PHANTOM MAPS AND PURITY IN MODULAR REPRESENTATION THEORY, III

D. J. BENSON

Abstract. In the third installment of this paper, we exhibit two phantom maps between kG-modules, whose composite is not projective, whenever k is an uncountable field of characteristic p and G is a finite group of p-rank at least two.

This paper continues the investigation begun in [1, 2]. In the first two parts, we investigated phantom maps in the modular representation theory of finite groups, and related them to purity. When those papers were written, we were unable to show the existence of phantom maps whose composite does not factor through a projective module, although we strongly expected them to exist. The purpose of this paper is to prove the existence of such nonprojective composites.

The example was already familiar to us; what was lacking was a proof that it really was an example of this phenomenon. Namely, if E is an elementary abelian p-group and k is a field of characteristic p, we let F (U) denote the Rickard F -idempotent module [6] in StMod(kG) corresponding to the thick subcategory U of stmod(kG) consisting of finitely generated kG-modules M such that V_{G}(M) is a proper subset of the cohomology variety V_{G}. Then Ĥ^{∗}(G, F (U)) is the localization of

H^{∗}(G, k) = \begin{cases}k[x_{1}, …, x_{r}], deg(x_{i}) = 1 & p = 2 \\ k[x_{1}, …, x_{r}] ⊗ Λ(y_{1}, …, y_{r}) & p odd \\ deg(x_{i}) = 2, deg(y_{i}) = 1, x_{i} = β(y_{i}) \end{cases}

obtained by inverting all homogeneous elements of positive degree in the polynomial subring k[x_{1}, …, x_{r}].

Theorem 1. The composite of two universal phantom maps out of F (U) does not factor through a projective module, provided k is uncountable and E has rank at least two.

The difficulty with proving this is that there are nontrivial maps from Ĥ^{∗}(G, F (U)) to negative Tate cohomology Ĥ^{−}(G, k). The device which we use in this paper in order to circumvent this difficulty is that we tensor with another Rickard idempotent module F (x_{1}). This has the effect of localizing Tate cohomology by inverting x_{1}, a process which kills all negative degree Tate cohomology. But enough survives the localization that the composite of the phantoms is still not projective.

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

It is worth rewriting this theorem using theorem 4.6.1 of [1].

Corollary 2. Provided E has rank at least two and k is uncountable, the image of the natural map

Pext_{kG}^{2}(F (U), −) → Ext_{kG}^{2}(F (U), −)

is nonzero.

It is also worth pointing out that by inducing from kE to kG, the theorem generalizes easily to any finite group G of p-rank at least two.

Corollary 3. Let k be an uncountable field of characteristic p, and let G be a finite group of p-rank at least two. Then there exists a pair of composable phantom maps between kG-modules, whose composite does not factor through a projective module.

We work throughout this paper in the stable category StMod(kG) of all kG-modules. The maps in this category are taken to be

\underline{Hom}_{kG}(M, N) = Hom_{kG}(M, N)/PHom_{kG}(M, N).

Here, PHom_{kG}(M, N) denotes the subspace of Hom_{kG}(M, N) consisting of homomorphisms which factor through some projective kG-module. It is well known that StMod(kG) is a triangulated category in which the triangles come from short exact sequences in the module category Mod(kG).

Since we shall be using the point of view developed in §4.4 of [1], we pause to point out that the proof theorem 4.4.1 of that paper is incorrect.^{1} In an appendix, we correct the proof.

Next, we show that phantom maps are still phantom after localization, or more generally, after tensoring.

Lemma 4. If ϕ : M → N is phantom and X is any kG-module then

ϕ ⊗ 1_{X} : M ⊗_{k} X → N ⊗_{k} X

is phantom.

Proof. If Y is a finite dimensional module and ρ : Y → M ⊗_{k} X is a homomorphism, then there is a finite dimensional submodule X_{0} of X such that ρ factors through the inclusion

Y → M ⊗_{k} X_{0} → M ⊗_{k} X.

Since X_{0} is finite dimensional, M ⊗_{k} X_{0} ≅ Hom_{k}(X_{0}^{∗}, M). The adjoint of Y → Hom_{k}(X_{0}^{∗}, M) is a map Y ⊗_{k} X_{0}^{∗} → M. Since ϕ is phantom, the composite Y ⊗_{k} X_{0}^{∗} → M → N is projective. But the adjunction preserves projective maps, so the composite Y → M ⊗_{k} X_{0} → N ⊗_{k} X_{0} is projective. Composing with the inclusions, the composite Y → M ⊗_{k} X → N ⊗_{k} X is also projective.

1I would like to thank Henning Krause for pointing out the error.

Now let E be an elementary abelian p-group of rank at least two, and let k be an uncountable field of characteristic p. Consider the Rickard idempotent module F (U) described earlier. By theorem 5.4.4 of [1], F (U) has pure projective dimension at least two. Consider the beginning of a pure projective resolution

0 → K_{0} ⟶^{α} R_{0} ⟶^{β} F (U) → 0  (1)
0 → K_{1} ⟶^{γ} R_{1} ⟶^{δ} K_{0} → 0  (2)

where R_{0} and R_{1} are formed from direct sums of modules of the form Ω^{n}k (n ∈ ℤ) and projective modules (see corollary 4.4.3 of [1]). Then a (weakly) universal phantom map out of F (U) is given by the connecting homomorphism η : F (U) → Ω^{−1}K_{0} for the short exact sequence (1), and a (weakly) universal phantom map out of Ω^{−1}K_{0} is obtained by applying Ω^{−1} to the connecting homomorphism ξ : K_{0} → Ω^{−1}K_{1} for (2), see Gnacadja [3] or the discussion following corollary 4.2.3 of [1]. Our goal is to prove that the com-

posite F (U) ⟶^{η} Ω^{−1}K_{0} ⟶^{Ω^{−1}ξ} Ω^{−2}K_{1} is not a projective homomorphism. So we assume that this composite is zero, and we need to obtain a contradiction. This will take some work, and the final contradiction will be to a theorem of Barbara Osofsky giving the projective dimensions of certain flat modules.

Now consider the following diagram in the triangulated stable module category StMod(kG).

\xymatrix{ &  & Ω^{−1}K_{1} \ar[d]^{Ω^{−1}γ} \\  &  & Ω^{−1}R_{1} \ar[d]^{Ω^{−1}δ} \\ R_{0} \ar[r]^{β} & F (U) \ar@{.>}[ur]^{θ} \ar[r]^{η} & Ω^{−1}K_{0} \ar[d]^{Ω^{−1}ξ} \ar[r]^{Ω^{−1}α} & Ω^{−1}R_{0} \\  &  & Ω^{−2}K_{1}}

If Ω^{−1}ξ ∘ η is projective (i.e., zero in StMod(kG)), then there is a map θ : F (U) → Ω^{−1}R_{1} such that Ω^{−1}δ ∘ θ = η.

Now tensor with the Rickard idempotent module F (x_{1}) and apply the Tate cohomology functor Ĥ^{∗}(G, −). By Lemma 4, tensoring the phantom maps η and Ω^{−1}ξ with F (x_{1}) gives phantom maps. We have Ĥ^{∗}(G, − ⊗_{k} F (x_{1})) = H^{∗}(G, −)_{x_{1}}, the localization of H^{∗}(G, −) (or equivalently of Ĥ^{∗}(G, −)) by inverting x_{1}. Since F (U)⊗_{k}F (x_{1}) is stably isomorphic to F (U), and phantom maps are zero in Tate cohomology, we get a diagram

[figure: 0; H^{∗}(G, Ω^{−1}K_{1})_{x_{1}}; Ω^{−1}γ_{∗}; H^{∗}(G, Ω^{−1}R_{1})_{x_{1}}; Ω^{−1}δ_{∗}; Ĥ^{∗}(G, F (U)) = H^{∗}(G, F (U))_{x_{1}}; 0; θ_{∗}; H^{∗}(G, Ω^{−1}K_{0})_{x_{1}}; 0]

Now in H^{∗}(G, F (U))_{x_{1}} = Ĥ^{∗}(G, F (U)), every element is divisible by (for example) x_{2}, while H^{∗}(G, Ω^{−1}R_{1})_{x_{1}} is a direct sum of shifted copies of H^{∗}(G, k)_{x_{1}}. So there are no nonzero H^{∗}(G, k)-module homomorphisms from H^{∗}(G, F (U))_{x_{1}} to H^{∗}(G, Ω^{−1}R_{1})_{x_{1}}. It follows that θ_{∗} = 0. Composing with β_{∗}, we see that (θ ∘ β)_{∗} = 0. But R_{0} is a direct sum of modules of the form Ω^{n}k (n ∈ ℤ) and projective modules, so (θ ∘ β)_{∗} = 0 implies that θ ∘ β is annihilated by some power of x_{1} in StMod(kG), say x_{1}^{j}(θ ∘ β) = 0. Regarding x_{1}^{j}θ = x_{1}^{j} ∘ θ = Ω^{−2j}θ ∘ x_{1}^{j} as a map from F (U) to Ω^{−2j−1}R_{1}, we see that there exists a homomorphism ψ: Ω^{−1}K_{0} → Ω^{−2j−1}R_{1} such that ψ ∘ η = x_{1}^{j}θ. [If p = 2, the 2j’s should be replaced by j’s in this argument.] Now

(x_{1}^{j}.1 − Ω^{−2j−1}δ ∘ ψ) ∘ η = x_{1}^{j}.1 ∘ (η − Ω^{−1}δ ∘ θ) = 0

and so there exists a homomorphism ϕ : Ω^{−1}R_{0} → Ω^{−2j−1}K_{0} such that

x_{1}^{j}.1 − Ω^{−2j−1}δ ∘ ψ = ϕ ∘ Ω^{−1}α.

But now we obtain

Ω^{−2j−1}ξ ∘ x_{1}^{j}.1 = Ω^{−2j−1}ξ ∘ (Ω^{−2j−1}δ ∘ ψ + ϕ ∘ Ω^{−1}α)
= Ω^{−2j−1}ξ ∘ ϕ ∘ Ω^{−1}α.

Now Ω^{−2j−1}ξ is phantom, so Ω^{−2j−1}ξ ∘ ϕ is phantom. But Ω^{−2j−1}R_{0} is pure projective, so by theorem 4.2.4 of [1], Ω^{−2j−1}ξ ∘ ϕ = 0 and hence Ω^{−2j−1}ξ ∘ x_{1}^{j}.1 = 0 in StMod(kG). So

Ω^{−2j−1}ξ ⊗ 1: Ω^{−2j−1}K_{0} ⊗_{k} F (x_{1}) → Ω^{−2j−2}K_{1} ⊗_{k} F (x_{1})

is also zero. So the sequence (2) splits after tensoring with F (x_{1}). So K_{1} ⊗_{k} F (x_{1}) is direct summand of R_{1} ⊗_{k} F (x_{1}), which is a direct sum of translates of F (x_{1}) and projective modules.

But now let us examine the sequence

0 → K_{1} ⊗_{k} F (x_{1}) → R_{0} ⊗_{k} F (x_{1}) → F (U) ⊗_{k} F (x_{1}) → 0.

Applying Ĥ^{∗}(G, −), we obtain an exact sequence of Ĥ^{∗}(G, k)_{x_{1}}-modules

0 → H^{∗}(G, K_{1})_{x_{1}} → H^{∗}(G, R_{0})_{x_{1}} → Ĥ^{∗}(G, F (U)) → 0.

So we end up with the information that Ĥ^{∗}(G, F (U)) has projective dimension at most one as an H^{∗}(G, k)_{x_{1}}-module.

Now, Ĥ^{∗}(G, F (U)) is the localization of H^{∗}(G, k)_{x_{1}} obtained by inverting all the nonzero homogeneous elements of positive degree. We are almost in the situation described in theorem 6.4 of Osofsky [4], except that instead of localization in a regular local ring, we are working in the graded situation, and everything is tensored with the exterior algebra Λ(y_{1}, …, y_{r}) in case p is odd. However, it is not hard to go through the proof of Osofsky’s theorem 6.4 and check that it works perfectly well in this case. It is also worth comparing with Osofsky [5], where the corresponding result for polynomial rings is proved, with essentially no change in the proof. The following is the version of Osofsky’s theorem that we need.

Theorem 5. Let k be a field of characteristic p and cardinality ℵ_{n} (n is allowed to be infinite), and let E be an elementary abelian p-group of rank r. Then the flat H^{∗}(G, k)_{x_{1}}-module Ĥ^{∗}(G, F (U)), which is the homogeneous localization of H^{∗}(G, k) given by inverting all the nonzero homogeneous elements of k[x_{1}, …, x_{r}] ⊆ H^{∗}(G, k) of positive degree, has projective dimension equal to min(r, n + 1).

Since part of our hypothesis was that r ≥ 2 and n ≥ 1, it follows that Ĥ(G, F (U)) has projective dimension at least two, contradicting the results of the above analysis. This completes the proof of Theorem 1.

Remark. It is tempting to suppose that the above proof generalizes to higher rank and cardinality. I have been unable to make this work, but I am willing to conjecture the following.

Conjecture 6. Let k be a field of characteristic p and cardinality ℵ_{n} (n is allowed to be infinite), and let E be an elementary abelian p-group of rank r. Then the composite of min(r, n + 1) universal phantom maps out of the kE-module F (U) does not factor through a projective module.

Appendix

This appendix corrects the proof of theorem 4.4.1 of [1]. This is used in the proof of Theorem 1, since we use corollary 4.4.3 of that paper.

The problem is that it is not true in general that if

0 → M_{1} → M_{2} → M_{3} → 0

is an exact sequence of kG-modules and N is another kG-module then the sequence of projective homomorphisms

0 → PHom_{kG}(N, M_{1}) → PHom_{kG}(N, M_{2}) → PHom_{kG}(N, M_{3}) → 0

is exact. It is certainly a complex, and is exact everywhere except the middle, but there may be homology at the middle term. A precise description of this homology is given by the following lemma.

Lemma 7. The homology

\frac{Ker(PHom_{kG}(N, M_{2}) → PHom_{kG}(N, M_{3}))}{Im(PHom_{kG}(N, M_{1}) → PHom_{kG}(N, M_{2}))}

is isomorphic to the image of the connecting homomorphism in Tate cohomology

Êxt_{kG}^{−1}(N, M_{3}) → Êxt_{kG}^{0}(N, M_{1}).

Proof. Consider the double complex

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

(the maps in the middle column have been negated to make the squares anticommute). Since the rows are exact, the total complex is exact. On the other hand, the E_{1} page of the corresponding spectral sequence can only be nonzero in four places: E_{1}^{0,1}, E_{1}^{2,0}, E_{1}^{1,2} and E_{1}^{2,2}. Now E_{1}^{1,2} and E_{1}^{2,2} are the images of

Hom_{kG}(N, M_{3}) → Ext_{kG}^{1}(N, M_{1})

and

\underline{Hom}_{kG}(N, M_{3}) → Ext_{kG}^{1}(N, M_{1})

respectively. These maps have the same image, and d_{1} : E_{1}^{1,2} → E_{1}^{2,2} is the identity map on this image. The only remaining differential, d_{2} : E_{2}^{0,1} → E_{2}^{2,0} is therefore an isomorphism. This is the isomorphism required to prove the lemma.

Now, in the proof of (iv) ⇒ (iii) in theorem 4.4.1 of [1], we are assuming that for all finitely generated kG-modules N,

0 → Ext_{kG}^{∗}(N, M_{1}) → Ext_{kG}^{∗}(N, M_{2}) → Ext_{kG}^{∗}(N, M_{3}) → 0

is exact. In particular, since Ω^{−2}N is also finitely generated, the connecting homomorphism

Ext_{kG}^{1}(Ω^{−2}N, M_{3}) → Ext_{kG}^{2}(Ω^{−2}N, M_{1})

is zero, and so by dimension shifting, the connecting homomorphism

Êxt_{kG}^{−1}(N, M_{3}) → Êxt_{kG}^{0}(N, M_{1})

is also zero. So by Lemma 7, the sequence

0 → PHom_{kG}(N, M_{1}) → PHom_{kG}(N, M_{2}) → PHom_{kG}(N, M_{3}) → 0

is exact. This is what is needed to make the rest of the proof work.

References

[1] D. J. Benson and G. Ph. Gnacadja, Phantom maps and purity in modular representation theory, I, Fundamenta Mathematicae 161 (1999), 37–91.

[2] , Phantom maps and purity in modular representation theory, II, Algebras and Representation Theory (to appear).

[3] G. Ph. Gnacadja, Phantom maps in the stable module category, J. Algebra 201 (1998), 686–702.

[4] B. L. Osofsky, Homological dimension and the continuum hypothesis, Trans. Amer. Math. Soc. 132 (1968), 217–230.

[5] , Homological dimensions of modules, Regional conference series in Mathematics, vol. 12, American Math. Society, 1973.

[6] J. Rickard, Idempotent modules in the stable category, J. London Math. Soc. 178 (1997), 149–170.

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

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