THE VARIETY OF SUBADDITIVE FUNCTIONS FOR FINITE GROUP SCHEMES

DAVE BENSON AND HENNING KRAUSE

Abstract. For a finite group scheme, the subadditive functions on finite dimensional representations are studied. It is shown that the projective variety of the cohomology ring can be recovered from the equivalence classes of subadditive functions. Using Crawley-Boevey’s correspondence between subadditive functions and endofinite modules, we obtain an equivalence relation on the set of point modules introduced in our joint work with Iyengar and Pevtsova. This corresponds to the equivalence relation on π-points introduced by Friedlander and Pevtsova.

1. Introduction

A theorem of Crawley-Boevey [5] gives a correspondence between endofinite modules for a ring A and subadditive functions on the finitely presented A-modules.

We examine this correspondence in the context of finite group schemes, for certain endofinite ‘point modules’ for a finite group scheme G introduced in our joint work with Iyengar and Pevtsova [3]. These point modules come from the π-points introduced by Friedlander and Pevtsova [6]. There is a natural equivalence relation on π-points, and it is proved in [6] that the equivalence classes of π-points can be used to reconstruct the variety Proj H^{∗}(G, k).

We translate the equivalence relation of Friedlander and Pevtsova into a corresponding equivalence relation on subadditive functions. This enables us to prove that one can recover Proj H^{∗}(G, k) from the subadditive functions on finite dimensional G-modules in a natural way (Theorem 4.1).

2. Subadditive functions and endofinite modules

We briefly review Crawley-Boevey’s correspondence between subadditive functions and endofinite modules.

We fix a ring A and consider the category of (right) A-modules. Let mod A denote the full subcategory of finitely presented A-modules. For an A-module M let ℓ_{A}(M) denote its composition length.

A subadditive function χ: mod A → ℕ assigns to each finitely presented Amodule a non-negative integer such that

(1) χ(X ⊕ Y) = χ(X) + χ(Y) for all X, Y ∈ mod A, and

(2) χ(X) + χ(Z) ≥ χ(Y) for each exact sequence X → Y → Z → 0 in mod A.

A subadditive function χ ≠ 0 is irreducible if χ cannot be written as a sum of two non-zero subadditive functions.

Date: April 5, 2016.

2010 Mathematics Subject Classification. 16G10 (primary); 20C20, 20G10 20J06 (secondary).

Key words and phrases. subadditive function, endofinite module, stable module category, finite group scheme.

An A-module M is called endofinite if it has finite composition length when viewed as a left module over its endomorphism ring End_{A}(M). An endofinite Amodule M gives rise to a subadditive function χ_{M} by setting

χ_{M}(X) := ℓ_{End_{A}(M)}(Hom_{A}(X, M)) for X ∈ mod A.

The following theorem of Crawley-Boevey [5, §5] provides the context for our study of subadditive functions.

Theorem 2.1. Any subadditive function mod A → ℕ can be written uniquely as a finite sum of irreducible subadditive functions. Sending an endofinite A-module M to χ_{M} induces a bijection between the isomorphism classes of indecomposable endofinite A-modules and the irreducible subadditive functions mod A → ℕ. □

Note that every endofinite module decomposes uniquely into indecomposable endofinite modules. We have χ_{M⊕M′} = χ_{M} + χ_{M′} when M and M′ have no common indecomposable summand, while χ_{M⊕M} = χ_{M} [5, §5].

3. The additive locus of a subadditive function

We fix a ring A such that mod A is an abelian category. For a subadditive function χ: mod A → ℕ we define the additive locus Adloc(χ) as the full subcategory of objects Z ∈ mod A such that for every exact sequence 0 → X → Y → Z → 0 in mod A

χ(X) − χ(Y) + χ(Z) = 0.

We collect some basic properties of the additive locus.

Lemma 3.1. Let M be an endofinite A-module and Z ∈ mod A. Then Z belongs to Adloc(χ_{M}) if and only if Ext_{A}^{1}(Z, M) = 0.

Proof. Applying Hom_{A}(−, M) to an exact sequence 0 → X → Y → Z → 0 in mod A induces a long exact sequence

0 → Hom_{A}(Z, M) → Hom_{A}(Y, M) → Hom_{A}(X, M) → Ext_{A}^{1}(Z, M) → ⋯

of End_{A}(M)-modules. Clearly, Ext_{A}^{1}(Z, M) = 0 implies

χ_{M}(X) − χ_{M}(Y) + χ_{M}(Z) = 0.

The converse follows by choosing for Y a projective A-module. □

Lemma 3.2. Let χ, χ′: mod A → ℕ be subadditive functions. Then

Adloc(χ + χ′) = Adloc(χ) ∩ Adloc(χ′).

Proof. Let η: 0 → X → Y → Z → 0 be an exact sequence in mod A. Then χ(η) := χ(X) − χ(Y) + χ(Z) is a non-negative integer. Thus (χ + χ′)(η) = 0 if and only if χ(η) = 0 = χ′(η). From this observation the assertion of the lemma follows. □

Lemma 3.3. Let χ: mod A → ℕ be a subadditive function. Then there exists an endofinite A-module M such that Adloc(χ) = Adloc(χ_{M}).

Proof. We apply Theorem 2.1. Write χ = ∑_{i} n_{i}χ_{i} with χ_{i} irreducible and χ_{i} ≠ χ_{j} for all i ≠ j. There are indecomposable endofinite A-modules M_{i} such that χ_{i} = χ_{M_{i}} for all i, and ∑_{i} χ_{i} = χ_{M} for M = ⨁_{i} M_{i}. Then Adloc(χ) = Adloc(χ_{M}) by Lemma 3.2. □

Let χ, χ′: mod A → ℕ be subadditive functions. We set

χ ≥ χ′ : ⟺ Adloc(χ) ⊆ Adloc(χ′)

and call χ and χ′ equivalent if Adloc(χ) = Adloc(χ′). Thus the equivalence classes of subadditive functions form a partially ordered set.

4. Subadditive functions for finite group schemes

Let G be a finite group scheme over a field k. Thus G is an affine group scheme such that its coordinate algebra k[G] is finite dimensional as a k-vector space. The k-linear dual of k[G] is a cocommutative Hopf algebra, called the group algebra of G, and denoted kG. We identify G-modules with modules over the group algebra kG. The category of finite dimensional G-modules is denoted by mod G.

The tensor product over k induces one for mod G via the diagonal action of G. A subcategory C of mod G is called tensor closed, if M ∈ C and N ∈ mod G imply that M ⊗_{k} N is in C.

We write H^{∗}(G, k) for the cohomology algebra of G and Proj H^{∗}(G, k) for the set of its homogeneous prime ideals not containing the unique maximal ideal of positive degree elements. Note that H^{∗}(G, k) acts on Ext_{G}^{∗}(M, N) for all G-modules M, N. The support of a finite dimensional G-module M is

supp_{G}(M) := {𝔭 ∈ Proj H^{∗}(G, k) | Ext_{G}^{∗}(M, M)_{𝔭} ≠ 0}.

The following theorem says that Proj H^{∗}(G, k) can be recovered from the equivalence classes of subadditive functions on mod G.

We call a subadditive function on mod G tensor closed if its additive locus is a tensor closed subcategory of mod G.

An element x of a poset is join irreducible if it is not the supremum of elements that are strictly smaller than x.

Theorem 4.1. Let G be a finite group scheme over a field k and let P (G, k) denote the partially ordered set of equivalence classes of tensor closed subadditive functions χ: mod G → ℕ. If the class of χ is join irreducible, then there exists a unique 𝔭 ∈ Proj H^{∗}(G, k) such that

Adloc(χ) = {M ∈ mod G | 𝔭 ∉ supp_{G}(M)}.

Sending χ to 𝔭 induces an order isomorphism between the set of join irreducible elements of P (G, k) and Proj H^{∗}(G, k).

The proof will be given in §7.

5. The additive locus of a tensor closed function

In this section we study the additive locus of a subadditive function that is tensor closed. We need the following result, which essentially comes from [1].

For G-modules M, N and n ∈ ℤ, we write Êxt_{G}^{n}(M, N) for the nth Tate extension group; it equals H^{n}(Hom_{G}(tM, N)) where tM denotes a Tate resolution of M. Note that

Êxt_{G}^{n}(M, N) ≅ Ext_{G}^{n}(M, N) for n > 0.

Theorem 5.1. Given a finite group scheme G over a field k, there exists a positive integer r such that for any G-modules M and N, if Êxt_{G}^{n}(M, N) = 0 for r consecutive values of n then Êxt_{G}^{n}(M, N) = 0 for all n positive and negative.

Proof. The proof for finite groups given in [1] works just as well for finite group schemes. The input is the finite generation of cohomology, which for finite group schemes was proved by Friedlander and Suslin [7]. □

A full subcategory C of mod G is said to be thick, if any direct summand of a module in C is also in C and for every exact sequence 0 → M′ → M → M′′ → 0 in mod G with two of M, M′, M′′ in C also the third is in C.

Corollary 5.2. Let χ: mod G → ℕ be a tensor closed subadditive function. Then Adloc(χ) is a thick subcategory of mod G. If χ = χ_{M} for some endofinite G-module M, then we have

Adloc(χ_{M}) = {X ∈ mod G | Êxt_{G}^{∗}(X, M) = 0}.

Proof. We may assume that χ = χ_{M} for some endofinite G-module M by Lemma 3.3. Fix X ∈ mod G. From Lemma 3.1 it follows that X is in Adloc(χ) if and only if Ext_{G}^{1}(X, M) = 0. Let Ω(X) denote the kernel of a projective cover of X, and observe that Ω(X) ≅ X ⊗_{k} Ω(k) up to projective direct summands. Using dimension shift and the fact that χ is tensor closed, it follows that X is in Adloc(χ) if and only if Ext_{G}^{n}(X, M) = 0 for every n > 0. Now Theorem 5.1 implies that X is in Adloc(χ) if and only if Êxt_{G}^{∗}(X, M) = 0. From this description of Adloc(χ) and the long exact sequence for Êxt_{G}^{∗}(−, M) it follows that Adloc(χ) is a thick subcategory. □

Remark 5.3. Given any subadditive function χ: mod G → ℕ, then

χ′ := ∑_{S simple} χ(− ⊗_{k} S)

is up to equivalence the unique minimal tensor closed subadditive function such that χ′ ≥ χ. In fact, Lemma 3.2 shows that

Adloc(χ′) = {X ∈ mod G | X ⊗_{k} Y ∈ Adloc(χ) for all Y ∈ mod G}.

6. Tensor closed thick subcategories and π-points

In this section we recall some of the results from our joint work with Iyengar and Pevtsova [2, 3]. The first result is a classification of tensor closed thick subcategories of mod G that has been anticipated in [6].

For a subcategory C of mod G we set

supp_{G}(C) := ⋃_{M∈C} supp_{G}(M).

A subset U of Proj H^{∗}(G, k) is called specialisation closed if whenever 𝔭 is in U so is any prime 𝔮 containing 𝔭.

Theorem 6.1. Let G be a finite group scheme over a field k. Then the assignments

C ↦ supp_{G}(C)  and  U ↦ {M ∈ mod G | supp_{G}(M) ⊆ U}

give mutually inverse and inclusion preserving bijections between the non-zero tensor closed thick subcategories of mod G and the specialisation closed subsets of Proj H^{∗}(G, k).

Proof. See Theorem 10.3 of [2]. □

In [6], Friedlander and Pevtsova introduced for a finite group scheme G over a field k of characteristic p > 0 the notion of a π-point. This is by definition a flat algebra homomorphism α: K[t]/(t^{p}) → KG for some field extension K/k such that α factors through the group algebra of a unipotent abelian subgroup scheme of G_{K}, where G_{K} denotes the group scheme over K with group algebra KG := kG ⊗_{k} K. Let α^{∗}: mod G_{K} → mod K[t]/(t^{p}) denote restriction along α. We set

Thick(α) := {M ∈ mod G | α^{∗}(M ⊗_{k} K) is projective}

and observe that Thick(α) is a tensor closed thick subcategory of mod G.

Two π-points α and β are equivalent if Thick(α) = Thick(β), and the equivalence classes are in natural bijection with the points of Proj H^{∗}(G, k); see [6, Theorem 3.6]. The following theorem makes this correspondence explicit.

Theorem 6.2. Let 𝔭 ∈ Proj H^{∗}(G, k). Then there exists a π-point α such that

Thick(α) = {M ∈ mod G | 𝔭 ∉ supp_{G}(M)}.

Proof. See Theorem 6.1 of [2]. □

7. Proof of the main theorem

This section provides the proof of the main theorem, and we begin with some preparation.

Let α: K[t]/(t^{p}) → KG be a π-point for G. This gives rise to a subadditive function χ_{α} on mod G by setting

χ_{α} := dim_{K}(Hom_{K[t]/(t^{p})}(α^{∗}(− ⊗_{k} K), K)).

We may think of χ_{α} as composite

mod G  mod G_{K}  mod K[t]/(t^{p}) ℕ. −⊗_{k}K α^{∗} χ_{K}

Lemma 7.1. We have Adloc(χ_{α}) = Thick(α).

Proof. Let Z ∈ mod G. If α^{∗}(Z ⊗_{k} K) is a projective K[t]/(t^{p})-module, then for any exact sequence 0 → X → Y → Z → 0 in mod G we have

χ_{α}(X) − χ_{α}(Y) + χ_{α}(Z) = 0.

For the converse, choose an exact sequence 0 → X → Y → Z → 0 in mod G with Y projective. Thus α^{∗}(Y ⊗_{k} K) is projective. If

χ_{α}(X) − χ_{α}(Y) + χ_{α}(Z) = 0,

then the sequence

0 → α^{∗}(X ⊗_{k} K) → α^{∗}(Y ⊗_{k} K) → α^{∗}(Z ⊗_{k} K) → 0

splits, and therefore α^{∗}(Z ⊗_{k} K) is projective. □

Lemma 7.2. Let X be a topological space which is a T_{0} space. Fix a set P of closed subsets of X that contains the closure \overline{\{x\}} for each x ∈ X, and view P as a poset via the inclusion order. Then the assignment x ↦ \overline{\{x\}} identifies X with the set of join irreducible elements of P.

Proof. Straightforward. □

Proof of Theorem 4.1. We consider Proj H^{∗}(G, k) with the Hochster dual of the Zariski topology. Thus the open subsets are precisely the specialisation closed subsets [8]. The assignment

χ ⟼ Proj H^{∗}(G, k) \ supp_{G}(Adloc(χ))

identifies the equivalence classes of tensor closed subadditive functions mod G → ℕ with certain closed subsets of Proj H^{∗}(G, k). This follows from Corollary 5.2 and Theorem 6.1. On the other hand, Theorem 6.2 provides for each 𝔭 ∈ Proj H^{∗}(G, k) a π-point α such that χ_{α} is a tensor closed subadditive function satisfying

Adloc(χ_{α}) = {M ∈ mod G | 𝔭 ∉ supp_{G}(M)}

by Lemma 7.1. Thus the function χ_{α} is sent to the closure \overline{\{𝔭\}} = \{𝔮 | 𝔮 ⊆ 𝔭\}. Moreover, given a π-point β corresponding to 𝔮 ∈ Proj H^{∗}(G, k) we have

χ_{α} ≥ χ_{β} ⟺ Adloc(χ_{α}) ⊆ Adloc(χ_{β}) ⟺ 𝔭 ⊇ 𝔮.

Now the assertion of the theorem follows from Lemma 7.2. □

8. π-points and point modules

Let G be a finite group scheme over a field k of characteristic p > 0. To each π-point α: K[t]/(t^{p}) → KG corresponds a point module

Δ_{G}(α) := res_{k}^{K}(Hom_{K[t]/(t^{p})}(KG, K)).

This is an endofinite G-module and plays a prominent role in recent work with Iyengar and Pevtsova [3].

Lemma 8.1. We have χ_{Δ_{G}(α)} = χ_{α}.

Proof. Adjunction gives for each M ∈ mod G a natural isomorphism

Hom_{G}(M, Δ_{G}(α)) ≅ Hom_{G_{K}} (M ⊗_{k} K, Hom_{K[t]/(t^{p})}(KG, K))
≅ Hom_{K[t]/(t^{p})}(α^{∗}(M ⊗_{k} K), K)

which restricts to submodules over the endomorphisms rings of Δ_{G}(α) and K respectively. □

Corollary 8.2. Let G be a finite group scheme over a field k. Given π-points α and β of G, the following conditions are equivalent:

(1) The π-points α and β are equivalent.

(2) The subadditive functions χ_{Δ_{G}(α)} and χ_{Δ_{G}(β)} are equivalent.

Proof. Combine Lemmas 7.1 and 8.1. □

References

[1] D. J. Benson, J. F. Carlson and G. R. Robinson, On the vanishing of group cohomology, J. Algebra 131 (1990), no. 1, 40–73

[2] D. J. Benson, S. B. Iyengar, H. Krause, and J. Pevtsova, Stratification for module categories of finite group schemes, preprint 2015; arXiv:1510.06773.

[3] D. J. Benson, S. B. Iyengar, H. Krause, and J. Pevtsova, Colocalising subcategories of modules over finite group schemes, preprint 2016; arXiv:1604.00524.

[4] W. Crawley-Boevey, Tame algebras and generic modules, Proc. London Math. Soc. (3) 63 (1991), no. 2, 241–265.

[5] W. Crawley-Boevey, Modules of finite length over their endomorphism rings, Representations of algebras and related topics (Tsukuba, 1990), 127–184, London Math. Soc. Lecture Note Ser., 168 Cambridge Univ. Press, Cambridge, 1992.

[6] E. M. Friedlander and J. Pevtsova, Π-supports for modules for finite groups schemes, Duke Math. J. 139 (2007), 317–368.

[7] E. M. Friedlander and A. Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), 209270.

[8] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43–60.

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

Henning Krause, Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany.