Finite good filtration dimension for modules over an algebra with good filtration.

Wilberd van der Kallen

Abstract

Let G be a connected reductive linear algebraic group over a field k of characteristic p > 0. Let p be large enough with respect to the root system. We show that if a finitely generated commutative k-algebra A with G-action has good filtration, then any noetherian A-module with compatible G-action has finite good filtration dimension.

1 Introduction

Consider a connected reductive linear algebraic group G defined over a field k of positive characteristic p. We say that G has the cohomological finite generation property (CFG) if the following holds: Let A be a finitely generated commutative k-algebra on which G acts rationally by k-algebra automorphisms. (So G acts on Spec(A).) Then the cohomology ring H^{∗}(G, A) is finitely generated as a k-algebra. Here, as in [9, I.4], we use the cohomology introduced by Hochschild, also known as ‘rational cohomology’.

In [13] we have shown that SL_{2} over a field of positive characteristic has property (CFG), and in [14] we proved that SL_{3} over a field of characteristic two has property (CFG). We conjecture that every reductive linear algebraic group has property (CFG). In this paper we show that this is at least a good heuristic principle: We derive one of the consequences of (CFG) for any simply connected semisimple linear algebraic group G that satisfies the following

Hypothesis 1.1 Assume that for every fundamental weight ϖ_{i} the symmetric algebra S^{∗}(∇_{G}(ϖ_{i})) on the fundamental representation ∇_{G}(ϖ_{i}) has a good filtration.

Recall that this hypothesis is satisfied if p ≥ max_{i}(dim(∇_{G}(ϖ_{i}))), by [1, 4.1(5) and 4.3(1)]. This inequality is not necessary. For instance, SL_{n} satisfies the hypothesis for n ≤ 5, by [13, Lemma 3.2]. When p = 2, the hypothesis does not hold for SL_{n} with n ≥ 6, by [13, 3.3].

In the sequel let G be a connected reductive linear algebraic group over a field k of characteristic p > 0 with simply connected commutator subgroup for which hypothesis 1.1 holds. Let A be a finitely generated commutative k-algebra on which G acts rationally by k-algebra automorphisms. Let M be a noetherian A-module on which G acts compatibly. This means that the structure map A ⊗ M → M is a G-module map. Our main result is

Theorem 1.2 If A has good filtration, then M has finite good filtration dimension and each H^{i}(G, M) is a noetherian A^{G}-module.

When A = k the theorem goes back to [5] and does not need hypothesis 1.1. Unlike the proofs in [13] and [14], the proof of our theorem does not involve any cohomology of finite group schemes and is thus independent of the work of Friedlander and Suslin [6]. But without their work we would not have guessed the theorem. For clarity we will pull some material of [13] free from finite group schemes.

2 Recollections

Some unexplained notations, terminology, properties, … can be found in [9]. We choose a Borel group B^{+} = T U^{+} and the opposite Borel group B^{−}. The roots of B^{+} are positive. If λ ∈ X(T) is dominant, then ind_{B^{−}}^{G}(λ) is the ‘dual Weyl module’ or ‘costandard module’ ∇_{G}(λ) with highest weight λ. The formula ∇_{G}(λ) = ind_{B^{−}}^{G}(λ) just means that ∇_{G}(λ) is obtained from the Borel-Weil construction: ∇_{G}(λ) equals H^{0}(G/B^{−}, L) for a certain line bundle on the flag variety G/B^{−}. In a good filtration 0 = V_{−1} ⊆ V_{0} ⊆ V_{1} … of a G-module V = ⋃_{i} V_{i} the nonzero layers V_{i}/V_{i−1} are of the form ∇_{G}(μ). As in [12] we will actually also allow a layer to be a direct sum of any number of copies of the same ∇_{G}(μ), cf. [9, II.4.16 Remark 1]. This is much more convenient when working with infinite dimensional G-modules. It is shown in [4] that a module of countable dimension that has a good filtration in our sense also has a filtration that is a good filtration in the old sense. Note that the module M in our theorem has countable dimension. It would do little harm to restrict to modules of countable dimension throughout.

If V is a G-module, and m ≥ −1 is an integer so that H^{m+1}(G, ∇_{G}(μ) ⊗ V) = 0 for all dominant μ, then we say as in [5] that V has good filtration dimension at most m. The case m = 0 corresponds with V having a good filtration. And for m ≥ 0 it means that V has a resolution

0 → V → N_{0} → ⋯ → N_{m} → 0

in which the N_{i} have good filtration, in our sense. We say that V has good filtration dimension precisely m, notation dim_{∇}(V) = m, if m is minimal so that V has good filtration dimension at most m. In that case H^{i+1}(G, ∇_{G}(μ) ⊗ V) = 0 for all dominant μ and all i ≥ m. In particular H^{i+1}(G, V) = 0 for i ≥ m. If there is no finite m so that dim_{∇}(V) = m, then we put dim_{∇}(V) = ∞.

2.1 Filtrations

For simplicity assume also that G is semisimple. (Until remark 3.1.) If V is a G-module, and λ is a dominant weight, then V_{≤λ} denotes the largest G-submodule all whose weights μ satisfy μ ≤ λ in the dominance partial order [9, II.1.5]. For instance, V_{≤0} is the module of invariants V^{G}. Similarly V_{<λ} denotes the largest G-submodule all whose weights μ satisfy μ < λ. As in [12], we form the X(T)-graded module

gr_{X(T)} V = ⨁_{λ∈X(T)} V_{≤λ}/V_{<λ}.

Each V_{≤λ}/V_{<λ}, or V_{≤λ/<λ} for short, has a B^{+}-socle (V_{≤λ/<λ})^{U} = V_{λ}^{U} of weight λ. We always view V^{U} as a B^{−}-module through restriction (inflation) along the homomorphism B^{−} → T. Then V_{≤λ/<λ} embeds naturally in its ‘good filtration hull’ hull_{∇}(V_{≤λ/<λ}) = ind_{B^{−}}^{G} V_{λ}^{U}. This good filtration hull has the same B^{+}-socle and by Polo it is the injective hull in the category C_{λ} of Gmodules N that satisfy N = N_{≤λ}. Compare [12, 3.1.10].

We convert the X(T)-graded module gr_{X(T)} V to a ℤ-graded module through an additive height function ht : X(T) → ℤ, defined by ht = 2 ∑_{α>0} α^{∨}, the sum being over the positive roots. (Our ht is twice the one used by Grosshans [7], because we prefer to get even degrees rather than just integer degrees.) The Grosshans graded module is now

gr V = ⨁_{i≥0} gr_{i} V,

with

gr_{i} V = ⨁_{ht(λ)=i} V_{≤λ/<λ}.

In other words, if one puts

V_{≤i} := ∑_{ht(λ)≤i} V_{≤λ},

then gr V is is the associated graded of the filtration V_{≤0} ⊆ V_{≤1} · · ·.

Let us apply the above to our finitely generated commutative k-algebra with G-action A. The Grosshans graded algebra gr A embeds in a good filtration hull, which Grosshans calls R, and which we call hull_{∇}(gr A),

hull_{∇}(gr A) := ind_{B^{−}}^{G} A^{U} = ⨁_{i} ⨁_{ht(λ)=i} hull_{∇}(A_{≤λ}/A_{<λ}).

Grosshans [7] shows that A^{U}, gr A, hull_{∇}(gr A) are finitely generated kalgebras with hull_{∇}(gr A) finite over gr A. Mathieu studied gr A and hull_{∇}(gr A) earlier in [11].

Example 2.2 Consider the multicone [10]

k[G/U] := ind_{U}^{G} k = ind_{B^{+}}^{G} ind_{U}^{B^{+}} k = ind_{B^{+}}^{G} k[T] = ⨁_{λ dominant} ∇_{G}(λ).

It is its own Grosshans graded ring. Recall [10] that it is generated as a k-algebra by the finite dimensional sum of the ∇_{G}(ϖ_{i}), where ϖ_{i} denotes the ith fundamental weight.

Lemma 2.3 Let A have a good filtration, so that gr A = hull_{∇}(gr A). Let R = ⊕_{i}R_{i} be a graded algebra with G-action such that R_{i} = (R_{i})_{≤i}. Then every T -equivariant graded algebra homomorphism R^{U} → (gr A)^{U} extends uniquely to a G-equivariant graded algebra homomorphism R → (gr A).

Proof Use that hull_{∇}(gr A) is an induced module. □

2.4 A graded polynomial G × D-algebra with good filtration

We now extract a construction from [13]. It is hidden in the study of a Hochschild-Serre spectral sequence which in the present situation would correspond with the case where as normal subgroup one takes the trivial subgroup!

As the algebra (gr A)^{U} is finitely generated, it is also generated by finitely many weight vectors. Consider one such weight vector v, say of weight λ. Clearly λ is dominant. If λ = 0, map a polynomial ring P_{v} := k[x] with trivial G-action to gr A by substituting v for x. Also put D_{v} := 1. Next assume λ ≠ 0. Let ℓ be the rank of G. Define a T -action on the X(T)-graded algebra

P = ⨂_{i=1}^{ℓ} S^{∗}(∇_{G}(ϖ_{i}))

by letting T act on ⨂_{i=1}^{ℓ} S^{m_{i}}(∇_{G}(ϖ_{i})) through weight ∑_{i} m_{i}ϖ_{i}. So now we have a G × T -action on P. Observe that by our key hypothesis 1.1 and the tensor product property [9, Ch. G] the polynomial algebra P has a good filtration for the G-action. Let D be the scheme theoretic kernel of λ. So D has character group X(D) = X(T)/ℤλ and D = Diag(X(T)/ℤλ) in the notations of [9, I.2.5]. The subalgebra P^{1×D} is a graded algebra with good filtration such that its subalgebra P^{U×D} contains a polynomial algebra on one generator x of weight λ × λ. In fact, this polynomial subalgebra contains all the weight vectors in P^{U×D} of weight μ×ν with ht(μ) ≥ ht(ν). The other weight vectors in P^{U×D} also have weight of the form μ × ν with ν a multiple of λ. These other weight vectors span an ideal in P^{U×D}. Now assume A has a good filtration. By lemma 2.3 one easily constructs a G-equivariant algebra homomorphism P^{1×D} → gr A that maps x to v. Write it as P_{v}^{1×D_{v}} → gr A, to stress the dependence on v.

As new P we take the tensor product of the finitely many P_{v} and as diagonalized group D we take the direct product of the D_{v}. Then we have a graded algebra map P^{D} → gr A. It is surjective because its image has good filtration ([9, Ch. A]) and contains (gr A)^{U}. The G × D-algebra P is an example of what we called in [13] a graded polynomial G × D-algebra with good filtration. We have proved

Lemma 2.5 If A has a good filtration, then there is a graded polynomial G × D-algebra P with good filtration and a graded G-equivariant surjection P^{D} → gr A.

Now recall M is a noetherian A-module on which G acts compatibly, meaning that the structure map A ⊗ M → M is a map of G-modules. Form the ‘semi-direct product ring’ A ⋉ M whose underlying G-module is A ⊕ M, with product given by (a_{1}, m_{1})(a_{2}, m_{2}) = (a_{1}a_{2}, a_{1}m_{2} +a_{2}m_{1}). By Grosshans gr(A ⊕ M) is a finitely generated algebra, so we get

Lemma 2.6 gr M is a noetherian gr A-module.

This is of course very reminiscent of the proof of the lemma [8, Thm. 16.9] telling that M^{G} is a noetherian module over the finitely generated k-algebra A^{G}. We will tacitly use its counterpart for diagonalized actions, cf. [2], [9, I.2.11].

Taking things together we learn that if A has a good filtration, then P ⊗_{P^{D}} gr M is what we called in [13] a finite graded P -module. Lemma [13, Lemma 3.7] then tells us

Lemma 2.7 Let A have good filtration. Then P ⊗_{P^{D}} gr M has finite good filtration dimension and each H^{i}(G, P ⊗_{P^{D}} gr M) is a noetherian P^{G}-module.

Extend the D-action on P to P ⊗_{P^{D}} gr M by using the trivial action on the second factor. Then we have a G × D-module structure on P ⊗_{P^{D}} gr M. As D is diagonalized, P^{D} is a direct summand of P as a P^{D}-module [9, I.2.11] and (P ⊗_{P^{D}} gr M)^{1×D} = gr M is a direct summand of the G-module P ⊗_{P^{D}} gr M. It follows that gr M also has finite good filtration dimension and it follows that each H^{i}(G, P ⊗_{P^{D}} gr M)^{1×D} = H^{i}(G, gr M) is a noetherian P^{G×D}-module. But the action of P^{G×D} on gr M factors through (gr A)^{G}, so we see that each H^{i}(G, gr M) is a noetherian (gr A)^{G}-module. And one always has (gr A)^{G} = (gr_{0} A)^{G} = A^{G}. We conclude

Lemma 2.8 Let A have good filtration. Then gr M has finite good filtration dimension and each H^{i}(G, gr M) is a noetherian A^{G}-module.

3 Degrading

We still have to get rid of the grading. The filtration M_{≤0} ⊆ M_{≤1} ⋯ induces a filtration of the Hochschild complex [9, I.4.14] whence a spectral sequence

E(M) : E_{2}^{ij} = H^{i+j}(G, gr_{−i} M) ⇒ H^{i+j}(G, M).

It lives in an unusual quadrant.

Assume that A has good filtration. Then by Lemma 2.8 E_{2}(M) is a finitely generated A^{G}-module. So the spectral sequence lives in only finitely many bidegrees (i, j). Thus there is the same kind of convergence as one would have in a more common quadrant.

Choose A^{G} as ring of operators to act on the spectral sequence E(M). As E_{2}(M) is a noetherian A^{G}-module, an easy case of the reasoning of Evens (compare the slogan [14, 3.12]) tells us that H^{∗}(G, M) is a noetherian A^{G}- module. To finish the proof of the theorem, we note that A ⊗ k[G/U] is also a finitely generated algebra with a good filtration and that M ⊗ k[G/U] is a noetherian module over it. So what we have just seen tells that H^{∗}(G, M ⊗ k[G/U]) is a noetherian (A ⊗ k[G/U])^{G}-module. In particular, there is an m ≥ −1 so that H^{m+1}(G, M ⊗ k[G/U]) = 0. □

Remark 3.1 Somewhere along the way we made the simplifying assumption that G is semisimple. So for the original G we have now proved that M has finite good filtration dimension with respect to the commutator subgroup H of G. But that is the same as having finite good filtration dimension with respect to G. Also, the fact that H^{i}(H, M) is a noetherian A^{H}-module implies that H^{i}(G, M) is a noetherian A^{G}-module by taking invariants under the diagonalizable center Z(G).

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