PHANTOM MAPS AND PURITY IN MODULAR REPRESENTATION THEORY, II

D. J. BENSON AND G. PH. GNACADJA

Abstract. In the second part of this paper, we use the theory described in the first part to construct an example of a counterintuitive phenomenon. We show how to produce examples of filtered systems in the stable module category stmod(kG) which do not lift to filtered systems in the module category mod(kG). Our main tool is the Extended Milnor Sequence, a five-term exact sequence which specializes to the classical Milnor sequence under certain countability conditions.

1. Introduction

In the first part of this paper, we established the machinery for understanding the connection between phantom maps and purity in modular representation theory. This second part is devoted to using this theory to provide examples of a phenomenon which it might be tempting to suppose could not exist.

Theorem 1.1. Let k be an uncountable field of characteristic p, and G be a finite group of p-rank at least two. Then there exists a filtered system of modules in the stable module category stmod(kG) which does not lift to a filtered system in the module category mod(kG).

Our main tool will be the spectral sequence

E_{2}^{pq} = lim_{\overleftarrow{α}}^{p} Ext_{kG}^{q}(M_{α}, N) ⟹ Ext_{kG}^{n}(M, N).

This spectral sequence has been studied by Christensen (see [2]) in relation to purity and phantom maps in the derived category of a ring. The result about the phantom filtration is from his work. One of the new features in our work is the Extended Milnor Sequence and the explicit description of the transgression d_{2}^{01}.

Theorem 1.2. There is an exact sequence

0⟶\underline{Ph}_{kG}(M, N)⟶\underline{Hom}_{kG}(M, N)⟶lim_{\overleftarrow{α}} \underline{Hom}_{kG}(M_{α}, N)

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

The first author is partly supported by a grant from the NSF.

This sequence formalizes the problem of lifting filtered systems from the stable module category to the module category. It shows that this problem can be reformulated in terms of the faithfulness of the natural (forgetful) functor Pext_{kG}^{2} → Ext_{kG}^{2}. And this in turn relates to the existence of twofold phantom maps.

Theorem 1.3. Suppose that k is a field of characteristic p and G is a finite group. Then a nonzero element Pext_{kG}^{2} gives rise either to a filtered system in stmod(kG) which does not lift to mod(kG), or to a nonzero twofold phantom map in \underline{Ph}_{kG}^{2}.

We have an explicit method of producing examples where we can show that the former occurs, and this leads to the proof of Theorem 1.1. We do not currently have any example where we can prove that there is a nonzero twofold phantom map. A more precise version of Theorem 1.1 is as follows.

Theorem 1.4. Suppose that M = \underrightarrow{lim}_{α}M_{α} is a kG-module of pure projective dimension at least two, written as a filtered colimit of finitely generated modules. Then there exists a kG-module Y such that

\underline{Hom}_{kG}(M, Y) → lim_{\overleftarrow{α}} \underline{Hom}_{kG}(M_{α}, Y)

is not surjective. Writing Y as a filtered colimit of finitely generated kGmodules Y = \underrightarrow{lim}_{β}Y_{β}, this gives rise to a filtered system in stmod(kG) {M_{α}; Y_{β}} which does not lift to mod(kG).

2. Extended Milnor Sequence

Let {M_{α}}_{α} be a filtered system of finitely generated kG-modules. Associated to it is the standard pure-projective resolution of the colimit M = \underrightarrow{lim}_{α} M_{α}

⋯ ⟶^{∂_{4}} R_{3} ⟶^{∂_{3}} R_{2} ⟶^{∂_{2}} R_{1} ⟶^{∂_{1}} R_{0} ⟶^{∂_{0}} M⟶0,

where

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

and M_{α_{0},α_{1},...,α_{n}} is a copy of M_{α_{0}}. See Section 2.6 of [1] for details. 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.

There is a spectral sequence

E_{2}^{pq} = lim_{\overleftarrow{α}}^{p} Ext_{kG}^{q}(M_{α}, N) ⟹ Ext_{kG}^{n}(M, N).

In Christensen [2, Section 9.1], this spectral sequence is constructed as the one associated to the exact couple with

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

and where 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 the maps resulting from applying Ext_{kG}^{q}(−, N) to 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.

This approach makes it apparent that for n ≥ 1, the resulting filtration of

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

is the phantom filtration

\underline{Hom}_{kG}(M, Ω^{−n}N) = \underline{Ph}_{kG}^{0}(M, Ω^{−n}N) ⊇ \underline{Ph}_{kG}(M, Ω^{−n}N) ⊇

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

See Christensen [2, Section 9.1] for details.

In Jensen [4, Théorème 4.2], the spectral sequence is constructed from a double complex. With this approach, we can explicitly describe the transgression d_{2}^{01} (see Proposition 3.1).

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_{kG}(R_{p}, J^{q})

where the vertical and horizontal differentials are

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

and

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

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

Using the five-term exact sequence with ΩN in place of N, we obtain the following exact sequence.

0⟶\underline{Ph}_{kG}(M, N)⟶\underline{Hom}_{kG}(M, N)⟶lim_{\overleftarrow{α}} \underline{Hom}_{kG}(M_{α}, N)

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

Here, we have replaced the usual last term in the five term sequence with its image, so that the last map is surjective. This proves Theorem 1.2. This sequence generalizes the three term Milnor exact sequence of [3, Theorem 2], obtained with the assumption that M is countably generated, which implies that the last two terms of the above sequence vanish.

Now an element of \underleftarrow{lim}_{α} \underline{Hom}_{kG}(M_{α}, N) gives rise to a filtered system {M_{α}; N} in StMod(kG) in which N is a terminal object. This filtered system lifts to a filtered system in Mod(kG) if and only if the given element is in the image of

lim_{\overleftarrow{α}} Hom_{kG}(M_{α}, N) → lim_{\overleftarrow{α}} \underline{Hom}_{kG}(M_{α}, N).

The commutative square

\xymatrix{Hom_{kG}(M, N) \ar[r]^{≅} \ar@{->>}[d] & \underleftarrow{lim}_{α} Hom_{kG}(M_{α}, N) \ar[d] \\ \underline{Hom}_{kG}(M, N) \ar[r] & \underleftarrow{lim}_{α} \underline{Hom}_{kG}(M_{α}, N)}

shows that this is equivalent to being in the image of

\underline{Hom}_{kG}(M, N) → lim_{\overleftarrow{α}} Hom_{kG}(M_{α}, N),

namely the kernel of d_{2}^{01}.

If the filtered system {M_{α}; N} does not lift from StMod(kG) to Mod(kG), this can be used to produce an example of a filtered system of finitely generated modules in stmod(kG) which does not lift to mod(kG) as follows. We write N as a filtered colimit of finitely generated modules, N = \underrightarrow{lim}_{β}N_{β}. Each map M_{α} → N factors through some N_{β}, and an easy transfinite induction shows that this can be used to construct a filtered system {M_{α}; N_{β}} in stmod(kG) in which each M_{α} maps to some N_{β}. A lifting of this system to mod(kG) gives rise in an obvious way to a lifting of {M_{α}; N} to Mod(kG).

If we are given a nonzero element of Pext_{kG}^{2}(M, ΩN), then there are two possibilities. The first is that it has nonzero image in \underline{Ph}_{kG}^{2}(M, Ω^{−1}N), in which case there is a nonzero twofold phantom map. The second is that it has zero image in \underline{Ph}_{kG}^{2}(M, Ω^{−1}N). In this case, it comes from an element of \underleftarrow{lim}_{α} \underline{Hom}_{kG}(M_{α}, N) which is not in the image of

\underline{Hom}_{kG}(M, N) → lim_{\overleftarrow{α}} Hom_{kG}(M_{α}, N),

and the construction described above gives us a filtered system in stmod(kG) which does not lift to mod(kG). This completes the proof of Theorem 1.3.

In Section 4, we show that if M has pure projective dimension at least two, then for a suitably chosen module N the map d_{2}^{01} has a nonzero kernel. We have not been able to determine whether N can be chosen so that the image of

Pext_{kG}^{2}(M, ΩN) → \underline{Ph}_{kG}^{2}(M, Ω^{−1}N)

is nonzero.

3. The transgression d_{2}^{01}

Let \{f_{α}\}_{α} be an element of \underleftarrow{lim}_{α} \underline{Hom}_{kG}(M_{α}, N). It can be represented by a module homomorphism f : R_{0} = _{⨁α}M_{α}⟶N. Let π_{N} : P_{N}⟶N be the projective cover of N. The map R_{1}⟶^{∂_{1}}R_{0}⟶^{f}N is projective. So 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.}

This diagram shows that d_{1}^{01}(f) = d_{0}^{10}(g) and h = d_{1}^{10}(g); here f and h should be regarded as maps R_{0}⟶N ⊆ J^{1} and R_{2}⟶ΩN ⊆ J^{0} = P_{N} respectively. Recalling that R_{2} = ⨁_{α≤β≤γ} M_{α,β,γ} with M_{α,β,γ} isomorphic to M_{α}, the map h consists of maps h_{αβγ} : M_{α,β,γ} → ΩN. So we have obtained:

Proposition 3.1. The module homomorphism h = {h_{αβγ}}_{α≤β≤γ} above represents the element d_{2}^{01}(\{f_{α}\}_{α}) of Pext_{kG}^{2}(M, ΩN) ≅ \underleftarrow{lim} _{α}^{2} Hom_{kG}(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 3.2).

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. The module Y is defined to be the cokernel in the middle row.

Apply the functor \underline{Hom}_{kG}(−, N) to the distinguished triangle

R_{1}⟶R_{0}⟶Y ⟶Ω^{−1}R_{1}

coming from the middle row. The space \underleftarrow{lim}_{α} \underline{Hom}_{kG}(M_{α}, N) is the kernel of

\underline{Hom}_{kG}(R_{0}, N)⟶\underline{Hom}_{kG}(R_{1}, N),

which is the same as the cokernel of the map

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

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

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

and hence also of the map

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

We note that in the notation of Proposition 3.1, the morphism h factors as

R_{2}⟶K_{2} ⟶^{h′} ΩN

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

Theorem 3.2. The following square is commutative:

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

where the vertical maps are the ones constructed above.

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}_{kG}(Y, N)⟶\underline{Hom}_{kG}(K_{2}, ΩN)⟶Pext_{kG}^{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}_{kG}(Y, N)⟶lim_{\overleftarrow{α}} \underline{Hom}_{kG}(M_{α}, N)⟶^{d_{2}^{01}}Pext_{kG}^{2}(M, ΩN).

4. Lifting filtered systems from StMod(kG) to Mod(kG)

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

Theorem 4.1. Suppose the module M has pure-projective dimension at least two and has no nonprojective phantom self-maps. Then there exists a filtered system of maps {M_{α} → Y }_{α} in StMod(kG) which does not lift to Mod(kG).

Proof. Such a system has to be an element of lim_{\overleftarrow{α}} \underline{Hom}_{kG}(M_{α}, Y) whose image

via the transgression d_{2}^{01} is a nonzero element of Pext_{kG}^{2}(M, ΩY). Consider the commutative diagram below, which is obtained by setting Y = N in Theorem 3.2.

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

Take the image of Id_{Y} in \underleftarrow{lim}_{α} \underline{Hom}_{kG}(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. We obtain that this image is nonzero by arguing that the map Pext_{kG}^{2}(M, K_{2}) → Pext_{kG}^{2}(M, ΩY) is injective. The sequence

0⟶Ω^{−1}K_{2} ⟶^{η} Y ⟶^{ζ} M⟶0

in the 3 × 3 diagram of Section 3 is pure-exact; the map M → Ω^{−2}K_{2} completing it to the right into a distinguished triangle is the universal 2-fold phantom map out of M. Applying the functor Pext_{kG}^{∗}(M, −) to it, we get the exact sequence

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

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

Applying the method described in Section 2, we can use the above theorem to produce a filtered system {M_{α}; Y_{β}} in stmod(kG) which does not lift to mod(kG). This completes the proof of Theorem 1.4.

If a module M has pure-projective dimension at least two and is pureinjective, then it satisfies the hypothesis of Theorem 4.1. Our paper [1] provides examples of such modules. By Theorems 5.4.4 and 5.4.6 of this paper, we can take for G the elementary abelian p-group of rank 2, for k an uncountable field of characteristic p, and for M Rickard’s module F (U), where U is the collection of all proper closed homogeneous subvarieties of the cohomology variety V_{G}. By Proposition 4.3.1 of [1], if E is an elementary abelian subgroup of rank two of a finite group G, we can induce the above example from E to G to obtain an example for G. This completes the proof of Theorem 1.1.

References

1. D. J. Benson and G. Ph. Gnacadja, Phantom maps and purity in modular representation theory, I, Fundamenta Mathematicae, to appear.

2. J. D. Christensen, Ideals in triangulated categories: phantoms, ghosts and skeleta, Adv. in Math., to appear.

3. G. Ph. Gnacadja, Phantom maps in the stable module category, to appear in Journal of Algebra.

4. C. U. Jensen, Les foncteurs dérivés de lim_{⟵} et leurs applications en théorie des modules, Lecture Notes in Mathematics, vol. 254, Springer-Verlag, Berlin/New York, 1972.

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

E-mail address: djb@byrd.math.uga.edu

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

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