RESOLUTIONS BY PERMUTATION MODULES

PAUL BALMER AND DAVE BENSON

Abstract. We prove that, up to adding a complement, every modular representation of a finite group admits a finite resolution by permutation modules.

Let G be a finite group and 𝕜 be a field of characteristic p > 0 dividing the order of G. It is well-known that if G has non-cyclic Sylow p-subgroups, the 𝕜linear representation theory of G is complicated. In particular, the Krull–Schmidt abelian category, 𝕜G-mod, of finite-dimensional 𝕜G-modules admits infinitely many isomorphism classes of indecomposable objects. On the other hand, there is a much simpler class of 𝕜G-modules, the permutation modules, i.e., those isomorphic to 𝕜X for X a finite G-set. The finite collection {𝕜(G/H)}_{H⩽G} additively generates all such modules.

For a 𝕜G-module M ∈ 𝕜G-mod, we want to analyze the existence of what we’ll call a permutation resolution for short, i.e., an exact sequence

(1)  0 → P_{n} → P_{n−1} → ⋯ → P_{1} → P_{0} → M → 0

where all P_{i} are permutation modules. Up to direct summands, it is always possible:

2. Theorem. Let G be a finite group and M ∈ 𝕜G-mod. Then there exists a 𝕜Gmodule N such that M ⊕ N admits a finite resolution (1) by permutation modules.

The related problem of resolutions (1) that are not only exact but remain exact under all fixed-point functors has been recently discussed in [BSW17]. Allowing p-permutation modules P_{i} (that is, direct summands of permutation modules), Bouc–Stancu–Webb prove that such resolutions exist for all M if and only if G has a Sylow subgroup that is either cyclic or dihedral (for p = 2).

Unsurprisingly, Theorem 2 reduces to a Sylow subgroup S of G, since every M is a direct summand of Ind_{S}^{G} Res_{S}^{G}(M) and since the functor Ind_{S}^{G} is exact and preserves permutation modules. So we focus on the case where G is a p-group.

For the proof, we shall consider a stronger property:

3. Definition. We say that a resolution (1) is free up to degree m ⩾ 0 if P_{i} is a free module for i = 0, …, m. We say that M admits good permutation resolutions if for every integer m ⩾ 0, there exists a finite resolution (1) by permutation modules that is free up to degree m.

4. Remark. Let G be a p-group. A 𝕜G-module M admits good permutation resolutions if and only if for all m ⩾ 1 the mth Heller loop Ω^{m}M admits a finite permutation resolution. Also, if Q is free and M ⊕ Q admits a permutation resolution as in (1) then the epimorphism P_{0} ↠ M ⊕ Q ↠ Q forces Q to be a direct summand of P_{0} and one can remove 0 → Q ⟶^{=} Q → 0 from the resolution. So if M ⊕ Q has a permutation resolution that is free up to degree m then so does M.

Date: 2020 March 9.

First-named author supported by NSF grant DMS-1901696.

An advantage of good permutation resolutions is the two out of three property:

5. Proposition. Let G be a p-group. Let 0 → L → M → N → 0 be an exact sequence of 𝕜G-modules. If two out of L, M and N have good permutation resolutions then so does the third.

Proof. If P ↠ N is a projective cover, we obtain by ‘rotation’ an exact sequence 0 → Ω^{1}N → L ⊕ P → M → 0. In view of Remark 4, we can rotate in this way and reduce to the case where L and M admit good permutation resolutions and then prove that N does. Let m ⩾ 0. Choose P_{•} → M a permutation resolution of M that is free up to degree m. Let ℓ ⩾ m be such that P_{i} = 0 for all i > ℓ. Now choose Q_{•} → L a permutation resolution of L that is free up to degree ℓ. We have the following picture (plain part) with exact rows:

(6)  \xymatrix{0 \ar[r] & Q_{n} \ar[r] \ar@{.>}[d] & ⋯ \ar[r] & Q_{ℓ+1} \ar[r] \ar@{.>}[d] & Q_{ℓ} \ar[r] \ar@{.>}[d] & ⋯ \ar[r] & Q_{0} \ar[r] \ar@{.>}[d] & L \ar[r] \ar[d] & 0 \\ 0 \ar[r] & 0 \ar[r] & ⋯ \ar[r] & 0 \ar[r] & P_{ℓ} \ar[r] & ⋯ \ar[r] & P_{0} \ar[r] & M \ar[r] & 0}

The standard lifting argument, using that Q_{j} is projective for j = 0, …, ℓ shows that there exists a lift f_{•}: Q_{•} → P_{•} of the morphism L → M. Then the mapping cone complex cone(f_{•}) yields a resolution of coker(L → M) = N and this complex cone(f_{•}) has free objects in degree 0, …, m since P_{•} and Q_{•} do. □

Let us discuss an example of Theorem 2, where we can even take N = 0.

7. Proposition. Let E = (C_{p})^{×r} = C_{p} × ⋯ × C_{p} be an elementary abelian group of rank r. Then every 𝕜E-module admits good permutation resolutions.

Proof. Consider for each 1 ⩽ i ⩽ r the (‘coordinate-wise’) subgroup

H_{i} = C_{p} × ⋯ × C_{p} × 1 × C_{p} × ⋯ × C_{p}

of rank r−1. Let m ⩾ 0. Inflating from E/H_{i} ≃ C_{p} the usual 2-periodic resolutions 0 → 𝕜 → 𝕜C_{p} → ⋯ → 𝕜C_{p} → 𝕜 → 0 of length at least m, we obtain quasiisomorphisms of 𝕜E-modules s_{i}: Q(i) → 𝕜[0] where the Q(i) are defined as follows:

\xymatrix{Q(i) := \ar[d]_{s_{i}} & 0 \ar[r] & 𝕜 \ar[r] \ar[d] & 𝕜(E/H_{i}) \ar[r] \ar[d] & ⋯ \ar[r] & 𝕜(E/H_{i}) \ar[r] \ar[d] & 𝕜(E/H_{i}) \ar[r] \ar[d] & 0 \\ 𝕜[0] = & 0 \ar[r] & 0 \ar[r] & 0 \ar[r] & ⋯ \ar[r] & 0 \ar[r] & 𝕜 \ar[r] & 0}

Tensoring all the above, we obtain a quasi-isomorphism

s_{1} ⊗ ⋯ ⊗ s_{r}: P_{•} := Q(1) ⊗ ⋯ ⊗ Q(r) → (𝕜[0])^{⊗r} ≅ 𝕜[0],

i.e., a permutation resolution P_{•} of 𝕜. In other words, we performed an ‘external tensor’ of all the periodic resolutions over each copy of C_{p} in E. Since the Mackey formula gives by induction 𝕜(E/H_{i_{1}}) ⊗ ⋯ ⊗ 𝕜(E/H_{i_{n}}) ≅ 𝕜(E/(H_{i_{1}} ∩ ⋯ ∩ H_{i_{n}})), we have produced a permutation resolution P_{•} of 𝕜 that is easily seen to be free up to degree m. As m ⩾ 0 was arbitrary, we proved that the trivial module 𝕜 admits good permutation resolutions. A general module M ∈ 𝕜E-mod admits a filtration whose successive quotients are trivial. We therefore conclude by induction, via Proposition 5. □

8. Remark. The proof of Proposition 7 shows that the stabilisers in the permutation resolution may be taken to be products of subsets with respect to the given decomposition of E. Applying the proposition to a module and its dual shows that given a module M we may form a finite exact complex of permutation modules with these stabilisers in such a way that the image of one of the maps is M. This should be compared with the main theorem of [BC] which shows that a finite exact sequence of permutation E-modules in which the set of stabilisers has no containment of index p necessarily splits, so that the image of every map is again a permutation module.

Proof of Theorem 2. As already mentioned, we can reduce to the case where G is a p-group. By [Car00], we know that for every 𝕜G-module M, there exists a 𝕜G-module N and a finite filtration 0 = L_{0} ⊂ L_{1} ⊂ ⋯ ⊂ L_{s} = M ⊕ N such that every L_{i}/L_{i−1} is induced from some elementary abelian subgroup E_{i} ⩽ G. Since the result holds for elementary abelian groups (Proposition 7) and is stable by induction, we see that all L_{i}/L_{i−1} admit good permutation resolutions. By Proposition 5, we conclude that so does M ⊕ N. In particular, M ⊕ N has a permutation resolution. □

Acknowledgements: The authors are grateful to Serge Bouc, Martin Gallauer and Peter Webb for useful discussions. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programmes ‘K-theory, algebraic cycles and motivic homotopy theory’ and ‘Groups, representations and applications: new perspectives’, where work on this paper was undertaken. The Isaac Newton Institute is supported by EPSRC grant no EP/R014604/1.

References

[BC] David J. Benson and Jon F. Carlson. Bounded complexes of permutation modules. In preparation.

[BSW17] Serge Bouc, Radu Stancu, and Peter Webb. On the projective dimensions of Mackey functors. Algebr. Represent. Theory, 20(6):1467–1481, 2017.

[Car00] Jon F. Carlson. Cohomology and induction from elementary abelian subgroups. Q. J. Math., 51(2):169–181, 2000.

Paul Balmer, Mathematics Department, UCLA, Los Angeles, CA 90095-1555, USA

   Email address: balmer@math.ucla.edu
   URL: http://www.math.ucla.edu/∼balmer

Dave Benson, Institute of Mathematics, University of Aberdeen, King’s College, Aberdeen AB24 3UE, Scotland U.K.

   Email address: d.j.benson@abdn.ac.uk
   URL: https://homepages.abdn.ac.uk/d.j.benson/pages/