Cohomology with Grosshans graded coefficients

Wilberd van der Kallen

Abstract Let the reductive group G act on the finitely generated commutative k-algebra A. We ask if the finite generation property of the ring of invariants extends to the full cohomology ring. We confirm this for G = SL_{2} and also when the action on A is replaced by the ‘contracted’ action on the Grosshans graded ring gr A, provided the characteristic of k is large.

1 Introduction

Consider a linear algebraic group or group scheme G defined over an algebraically closed field k of positive characteristic p. Let A be a finitely generated commutative k-algebra on which G acts rationally by k-algebra automorphisms. So G acts on Spec(A). One may then ask if the cohomology ring H^{∗}(G, A) is finitely generated as a k-algebra.

Say G is geometrically reductive. Then we know by Nagata that at least the ring of invariants A^{G} = H^{0}(G, A) is finitely generated. (As explained in [4], Nagata’s proof [19] extends to group schemes.)

By Waterhouse [25] a finite group scheme is geometrically reductive. If G is a finite group scheme, then A is a finite module over A^{G} and hence the cohomology ring H^{∗}(G, A) is indeed finitely generated by Friedlander and Suslin [10]. (If G is finite reduced, see Evens [8, Thm. 8.1]. If G is finite and connected, take C = A^{G} in [10, Theorem 1.5, 1.5.1]. If G is not connected, one finishes the argument by following [8] as on pages 220–221 of [10].)

Note that if the geometrically reductive G is a subgroup of GL_{n}, then GL_{n} /G is affine, ind_{G}^{GL_{n}} A is finitely generated, and H^{∗}(G, A) = H^{∗}(GL_{n}, ind_{G}^{GL_{n}} A). (Compare [19], [12, Ch. II], [14, I 4.6, I 5.13].) Therefore let us now assume G = GL_{n}, or rather G = SL_{n} to keep it semi-simple. (The SL_{n} case suffices, as H^{∗}(GL_{n}, A) = H^{∗}(SL_{n}, A)^{T} for a GL_{n}-algebra A.)

Grosshans [11] has introduced a filtration on A. The associated graded ring gr A is finitely generated [11, Lemma 14]. There is a flat family with general fibre A and special fibre gr A [11, Theorem 13]. Its counterpart in characteristic zero was introduced by Popov [20]. Our first main result says

Theorem 1.1 If n < 6 or p > 2^{n}, then H^{∗}(SL_{n}, gr A) is finitely generated as a k-algebra.

Remark 1.2 In problems 3.10 and 3.11 we discuss how one could try to remove the gr in the conclusion of the theorem.

Our second main result concerns the case n = 2, where we succeed in removing the gr, using a family of characteristic classes which behaves as if it is obtained by taking divided powers of the class e_{1} of Friedlander and Suslin [10].

Theorem 1.3 (Cohomological invariant theory in rank one) Let A be a finitely generated commutative k-algebra on which SL_{2} acts rationally by algebra automorphisms. Then H^{∗}(SL_{2}, A) is finitely generated as a k-algebra.

Remark 1.4 We know very little about the size of H^{∗}(G, A), even in simple examples. For instance, let p = 2 and consider the second Steinberg module V = St_{2} of SL_{2}. The dimension of V is four. Its highest weight is three times the fundamental weight. By the theorem H^{∗}(SL_{2}, S^{∗}(V)) is finitely generated, where S^{∗}(V) denotes the symmetric algebra. But that is all we know about the size of H^{∗}(SL_{2}, S^{∗}(V)). Note that S^{∗}(V) does not have a good filtration [5, p.71 Example], as H^{1}(SL_{2}, S^{2}(V)) ≠ 0.

2 Recollections

For simplicity we stay with the important case G = SL_{n} until 3.13. We choose a Borel group B^{+} = T U^{+} of upper triangular matrices 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 ∇_{G}(λ) with highest weight λ. In a good filtration of a G-module the layers are of the form ∇_{G}(μ). As in [24] we will actually also allow a layer to be a direct sum of any number of copies of the same ∇_{G}(μ). If M is a G-module, and m ≥ −1 is an integer so that H^{m+1}(G, ∇_{G}(μ) ⊗ M) = 0 for all dominant μ, then we say as in [9] that M has good filtration dimension at most m. The case m = 0 corresponds with M having a good filtration. We say that M has good filtration dimension precisely m, notation dim_{∇}(M) = m, if m is minimal so that M has good filtration dimension at most m. In that case H^{i+1}(G, ∇_{G}(μ) ⊗ M) = 0 for all dominant μ and all i ≥ m. In particular H^{i+1}(G, M) = 0 for i ≥ m. If there is no finite m so that dim_{∇}(M) = m, then we put dim_{∇}(M) = ∞.

Lemma 2.1 Let 0 → M′ → M → M′′ → 0 be exact. Then

1. dim_{∇}(M) ≤ max(dim_{∇}(M′), dim_{∇}(M′′)),

2. dim_{∇}(M′) ≤ max(dim_{∇}(M), dim_{∇}(M′′) + 1),

3. dim_{∇}(M′′) ≤ max(dim_{∇}(M), dim_{∇}(M′) − 1),

4. dim_{∇}(M′ ⊗ M′′) ≤ dim_{∇}(M′) + dim_{∇}(M′′).

2.2 Filtrations

If M is a G-module, and λ is a dominant weight, then M_{≤λ} denotes the largest G-submodule all whose weights μ satisfy μ ≤ λ in the usual partial order [14, II 1.5]. Similarly M_{<λ} denotes the largest G-submodule all whose weights μ satisfy μ < λ. As in [24], we form the X(T)-graded module

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

Each M_{≤λ}/M_{<λ}, or M_{≤λ/<λ} for short, has a B^{+}-socle (M_{≤λ/<λ})^{U} = M_{λ}^{U} of weight λ. We view M^{U} as a B^{−}-module through restriction (inflation) along the homomorphism B^{−} → T. Then M_{≤λ/<λ} embeds naturally in its ‘good filtration hull’ hull_{∇}(M_{≤λ/<λ}) = ind_{B^{−}}^{G} M_{λ}^{U}. This good filtration hull has the same B^{+}-socle and is the injective hull in the category C_{λ} of G-modules N that satisfy N = N_{≤λ}. Compare [24, 3.1.10].

Let us apply this in particular to our finitely generated commutative kalgebra with G action A. We get an X(T)-graded algebra gr_{X(T)} A. We convert it to a ℤ-graded algebra through an additive height function ht : X(T) → ℤ, defined by ht(γ) = 2 ∑_{α>0}⟨γ, α̂⟩, the sum being over the positive roots. (Our ht is twice the one used by Grosshans, because we prefer to get even degrees rather than just integer degrees.) The Grosshans graded algebra is now

gr A = ⨁_{i≥0} gr_{i} A,

with

gr_{i} A = ⨁_{ht(λ)=i} A_{≤λ/<λ}.

It 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 shows that A^{U}, gr A, hull_{∇}(gr A) are finitely generated with hull_{∇}(gr A) finite over gr A. Mathieu did a little better in [17]. His argument shows that in fact hull_{∇}(gr A) is a p-root closure of gr A. That is,

Lemma 2.3 For every x ∈ hull_{∇}(gr A), there is an integer r ≥ 0, so that x^{p^{r}} ∈ gr A.

Proof It suffices to take x ∈ hull_{∇}(A_{≤λ}/A_{<λ}) for some λ. If λ = 0, then hull_{∇}(A_{≤λ}/A_{<λ}) = A^{G} = gr_{0} A. So say λ > 0 and consider the subalgebra S = k ⊕ ⨁_{i>0} hull_{∇}(A_{≤iλ}/A_{<iλ}) of hull_{∇}(gr A), with its subalgebra S ∩ gr A. Apply [24, Thm 4.2.3] to conclude that the p-root closure of S ∩ gr A in S has a good filtration. As it contains all of S^{U}, it must be S itself. □

Example 2.4 Consider the multicone [15]

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 [15] that it is finitely generated by the sum of the ∇(ϖ_{i}), where ϖ_{i} denotes the ith fundamental weight. Together with the transfer principle A^{U} ≅ (k[G/U] ⊗ A)^{G}, see [12, Ch Two], this gives finite generation of A^{U}.

3 Proof of Theorem 1.1

Choose r so big that x^{p^{r}} ∈ gr A for all x ∈ hull_{∇}(gr A). We may view gr A as a finite hull_{∇}(gr A)^{(r)}-module, where the exponent (r) denotes an rth Frobenius twist [14, I 9.2].

3.1 Key hypothesis

We assume that for every fundamental weight ϖ_{i} the symmetric algebra S^{∗}(∇(ϖ_{i})) has a good filtration.

The hypothesis in the theorem is explained by

Lemma 3.2 If n < 6 or p > 2^{n}, then the key hypothesis is satisfied.

Proof We follow [1, section 4]. If p > 2^{n}, then p > ∑_{i} dim(∇(ϖ_{i})), so ∧^{j}(∇(ϖ_{i})) has a good filtration for all j by [1], so S^{∗}(∇(ϖ_{i})) has a good filtration by [1]. If n < 6, then by symmetry of the Dynkin diagram (contragredient duality) it suffices to consider ϖ_{1} and ϖ_{2}. But S^{∗}(∇(ϖ_{1})) has a good filtration for every n because ∧^{j}(∇(ϖ_{1})) has a good filtration for all j. And S^{∗}(∇(ϖ_{2})) has a good filtration for every n by Boffi [3]. (Thanks to J. Weyman and T. Jozefiak for pointing this out.) □

Example 3.3 (J. Weyman) If p = 2 and n ≥ 6, there is a submodule with highest weight ϖ_{6} in S^{2}(∇(ϖ_{3})). (If n = 6, read zero for ϖ_{6}, as we are working with SL_{6} rather than GL_{6}.) With a character computation this implies that S^{2}(∇(ϖ_{3})) does not have a good filtration. The submodule in question is generated by a highest weight vector which is a sum of ten terms (e_{1} ∧ e_{σ(2)} ∧ e_{σ(3)})(e_{σ(4)} ∧ e_{σ(5)} ∧ e_{σ(6)}). One sums over the ten permutations σ of 2, 3, 4, 5, 6 that satisfy σ(2) < σ(3) and σ(4) < σ(5) < σ(6).

We want to view S^{∗}(∇(ϖ_{i})) as a graded polynomial G-algebra with good filtration. Let us collect the properties that we will use in a rather artificial definition.

Definition 3.4 Let D be a diagonalizable group scheme [14, I 2.5]. We say that P is a graded polynomial G × D-algebra with good filtration, if the following holds. First of all P is a polynomial algebra over k in finitely many variables. Secondly, these variables are homogeneous of non-negative integer degree, thus making P into a graded k-algebra. There is also given an action of G×D on P by algebra automorphisms that are compatible with the grading, with G × D acting trivially on the degree zero part P_{0} of P. This P_{0} is thus the polynomial algebra generated by the variables of degree zero. The variables of positive degree generate their own polynomial algebra P^{c}, which we also assume to be G × D invariant. Thus P = P_{0} ⊗_{k} P^{c}. And finally, P^{c} is an algebra with a good filtration for the action of G. Then of course, so is P.

Example 3.5 Our key hypothesis makes that one gets a graded polynomial G × D-algebra with good filtration by taking for D any subgroup scheme of T, with T acting on P = S^{∗}(∇(ϖ_{i})) through its natural X(T)-grading: On S^{j}(∇(ϖ_{i})) we make T act with weight jϖ_{i}.

If P_{i} are graded polynomial G×D_{i}-algebras with good filtration for i = 1, 2, then P_{1} ⊗ P_{2} is a graded polynomial G × (D_{1} × D_{2})-algebra with good filtration.

Definition 3.6 If P is a graded polynomial G × D-algebra with good filtration, then by a finite graded P -module M we mean a finitely generated ℚ-graded module for the graded polynomial ring, together with a G × D action on M which is compatible with the grading and with the action on P. It is not required that the action is trivial on the degree zero part of M. We call M free if it is free as a module over the polynomial ring. We call M extended if there is a finitely generated ℚ-graded P_{0}-module V with G × D action, so that M = V ⊗_{P_{0}} P as graded P -modules.

Lemma 3.7 Let P be a graded polynomial G×D-algebra with good filtration, and let M be a finite graded P -module.

1. There is a finite free resolution

0 → F_{s} → F_{s−1} → ⋯ → F_{0} → M.

2. Every finite free P -module has a finite filtration whose quotients are extended.

3. dim_{∇}(M) < ∞.

4. H^{i}(G, M) is a finite P^{G}-module for every i ≥ 0.

Proof Take a finite dimensional graded G × D-submodule V of M that generates M as a P -module. Then F_{0} = V ⊗_{k} P is free and it maps onto M. As M has finite projective dimension [18, 18C], we may repeat until the syzygy is projective. But a projective module over a polynomial ring is free by Quillen and Suslin [16].

Now consider a free P -module F. Let P_{+} be the ideal generated by the P_{i} with i > 0. If F is free of rank t, then F/P_{+}F is free of rank t over P_{0}. Choose homogeneous elements e_{1}, …, e_{t} in F so that their classes form a basis of F/P_{+}F. Then the e_{i} generate F (cf. [13, 6.13 Lemma 5]), so they also form a basis of F. Let F_{m} be the component of lowest degree in F, assuming F ≠ 0. The e_{i} that lie in F_{m} form a basis of F_{m} over P_{0}. So F_{m} generates a free, extended submodule P F_{m} of F and the quotient F/P F_{m} is free of lower rank.

To show that dim_{∇}(M) < ∞, it suffices to consider the case of an extended module M = V ⊗_{P_{0}} P. As V ⊗_{P_{0}} P = V ⊗_{k} P^{c}, it suffices to check that V has finite good filtration dimension. But V is a finitely generated P_{0}-module and thus has only finitely many weights. Therefore the argument used in [9] to show that finite dimensional G modules have finite good filtration dimension, applies to V.

Finally let M be any finite graded P -module again. As G is reductive by Haboush [14, 10.7], it is well known that H^{0}(G, M) is a finite P^{G}-module, because (S_{P}^{∗} (M))^{G} is finitely generated. So we argue by dimension shift. We claim that for s sufficiently large the module M ⊗ St_{s} ⊗ St_{s} has a good filtration. It suffices to check this for the extended case M = V ⊗_{P_{0}} P and then one can use again that V has only finitely many weights, so that one may choose s so large that all weights of V ⊗ k_{−(p^{s}−1)ρ} are anti-dominant. Then V ⊗St_{s} = ind_{B^{+}}^{G} V ⊗k_{−(p^{s}−1)ρ} has a good filtration and so does M ⊗St_{s} ⊗ St_{s}. Then H^{i}(G, M) is the cokernel of H^{i−1}(G, M ⊗ St_{s} ⊗ St_{s}) → H^{i−1}(G, M ⊗ St_{s} ⊗ St_{s} /M) for i ≥ 1. □

Recall that we choose r so big that x^{p^{r}} ∈ gr A for all x ∈ hull_{∇}(gr A). Let G_{r} denote the r-th Frobenius kernel. We will need multiplicative structure on a Hochschild–Serre spectral sequence. See for instance [2], translating from modules over a Hopf algebra to comodules over a Hopf algebra.

Proposition 3.8 Assume the key hypothesis 3.1. With r as indicated, the spectral sequence

E_{2}^{ij} = H^{i}(G/G_{r}, H^{j}(G_{r}, gr A)) ⇒ H^{i+j}(G, gr A)

stops, i.e. E_{s} = E_{∞} for some s < ∞. In fact, H^{∗}(G_{r}, gr A)^{(−r)} has finite good filtration dimension. Moreover, E_{2}^{∗∗} is a finite module over the even part of E_{2}^{0∗}.

Proof By [10, Th 1.5, 1.5.1] the ring H^{∗}(G_{r}, gr A)^{(−r)} is a finite module over the algebra

⨂_{a=1}^{r} S^{∗}((𝔤𝔩_{n})^{#}(2p^{a−1})) ⊗ hull_{∇}(gr A).

Here (𝔤𝔩_{n})^{#}(2p^{a−1}) denotes the dual of 𝔤𝔩_{n} placed in degree 2p^{a−1}. As G = SL_{n}, it follows from [1, 4.3] that the algebra ⨂_{a=1}^{r} S^{∗}((𝔤𝔩_{n})^{#}(2p^{a−1})) is a graded polynomial G × {1}-algebra with good filtration. So we now try to replace hull_{∇}(gr A) by a similarly nice algebra. That is, we seek a graded polynomial G × D-algebra P with good filtration and a surjection P^{D} → hull_{∇}(gr A) of graded G-algebras. This is where the key hypothesis comes in. Let v be a weight vector in hull_{∇}(gr A)^{U} = A^{U}. If v has weight zero, we may map x to v, where x is the variable in a polynomial ring k[x] with trivial grading and trivial G × T action. If v has weight λ ≠ 0, take for D the scheme theoretic kernel of λ and observe that by our hypothesis the X(T)-graded algebra

P = ⨂_{i=1}^{n−1} S^{∗}(∇(ϖ_{i}))

is a graded polynomial G × D-algebra with good filtration, if we give ∇(ϖ_{i}) degree ht(ϖ_{i}). We have P^{D} = ⨁_{j} P_{jλ}, where P_{jλ} denotes the summand ⨂_{i=1}^{n−1} S^{m_{i}}(∇(ϖ_{i})) with ∑_{i} m_{i}ϖ_{i} = jλ. Choose a weight vector x of weight λ in P_{λ} (for the G action). The map from the polynomial ring k[x] to hull_{∇}(gr A)^{U} which sends x to v extends uniquely to a G equivariant algebra map P^{D} → hull_{∇}(gr A) because P_{jλ} = (P_{jλ})_{≤jλ}. (The first subscript in the right hand side refers to a T action associated with the X(T)-grading, the second to the G action on P.) Compare [24, 4.2.4]. As A^{U} is finitely generated, we may combine finitely many such maps P (i)^{D(i)} → hull_{∇}(gr A) into one surjective map P^{D} → hull_{∇}(gr A), with D the product of the D(i) and P the tensor product of the P (i). If we let the linearly reductive D act on P ⊗_{P^{D}} H^{∗}(G_{r}, gr A)^{(−r)} through its action on P, then H^{∗}(G_{r}, gr A)^{(−r)} is just the direct summand, as a G-module, consisting of the D-invariants. So in order to show that H^{∗}(G_{r}, gr A)^{(−r)} has finite good filtration dimension, it suffices to show that P ⊗_{P^{D}} H^{∗}(G_{r}, gr A)^{(−r)} has finite good filtration dimension. We now view P ⊗⨂_{a=1}^{r} S^{∗}((𝔤𝔩_{n})^{#}(2p^{a−1})) as a graded polynomial G×D-algebra with good filtration. (Collect the bigrading into a single total grading.) The algebra P ⊗_{P^{D}} H^{∗}(G_{r}, gr A)^{(−r)} is a finite graded P ⊗⨂_{a=1}^{r} S^{∗}((𝔤𝔩_{n})^{#}(2p^{a−1}))- module. (We need a ℚ-grading on P ⊗_{P^{D}} H^{∗}(G_{r}, gr A)^{(−r)} because of the twist.) By Lemma 3.7 such a P -module has finite good filtration dimension. Therefore there are only finitely many i with E_{2}^{i∗} ≠ 0 and the spectral sequence stops. By the same lemma H^{i}(G, P ⊗_{P^{D}} H^{∗}(G_{r}, gr A)^{(−r)}) is finite over H^{0}(G, P ⊗ ⨂_{a=1}^{r} S^{∗}((𝔤𝔩_{n})^{#}(2p^{a−1}))). Taking D-invariants again, we see that E_{2}^{∗∗} is a finite module over H^{0}(G/G_{r}, (P^{D}⊗⨂_{a=1}^{r} S^{∗}((𝔤𝔩_{n})^{#}(2p^{a−1})))^{(r)}). But this ring acts by way of the the even part of E_{2}^{0∗}. □

3.9 End of proof of theorem 1.1

Look some more at the spectral sequence of Proposition 3.8. We argue as in Evens’ proof of his finite generation theorem [2, 4.2], [8]. As we have already observed, it follows from [10] that H^{∗}(G_{r}, gr A) is noetherian over its finitely generated even degree part. By the proposition E_{2} is noetherian over the even degree part of E_{2}^{0∗}, which is finitely generated because G/G_{r} is reductive. Or, recall from the proof of the proposition that the E_{2} term is finite over the finitely generated k-algebra H^{0}(G/G_{r}, (P^{D}⊗⨂_{a=1}^{r} S^{∗}((𝔤𝔩_{n})^{#}(2p^{a−1})))^{(r)}). Now the E_{2} term is a differential graded algebra in characteristic p, so the pth power of an element of the even part passes to the next term in the spectral sequence. Therefore E_{3} is also noetherian over its finitely generated even degree part. As the spectral sequence stops, we get by repeating this argument that E_{∞} is finitely generated. But then so is the abutment. □

Problem 3.10 There is a spectral sequence

E_{2}^{ij} = H^{i+j}(G, gr_{−i} A) ⇒ H^{i+j}(G, A),

and the theorem says that E_{2} is finitely generated. The generators are in bidegree (0, 0) or (i, j) with i + j > 0, i < 0. We would like this spectral sequence to stop too. It would suffice to know that if J is a G-stable ideal in a ring like A, then the even part of H^{∗}(G, A/J) is integral over the image of the even part of H^{∗}(G, A). Of course this integrality would be implied by finite generation of H^{∗}(G, S_{A}^{∗}(A/J)). But recall that Nagata proves first that H^{0}(G, A/J) is integral over H^{0}(G, A). So one could hope for a direct proof.

Problem 3.11 It is probably too much to ask for a divided power structure on the even part of the bigraded algebra H^{∗}(GL_{n}, Γ^{∗}((𝔤𝔩_{n})(2)^{(1)})), extending the divided power structure [7, Appendix A2], [21] on the graded algebra H^{0}(GL_{n}, Γ^{∗}((𝔤𝔩_{n})(2)^{(1)})). But such a structure would be very helpful, as it would explain and enrich the supply of characteristic classes from [10]. And H^{i+j}(G, gr_{−i} A) ⇒ H^{i+j}(G, A) would then undoubtedly stop. See our treatment of the rank one case below.

Problem 3.12 To improve on the conditions of theorem 1.1, rather than on its conclusion, one should try and prove the following. If our finitely generated A has a good filtration and M is a finite A-module on which G acts compatibly, then H^{∗}(G, M) is finite over A^{G}. Of course this would again be implied by finite generation of H^{∗}(G, S_{A}^{∗}(M)).

Remark 3.13 Let G be a semi-simple group defined over k, and let V be a tilting G-module [6] of dimension n. Choose a basis in V and assume that the representation is faithful, so that G can be identified with a subgroup of SL_{n}. Assume p ≥ n/2. Then S^{∗}((𝔤𝔩_{n})^{#}) has a good filtration as a G-module [1, 4.4]. Also assume for every fundamental weight ϖ_{i} of G that the symmetric algebra S^{∗}(∇(ϖ_{i})) has a good filtration. This happens for instance when p exceeds the dimensions of the fundamental representations of G. Let A be a finitely generated commutative k-algebra on which G acts rationally by k-algebra automorphisms. Then again H^{∗}(G, gr A) is finitely generated as a k-algebra. The proof is the same as the proof of theorem 1.1.

4 Divided powers

Let W_{2}(k) = W (k)/p^{2}W (k) be the ring of Witt vectors of length two over k, see [22, II §6]. One has an extension of algebraic groups

1 → 𝔤𝔩_{n}^{(1)} → GL_{n}(W_{2}(k)) → GL_{n}(k) → 1,

whence a cocycle class e_{1} ∈ H^{2}(GL_{n}, 𝔤𝔩_{n}^{(1)}). We call it the Witt vector class for GL_{n}. Analogously there are Witt vector extensions and Witt vector classes for other groups that are originally defined over the integers, and for Frobenius kernels in them.

Observe that if x_{α} : 𝔾_{a} → SL_{n} is a root homomorphism corresponding to the root α, then the restriction of e_{1} to 𝔾_{a} is non-trivial. In fact one may restrict further to the Frobenius kernel 𝔾_{a}_{1}. This 𝔾_{a}_{1} acts trivially on 𝔤𝔩_{n}^{(1)}, and the T -equivariant projection of 𝔤𝔩_{n}^{(1)} onto its pα weight space is thus also 𝔾_{a}_{1}-equivariant. Altogether we get an image β of e_{1} in H^{2}(𝔾_{a}_{1}, (𝔤𝔩_{n}^{(1)})_{pα})^{T}. It is just the Witt vector class of 𝔾_{a}_{1}. It is well known that this class is non-trivial, compare [14, I 4.22, I 4.25], [10, §6]. Also note that the image of β in H^{2}(𝔾_{a}_{1}, 𝔤𝔩_{n}^{(1)}) under the map induced by the inclusion of (𝔤𝔩_{n}^{(1)})_{pα} into 𝔤𝔩_{n}^{(1)} is the same as the restriction of e_{1} to 𝔾_{a}_{1}.

Lemma 4.1 The Witt vector class for GL_{n} coincides with the characteristic class e_{1} of Friedlander and Suslin, up to a non-zero scalar factor.

Proof Using [10, Remark 1.2.1, Corollary 3.13] we see it suffices to take r = j = 1 and q = 2 in [10, Theorem 4.5]. □

4.2 Divided powers in rank one.

If R is a commutative k-algebra, and V is a vector space over k, then R ⊗ Γ^{∗}(V) is a ring with divided powers ([7, Appendix A2]). We write Δ_{i,j} for the component Γ^{i+j}(V) → Γ^{i}(V) ⊗ Γ^{j}(V) of the comultiplication map Γ^{∗}(V) → Γ^{∗}(V) ⊗ Γ^{∗}(V). So Δ_{i,j} is the dual of the multiplication map S^{i}(M^{#}) ⊗ S^{j}(M^{#}) → S^{i+j}(M^{#}). Put G = GL_{n} and define T, B^{+}, B^{−} as usual. Actually n will be two, but we optimistically keep writing n. Let r ≥ 1. As G_{r} acts trivially on 𝔤𝔩_{n}^{(r)}, there is a divided power structure on H^{even}(G_{r}, Γ^{∗}(𝔤𝔩_{n}^{(r)})) = H^{even}(G_{r}, k) ⊗ Γ^{∗}(𝔤𝔩_{n}^{(r)}), with the m-th divided power operation going from H^{2a}(G_{r}, Γ^{b}(𝔤𝔩_{n}^{(r)})) to H^{2am}(G_{r}, Γ^{bm}(𝔤𝔩_{n}^{(r)})) for b ≥ 1.

It would be nice to extend the next theorem to arbitrary n. We do not know how to put a divided power structure on ⨁_{m} H^{2m}(GL_{n}, Γ^{m}(𝔤𝔩_{n}^{(1)})). Nevertheless we feel the theorem is named appropriately.

Theorem 4.3 (Divided powers in rank one) Let n = 2. There are classes c[m] ∈ H^{2m}(GL_{n}, Γ^{m}(𝔤𝔩_{n}^{(1)})) so that 1. c[1] is the Witt vector class e_{1},

2. (Δ_{i,j})_{∗}(c[i + j]) = c[i] ∪ c[j] for i, j ≥ 1.

Proof Let α be the negative root, and let x_{α} : 𝔾_{a} → GL_{n} be its root homomorphism. By Kempf vanishing H^{2m}(GL_{n}, Γ^{m}(𝔤𝔩_{n}^{(1)})) = H^{2m}(𝔾_{a}, Γ^{m}(𝔤𝔩_{n}^{(1)}))^{T}, see [14, II 4.7c]. So we restrict to 𝔾_{a}. As 𝔾_{a} acts trivially on Γ^{∗}((𝔤𝔩_{n}^{(1)})_{pα}), we also have a divided power structure on H^{even}(𝔾_{a}, Γ^{∗}((𝔤𝔩_{n}^{(1)})_{pα})). Take the m-th divided power in H^{even}(𝔾_{a}, Γ^{∗}((𝔤𝔩_{n}^{(1)})_{pα}))^{T} of the Witt vector class of 𝔾_{a} and map it to H^{2m}(𝔾_{a}, Γ^{m}(𝔤𝔩_{n}^{(1)}))^{T} using the inclusion of (𝔤𝔩_{n}^{(1)})_{pα} into 𝔤𝔩_{n}^{(1)}. One lands in H^{2m}(B^{−}, Γ^{m}(𝔤𝔩_{n}^{(1)})) ≅ H^{2m}(G, Γ^{m}(𝔤𝔩_{n}^{(1)})). This gives the desired class c[m]. Indeed (Δ_{i,j})_{∗}(c[i + j]) = c[i] ∪ c[j] holds in the context of the divided power algebra H^{even}(𝔾_{a}, Γ^{∗}((𝔤𝔩_{n}^{(1)})_{pα})) = H^{even}(𝔾_{a}, k) ⊗ Γ^{∗}((𝔤𝔩_{n}^{(1)})_{pα}) and thus the result follows by functoriality of Δ_{i,j} and of cohomology. □

4.4 Other characteristic classes

If M is a finite dimensional vector space over k and r ≥ 1, we have a natural homomorphism of symmetric algebras S^{∗}(M^{#(r)}) → S^{∗}(M^{#(1)}) induced by the map M^{#(r)} → S^{p^{r−1}}(M^{#(1)}) which raises an element to the power p^{r−1}. Dually we have the algebra map π^{r−1} : Γ^{p^{r−1}∗}(M(1)) → Γ^{∗}(M(r)) whose kernel is the ideal generated by Γ^{1}(M^{(1)}) through Γ^{p^{r−1}−1}(M^{(1)}). So π^{r−1} maps Γ^{p^{r−1}a}(M(1)) onto Γ^{a}(M(r)).

Notation 4.5 We now introduce analogues of the classes e_{r} and e_{r}^{(j)} of Friedlander and Suslin [10, Theorem 1.2, Remark 1.2.2]. We write π_{∗}^{r−1}(c[ap^{r−1}]) ∈ H^{2ap^{r−1}}(G, Γ^{a}(𝔤𝔩_{n}^{(r)})) as c_{r}[a]. By Frobenius twist we get c_{r}[a]^{(j)} ∈ H^{2ap^{r−1}}(G, Γ^{a}(𝔤𝔩_{n}^{(r+j)})).

Lemma 4.6 The c_{i}[a]^{(r−i)} enjoy the following properties (r ≥ i ≥ 1)

1. There is a homomorphism of algebras S^{∗}(𝔤𝔩_{n}^{#(r)}) → H^{2p^{i−1}∗}(G_{r}, k) given on S^{a}(𝔤𝔩_{n}^{#(r)}) = H^{0}(G_{r}, S^{a}(𝔤𝔩_{n}^{#(r)})) by cup product with the restriction of c_{i}[a]^{(r−i)}.

2. For r ≥ 1 the restriction of c_{r}[1] to H^{2p^{r−1}}(G_{1}, 𝔤𝔩_{n}^{(r)}) is nontrivial, so that c_{r}[1] may serve as characteristic class e_{r} in [10, Thm 1.2].

Proof 1. By theorem 4.3 we have (Δ_{a,b})_{∗}(c_{i}[a + b]) = c_{i}[a] ∪ c_{i}[b] and thus (Δ_{a,b})_{∗}(c_{i}[a + b]^{(r−i)}) = c_{i}[a]^{(r−i)} ∪ c_{i}[b]^{(r−i)} by pull back along a Frobenius homomorphism.

2. In fact the restriction to H^{2p^{r−1}}(𝔾_{a}_{1}, (𝔤𝔩_{n}^{(r)})_{p^{r}α}) is nontrivial, where x_{α} : 𝔾_{a} → G is our root homomorphism. That is because the Witt vector class generates the polynomial ring H^{even}(𝔾_{a}_{1}, k), see [14, I 4.26]. □

Corollary 4.7 Let n = 2, r ≥ 1. Further let A be a commutative k-algebra with SL_{n} action and J an invariant ideal in A so that the algebra A/J is finitely generated. Then H^{0}(SL_{n}, H^{∗}((SL_{n})_{r}, A/J)) is a noetherian module over a finitely generated subalgebra of H^{even}(SL_{n}, A).

Proof We may assume A is finitely generated. Put C = H^{0}((SL_{n})_{r}, A)). Then C contains the elements of A raised to the power p^{r}, so C is also a finitely generated algebra and A/J is a noetherian module over it. By [10, Thm 1.5] H^{∗}((SL_{n})_{r}, A/J) is a noetherian module over the finitely generated algebra

R = ⨂_{a=1}^{r} S^{∗}((𝔤𝔩_{n}^{(r)})^{#}(2p^{a−1})) ⊗ C.

Then, by invariant theory [12, Thm. 16.9], H^{0}(SL_{n}, H^{∗}((SL_{n})_{r}, A/J)) is a noetherian module over the finitely generated algebra H^{0}(SL_{n}, R). By lemma 4.6 we may take the algebra homomorphism R → H^{∗}((SL_{n})_{r}, A/J) to be based on cup product with our c_{i}[a]^{(r−i)}. But then the map H^{0}(SL_{n}, R) → H^{∗}((SL_{n})_{r}, A) factors, as a linear map, through H^{even}(SL_{n}, A). □

4.8 The contraction

Let A be a finitely generated commutative k-algebra on which SL_{n} acts rationally. Recall that A comes with an increasing filtration A_{≤i} = ∑_{ht(λ)≤i} A_{≤λ} whose associated graded is the ring gr A. Let A be the subring of the polynomial ring A[t] generated by the subsets t^{i} ∑_{ht(λ)≤i} A_{≤λ}. This ring A, denoted D by Grosshans, is the coordinate ring of a flat family with general fibre A and special fibre gr A [11, Theorem 13].

4.9 The special fibre

There is a homomorphism of graded algebras A → gr A with kernel tA, mapping t^{i} ∑_{ht(λ)=i} A_{≤λ} onto ∑_{ht(λ)=i} A_{≤λ}/A_{<λ} by ‘dropping t^{i}’. By corollary 4.7 the even part of E_{2}^{0∗} in proposition 3.8 is noetherian over a finitely generated subalgebra R of H^{even}(SL_{n}, A), which implies in the usual way ([10, Lemma 1.6]) that in fact the abutment H^{∗}(SL_{n}, gr A) is noetherian over R.

4.10 The general fibre

One gets a homomorphism A → A by substituting a nonzero scalar for t. Let us use the substitution t ↦ 1. On A we put the filtration with A_{≤m} = ∑_{i≤m} ∑_{ht(λ)≤i} t^{i}A_{≤λ}. Then the associated graded gr A is just A itself. The map A → A is compatible with the filtrations, so we get a map of spectral sequences from

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

to

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

Note that by the construction of the Grosshans filtration H^{0}(G, gr_{−i} A) vanishes for i ≠ 0. Further E_{2}^{∗∗}(A) is finitely generated by theorem 1.1 and therefore there are for each m only finitely many non-zero E_{2}^{m−i,i}(A). This makes that, even though the spectral sequence is a second quadrant spectral sequence, the abutment will be finitely generated as soon as E_{∞}^{∗∗}(A) is.

Theorem 4.11 (Cohomological invariant theory in rank one) Let A be a finitely generated commutative k-algebra on which SL_{2} acts rationally by algebra automorphisms. Then H^{∗}(SL_{2}, A) is finitely generated as a k-algebra.

Proof We combine the above. The spectral sequence

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

is pleasantly boring: its E_{2} is the same as the abutment. The spectral sequence E(A) is a module over it. In particular, E(A) is a module over the finitely generated subring R of H^{even}(SL_{n}, A) over which E_{2}^{∗∗}(A) is noetherian by 4.9. (Yes, the homomorphism R → E_{2}^{∗∗}(A) agrees with the homomorphism E_{2}^{∗∗}(A) → E_{2}^{∗∗}(A).) So the usual argument (see proof of [10, Lemma 1.6]) shows that E(A) stops and that E_{∞}^{∗∗}(A) is noetherian over R. □

Corollary 4.12 Let A be a finitely generated commutative k-algebra on which SL_{2} acts rationally by algebra automorphisms. Assume that A has a good filtration. Let M be a finitely generated A module with a compatible SL_{2} action. Then M has finite good filtration dimension.

Proof Write G = SL_{2}. If we tensor A with the multicone k[G/U] of example 2.4, then the result is a finitely generated G-acyclic k-algebra A ⊗ k[G/U] over which we have a finitely generated module M ⊗ k[G/U]. As H^{∗}(G, S_{A⊗k[G/U]}^{∗}(M ⊗ k[G/U])) is finitely generated, H^{i}(G, M ⊗ k[G/U]) vanishes for i >> 0. But this means that M has finite good filtration dimension. □

References

[1] H. H. Andersen, J.-C. Jantzen, Cohomology of induced representations for algebraic groups, Math. Ann. 269 (1984), 487–525.

[2] D. J. Benson, Representations and cohomology. II. Cohomology of groups and modules. Second edition. Cambridge Studies in Advanced Mathematics, 31. Cambridge University Press, Cambridge, 1998.

[3] G. Boffi, On some plethysms. Adv. Math. 89 (1991), 107–126.

[4] H. Borsari, W. Ferrer Santos, Geometrically reductive Hopf algebras, J. Algebra 152 (1992), 65–77.

[5] S. Donkin, Good filtrations of rational modules for reductive groups, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), 69–80, Proc. Sympos. Pure Math., 47, Part 1, Amer. Math. Soc., Providence, RI, 1987.

[6] S. Donkin, On tilting modules for algebraic groups, Math. Z. 212 (1993), 39–60.

[7] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995.

[8] L. Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), 224–23.

[9] E. M. Friedlander, B. J. Parshall, Cohomology of Lie algebras and algebraic groups, Amer. J. Math. 108 (1986), 235–253 (1986).

[10] E. M. Friedlander, A. A. Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), 209–270.

[11] F. D. Grosshans, Contractions of the actions of reductive algebraic groups in arbitrary characteristic, Invent. Math. 107 (1992), 127–133.

[12] F. D. Grosshans, Algebraic homogeneous spaces and invariant theory, Lecture Notes in Mathematics, 1673. Springer-Verlag, Berlin, 1997.

[13] N. Jacobson, Basic algebra. II. Second edition. W. H. Freeman and Company, New York, 1989.

[14] J.-C. Jantzen, Representations of Algebraic Groups, Pure and Applied Mathematics v. 131, Boston: Academic Press 1987.

[15] G. Kempf, A. Ramanathan, Multicones over Schubert varieties, Invent. Math. 87 (1987), 353–363.

[16] T.-Y. Lam, Serre’s conjecture, Lecture Notes in Math., Vol. 635, Springer, Berlin, 1978.

[17] O. Mathieu, Filtrations of G-modules, Ann. Sci. École Norm. Sup. 23 (1990), 625–644.

[18] H. Matsumura, Commutative algebra. W. A. Benjamin, Inc., New York 1970.

[19] M. Nagata, Invariants of a group in an affine ring, J. Math. Kyoto Univ. 3 (1963/1964), 369–377.

[20] V. L. Popov, Contractions of actions of reductive algebraic groups. (Russian) Mat. Sb. (N.S.) 130(172) (1986), no. 3, 310–334, 431. Translated in Math. USSR-Sb. 58 (1987), no. 2, 311–335.

[21] N. Roby, Construction de certaines algèbres à puissances divisées. Bull. Soc. Math. France 96 (1968), 97–113.

[22] J.-P. Serre, Local fields. Translated from the French by Marvin Jay Greenberg. Graduate Texts in Mathematics, 67. Springer-Verlag, New York-Berlin, 1979.

[23] T. A. Springer, Invariant theory. Lecture Notes in Mathematics, 585. Springer-Verlag, Berlin-New York, 1977.

[24] W. van der Kallen, Lectures on Frobenius splittings and B-modules. Notes by S. P. Inamdar, Tata Institute of Fundamental Research, Bombay, and Springer-Verlag, Berlin, 1993.

[25] W. C. Waterhouse, Geometrically reductive affine group schemes, Arch. Math. 62 (1994), 306–307.