. we have

∂_{n;α}∂_{n+1;α} = 0

for every n ≥ 0. In addition, it admits the sequence of module homomorphisms s_{n;α} : R_{n;α} ⟶ R_{n+1;α}, n ≥ −1, as a contracting homotopy. In other words, we have

s_{n−1;α}∂_{n;α} + ∂_{n+1;α}s_{n;α} = Id_{R_{n;α}}

for every n ≥ 0.

The real difficulty in the proof is to carefully keep track of multicomponent indices. We will use the following notation. If x ∈ M_{α_{0}α_{1}...α_{n}α}, we will denote by ⟨x⟩_{α_{0}α_{1}...α_{n}α} the element of R_{n;α} whose component of index (α_{0}, α_{1}, …, α_{n}, α) is x and other components are zero.

Proof. We start with the proof that the sequence is a complex. We show that the composite ∂_{n;α}∂_{n+1;α} is zero on every summand M_{α_{0}α_{1}...α_{n}α_{n+1}α} of R_{n+1;α}. Let x ∈ M_{α_{0}α_{1}...α_{n}α_{n+1}α}.

∂_{n+1;α}(⟨x⟩_{α_{0}α_{1}...α_{n}α_{n+1}α}) = ⟨f_{α_{0}α_{1}}(x)⟩_{α_{1}...α_{n}α_{n+1}α} + ∑_{s=1}^{n+1}(−1)^{s}⟨x⟩_{α_{0}α_{1}...α̂_{s}...α_{n}α_{n+1}α}.

∂_{n;α}∂_{n+1;α}(⟨x⟩_{α_{0}α_{1}...α_{n}α_{n+1}α})

= ∂_{n;α}(⟨f_{α_{0}α_{1}}(x)⟩_{α_{1}...α_{n}α_{n+1}α}) + ∑_{s=1}^{n+1}(−1)^{s}∂_{n;α}(⟨x⟩_{α_{0}α_{1}...α̂_{s}...α_{n}α_{n+1}α})

= [⟨f_{α_{1}α_{2}}(f_{α_{0}α_{1}}(x))⟩_{α_{2}...α_{n}α_{n+1}α} + ∑_{s=2}^{n+1}(−1)^{s−1}⟨f_{α_{0}α_{1}}(x)⟩_{α_{1}α_{2}...α̂_{s}...α_{n}α_{n+1}α}]

+ [−∂_{1;α}(⟨x⟩_{α_{0}α_{2}...α_{n}α_{n+1}α}) + ∑_{s=2}^{n+1}(−1)^{s}∂_{n;α}(⟨x⟩_{α_{0}α_{1}α_{2}...α̂_{s}...α_{n}α_{n+1}α})]

= ⟨f_{α_{0}α_{2}}(x)⟩_{α_{2}...α_{n}α_{n+1}α} + ∑_{s=2}^{n+1}(−1)^{s−1}⟨f_{α_{0}α_{1}}(x)⟩_{α_{1}α_{2}...α̂_{s}...α_{n}α_{n+1}α}

− [⟨f_{α_{0}α_{2}}(x)⟩_{α_{2}...α_{n}α_{n+1}α} + ∑_{r=2}^{n+1}(−1)^{r−1}⟨x⟩_{α_{0}α_{2}...α̂_{r}...α_{n+1}α}]

+ ∑_{s=2}^{n+1}(−1)^{s}⟨f_{α_{0}α_{1}}(x)⟩_{α_{1}α_{2}...α̂_{s}...α_{n}α_{n+1}α}

+ ∑_{s=2}^{n+1}(−1)^{s} ∑_{r=1}^{s−1}(−1)^{r}⟨x⟩_{α_{0}α_{1}...α̂_{r}...α̂_{s}...α_{n+1}α}

+ ∑_{s=2}^{n+1} ∑_{r=s+1}^{n+1}(−1)^{r−1}⟨x⟩_{α_{0}α_{1}α_{2}...α̂_{s}...α̂_{r}...α_{n+1}α}

= ∑_{s=2}^{n+1}(−1)^{s} ∑_{r=s+1}^{n+1}(−1)^{r−1}⟨x⟩_{α_{0}α_{1}α_{2}...α̂_{r}...α̂_{s}...α_{n+1}α}

− ∑_{r=2}^{n+1}(−1)^{r−1}⟨x⟩_{α_{0}α_{2}...α̂_{r}...α_{n+1}α}

+ ∑_{s=2}^{n+1}(−1)^{s} ∑_{r=s+1}^{n+1}(−1)^{r−1}⟨x⟩_{α_{0}α_{1}α_{2}...α̂_{s}...α̂_{r}...α_{n+1}α}

= − ∑_{s=2}^{n} ∑_{r=s+1}^{n+1}(−1)^{r+s}⟨x⟩_{α_{0}α_{1}α_{2}...α̂_{r}...α̂_{s}...α_{n+1}α}

− ∑_{s=2}^{n+1}(−1)^{s−1}⟨x⟩_{α_{0}α_{2}...α̂_{s}...α_{n+1}α}

+ ∑_{r=1}^{n} ∑_{s=r+1}^{n+1}(−1)^{r+s}⟨x⟩_{α_{0}α_{1}α_{2}...α̂_{r}...α̂_{s}...α_{n+1}α}

= − ∑_{s=2}^{n} ∑_{r=s+1}^{n+1}(−1)^{r+s}⟨x⟩_{α_{0}α_{1}α_{2}...α̂_{r}...α̂_{s}...α_{n+1}α}

+ ∑_{r=2}^{n} ∑_{s=r+1}^{n+1}(−1)^{r+s}⟨x⟩_{α_{0}α_{1}α_{2}...α̂_{r}...α̂_{s}...α_{n+1}α}

= 0.

Now we prove that the sequence of module homomorphisms s_{n;α} : R_{n;α} ⟶ R_{n+1;α}, n ≥ −1, is a contracting homotopy. We show that for each n ≥ 0, the map

s_{n−1;α}∂_{n;α} + ∂_{n+1;α}s_{n;α}

acts as the identity on each summand M_{α_{0}α_{1}...α_{n}α} of R_{n;α}. We first deal with the case n = 0. So let x ∈ M_{α_{0}α}. We have

∂_{0;α}(⟨x⟩_{α_{0}α}) = ⟨f_{α_{0}α}(x)⟩_{α},
s_{−1;α}∂_{0;α}(⟨x⟩_{α_{0}α}) = ⟨f_{α_{0}α}(x)⟩_{αα},

and

s_{0;α}(⟨x⟩_{α_{0}α}) = −⟨x⟩_{α_{0}αα},
∂_{1;α}s_{0;α}(⟨x⟩_{α_{0}α}) = −∂_{1;α}(⟨x⟩_{α_{0}αα})
= −(⟨f_{α_{0}α}(x)⟩_{αα} − ⟨x⟩_{α_{0}α}).

So

s_{−1;α}∂_{0;α}(⟨x⟩_{α_{0}α}) + ∂_{1;α}s_{0;α}(⟨x⟩_{α_{0}α}) = ⟨x⟩_{α_{0}α}.

Now let n ≥ 1 and x ∈ M_{α_{0}α_{1}...α_{n}α}.

∂_{n;α}(⟨x⟩_{α_{0}α_{1}...α_{n}α}) = ⟨f_{α_{0}α_{1}}(x)⟩_{α_{1}...α_{n}α} + ∑_{i=1}^{n}(−1)^{i}⟨x⟩_{α_{0}α_{1}...α̂_{i}...α_{n}α},
s_{n−1;α}∂_{n;α}(⟨x⟩_{α_{0}α_{1}...α_{n}α}) = (−1)^{n}⟨f_{α_{0}α_{1}}(x)⟩_{α_{1}...α_{n}αα} + ∑_{i=1}^{n}(−1)^{n+i}⟨x⟩_{α_{0}α_{1}...α̂_{i}...α_{n}αα},

and

s_{n;α}(⟨x⟩_{α_{0}α_{1}...α_{n}α}) = (−1)^{n−1}⟨x⟩_{α_{0}α_{1}...α_{n}αα},
∂_{n+1;α}s_{n;α}(⟨x⟩_{α_{0}α_{1}...α_{n}α}) = (−1)^{n−1}∂_{n+1;α}(⟨x⟩_{α_{0}α_{1}...α_{n}αα})

= (−1)^{n−1} [⟨f_{α_{0}α_{1}}(x)⟩_{α_{1}...α_{n}αα} + ∑_{i=1}^{n} (−1)^{i}⟨x⟩_{α_{0}α_{1}...α̂_{i}...α_{n}αα}

+ (−1)^{n+1}⟨x⟩_{α_{0}α_{1}...α_{n}α}]

= −(−1)^{n}⟨f_{α_{0}α_{1}}(x)⟩_{α_{1}...α_{n}αα} − ∑_{i=1}^{n}(−1)^{n+i}⟨x⟩_{α_{0}α_{1}...α̂_{i}...α_{n}αα}

+ ⟨x⟩_{α_{0}α_{1}...α_{n}α}.

So

s_{n−1;α}∂_{n;α}(⟨x⟩_{α_{0}α_{1}...α_{n}α}) + ∂_{n+1;α}s_{n;α}(⟨x⟩_{α_{0}α_{1}...α_{n}α}) = ⟨x⟩_{α_{0}α_{1}...α_{n}α}.

Appendix B

Artinian x-torsion k[x]-modules

The main point of this appendix is Proposition B.3. The property it states was used for Example 3.6.6 and Proposition 3.6.7.

Let k be an arbitrary field, and k[x] is the polynomial ring in one indeterminate over k. We consider the k[x]-module

L = k[x, x^{−1}]/(x) = k[x, x^{−1}]/k[x, x^{−1}](x) .

As a k-vector space, L admits the set {x^{0}, x^{−1}, x^{−2}, x^{−3}, … } as a basis on which the action of x is :

x·x^{−i} = x^{−i+1} for i ≥ 1; x·x^{0} = 0.

We may identify L with k[x^{−1}], with the understanding the x·1_{k[x^{−1}]} = 0. We have in L the (i+1)-dimensional k[x]-submodule L_{i} with basis {x^{0}, x^{−1}, …, x^{−i}}. Regarding k as a k[x]-module with x acting as zero, we have the canonical k[x]-module embedding

k ⟶ L, 1 ⟼ x^{0}

of k into L, onto L_{0}.

Proposition B.1. The ascending chain

L_{0} ⊂ L_{1} ⊂ L_{2} ⊂ ⋯ ⊂ L_{i} ⊂ L_{i+1} ⊂ ⋯

is the complete lattice of nonzero finite-dimensional k[x]-submodules of L, and L is the only infinite-dimensional k[x]-submodule in L. In particular, Soc(L) = L_{0} ≅ k and L is Artinian.

This result holds in the more general setting where k is a commutative Artinian ring; see Krause [44, Lemma 8.5].

Proof. We will prove that for any k[x]-submodule V of L, we have

L_{i} ⫋ V ⇒ L_{i+1} ⊆ V.

Let us first assume this. Every nonzero element of L is taken by some power of x to a nonzero element of L_{0}. So every nonzero k[x]-submodule of L contains L_{0}. Let V be a k[x]-submodule of L with dim V = i + 1. By applying the implication above inductively, we obtain that V = L_{i}. If V is infinite-dimensional, the same implication shows that V contains the submodules L_{i} for all i ≥ 1, whence that V = L. Now let V be a k[x]-submodule V of L such that L_{i} ⫋ V. Then V contains an element r = x^{−j} + ∑_{m=0}^{j−1} r_{m}x^{−m} with j > i. We have x^{j−i−1}·r ∈ V, and x^{j−i−1}·r = x^{−i−1} + r′, where r′ = x^{j−i−1}· ∑_{m=0}^{j−1} r_{m}x^{−m} = ∑_{n=0}^{i} r_{n+j−i+1}x^{−n} ∈ L_{i} ⊂ V. So x^{−i−1} ∈ V, whence L_{i+1} ⊆ V.

Proposition B.2. The k[x]-module L is divisible, and the canonical k[x]-embedding

k ⟶ L, 1 ⟼ x^{0}

of k into L, onto L_{0} is the injective hull of the trivial module k.

Proof. In order to show that L is divisible, we first show that

( a ∈ k[x] and a·L = 0 ) ⇒ a = 0.

We proceed by induction on the number of monomials in a. First suppose that a is a monomial, a = a_{m}x^{m}, such that a·L = 0. Then in particular 0 = a·x^{−m} = a_{m}x^{m}·x^{−m} = a_{m}x^{0}, so a_{m} = 0, a = 0. Now suppose that a is a sum of (i + 1) monomials, a = ∑_{m=j}^{j+i} a_{m}x^{m}, such that a·L = 0. Then in particular 0 = a·x^{−j} = ∑_{m=j}^{j+i} a_{m}x^{m}·x^{−j} = a_{j}x^{0}, so a_{j} = 0, and a is a sum of i monomials, whence by induction a = 0. Next we show that if a ∈ k[x] and a·L is finite-dimensional, then a = 0. By Proposition B.1, we have a·L = L_{i} for some i ≥ −1 (L_{−1} = 0). Then 0 = x^{i+1}·(a·L) = (x^{i+1}a)·L, whence x^{i+1}a = 0 in k[x], a = 0. Finally, for 0 ≠ a ∈ k[x], a·L is infinite-dimensional, a·L = L; L is divisible. Since the ring k[x] is a principal ideal domain, a k[x]-module is divisible if and only if it is injective. We show that any k[x]-morphism f : k ⟶ M with M a divisible k[x]-module factors through the canonical monomorphism k ⟶ L, 1 ⟼ x^{0}. Let y_{0} = f(1). Let y_{1}, y_{2}, … ∈ M be chosen inductively such that x·y_{i+1} = y_{i}. The assignment x^{−i} ↦ y_{i} defines a vector space homomorphism g : L ⟶ M which is a k[x]-morphism. And of course, f factors as k ⟶ L ⟶^{g} M.

Proposition B.3. Let M be a k[x]-module which is x-torsion, in the sense that the action of x on M is locally nilpotent (i.e. every element of M is annihilated by some power of x). Then

Soc(M) = Ker(xId_{M} : M ⟶^{x·} M).

Let I_{M} be the injective hull of M and {b_{α}}_{α} be a k-basis for Soc(M). Then

I_{M} ≅ ⨁_{α} L.

In particular, the k[x]-module M is Artinian if and only if its socle is finitedimensional.

Proof.

(1) Every one-dimensional k-subspace of Ker(xId_{M}) of M is a k[x]-submodule of M, and hence has to be a simple submodule of M. Thus Ker(xId_{M}) ⊆ Soc(M). Now the fact that the module M is x-torsion implies that the submodule Ker(xId_{M}) is essential in M. This forces Soc(M) = Ker(xId_{M}).

(2) The injective hull of M is the injective hull of its socle. The ring k[x] is Nœtherian, so arbitrary direct sums of injective modules are injective. So, by Proposition B.2, the injective hull of Soc(M) is as stated in the Proposition.

(3) If Soc(M) is finite-dimensional, then the module I_{M} is Artinian, being a direct sum of finitely many Artinian modules. And so its submodule M must be Artinian. Conversely, if M is Artinian, then so is its submodule Soc(M), which must then be finite-dimensional because every k-subspace of Soc(M) is a k[x]-submodule.

Appendix C

Lifting tubes from the stable

category

In this appendix, we collect the definitions and results about tubes. This notion was used in Proposition 3.6.8 and Section 6.2. The terminology is inspired by Krause [44, Section 8].

Throughout, Λ will denote a finite-dimensional self-injective algebra over a field k, and all modules are left Λ-modules.

Let (M_{i})_{i≥1} be a sequence of finite-dimensional modules and σ_{i} : M_{i} ⟶ M_{i+1} and τ_{i+1} : M_{i+1} ⟶ M_{i}, i ≥ 1 be module homomorphisms. Let

σ_{i}^{j} : M_{i} ⟶^{σ_{i}} M_{i+1} ⟶^{σ_{i+1}} ⋯ ⟶^{σ_{i+j−1}} M_{i+j},

τ_{i+j}^{j} : M_{i+j} ⟶^{τ_{i+j}} M_{i+j−1} ⟶^{τ_{i+j−1}} ⋯ ⟶^{τ_{i+1}} M_{i}.

Definition C.1.

• We say that the data (M_{i}, σ_{i}, τ_{i})_{i≥1} form a tube in mod(Λ) provided that we have exact sequences

0 ⟶ M_{i} ⟶^{σ_{i}^{j}} M_{i+j} ⟶^{τ_{i+j}^{i}} M_{j} ⟶ 0

and commutative squares

\xymatrix{M_{i} \ar[r]^{τ_{i}} \ar[d]_{σ_{i}} & M_{i−1} \ar[d]^{σ_{i−1}} \\ M_{i+1} \ar[r]_{τ_{i+1}} & M_{i}}

The direct limit M in Mod(Λ) of the direct sequence

M_{1} ⟶^{σ_{1}} M_{2} ⟶^{σ_{2}} M_{3} ⟶^{σ_{3}} ⋯

will be called the direct limit of the tube.

• Regard the modules and maps under consideration as objects and morphisms in stmod(Λ). We say that the data (M_{i}, σ_{i}, τ_{i})_{i≥1} form a tube in stmod(Λ) provided that we have exact triangles

M_{i} ⟶^{σ_{i}^{j}} M_{i+j} ⟶^{τ_{i+j}^{i}} M_{j} ⟶ Ω^{−1}M_{i}

and (stably) commutative squares

\xymatrix{M_{i} \ar[r]^{τ_{i}} \ar[d]_{σ_{i}} & M_{i−1} \ar[d]^{σ_{i−1}} \\ M_{i+1} \ar[r]_{τ_{i+1}} & M_{i}}

“The” (homotopy) direct limit M in StMod(Λ) of the direct sequence

M_{1} ⟶^{σ_{1}} M_{2} ⟶^{σ_{2}} M_{3} ⟶^{σ_{3}} ⋯

will be called the direct limit of the tube.

Theorem C.2. Suppose that the sequences

0 ⟶ M_{i} ⟶^{σ_{i}} M_{i+1} ⟶^{τ_{i+1}^{i}} M_{1} ⟶ 0, i ≥ 1,

are exact, or else that the sequences

0 ⟶ M_{1} ⟶^{σ_{1}^{j}} M_{j+1} ⟶^{τ_{j+1}} M_{j} ⟶ 0, j ≥ 1,

are exact. Suppose also that the squares

\xymatrix{M_{i} \ar[r]^{τ_{i}} \ar[d]_{σ_{i}} & M_{i−1} \ar[d]^{σ_{i−1}} \\ M_{i+1} \ar[r]_{τ_{i+1}} & M_{i}}

are commutative. Then (M_{i}, σ_{i}, τ_{i})_{i≥1} is a tube in mod(Λ). Proof. If we are given that the sequences

0 ⟶ M_{i} ⟶^{σ_{i}} M_{i+1} ⟶^{τ_{i+1}^{i}} M_{1} ⟶ 0, i ≥ 1,

are exact, we show by induction on j that for every i ≥ 1, the sequence

0 ⟶ M_{i} ⟶^{σ_{i}^{j}} M_{i+j} ⟶^{τ_{i+j}^{i}} M_{j} ⟶ 0

is exact. We do this by using the following commutative diagram.

[figure: 0; 0; 0; M_{i}; σ_{i}^{j}; M_{i+j}; τ_{i+j}^{i}; σ_{i+j}; M_{j}; σ_{j}; 0; 0; M_{i}; σ_{i}^{j+1}; M_{i+j+1}; τ_{i+j+1}^{i}; τ_{i+j+1}^{i+j}; M_{j+1}; τ_{j+1}^{j}; 0; M_{1}; M_{1}; 0; 0]

By assumption, the two vertical sequences are exact, and by induction hypothesis, the top horizontal sequence is exact. So the middle horizontal sequence is exact by the 3×3 Lemma. If we are given that the sequences

0 ⟶ M_{1} ⟶^{σ_{1}^{j}} M_{j+1} ⟶^{τ_{j+1}} M_{j} ⟶ 0, j ≥ 1,

are exact, we show by induction on i that for every j ≥ 1, the sequence

0 ⟶ M_{i} ⟶^{σ_{i}^{j}} M_{i+j} ⟶^{τ_{i+j}^{i}} M_{j} ⟶ 0

is exact. The procedure is similar to the previous one. The relevant diagram is the following.

[figure: 0; 0; M_{1}; σ_{1}^{i}; M_{1}; σ_{1}^{i+j}; 0; M_{i+1}; σ_{i+1}^{j}; τ_{i+1}; M_{i+j+1}; τ_{i+j+1}^{i+1}; τ_{i+j+1}; M_{j}; 0; 0; M_{i}; σ_{i}^{j}; M_{i+j}; τ_{i+j}^{i}; M_{j}; 0; 0; 0]

We will need the following result on lifting exact triangles in StMod(Λ) to short exact sequences in Mod(Λ).

Lemma C.3. Let f : A ⟶ B and g : B ⟶ C be morphisms in Mod(Λ). Suppose that f is monomorphic, g is epimorphic, gf = 0, and that

A ⟶^{f} B ⟶^{g} C

is an exact triangle is StMod(Λ). Then there are submodules B′ and Q in B, with B′ ⊇ Im f, Q projective, and B = B′⊕Q, such that the sequence

0 ⟶ A ⟶^{f′} B′ ⟶^{g′} C ⟶ 0,

where f′ is the corestriction of f to B′ and g′ is the restriction of g to B′, is exact.

Proof. Let f′′ : A ⟶ Ker g is the corestriction of f to Ker g. The commutative diagram

[figure: A; f; f′′; B; g; C; Ker g; ⊆; B; g; C]

displays an isomorphism of exact triangles. In particular, the monomorphism f′′ : A ⟶ Ker g is a stable isomorphism. Therefore its cokernel Ker g/ Im f is projective. So the epimorphism Ker g ⟶ Ker g/ Im f has a splitting monomorphism Ker g/ Im f ⟶ Ker g whose image is a projective submodule Q of B contained in Ker g. On another hand, the module Ker g/ Im f is also injective, so the epimorphism Ker g ⟶ Ker g/ Im f extends to an epimorphism h : B ⟶ Ker g/ Im f. Set B′ = Ker h. We have B = B′⊕Q. Since Q ⊆ Ker g, the restriction g′ : B′ ⟶ C of the surjection g : B ⟶ C is surjective. We have

Ker g′ = B′ ∩ Ker g = Ker h ∩ Ker g = Ker(h_{| Ker g}) = Im f.

So we do have B′ ⊇ Im f, and the short exact sequence

0 ⟶ A ⟶^{f′} B′ ⟶^{g′} C ⟶ 0.

Lemma C.4. Consider the following diagram in Mod(Λ).

[figure: A′; f′; a; B̄′; ḡ′; b̄; C; 0; A; f; B; g; C; 0]

Suppose that the top row is an exact triangle in StMod(Λ), the bottom row is an exact sequence in Mod(Λ), and the diagram is commutative in StMod(Λ). Then there are a module B′ stably isomorphic to B̄′, and module homomorphisms g′ : B′ ⟶ C and b : B′ ⟶ B stably equivalent to ḡ′ : B̄′ ⟶ C and b̄ : B̄′ ⟶ B respectively, such that the diagram

[figure: 0; A′; f′; a; B′; g′; b; C; 0; 0; A; f; B; g; C; 0]

is commutative in Mod(Λ) and the rows are exact sequences.

Proof. By enlarging the module B̄′ with an appropriate projective summand and appropriatly co-extending the map A′ ⟶ B̄′ and extending the map B̄′ ⟶ C, we may assume that the square

\xymatrix{A′ \ar[r]^{f′} \ar[d]^{a} & B̄′ \ar[d]_{b̄} \\ A \ar[r]^{f} & B}

is commutative in Mod(Λ) and that the map f′ : A′ ⟶ B̄′ is injective and the map b̄ : B̄′ ⟶ B is surjective. We may also assume that ḡ′ denotes the composite B̄′ ⟶^{b̄}B ⟶^{g}C. So we have the commutative diagram

[figure: A′; f′; a; B̄′; ḡ′; b̄; C; 0; A; f; B; g; C; 0]

in Mod(Λ) with the bottom row an exact sequence and the top row an exact triangle. Moreover, the map f′ : A′ ⟶ B̄′ is injective, the map ḡ′ is surjective, and the composite ḡ′f′ is zero. So we obtain the required result by applying Lemma C.3.

Theorem C.5. A tube (M̄_{i}, σ̄_{i}, τ̄_{i})_{i≥1} in stmod(Λ) lifts to a tube (M_{i}, σ_{i}, τ_{i})_{i≥1} in mod(Λ).

Proof. We will inductively construct finite-dimensional modules M_{i} and module homomorphisms σ_{i} : M_{i} ⟶ M_{i+1} and τ_{i+1} : M_{i+1} ⟶ M_{i} such that M_{i} and M̄_{i} are stably isomorphic, σ_{i} and σ̄_{i} are stably equivalent, τ_{i} and τ̄_{i} are stably equivalent, and the sequences

0 ⟶ M_{i} ⟶^{σ_{i}} M_{i+1} ⟶^{τ_{i+1}^{i}} M_{1} ⟶ 0

are exact, and squares

\xymatrix{M_{i} \ar[r]^{τ_{i}} \ar[d]_{σ_{i}} & M_{i−1} \ar[d]^{σ_{i−1}} \\ M_{i+1} \ar[r]_{τ_{i+1}} & M_{i}}

are commutative. Then we will have obtained a tube (M_{i}, σ_{i}, τ_{i})_{i≥1} in mod(Λ) – by Theorem C.2 – which lifts the tube (M̄_{i}, σ̄_{i}, τ̄_{i})_{i≥1}. The required stable isomorphisms of modules and stable equivalences of maps will be refered to as the lifting conditions.

We start by choosing the module M_{1}, and then the module M_{2}, and maps σ_{1} : M_{1} ⟶ M_{2} and τ_{2} : M_{2} ⟶ M_{1} such that the lifting conditions are satisfied and the sequence

0 ⟶ M_{1} ⟶^{σ_{1}} M_{2} ⟶^{τ_{2}} M_{1} ⟶ 0

is exact. Then we assume for induction that we have the sequence

0 ⟶ M_{i} ⟶^{σ_{i}} M_{i+1} ⟶^{τ_{i+1}^{i}} M_{1} ⟶ 0

is exact. We apply Lemma C.4 to the diagram

\xymatrix{ & M_{i+1} \ar[r]^{σ_{i+1}} \ar[d]^{τ_{i+1}} & M̄_{i+2} \ar[d]_{τ̄_{i+2}} \\ 0 \ar[r] & M_{i} \ar[r]^{σ_{i}} & M_{i+1} \ar[r]^{τ_{i+1}^{i}} & M_{1} \ar[r] & 0}

to obtain a module M_{i+2} and maps σ_{i+1} : M_{i+1} ⟶ M_{i+2} and τ_{i+2} : M_{i+2} ⟶ M_{i+1} which satisfy the lifting conditions, and such that the sequence

0 ⟶ M_{i+1} ⟶^{σ_{i+1}} M_{i+2} ⟶^{τ_{i+2}^{i+1}} M_{1} ⟶ 0

is exact and the square

\xymatrix{M_{i+1} \ar[r]^{τ_{i+1}} \ar[d]_{σ_{i+1}} & M_{i} \ar[d]^{σ_{i}} \\ M_{i+2} \ar[r]_{τ_{i+2}} & M_{i+1}}

is commutative.

One can of course define tubes in Mod(Λ) and StMod(Λ), and tubes in StMod(Λ) lift to tubes in Mod(Λ).

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