PHANTOM MAPS IN THE STABLE MODULE CATEGORY

GILLES PH. GNACADJA

Abstract. We study phantom maps in the stable module category StMod(kG), where k is a field and G is a finite group. In this paper we almost exclusively deal with maps out of countably generated modules. We show that the space \underline{Ph}_{kG}(M, N) of stabilized phantom maps M→N has an expression as a \underleftarrow{lim}^{1} space which allows some control on its vanishing. Then we present a situation where all maps are phantom and need not be trivial. We provide explicit details for a particular such situation. Finally we construct a universal phantom map. We use it to show that the composite of two phantom maps is trivial, and to characterize the modules with no nontrivial outbound phantom maps.

Introduction

The stable module category StMod(kG) appears in two aspects of the modular representation theory and the cohomology of finite groups that have recently gained much interest: the study of infinite dimensional representations and the categorical approach. The understanding of infinite dimensional representations of finite groups could extend the scope of representation theory to classes of infinite groups. In Section 1, where we set up the terminology and notations for this paper, we recall the definition of StMod(kG) and give some reasons why it is relevant to the cohomology of G and homological algebra over kG. The notion of phantom map is defined. We will be concerned with phantom maps out of countably generated modules. Phantom maps in a greater generality will be the subject of a later publication. In Section 2, we express the space of phantom maps as a \underleftarrow{lim}^{1} space and put it in a short exact sequence. The sequence is similar to the Milnor exact sequence of pointed sets in homotopy theory. The expression can be used to decide on the existence of nontrivial phantom maps. As a first application we show that there are no nontrivial phantom maps into a finitely generated module, and more generally into a profinite module (by which we mean the inverse limit of an inverse system of finitely generated modules). In Section 3 we present a situation where all maps are phantom, using the interplay between the product in the cohomology ring H^{∗}(G, k) and the composition law in the category StMod(kG). We show that under some conditions there are nontrivial (phantom) maps, by testing the Mittag-Leffler condition on the \underleftarrow{lim}^{1} expression established in Section 2. We provide an explicit description of the trivial maps in a particular case. The modules involved are represented by diagrams. Finally in Section 4 we construct a weakly universal outbound phantom map. We use it to show that the composite of two phantom maps is trivial, and to characterize the modules with no nontrivial outbound phantom maps as the pure-projective modules.

Date: November 1996.

1991 Mathematics Subject Classification. Primary: 20C20; Secondary: 20J06.

Key words and phrases. Phantom Map, Stable Module Category.

1. Terminology and Notations

We fix a finite group G and a field k. We assume that k has prime characteristic dividing the order of G; this is not necessary, but otherwise every theory becomes trivial. A map M→N of kG-modules is said to be projective if it factors through a projective module. Note that the notions of projectivity and injectivity are equivalent for kG-modules (and more generally for modules over a finite dimensional self-injective algebra over a field), see [1, Section 1.6]. The projective maps M→N form a kvector space PHom_{kG}(M, N). By definition, the stable module category StMod(kG) has the same class of objects as the category Mod(kG) of all (not necessarily finitely generated) left kG-modules, and the space of maps M→N is

\underline{Hom}_{kG}(M, N) := Hom^{kG}(M, N)/PHom_{kG}(M, N) ;

two maps M→N in Mod(kG) are identified in StMod(kG) if their difference factors through a projective module. The category StMod(kG) is additive but not abelian. If M is a kG-module, ΩM denotes the kernel of its projective cover. Ω is a self-equivalence on the category StMod(kG), and StMod(kG) is an additive triangulated category with translation functor Ω^{−1}. A recent treatment of the theory of triangulated categories can be found in [15, Section 10.2]. See also [4, Section 5] for a treatment on the stable category stmod(kG) of finitely generated modules. StMod(kG) is an appropriate setting for studying homological algebra over kG. For instance the positive degree H^{∗}(G, −) spaces and Ext_{kG}^{∗}(−, −) spaces, and more generally the Ĥ^{∗}(G, −) spaces and Êxt_{kG}^{∗}(−, −) spaces are \underline{Hom}_{kG}(−, −) spaces. The products in the graded rings H^{∗}(G, k) and Ĥ^{∗}(G, k), and the composition for the Êxt_{kG}^{∗}(−, −) spaces have a natural interpretation in terms of the composition of maps in StMod(kG), the Yoneda composition; see [1, Section 2.6] and [4, Section 6]. We are concerned with the “phantom phenomena” that occur in StMod(kG) : a map M→N in StMod(kG) need not be determined by its restrictions to the finitely generated submodules of M. We are thus led to study phantom maps.

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

Of course phantom maps M→N are of interest only if M is not finitely generated.

There is an analogous notion in homotopy theory (where we borrowed the name from). If X = ⋃_{n=1}^{∞} X^{n} and Y are CW-complexes, 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. For a survey on homotopy theoretic phantom maps, see [12]. Phantom maps in a stable homotopy category are studied in [5].

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

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

\underline{Ph}_{kG}(M, N) = Ph^{kG}(M, N)/PHom_{kG}(M, N) .

2. A Milnor isomorphism. Applications.

We begin by recording some characterizations of projective maps. First recall that if X is a kG-module with invariant subspace X^{G}, we have the trace map

Tr_{G} : X → X^{G}, x ↦ Tr_{G}x = ∑_{g∈G} gx.

In particular if M, N are kG-modules, we have the map

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

gv is defined by (gv)(x) = g((v(g^{−1}x))) for x ∈ M.

Proposition 1. For a map u : M → N of kG-modules, the following are equivalent:

(i) u is projective; (ii) u factors through any epimorphism P ↠ N with P a projective module, in

particular through the projective cover PN ↠ N of N; (iii) u factors through any monomorphism M ↪ I with I an injective module, in

particular through the injective hull M ↪ IM of M; (iv) u is the trace of some v ∈ Hom_{k}(M, N).

Properties (ii) and (iii) will be used in the sequel without explicit reference thereto. The equivalences (i) ⇔ (iv) ⇔ (ii) can be found in [14, Lemma 32.2]; modules are assumed to be finitely generated but this is not necessary. Similar arguments show the equivalence of (iii) with the other statements.

We see in the following theorem that if M is countably generated, there is a short exact sequence similar to the Milnor exact sequence of pointed sets in homotopy theory (see [3, Chapter IX, Corollary 3.3]). \underline{Ph}_{kG}(M, N) is isomorphic to the first term of this sequence.

Theorem 2. Let M, N be kG-modules with M countably generated, let (M_{i})_{i≥1} be an ascending chain of finitely generated submodules of M with M = ⋃_{i=1}^{∞} M_{i}. Then we have the k-vector spaces isomorphisms

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

and the Milnor short exact sequence of k-vector spaces and k-linear maps

0 → \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}\underline{Hom}_{kG}(M_{i}, ΩN) → \underline{Hom}_{kG}(M, N) → \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}\underline{Hom}_{kG}(M_{i}, N) → 0.(2)

Proof. We have the following commutative diagram with exact rows.

\xymatrix{ & ⋮ \ar[d] & ⋮ \ar[d] & ⋮ \ar[d] \\ 0 \ar[r] & Hom_{kG}(M_{3}, ΩN) \ar[r] \ar[d] & Hom_{kG}(M_{3}, PN) \ar[r] \ar[d] & PHom_{kG}(M_{3}, N) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Hom_{kG}(M_{2}, ΩN) \ar[r] \ar[d] & Hom_{kG}(M_{2}, PN) \ar[r] \ar[d] & PHom_{kG}(M_{2}, N) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Hom_{kG}(M_{1}, ΩN) \ar[r] & Hom_{kG}(M_{1}, PN) \ar[r] & PHom_{kG}(M_{1}, N) \ar[r] & 0}

The maps in the middle inverse system (and in the RHS one) are surjective. So \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}Hom_{kG}(M_{i}, PN) = 0 (and \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}PHom_{kG}(M_{i}, N) = 0), and the six-term exact sequence for the \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1} functor yields the exact sequence

0 → Hom_{kG}(M, ΩN) → Hom_{kG}(M, PN) → Ph_{kG}(M, N) → \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}Hom_{kG}(M_{i}, ΩN) → 0.

Reducing modulo Hom_{kG}(M, ΩN), we get the short exact sequence

0 → PHom_{kG}(M, N) → Ph_{kG}(M, N) → \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}Hom_{kG}(M_{i}, ΩN) → 0.

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

Now consider the following commutative diagram with exact rows.

\xymatrix{ & ⋮ \ar[d] & ⋮ \ar[d] & ⋮ \ar[d] \\ 0 \ar[r] & PHom_{kG}(M_{3}, N) \ar[r] \ar[d] & Hom_{kG}(M_{3}, N) \ar[r] \ar[d] & \underline{Hom}_{kG}(M_{3}, N) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & PHom_{kG}(M_{2}, N) \ar[r] \ar[d] & Hom_{kG}(M_{2}, N) \ar[r] \ar[d] & \underline{Hom}_{kG}(M_{2}, N) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & PHom_{kG}(M_{1}, N) \ar[r] & Hom_{kG}(M_{1}, N) \ar[r] & \underline{Hom}_{kG}(M_{1}, N) \ar[r] & 0}

Since \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}PHom_{kG}(M_{i}, N) = 0, the six-term exact sequence yields the isomor-

phism \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}Hom_{kG}(M_{i}, N) ≅ \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}\underline{Hom}_{kG}(M_{i}, N) (so \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}Hom_{kG}(M_{i}, ΩN) ≅

\begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}\underline{Hom}_{kG}(M_{i}, ΩN) ), and the short exact sequence

0 → Ph_{kG}(M, N) → Hom_{kG}(M, N) → \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}\underline{Hom}_{kG}(M_{i}, N) → 0.

Reducing modulo PHom_{kG}(M, N), we get the short exact sequence

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

which is the Milnor short exact sequence. Corollary 3. Let M, N be kG-modules with M countably generated and N finitely generated. All phantom maps M → N are trivial.

Proof. An inverse system of finite dimensional vector spaces has vanishing \begin{matrix}lim \\ ←------- \end{matrix}^{1}; see for instance [11, Corollaire 7.2].

This result can be generalized in two ways. The argument of the proof can be used to show that phantom maps from any module into a finitely generated module are trivial. Corollary 6 at the end of this section asserts that the conclusion of Corollary 3 still holds if N is merely assumed to be profinite. Corollary 3 is a contrast with homotopy theory. For instance B.I. Gray has shown in [10] that there are (uncountably many) non-nullhomotopic phantom maps ℂP^{∞} → S^{3}.

The surjective map Ph_{kG}(M, N) ↠ \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}Hom_{kG}(M_{i}, ΩN) which gives rise to

the isomorphism \underline{Ph}_{kG}(M, N) ⟶^{≅} \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}Hom_{kG}(M_{i}, ΩN) is actually a connecting homomorphism occuring in an application of the Snake Lemma. It can be explicitly described as follows. Let f ∈ Ph_{kG}(M, N). Let, for each i ≥ 1, g_{i} : M_{i} → PN such that f_{|M_{i}} is the composite M_{i} ⟶^{g_{i}} PN → N. Set h_{i} := g_{i} − g_{i+1}_{|M_{i}}. Then (PN → N) ∘ h_{i} = f_{i} − f_{i+1}_{|M_{i}} = 0, so h_{i} ∈ Hom_{kG}(M_{i}, ΩN). The map

Ph_{kG}(M, N) ↠ \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}Hom_{kG}(M_{i}, ΩN) does f ↦ class of (h_{i})_{i≥1}.

It happens that modules are given just as colimits. For such situations we have the following.

Theorem 4. Let (M_{i})_{i≥1} be a direct sequence of finitely generated kG-modules, with direct limit M = \begin{matrix}lim \\ -------→ \end{matrix}_{i≥1}M_{i}. Let N be a kG-module. We have

\underline{Ph}_{kG}(M, N) ≅ \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}\underline{Hom}_{kG}(M_{i}, ΩN).  (3)

By the isomorphism (3), if \underline{Ph}_{kG}(M, N) is nonzero then its k-dimension is larger than the countable cardinal ℵ_{0}. If N is countably kG-generated then each \underline{Hom}_{kG}(M_{i}, ΩN) is a countable dimensional k-vector space, so the Mittag-Leffler condition can be used as a criterion to decide on the vanishing of \underline{Ph}_{kG}(M, N); see [7, Proposition A.3] or [8]. We will use this in the next section to assert the existence of nontrivial phantom maps.

Proof. We produce an ascending tower of finitely generated modules (M′_{i})_{i≥1} which is stable isomorphic to the direct sequence (M_{i})_{i≥1}. We denote φ_{i} the map M_{i} → M_{i+1}. Let for each i ≥ 1, J_{i} be an injective module containing M_{i}. Set M′_{i} := J_{1} ⊕ ⋯ ⊕ J_{i−1} ⊕ M_{i}. We define the map

φ′_{i} : M′_{i} → M′_{i+1}, (x_{1}, …, x_{i−1}, x_{i}) ↦ (x_{1}, …, x_{i−1}, x_{i}, φ_{i}(x_{i})).

We have the following commutative diagram where the columns are exact; the unnamed maps are the obvious ones.

[figure: 0; 0; ⋯; J_{1} ⊕ ⋯ ⊕ J_{i−1}; J_{1} ⊕ ⋯ ⊕ J_{i−1} ⊕ J_{i}; ⋯; ⋯; φ′_{i−1}; M′_{i}; φ′_{i}; M′_{i+1}; φ′_{i+1}; ⋯; ⋯; φ_{i−1}; M_{i}; φ_{i}; M_{i+1}; φ_{i+1}; ⋯; 0; 0]

Set J := ⨁_{i=1}^{∞} J_{i} and M′ := \begin{matrix}lim \\ -------→ \end{matrix}_{i≥1}M′_{i}. J is injective, and we have the short exact sequence

0 → J → M′ → M → 0.

The inverse systems of vector spaces (\underline{Hom}_{kG}(M′_{i}, ΩN))_{i≥1} and (\underline{Hom}_{kG}(M_{i}, ΩN))_{i≥1} are isomorphic, so they have isomorphic \begin{matrix}lim \\ ←------- \end{matrix}^{1}. Now the maps φ_{i} are monomorphisms, so it follows from (1) that \underline{Ph}_{kG}(M′, N) ≅ \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}\underline{Hom}_{kG}(M′_{i}, ΩN). As M′ and M are stable isomorphic, we have \underline{Ph}_{kG}(M, N) ≅ \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}\underline{Hom}_{kG}(M_{i}, ΩN) as claimed.

The previously announced generalization of Corollary 3 will be a consequence of the following theorem.

Theorem 5. Let A be a finitely generated kG-module and {B_{α}}_{α} be an inverse system of finitely generated kG-modules. Let Y := \begin{matrix}lim \\ ←------- \end{matrix}_{α}B_{α}. We have the isomorphisms

PHom_{kG}(A, Y) ⟶^{≅} \begin{matrix}lim \\ ←------- \end{matrix}_{α}PHom_{kG}(A, B_{α}) ,  (4)

and

\underline{Hom}_{kG}(A, Y) ⟶^{≅} \begin{matrix}lim \\ ←------- \end{matrix}_{α}\underline{Hom}_{kG}(A, B_{α}) .  (5)

Proof. Consider the inverse system of short exact sequences

0 → Hom_{kG}(Ω^{−1}A, B_{α}) → Hom_{kG}(IA, B_{α}) → PHom_{kG}(A, B_{α}) → 0.

\begin{matrix}lim \\ ←------- \end{matrix}_{α}^{1}Hom_{kG}(Ω^{−1}A, B_{α}) = 0 since each Hom_{kG}(Ω^{−1}A, B_{α}) is finite dimensional. So we get the short exact sequence

0 → Hom_{kG}(Ω^{−1}A, Y) → Hom_{kG}(IA, Y) → \begin{matrix}lim \\ ←------- \end{matrix}_{α}PHom_{kG}(A, B_{α}) → 0,

whence the isomorphism (4). Now consider the inverse system of short exact sequences

0 → PHom_{kG}(A, B_{α}) → Hom_{kG}(A, B_{α}) → \underline{Hom}_{kG}(A, B_{α}) → 0.

\begin{matrix}lim \\ ←------- \end{matrix}_{α}^{1}PHom_{kG}(A, B_{α}) = 0 so we have the short exact sequence

0 → \begin{matrix}lim \\ ←------- \end{matrix}_{α}PHom_{kG}(A, B_{α}) → Hom_{kG}(A, Y) → \begin{matrix}lim \\ ←------- \end{matrix}_{α}\underline{Hom}_{kG}(A, B_{α}) → 0,

or, by (4),

0 → PHom_{kG}(A, Y) → Hom_{kG}(A, Y) → \begin{matrix}lim \\ ←------- \end{matrix}_{α}\underline{Hom}_{kG}(A, B_{α}) → 0,

and so the isomorphism (5).

Corollary 6. Let M be a countably generated module and {B_{α}}_{α∈A} an inverse system of finitely generated modules with Y = \begin{matrix}lim \\ ←------- \end{matrix}_{α∈A}B_{α}. Then \underline{Ph}_{kG}(M, Y) = 0; all phantom maps M → Y are trivial. In particular, let N be a module and N̂ be the inverse limit of the inverse system of its finitely generated quotients. All phantom maps M → N̂ are trivial.

Proof. Let (M_{i})_{i≥1} be an ascending chain of finitely generated submodules of M with M = ⋃_{i=1}^{∞} M_{i}. We have

\underline{Ph}_{kG}(M, Y) ≅ \begin{matrix}lim \\ ←------- \end{matrix}_{i≥1}^{1}\underline{Hom}_{kG}(Ω^{−1}M_{i}, Y) ≅ \begin{matrix}lim \\ ←------- \end{matrix}_{i∈ℤ_{≥1}}^{1} \begin{matrix}lim \\ ←------- \end{matrix}_{α∈A}\underline{Hom}_{kG}(Ω^{−1}M_{i}, B_{α}).

\begin{matrix}lim \\ ←------- \end{matrix}_{i∈ℤ_{≥1}}^{1} \begin{matrix}lim \\ ←------- \end{matrix}_{α∈A}\underline{Hom}_{kG}(Ω^{−1}M_{i}, B_{α}) embeds into \begin{matrix}lim \\ ←------- \end{matrix}_{(i,α)∈ℤ_{≥1}×A}^{1}\underline{Hom}_{kG}(Ω^{−1}M_{i}, B_{α});

this monomorphism is the first map of the five-term exact sequence for a Grothendieck spectral sequence. For more details on this see [15, Section 5.8]. But

\begin{matrix}lim \\ ←------- \end{matrix}_{(i,α)∈ℤ_{≥1}×A}^{1}\underline{Hom}_{kG}(Ω^{−1}M_{i}, B_{α}) = 0 since each \underline{Hom}_{kG}(Ω^{−1}M_{i}, B_{α}) is finite di-

mensional.

Note that by Corollary 6 if a map f : M → N is phantom then the composite M ⟶^{f} N ⟶ N̂ is projective. One may ask if this latter property characterizes phantom maps M → N. It would be sufficient (and necessary) to show that if a map f : A → N with A finitely generated is such that the composite A ⟶^{f} N ⟶ N̂ is projective then f : A → N is projective. This is trivially true if N is semisimple.

3. Examples

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 system

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

F(α) is defined to be its colimit. Actually F(α) is just a homotopy colimit in StMod(kG), but two such colimits are (non uniquely) stable isomorphic. The construction and many relevant questions are studied in [13]. 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 7. 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 8. 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 7) 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. We successively have three commutative diagrams as follows.

[figure: Ω^{t−sm}k; Ω^{−rm}k; u; α^{i}; f_{r}; Ω^{t−sm}k; Ω^{t}F(β); Ω^{−rm}k; u; f_{r}; Ω^{t}F(β); Ω^{−(r+i)m}k; f_{r+i}]

[figure: Ω^{−rm}k; u; α^{i}; Ω^{t−sm}k; β^{j}; Ω^{−(r+i)m}k; v; Ω^{t−(s+j)m}k]

s and j may be chosen large enough so that 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 Theorem 4 and the subsequent comments.)

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.

[figure: k; f_{i}; α^{i+1}; Ω^{c}F(β); k; γα^{i}; α^{i+1}; Ω^{c−(i+1)m}k; β^{j}; Ω^{−(i+1)m}k; h; Ω^{−(i+1)m}k; 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 [4, Section 7] for more on the use of diagrams to produce examples of modules over group algebras. Refer to [2] 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 9. 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 by Corollary 8 f is phantom. We will show that Kerf ⊇ ⨁_{s=0}^{∞} ka_{2s}. It will follow that for every s ≥ 0, f(a_{2s+1}) is annihilated by u and v, hence is in the invariant subspace F(β)^{G} = ⨁_{t=0}^{∞} kb_{2t}. Let {b_{j}^{∗}}_{j≥0} be the family of k-linear forms dual to the k-basis {b_{j}}_{j≥0} of F(β). For j < 0 set b_{j}^{∗} := 0. One readily sees that b_{2t}^{∗} ∘ u = b_{2t−1}^{∗} and b_{2t}^{∗} ∘ v = b_{2t+1}^{∗} for every t ∈ ℤ. One can then get that b_{2t−2s}^{∗}(f(a_{2s})) = b_{2t}^{∗}(f(a_{0})), or equivalently that b_{2t}^{∗}(f(a_{2s})) = b_{2t+2s}^{∗}(f(a_{0})), for every t ∈ ℤ and s ∈ ℤ_{≥0}. Fix t, and let s > t. We have b_{2t}^{∗}(f(a_{0})) = b_{2t−2s}^{∗}(f(a_{2s})) = 0. But f(a_{0}) ∈ ⨁_{t=0}^{∞} kb_{2t}, so f(a_{0}) = 0. Then for every s ≥ 0 and t ≥ 0 we have b_{2t}^{∗}(f(a_{2s})) = b_{2t+2s}^{∗}(f(a_{0})) = 0. f(a_{2s}) ∈ ⨁_{t=0}^{∞} kb_{2t}, so f(a_{2s}) = 0.

The injective hull of F(α) is j : F(α) ↪ J where J is the free kG-module with kGbasis {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

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

We close this section with 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 [9, Theorem 10.3.1], which in turn is an adaptation-generalization to group cohomology of a theorem of J. Duflot in equivariant cohomology (see [6]).

Proposition 10. 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.

4. Universal Phantom Maps

Let M be a countably generated kG-module, let (M_{i})_{i≥1} be an ascending chain of finitely generated submodules of M with M = ⋃_{i=1}^{∞} M_{i}. Consider the maps

δ : ⨁_{i=1}^{∞} M_{i} → ⨁_{i=1}^{∞} M_{i}, x = (x_{i})_{i≥1} ↦ δ(x) = (x_{1}, x_{2} − x_{1}, x_{3} − x_{2}, x_{4} − x_{3}, …) ,

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

We have a commutative diagram as follows, where the rows are exact.

[figure: 0; ⨁_{i=1}^{∞} M_{i}; δ; ⨁_{i=1}^{∞} M_{i}; π; M; θ; 0; 0; ⨁_{i=1}^{∞} M_{i}; ⨁_{i=1}^{∞} IM_{i}; ⨁_{i=1}^{∞} ΩM_{i}; 0]

Theorem 11. The map θ : M → ⨁_{i=1}^{∞} ΩM_{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(kG) through θ.

Proof. We argue in StMod(kG). The composite θπ is zero, so for every i ≥ 1 the restriction of θ to M_{i} is zero. Hence θ is phantom. Now the sequence

[figure: ⨁_{i=1}^{∞} M_{i}; π; M; θ; ⨁_{i=1}^{∞} ΩM_{i}]

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

[figure: ⨁_{i=1}^{∞} M_{i}; π; zero; M; θ; f; ⨁_{i=1}^{∞} ΩM_{i}; f̃; N]

Corollary 12. Let M, M′, M′′ be kG-modules with M countably generated. If f : M → M′ and f′ : M′ → M′′ are phantom maps then the composite f′f is trivial.

Proof. We make use of the notations of Theorem 11 and its proof, and argue in StMod(kG). We have f = f̃θ so f′f factors through f′f̃. This latter map is phantom out of a direct sum of finitely generated modules, and hence is trivial.

The construction of the universal map θ and its use to prove Corollary 12 have their analogues in the theory of spectra; see [5].

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. It appears that the pure-projective dimension of a module M provides an upper bound for the number of phantom maps one can compose, starting with a map out of M, and not necessarily get the trivial map. But it remains unclear whether this is the least upper bound.

The following theorem characterizes modules out of which all phantom maps are trivial.

Theorem 13. The following are equivalent:

(i) All phantom maps out of M are trivial. (ii) The universal phantom map θ : M → ⨁_{i=1}^{∞} ΩM_{i} is trivial. (iii) M is a direct summand of a direct sum of finitely generated kG-modules, i.e.

M is pure-projective.

Proof. (i) and (ii) are equivalent by the universality of θ. The triviality of θ is equivalent to the existence of a split short exact sequence

[figure: 0; ⨁_{i=1}^{∞} M_{i}; δ̃; (proj.) ⊕ ⨁_{i=1}^{∞} M_{i}; π̃; M; 0]

where δ̃ and π̃ are stable equivalent to δ and π respectively.

Acknowledgements

This work is part of Ph.D. research in progress. I would like to thank my advisor David J. Benson for many inspirational conversations. A preliminary report on a portion a this paper was presented at the American Mathematical Society 1996 Summer Research Institute. I would like to thank the organizers of the meeting.

References

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[9] L. Evens, “The cohomology of groups,” Oxford Mathematical Monographs, Clarendon Press, Oxford - New York - Tokyo, 1991.

[10] B.I. Gray, Spaces of the same n-type, for all n, Topology 5 (1966), 241-243.

[11] C.U. Jensen, Sur les foncteurs dérivés de \begin{matrix}lim \\ ←------- \end{matrix} et leurs applications en théorie des modules, Lecture Notes in Mathematics 254, Springer-Verlag, Berlin - Heidelberg - New York, 1972.

[12] C.A. McGibbon, Phantom maps, in “Handbook of algebraic topology,” 1209-1257,Elsevier Science B.V., Amsterdam, 1995.

[13] J. Rickard, Idempotent modules in the stable category, J. London Math. Soc., to appear.

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[15] C.A. Weibel, “An introduction to homological algebra,” Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge - New York - Melbourne, 1994.

Department of Mathematics, University of Georgia, Athens, GA 30602-7403, USA

   E-mail address: gnacadja@math.uga.edu