A Friedlander-Suslin theorem over a noetherian base ring

Let $k$ be a noetherian commutative ring and let $G$ be a finite flat group scheme over $k$. Let $G$ act rationally on a finitely generated commutative $k$-algebra $A$. We show that the cohomology algebra $H^*(G,A)$ is a finitely generated $k$-algebra. This unifies some earlier results.


Introduction
In view of [12] the following theorem will be the key.
Theorem 1 Let k be a noetherian commutative ring and let G be a finite flat group scheme over k.There is a positive integer n that annihilates H i (G, M ) for all i > 0 and all G-modules M .

Remark 1
If k does not contain Z, then one may clearly take n to be the additive order of 1 ∈ k.

Remark 2
If G is a constant group scheme, then it is well known that one may take n to be the order of the group [15,Theorem 6.5.8].(A proof is also implicit in the proof of Theorem 1 below.)Theorem 2 (Friedlander-Suslin theorem over noetherian base ring) Let k be a noetherian commutative ring and let G be a finite flat group scheme over k.Let G act rationally on a finitely generated commutative k-algebra A. Then the cohomology algebra H * (G, A) is a finitely generated k-algebra.

Remark 4
If G is a constant group scheme, then Theorem 2 follows from [2,Theorem 8.1].The original proof in [2] is much more efficient than a proof that relies on [12].

Remark 5
If k is a field of finite characteristic, then Theorem 2 is implicit in [4].(If G is finite reduced, see Evens.If G is finite and connected, take C = A G in [4, Theorem 1.5, 1.5.1].If G is not connected, one finishes the argument by following [2] as on pages 220-221 of [4].)

Remark 6
The flatness assumption is essential for this kind of representation theory.One needs it to ensure that taking invariants is left exact [6, I 2.10 (4)] and that the category of comodules is abelian [6, I 2.9].

Conventions
The  Chapter 8] it is assumed that k is a field, so we will refer to Pareigis [8] for some needed facts.
We say that G acts rationally on a commutative k-algebra A, if A is also a G-module and the multiplication map An abelian group L is said to have bounded torsion if there is a positive integer n so that nL tors = 0, where L tors is the torsion subgroup of L. By [12, Theorem 10.5] bounded torsion is intimately related to finite generation of cohomology algebras.

Proofs
Proof of Theorem 2 assuming Theorem 1.First embed G into some Lemma 1 This defines a closed embedding G ⊂ GL N .

Proof of Lemma
is determined by its values f i (g).Let G(R) be the subset of R r consisting of the a = (a 1 , • • • , a r ) such that f i → a i extends to an algebra homomorphism, denoted g a , from k[G] to R. In other words, G is the subfunctor associated with the closed embedding G ⊂ A r given by the f i .Note that g a ∈ G(R) for a ∈ G(R).
We seek equations on GL N that cut out the image of G in GL N .It turns out to be more convenient to cut out an intermediate subscheme • g = g a .The first two properties define a k-closed subscheme X of GL N and the last property shows that G is a retract of X. Therefore the embedding is closed.
Alternative proof of the Lemma: If k[G] and Q are free k-modules, then one may ignore Q and use the proof of [14, Theorem 3.4], taking V = A. Now use that G → GL N is a closed embedding if there is a cover by affine opens of Spec(k) over which it is a closed embedding.
We may thus view GL N as a group scheme over k with G as k-subgroup scheme.Notice that GL N /G is affine [6, I 5.5 (6)] so that ind GLN Proof of Theorem 1. (•) Observe that the problem is local in the Zariski topology on Spec(k), by the following Lemma.

Proof of Lemma Recall that the f i generate the unit ideal if and only if the principal open subsets D(f
contains a power of f i for each i.These powers generate the unit ideal. Let H be the Hopf algebra k[G].In the notations of Pareigis [8] we have a rank 1 projective k-module P (H * ) that is a direct summand of the k-module H * = M (G) [8, Lemma 2, Proposition 3].If that projective module is free, then Pareigis shows that M (G) G ℓ is a direct summand of H * = M (G), free of rank one [8, Lemma 3].
By the observation (•) we may and shall assume that P (H * ) is indeed free.Take a generator ψ of M (G) G ℓ .By remark 1 we may also assume that k contains Z, so that tensoring with Q does not kill everything.
We claim that ψ( 1) is now a unit in k 1 = Q ⊗ k.It suffices to check this at a geometric point x = Spec(F ) of Spec(k 1 ).As F is an algebraically closed field of characteristic zero, G is a constant group scheme at x by Cartier's Theorem [14, 11.4, 6.4].The coordinate ring F [G] is now the F -algebra of maps from the finite group G(F ) to F .Evaluation at an element g of G(F ) defines a Dirac measure δ g : F [G] → F and ψ is a nonzero scalar multiple of the sum ψ 0 = g∈G(F ) δ g of the Dirac measures.Evaluating ψ 0 at 1 yields the order of G(F ), which is indeed invertible in F . Put Then ψ 1 (1) = 1 and we conclude that 1 ∈ Q ⊗ kψ(1).Then there is a ∈ k so that aψ(1) is a positive integer n.Put φ = aψ.We now observe that φ − n annihilates k in k[G], and thus annihilates all invariants in G-modules.And for any G-module M we have φ(M ) ⊆ M G because φ is left invariant.
Consider a short exact sequence of G-modules If m ′′ ∈ M ′′G , let m ∈ M be a lift.One has 0 = (φ − n)m ′′ = πφ(m) − nm ′′ .As φ(m) ∈ M G , we conclude that n annihilates the cokernel of M G → M ′G .Taking M injective, we see that n annihilates H 1 (G, M ′ ).This applies to arbitrary G-modules M ′ .By dimension shift we get Theorem 1.

Remark 7
One does not need to use (•), because actually Q ⊗ M (G) G ℓ maps onto k 1 , even when M (G) G ℓ is not free over k.Indeed, consider the map v : Q ⊗ M (G) G ℓ → k 1 induced by χ → χ(1) : M (G) G ℓ → k.To see that v is surjective, it suffices again to check at the arbitrary geometric point x = Spec(F ) of Spec(k 1 ).
In fact Q ⊗ M (G) G ℓ is always free over k 1 = Q ⊗ k.

COI
The author declares that he has no conflict of interest.
coordinate ring k[G] is a Hopf algebra [6, Part I, Chapter 2].The dual Hopf algebra M (G) is the algebra of measures on G. Recall that over a noetherian commutative ring finite flat modules are finitely generated projective.Thus both k[G] and M (G) are finitely generated projective k-modules.Following [6, Part I, Chapter 8] we denote by M (G) G ℓ the k-module of left invariant measures.Any G-module V may be viewed as a left M (G)-module and one has

G
is exact [6, Corollary I 5.13].Thus by [6, I 4.6] we may rewrite H * (G, A) as H * (GL N , ind GLN G (A)), with ind GLN G (A) a finitely generated k-algebra, by invariant theory.As A G is noetherian, it has bounded torsion, and by Theorem 1 H >0 (GL N , ind GLN G (A)) = H >0 (G, A) also has bounded torsion.Theorem 2 now follows from Theorem 10.5 in [12].Remains to prove Theorem 1.