SUPPORT VARIETIES FOR INFINITESIMAL GROUP SCHEMES

Andrei Suslin^{∗}, Eric M. Friedlander^{∗∗}, and Christopher P. Bendel

The representation theory of a connected smooth affine group scheme over a field k of characteristic p > 0 is faithfully captured by that of its family of Frobenius kernels. Such Frobenius kernels are examples of infinitesimal group schemes, affine group schemes G whose coordinate (Hopf) algebra k[G] is a finite dimensional local k-algebra. This paper presents a study of the cohomology algebra H^{∗}(G, k) of an arbitrary infinitesimal group scheme over k.

We provide a geometric determination of the “cohomological support variety” |G| ≡ Spec H^{ev}(G, k) analogous to that given by D. Quillen for the cohomology of finite groups [Q]. We further study finite dimensional rational G-modules M for arbitrary infinitesimal group schemes G over k. In a manner initiated by J. Alperin and L. Evens [A-E] and J. Carlson [C1] for finite groups, we consider the variety |G|_{M} ⊂ |G| of the ideal I_{M} = ker{H^{ev}(G, k) → Ext_{G}^{∗}(M, M)} and provide a geometric description of this variety which is analogous to that given by G. Avrunin and L. Scott for finite dimensional modules for finite groups [A-S].

This paper is a continuation of our recent work establishing the finite generation of H^{∗}(G, k) [F-S] and investigating the infinitesimal 1-parameter subgroups of G [S-F-B]. Earlier work of E. Friedlander and B. Parshall [FP1], [FP2], [FP3], [FP4] and J. Jantzen [J1] concerning the cohomology of restricted Lie algebras are forerunners of the results presented here: finite dimensional restricted Lie algebras are in 1-1 correspondence with infinitesimal group schemes of height ≤ 1. Our main theorems (Theorems 5.2 and 6.7 below) when restricted to infinitesimal group schemes of height ≤ 1 significantly strengthen previously known cohomological information for restricted Lie algebras.

An interesting aspect of our work is the extent to which infinitesimal 1-parameter subgroups ν : 𝔾_{a(r)} → G for infinitesimal group schemes G of height ≤ r play the role of elementary abelian p-subgroups (and their generalizations, shifted subgroups) for finite groups. Indeed, much of our effort is dedicated to proving that cohomology classes are detected (modulo nilpotence) by such 1-parameter sub groups. This is first done in §1 for unipotent infinitesimal group schemes, using an induction argument made possible by a structure theorem present in §1. This structure theorem is the analogue in our context of a theorem of J.-P. Serre characterizing elementary abelian p-groups [S].

The proof of the detection theorem for arbitrary infinitesimal group schemes over k relies upon a generalization of a spectral sequence introduced by H. Andersen and J. Jantzen [A-J] which presents the cohomology of an infinitesimal kernel G_{(r)} of a reductive algebraic

Typeset by AMS-TEX

∗∗Partially supported by the N.S.F. and the Humboldt Foundation .

2 ANDREI SUSLIN^{∗}, ERIC M. FRIEDLANDER^{∗∗}, AND CHRISTOPHER P. BENDEL

group in terms of the cohomology of the infinitesimal kernel of a Borel subgroup. Our generalized spectral sequence is presented in §3, enabling the proof in §4 of the general detection theorem (Theorem 4.3).

The detection theorem demonstrates the essential injectivity of the natural map considered in [S-F-B]

ψ : H^{ev}(G, k) → k[V_{r}(G)],

where V_{r}(G) is the scheme of infinitesimal 1-parameter subgroup schemes ν : 𝔾_{a(r)} → G of an infinitesimal group scheme G over k of height ≤ r. The essential surjectivity of ψ (more precisely, surjectivity onto p^{r}-th powers) is a main result of [S-F-B]. This is formalized in Theorem 5.2 which presents a geometric, non-cohomological description of the cohomological support variety |G| of G. Corollary 6.8 gives a similarly geometric, noncohomological identification of |G|_{M} ⊂ |G| for any finite dimensional rational G-module M. We conclude in §7 with a few applications of these descriptions, applications analogous to results obtained previously for the cohomology of finite groups.

E. Friedlander gratefully acknowledges the hospitality of the University of Heidelberg.

§1. 1-parameter cohomomorphisms.

In this section, we investigate properties associated to homomorphisms G → 𝔾_{a(s)}, homomorphisms which we call 1-parameter cohomomorphisms. Even though the main theorems of this paper concern the interpretation of the cohomology of an (infinitesimal) group scheme G in terms of 1-parameter homomorphisms 𝔾_{a(r)} → G, such 1-parameter cohomomorphisms arise in our inductive analysis of a unipotent group scheme G.

We remind the reader that an affine group scheme is said to be unipotent if it admits an embedding as a closed subgroup scheme of some U_{N}, the group scheme of strictly upper triangular matrices of GL_{N}.

Lemma 1.1. Let G be an affine group scheme over k. a. H^{1}(G, k) = Hom_{Gr/k}(G, 𝔾_{a}) (as abelian groups). b. If G is infinitesimal of height ≤ r, then H^{1}(G, k) = Hom_{Gr/k}(G, 𝔾_{a(r)}). c. Hom_{Gr/k}(G, 𝔾_{a(1)}) = ker{F : H^{1}(G, k) → H^{1}(G, k)}, where F is induced by the

Frobenius map F : 𝔾_{a} → 𝔾_{a} using the identification of (a.). d. If G is a non-trivial unipotent group scheme, then H^{1}(G, k) ≠ 0.

Proof. Part (a.) can be seen directly from the standard cobar resolution used to compute H^{∗}(G, k). Part (b.) then follows formally from (a.). Part (c.) follows from the short exact sequence 0 → 𝔾_{a(1)} → 𝔾_{a} →^{F} 𝔾_{a} → 0. Finally, to establish part (d.), embed G as a closed subgroup of some U_{N} ⊂ GL_{N}. Denote by X^{i,j} (1 ≤ i, j ≤ N) the standard coordinate functions on GL_{N}. Find an integer n > 0 such that all functions X^{i,j} with 0 < j − i < n vanish on G and such that there exists some X^{i_{0},j_{0}} with j_{0} − i_{0} = n which is non trivial on G. It is clear that X^{i_{0},j_{0}} : G → 𝔾_{a} is a desired non trivial group homomorphism.

We shall employ the following familiar result.

Lemma 1.2. Let ψ : G → H be a surjective homomorphism of unipotent group schemes. If ψ^{∗} : H^{1}(H, k) → H^{1}(G, k) is an isomorphism and if ψ^{∗} : H^{2}(H, k) → H^{2}(G, k) is injective, then ψ is an isomorphism.

Proof. The Hochschild-Serre spectral sequence assoicated to the extension of group schemes

1 → N → G ⟶^{ψ} H → 1

gives the exact sequence

0 → H^{1}(H, k) ⟶^{ψ^{∗}} H^{1}(G, k) → H^{1}(N, k)^{H} → H^{2}(H, k) ⟶^{ψ^{∗}} H^{2}(G, k).

Thus, our hypotheses imply that H^{1}(N, k)^{H} = 0. Since H is unipotent, we conclude that H^{1}(N, k) = 0 and thus by Lemma 1.1.d that N is the trivial group scheme.

For future reference, we recall the following computation of the cohomology of 𝔾_{a} and its infinitesimal subgroups 𝔾_{a(r)}.

Theorem 1.3 [CPSvdK].

1. Assume that p ≠ 2. Then the cohomology algebra H^{∗}(𝔾_{a}, k) is a tensor product of a polynomial algebra k[x_{1}, x_{2}, ...] in generators x_{i} of degree 2 and an exterior algebra Λ(λ_{1}, λ_{2}, ...) in generators λ_{i} of degree one. If p = 2, then H^{∗}(𝔾_{a}, k) = k[λ_{1}, λ_{2}, ...] is a polynomial algebra in generators λ_{i} of degree 1; in this case, we set x_{i} = λ_{i}^{2}.

2. Let F : 𝔾_{a} → 𝔾_{a} denote the Frobenius endomorphism, then F^{∗}(x_{i}) = x_{i+1}, F^{∗}(λ_{i}) = λ_{i+1}.

3. Let s be an element of k and use the same notation s for the endomorphism (multiplication by s) of 𝔾_{a}. Then s^{∗}(x_{i}) = s^{p^{i}}x_{i}, s^{∗}(λ_{i}) = s^{p^{i−1}}λ_{i}.

4. Restriction of x_{i} and λ_{i} to 𝔾_{a(r)} is trivial for i > r. Denoting the restrictions of x_{i} and λ_{i} (for i ≤ r) to 𝔾_{a(r)} by the same letter we have

H^{∗}(𝔾_{a(r)}, k) =k[x_{1}, ..., x_{r}] ⊗ Λ(λ_{1}, ..., λ_{r})  p ≠ 2
H^{∗}(𝔾_{a(r)}, k) =k[λ_{1}, ..., λ_{r}]  p = 2.

Lemma 1.4. Denote by m : 𝔾_{a} × 𝔾_{a} → 𝔾_{a} the addition homomorphism. Then a. m^{∗}(λ_{i}) = λ_{i} ⊗ 1 + 1 ⊗ λ_{i}. b. m^{∗}(x_{i}) = x_{i} ⊗ 1 + 1 ⊗ x_{i}.

Proof. The first statement is evident. In proving the second one it suffices (in view of Theorem 1.3.2) to consider the case i = 1. Moreover we may assume, extending scalars if necessary, that the field k is infinite. Theorem 1.3.3 shows that the element m^{∗}(x_{1}) − x_{1} ⊗ 1 − 1 ⊗ x_{1} ∈ H^{1}(𝔾_{a}, k) ⊗ H^{1}(𝔾_{a}, k) has weight p with respect to the diagonal action of 𝔾_{m}. On the other hand, the weights appearing in H^{1}(𝔾_{a}, k) ⊗ H^{1}(𝔾_{a}, k) are of the form p^{i} + p^{j} with 0 ≤ i, j. This implies immediately that the above element is trivial unless p = 2. However in the case p = 2 our statement is trivial since in this case x_{1} = λ_{1}^{2}.

Remark 1.4.1 Alternatively in case p ≠ 2 one could use the formula x_{i} = −βP_{0}(λ_{i}) [M] and the properties of the Steenrod operations.

Corollary 1.5. For any ϕ : G → 𝔾_{a(1)}, set

x_{ϕ} ≡ ϕ^{∗}(x_{1}) ∈ H^{2}(G, k).

Then a. x_{tϕ} = t^{p}x_{ϕ} for any t ∈ k. b. x_{ϕ+ϕ′} = x_{ϕ} + x_{ϕ′} for any ϕ, ϕ′ : G → 𝔾_{a(1)}.

Proof. Part (a.) follows immediately from Theorem 1.3.3. Part (b.) follows from Lemma 1.4.b.).

The proof of the following theorem is inspired by the original argument of J.-P. Serre characterizing elementary abelian p-groups in terms of their cohomology algebras [S].

Theorem 1.6. Let G be an infinitesimal unipotent group scheme of height ≤ r with the property that Hom_{Gr/k}(G, 𝔾_{a(1)}) is 1-dimensional spanned by some ϕ : G → 𝔾_{a(1)}. Then either G = 𝔾_{a(s)} for some s ≤ r or x_{ϕ} ∈ H^{2}(G, k) is nilpotent.

Proof. Let s be the least integer such that F^{s} : H^{1}(G, k) → H^{1}(G, k) is 0. Then ϕ can be written as ϕ = F^{s−1}(ψ) for some ψ ∈ H^{1}(G, k). Moreover it’s clear that s ≤ r and the elements ψ, F (ψ), ..., F^{s−1}(ψ) = ϕ form a basis of H^{1}(G, k). The homomorphism ψ : G → 𝔾_{a(s)} induces an isomorphism ψ^{∗} : H^{1}(𝔾_{a(s)}, k) → H^{1}(G, k): the basis λ_{1}, λ_{2}, …, λ_{s} is mapped to the basis ψ, F (ψ), …, F^{s−1}(ψ). Lemma 1.2 implies that either ψ is an isomorphism or the ideal

I = Ker{ψ^{∗} : H^{∗}(𝔾_{a(s)}, k) → H^{∗}(G, k)}

contains a non-zero element of degree 2. Since ψ^{∗} is a map of modules for the Steenrod algebra [D], [P], I is stable with respect to action of the Steenrod operations P^{i}, βP^{i} for p ≠ 2 and Sq^{i} for p = 2.

Observe that x_{ϕ} = ψ^{∗}(x_{s}) ∈ H^{2}(G, k). Thus, the theorem follows from the following proposition.

Proposition 1.7. Let I ⊂ H^{∗}(𝔾_{a(s)}, k) be a k-submodule stable with respect to the action of the Steenrod operations P^{i}, βP^{i} for p ≠ 2 and Sq^{i} for p = 2. If I contains a non-zero element of degree 2, then some power of x_{s} lies in I.

Proof. We first assume that p ≠ 2. Here is a tabulation of the actions of P^{i}, βP^{i} on the generators x_{1}, …, x_{s}, λ_{1}, …, λ_{s}. a. P^{0}(x_{i}) = F^{∗}(x_{i}); P^{0}(λ_{i}) = F^{∗}(λ_{i}) b. P^{j}(x_{i}) = 0, j > 1, βP^{j}(x_{i}) = 0, j ≥ 1; P^{j}(λ_{i}) = 0 = βP^{j}(λ_{i}), j ≥ 1 c. P^{1}(x_{i}) = x_{i}^{p} d. βP^{0}(λ_{i}) = −x_{i}; βP^{0}(x_{i}) = 0 e. P^{m}(x_{s}^{k}) = x_{s}^{pk} if m = k; P^{m}(x_{s}^{k}) = 0 if m ≠ k. where (a.) and (c.) are among the defining axioms of the Steenrod algebra action and (b.) follows from the dimension axiom. The first part of (d.) is proved in [M] using the theory of Massey products, whereas the second statement follows by applying the Adem relation. Finally, (e.) follows from the Cartan formula using induction on m.

Let x = ∑_{i<j} a_{i,j}λ_{i}λ_{j} + ∑_{b} a_{b}x_{b} be a non-zero element in I ∩ H^{2}(𝔾_{a(s)}, k). Assume first that all coefficients a_{i,j} are 0. Find the smallest ℓ such that a_{ℓ} ≠ 0. Applying (P^{0})^{s−ℓ} = F^{∗s−ℓ} to x, we conclude that x_{s} = a_{ℓ}^{−p^{s−ℓ}}(P^{0})^{s−ℓ}(x) ∈ I.

If some coefficient a_{i,j} ≠ 0, then by applying to x an appropriate power of P^{0} we may assume that a_{i,j} = 0 for j < s; thus,

x = ∑_{i<s} a_{i,s}λ_{i}λ_{s} + ∑_{b} a_{b}x_{b}.

Now apply the operation βP^{1}βP^{0}. Using the Cartan formula

βP^{i}(u · v) = ∑_{j=0}^{i} βP^{j}(u) · P^{i−j}(v) + (−1)^{dimu}P^{j}(u) · βP^{i−j}(v)

and the above tabulation, we easily compute that

−βP^{1}βP^{0}(x) = ∑_{i<s} a_{i,s}^{p^{2}}x_{i+1}x_{s}^{p}.

Using the Cartan formula and (e.) above, we conclude that P^{p} applied to the right hand side of the above equality yields

P^{p}((∑_{i<s} a_{i,s}^{p^{2}}x_{i+1})x_{s}^{p}) = P^{0}(∑_{i<s} a_{i,s}^{p^{2}}x_{i+1}) · P^{p}(x_{s}^{p}) = (∑_{i<s} a_{i,s}^{p^{3}}x_{i+2}) · x_{s}^{p^{2}}.

Successively applying the operations P^{p^{2}}, P^{p^{3}}, …, we conclude that x_{s}^{p^{s−i}+1} ∈ I provided that i is the least integer for which a_{i,s} ≠ 0.

For p = 2, the proof has exactly the same form with P^{p^{k}} replaced by Sq^{2^{k+1}} and βP^{p^{k}} replaced by Sq^{2^{k+1}+1}.

§2. The detection theorem for unipotent infinitesimal group schemes

We prove in Theorem 2.5 below for G/k a unipotent infinitesimal group scheme of height ≤ r and z ∈ H^{∗}(G, k) that z is nilpotent provided that its restrictions via all 1-parameter homomorphims ν : 𝔾_{a(r)} ⊗_{k} K → G ⊗_{k} K are nilpotent where K is an arbitrary field extension of k. We prove this theorem for the cohomology algebra H^{∗}(G, Λ), where Λ is an associative unital rational G-algebra. Consideration of such Λ of the form Ind_{G}^{GL_{n(r)}}k will be required in our proof (in §4) of the detection theorem for H^{∗}(G, k) for arbitrary infinitesimal group schemes G.

Throughout this section, G will denote an infinitesimal group scheme over k and Λ an associative, unital rational G-algebra (i.e., Λ has structures of an associative unital k-algebra and of a rational G-module which are compatible in the sence that 1_{Λ} ∈ Λ^{G} and the multiplication map Λ ⊗_{k} Λ → Λ is a homomorphism of rational G-modules).

The following lemma is an explicit statement of the usual graded commutativity of the cohomology of a Hopf algebra.

Lemma 2.1. H^{∗}(G, Λ) is a graded H^{∗}(G, k)-algebra with the property that

ρ_{Λ}(x) · y = (−1)^{nm}y · ρ_{Λ}(x) , x ∈ H^{n}(G, k), y ∈ H^{m}(G, Λ)

where ρ_{Λ} : H^{∗}(G, k) → H^{∗}(G, Λ) is induced by the unit 1_{Λ} : k → Λ. In particular, ρ_{Λ}(H^{ev}(G, k)) is contained in the center of H^{∗}(G, Λ).

We next give an explicit statement of the multiplicative nature of the Hochschild-Serre spectral sequence (cf. [E]).

Lemma 2.2. Let

1 → N → G → H → 1

be an extension of affine group schemes. Then the Hochschild-Serre spectral sequence

E_{2}^{p,q} = H^{p}(H, H^{q}(N, Λ)) ⟹ H^{p+q}(G, Λ)

has a natural algebra structure. In particular,

F^{i}(H^{n}(G, Λ)) · F^{j}(H^{m}(G, Λ)) ⊂ F^{i+j}(H^{n+m}(G, Λ))

where F^{i}(H^{n}(G, Λ)) is the (decreasing) filtration associated to this spectral sequence.

The following proposition is a reformulation of the “Quillen-Venkov Lemma” [Q-V] in our context.

Proposition 2.3. Let f : G → 𝔾_{a(1)} be a non-trivial homomorphism of affine group schemes, and let z ∈ H^{n}(G, Λ) satisfy z_{|Ker f} = 0. Then z^{2} is divisible by ρ_{Λ}(x_{f}) ∈ H^{2}(G, Λ).

Proof. Denote Kerf by N and consider the Hochschild-Serre spectral sequence for the group scheme extension

1 → N → G ⟶^{f} 𝔾_{a(1)} → 1

E_{2}^{p,q} = H^{p}(𝔾_{a(1)}, H^{q}(N, Λ)) ⟹ H^{p+q}(G, Λ).

This spectral sequence admits an action of H^{∗}(𝔾_{a(1)}, k) compatible with the action of H^{∗}(𝔾_{a(1)}, k) on H^{∗}(G, Λ).

The periodicity of the cohomology of 𝔾_{a(1)} with respect to multiplication by x_{1} ∈ H^{2}(𝔾_{a(1)}, k) shows that the natural map x_{1} : E_{2}^{p,q} → E_{2}^{p+2,q} is surjective for p ≥ 0 and bijective for p ≥ 1. An easy induction on r shows further that x_{1} : E_{r}^{p,q} → E_{r}^{p+2,q} is surjective for p ≥ 0 and bijective for p ≥ r − 1. Thus x_{1} : E_{∞}^{p,q} → E_{∞}^{p+2,q} is surjective for all p ≥ 0. In other words,

F^{p+2}(H^{n+2}(G, Λ)) = ρ_{Λ}(x_{f}) · F^{p}(H^{n}(G, Λ)) + F^{p+3}(H^{n+2}(G, Λ)).

Using descending induction on p, we conclude that for p ≥ 0

F^{p+2}(H^{n+2}(G, Λ)) = ρ_{Λ}(x_{f}) · F^{p}(H^{n}(G, Λ)).

Finally, the kernel of H^{n}(G, Λ) → H^{n}(N, Λ) coincides with F^{1}(H^{n}(G, Λ)), so that z ∈ F^{1}(H^{n}(G, Λ)). Lemma 2.2 implies now that

z^{2} ∈ F^{2}(H^{2n}(G, Λ)) = ρ_{Λ}(x_{f}) · H^{2n−2}(G, Λ).

Lemma 2.4. Assume that r ≥ s and consider the canonical projection 𝔾_{a(r)} → 𝔾_{a(s)}. For any rational 𝔾_{a(s)}-module M, the induced map in cohomology H^{∗}(𝔾_{a(s)}, M) → H^{∗}(𝔾_{a(r)}, M) is injective.

Proof. Let u_{i} ∈ k[𝔾_{a(r)}]^{#} = (k[t]/t^{p^{r}})^{#} (0 ≤ i ≤ r − 1) be a linear function defined by u_{i}(t^{j}) = δ_{p^{i},j}. The algebra k[𝔾_{a(r)}]^{#} is equal to k[u_{0}, ..., u_{r−1}]/(u_{0}^{p}, ..., u_{r−1}^{p}) -see [S-F-B] Corollary 1.4. In the same way, k[𝔾_{a(s)}]^{#} = k[u_{0}, ..., u_{s−1}]/(u_{0}^{p}, ..., u_{s−1}^{p}). The homomorphism k[𝔾_{a(r)}]^{#} → k[𝔾_{a(s)}]^{#} takes u_{i} to the generator denoted by the same letter if i ≤ s−1 and to zero otherwise. The statement follows since there is an evident splitting of k-algebras k[𝔾_{a(s)}]^{#} → k[𝔾_{a(r)}]^{#} (which is not a homomorphism of Hopf algebras).

Theorem 2.5. Let G be a unipotent infinitesimal group scheme over k of height ≤ r and let Λ be an associative unital rational G-algebra. If z ∈ H^{n}(G, Λ) satisfies the property that for any field extension K/k and any group scheme homomorphism ν : 𝔾_{a(r)}⊗_{k}K → G⊗_{k}K over K the cohomology class ν^{∗}(z_{K}) ∈ H^{n}(𝔾_{a(r)} ⊗_{k} K, Λ_{K}) is nilpotent, then z is itself nilpotent.

Proof. Observe that if z_{L} ∈ H^{n}(G_{L}, Λ_{L}) is nilpotent for some field extension L/k then z is itself nilpotent. Thus we may assume, extending scalars, that the field k is algebraically closed.

The theorem is trivially valid if dim_{k}k[G] = 1. Proceeding by induction on dim_{k}k[G], we may assume that the theorem has been proved for all unipotent infinitesimal group schemes H over fields K/k with dim_{K}K[H] < dim_{k}k[G]. For notational convenience, we shall abuse notation in the remainder of this proof by writing x_{ϕ} for ρ_{Λ}(x_{ϕ}).

Assume first that dim_{k}Hom_{Gr/k}(G, 𝔾_{a(1)}) = 1 and let ϕ : G → 𝔾_{a(1)} be a non-trivial homomorphism. If G = 𝔾_{a(s)} for some s ≤ r, then the theorem follows immediately by employing the canonical projection ν : 𝔾_{a(r)} → 𝔾_{a(s)} - see Lemma 2.4. If G is not isomorphic to 𝔾_{a(s)} for any s ≤ r, Theorem 1.6 implies that x_{ϕ} ∈ H^{2}(G, k) is nilpotent. Let H = Ker ϕ ⊂ G. By induction, z_{|H} ∈ H^{n}(H, Λ) is nilpotent; say (z_{|H})^{N} = 0 ∈ H^{nN}(H, Λ). Proposition 2.3 now implies that x_{ϕ} divides z^{2N}, so that z is nilpotent.

Assume now that dim_{k}Hom_{Gr/k}(G, 𝔾_{a(1)}) > 1 and let ϕ_{1}, ϕ_{2} : G → 𝔾_{a(1)} be linearly independent homomorphisms. Set ϕ equal to

ϕ ≡ Y_{1}^{1/p}ϕ_{1} + Y_{2}^{1/p}ϕ_{2} : G ⊗_{k} K → 𝔾_{a(1)} ⊗_{k} K

where K = k(Y_{1}^{1/p}, Y_{2}^{1/p}) is a finite purely inseparable extension of the rational function field k(Y_{1}, Y_{2}). Our induction hypothesis implies that there exists N > 0 such that (z_{K}^{N})_{|Ker ϕ} = 0 and also (z^{N})_{|Ker ϕ_{i}} = 0. Proposition 2.3 shows now that x_{ϕ} divides z_{K}^{2N} and also x_{ϕ_{i}} divides z^{2N}.

We write z_{K}^{2N} ∈ H^{2nN}(G ⊗_{k} K, Λ_{K}) as

z_{K}^{2N} = x_{ϕ} · (∑_{i=1}^{m} a_{i}v_{i}) = ∑_{i=1}^{m} a_{i} · {(Y_{1}x_{ϕ_{1}} + Y_{2}x_{ϕ_{2}}) · v_{i}}

and write

z^{2N} = x_{ϕ_{1}} · (∑_{i=1}^{m} b_{i}v_{i})

for some elements a_{i} ∈ K, b_{i} ∈ k and some linearly independent elements v_{1}, …, v_{m} ∈ H^{2nN−2}(G, Λ). We may consider the first of the above formulas as a system of linear equations in variables a_{1}, ..., a_{m} with coefficients in the field k(Y_{1}, Y_{2}). Since this system has a solution in K, it really has a solution in k(Y_{1}, Y_{2}) as well (i.e. the a_{i} may be chosen in k(Y_{1}, Y_{2})). Eliminating denominators, we conclude that there exist polynomials P_{1}, ..., P_{m}, Q ∈ k[Y_{1}, Y_{2}], Q ≠ 0 such that

Q(Y_{1}, Y_{2}) · z^{2N} = ∑_{i=1}^{m} P_{i}(Y_{1}, Y_{2}) · {(Y_{1}x_{ϕ_{1}} + Y_{2}x_{ϕ_{2}}) · v_{i}} ∈ H^{2nN}(G, Λ) ⊗_{k} k[Y_{1}, Y_{2}]. (2.5.1)

Replacing P_{i} and Q by their appropriate homogenous components we may even assume that P_{i} and Q are homogenous and deg P_{i} + 1 = deg Q for all i.

Define the k-vector space

W = {w ∈ H^{2nN}(G, Λ) : z^{ℓ}w = 0 for some ℓ > 0}.

To prove that z is nilpotent, it suffices to show that z^{2N} ∈ W. Since z^{2N} is a linear combination of x_{ϕ_{1}} · v_{i}, it suffices to show that x_{ϕ_{1}} · v_{i} ∈ W for all i = 1, ..., m.

If x_{ϕ_{1}} · v, x_{ϕ_{2}} · v are linearly dependent modulo W for some v ∈ H^{2nN−2}(G, Λ), then t_{1}x_{ϕ_{1}} · v + t_{2}x_{ϕ_{2}} · v = (x_{t_{1}^{1/p}ϕ_{1}+t_{2}^{1/p}ϕ_{2}}) · v ∈ W

for some 0 ≠ (t_{1}, t_{2}) ∈ k^{2}. Applying our induction hypothesis to Ker ψ, where

ψ = t_{1}^{1/p}ϕ_{1} + t_{2}^{1/p}ϕ_{2} : G → 𝔾_{a(1)}

and using Proposition 2.3 we conclude that x_{ψ} divides an appropriate power of z. Since x_{ψ} ·v ∈ W, we conclude further that z^{ℓ} ·v = 0 for some ℓ > 0 and hence x_{ϕ_{1}} ·v, x_{ϕ_{2}} ·v ∈ W. In other words, if x_{ϕ_{1}} · v, x_{ϕ_{2}} · v are linearly dependent modulo W, then both x_{ϕ_{1}} · v and x_{ϕ_{2}} · v belong to W.

Modifying the linearly independent set v_{1}, …, v_{m} of H^{2nN−2}(G, Λ), we may assume that x_{ϕ_{1}} ·v_{1}, …, x_{ϕ_{1}} ·v_{j} are linearly independent modulo W whereas x_{ϕ_{1}} ·v_{j+1}, …, x_{ϕ_{1}} ·v_{m} belong to W (and hence x_{ϕ_{2}} ·v_{j+1}, ..., x_{ϕ_{2}} ·v_{m} ∈ W). We want to show that j = 0. Assume to the contrary that j > 0. Reducing the equation (2.5.1) modulo W we obtain the following equation in H^{2nN}(G, Λ)/W ⊗_{k} k[Y_{1}, Y_{2}]:

Q · \overline{z}^{2N} = ∑_{i=1}^{j} P_{i} · \{Y_{1}(\overline{x_{ϕ_{1}} · v_{i}}) + Y_{2}(\overline{x_{ϕ_{2}} · v_{i}})\}.  (2.5.2)

Dividing both sides of (2.5.2) by the greatest common divisor of Q, P_{1}, …, P_{j}, we may assume that g.c.d.(Q, P_{1}, …, P_{j}) = 1. If deg Q = 0 so that each P_{i} = 0, then z^{2N} ∈ W; this implies that z is nilpotent and hence W = H^{2nN}(G, Λ), thereby contradicting the hypothesis that j > 0. If deg Q > 0, we use the fact that k is algebraically closed to find 0 ≠ (t_{1}, t_{2}) ∈ k^{2} such that Q(t_{1}, t_{2}) = 0 and P_{i}(t_{1}, t_{2}) ≠ 0 for some i (1 ≤ i ≤ j). Then

∑_{i=1}^{j} P_{i}(t_{1}, t_{2})\{t_{1}\overline{x_{ϕ_{1}} · v_{i}} + t_{2}\overline{x_{ϕ_{2}} · v_{i}}\} = t_{1}\overline{x_{ϕ_{1}} · v} + t_{2}\overline{x_{ϕ_{2}} · v} = 0 ∈ H^{2nN}(G, Λ)/W

where v = ∑_{i=1}^{j} P_{i}(t_{1}, t_{2})v_{i}. As seen above, this implies that x_{ϕ_{1}} · v ∈ W. On the other hand, this contradicts the assumed linear independence of x_{ϕ_{1}} · v_{1}, …, x_{ϕ_{1}} · v_{j} modulo W.

§3. A spectral sequence for Frobenius kernels

In [A-J], H. Andersen and J. Jantzen introduced a spectral sequence for a reductive (smooth) group scheme G and a rational G-module M

E_{2}^{p,q} = H^{p}(G^{(r)}/B^{(r)}, L(H^{q}(B_{(r)}, M))) ⟹ H^{p+q}(G_{(r)}, M)

which relates the cohomology of the Frobenius kernel G_{(r)} to that of the Frobenius kernel of the Borel subgroup B ⊂ G. As we observe below, this immediately generalizes to arbitrary smooth group schemes. The purpose of this section is to further generalize this spectral sequence so that it applies to rational G_{(r)}-modules M which are not necessarily modules for G.

Let G/k be a reduced (=smooth) affine group scheme. We recall that when k is algebraically closed the Borel subgroup B ⊂ G is defined to be a closed, reduced, connected, solvable subgroup scheme of maximal possible dimension. In general, we will say that B ⊂ G is a Borel subgroup iff B ⊗_{k} \overline{k} is a Borel subgroup of G ⊗_{k} \overline{k} (where \overline{k} denotes the algebraic closure of k). Note that when k is not algebraically closed not every reduced group scheme has Borel subgroups. Throughout this section (if not specified otherwise) G will denote a connected reduced affine group scheme over k and B will denote its Borel subgroup (thus we assume that Borel subgroups in G exist).

The following proposition is a straight-forward generalization of a fundamental theorem of E. Cline, B. Parshall, L. Scott, and W. van der Kallen [CPSvdK] (see also [K]).

Proposition 3.1. For any rational G-module M, the restriction map

H^{∗}(G, M) → H^{∗}(B, M)

is an isomorphism.

Proof. Since rational cohomology commutes with flat base change , it suffices to assume that k is algebraically closed. If G is semisimple, then our statement is precisely the theorem proved in [CPSvdK]. In the general case, let R(G) denote the radical of G and compare the two Hochschild-Serre spectral sequences

E_{2}^{p,q} = H^{p}(G/R(G), H^{q}(R(G), M)) ⟹ H^{p+q}(G, M)
E_{2}^{p,q} = H^{p}(B/R(G), H^{q}(R(G), M)) ⟹ H^{p+q}(B, M).

The proposition now follows since B/R(G) is a Borel subgroup of the semisimple group G/R(G).

Proposition 3.2. Let G be an arbitrary affine group scheme over k. Let further H ⊂ G be a closed subgroup scheme and let X = G/H denote the quotient scheme with quotient map p : G → X. Then the functor M ↦ Γ(G, p^{∗}M) defines an equivalence of categories between the category of quasi-coherent O_{X}-modules M and the category of k-vector spaces M provided with the structures of a left k[G]-module and of a rational H-module which are compatible in the sense that the multiplication map k[G] ⊗ M → M is a homomorphism of rational H-modules (H acts on k[G] via the right regular representation).

Proof. Since p : G → X is a faithfully flat, quasi-compact morphism, the theory of faithfully flat descent [SGA1] tells us that the functor p^{∗} is an equivalence of the category of quasi-coherent O_{X}-modules and the category of quasi-coherent O_{G}-modules N equipped with descent data: an isomorphism ψ : q_{1}^{∗}(N) ⟶^{∼} q_{2}^{∗}(N) of O_{G×_{X}G}-modules (where q_{1}, q_{2} : G ×_{X} G → G are the projections) satisfying the usual compatibility property: q_{23}^{∗}(ψ)q_{12}^{∗}(ψ) = q_{13}^{∗}(ψ). Moreover, since the scheme G is affine, the category of quasi-coherent O_{G}-modules is equivalent (via the functor M ↦ M̃) to the category of left k[G]-modules.

The cartesian square (cf. [D-G;III.1.2.4])

G × H ⟶^{pr_{1}} G
m↓  ↓^{p}
G ⟶^{p} X

enables us to identify q_{1}, q_{2} : G ×_{X} G → G with m, pr_{1} : G × H → G. This shows that descent data on M̃ is equivalent to the data of a homomorphism

ϕ : M → M ⊗ k[H]

satisfying the following properties: (a.) ϕ is a homomorphism of k[G]-modules provided that M ⊗ k[H] is made into a k[G]- module via the homomorphism m^{∗} : k[G] → k[G] ⊗ k[H]. (b.) The induced homomorphism of k[G] ⊗ k[H]-modules

(k[G] ⊗ k[H]) ⊗_{m^{∗}} M → M ⊗ k[H]

is an isomorphism. (c.) The following diagram (in which Δ denotes the diagonal map of the Hopf algebra k[H]) commutes

M  ⟶^{ϕ}  M ⊗ k[H]
ϕ↓  ^{1_{M} ⊗Δ}↓
M ⊗ k[H] ⟶^{ϕ⊗1_{k[H]}} M ⊗ k[H] ⊗ k[H].

Finally it is not hard to see that conditions (a.),(b.),(c.) are equivalent to the fact that ϕ gives M a structure of a rational H-module compatible with its structure as a k[G]-module.

Remark 3.3. Let N be a rational H-module. The k[G]-module k[G]⊗_{k} N has a natural structure of a rational H-module (H acts in the given way on N and right regularly on k[G]). This H-module structure is clearly compatible with the k[G]-module structure in the sence described above and hence we get a quasicoherent sheaf on G/H which we’ll denote by L(N). The theory of faithfully flat descent shows that L(N) is a locally free O_{G/H}-module and it is coherent if and only if N is finite dimensional. It’s easy to see that this construction gives the same result as the one used in [J].

Proposition 3.4. Assume the notation and conventions of (3.2). Then for any quasicoherent O_{X}-module M, the Zariski cohomology H_{Zar}^{∗}(X, M) is naturally isomorphic to the rational cohomology H^{∗}(H, Γ(G, p^{∗}(M))).

Proof. It is well known (cf. [Mi,III.3.7]) that Zariski cohomology with coefficients in a quasi-coherent sheaf coincides with fppf-cohomology. Starting with the fppf-covering p : G → X of a scheme X, we may construct its nerve (i.e. the simplicial scheme X_{•} with X_{n} = G ×_{X} ... ×_{X} G (n + 1 times) and the standard face and degeneracy operators). In a familiar way, this nerve determines a spectral sequence

E_{2}^{pq} = H_{fppf}^{p}(X_{q}, M) = H_{Zar}^{p}(X_{q}, M) ⟹ H_{fppf}^{p+q}(X, M) = H_{Zar}^{p+q}(X, M).

The scheme X_{q} may be identified with G × H^{q} and in particular is affine. Thus the above spectral sequence degenerates, thereby identifying H_{Zar}^{∗}(X, M) with the cohomology of the complex

Γ(G, p^{∗}(M) → Γ(G, p^{∗}(M)) ⊗ k[H] → Γ(G, p^{∗}(M)) ⊗ k[H]^{⊗2} → ...  (3.4.1)

Finally (3.4.1) is the Hochschild complex computing the rational cohomology groups of H with coefficients in Γ(G, p^{∗}(M)).

Let F^{r} : G → G^{(r)} be the r-th power of the Frobenius map, where G^{(r)} is the r-th Frobenius twist of G (see, for example, [F-S;§1]). The kernel of F^{r}, G_{(r)}, is an infinitesimal group scheme of height r and F^{r} induces an isomorphism

F^{r} : G/G_{(r)} ˜⟶G^{(r)}.

Proposition 3.5. For any rational G_{(r)}-module M and any q ≥ 0, the cohomology group H^{q}(B_{(r)}, Ind_{G_{(r)}}^{G}(M)) has the natural structures of a rational B^{(r)}-module and of a k[G^{(r)}]- module which satisfy the compatibility condition of (3.2).

Proof. Recall that the induced module Ind_{G_{(r)}}^{G}(M) is defined to be (k[G] ⊗ M)^{G_{(r)}}, where G_{(r)} acts via the given action on M and via the right regular representation on k[G] and where Ind_{G_{(r)}}^{G}(M) is given the structure of a rational G-module via the left regular representation of G on k[G] and the trivial action on M. For any q ≥ 0, H^{q}(B_{(r)}, Ind_{G_{(r)}}^{G}(M)) has the structure of a rational B-module on which B_{(r)} ⊂ B acts trivially, hence the structure of a rational B^{(r)}-module. Moreover, the action of k[G^{(r)}] = k[G]^{G_{(r)}} on Ind_{G_{(r)}}^{G}(M) (by B_{(r)}-linear transformations) determines an action on H^{q}(B_{(r)}, Ind_{G_{(r)}}^{G}(M)). One readily verifies that the multiplication map

k[G^{(r)}] ⊗ H^{q}(B_{(r)}, Ind_{G_{(r)}}^{G}(M)) → H^{q}(B_{(r)}, Ind_{G_{(r)}}^{G}(M))

is a homomorphism of rational B^{(r)}-modules provided that B^{(r)} acts on k[G^{(r)}] via the left regular representation. To get the compatibility we need to change the k[G^{(r)}]-module structure using the automorphism of k[G^{(r)}] induced by the automorphism of the scheme G^{(r)}

σ : G^{(r)} → G^{(r)} : σ(x) = x^{−1}.

Definition 3.5.1. We denote by H^{q}(B_{(r)}, M) the quasicoherent sheaf on X^{(r)} = G^{(r)}/B^{(r)} associated in view of Proposition 3.2 to H^{q}(B_{(r)}, Ind_{G_{(r)}}^{G}(M)) with the above structures.

Theorem 3.6. For any rational G_{(r)}-module M there is a natural spectral sequence

E_{2}^{p,q} = H^{p}(X^{(r)}, H^{q}(B_{(r)}, M)) ⟹ H^{p+q}(G_{(r)}, M)

where X^{(r)} = G^{(r)}/B^{(r)}.

Proof. Following [FP4], we consider the Hochschild-Serre spectral sequence

E_{2}^{p,q} = H^{p}(B^{(r)}, H^{q}(B_{(r)}, Ind_{G_{(r)}}^{G}(M))) ⟹ H^{p+q}(B, Ind_{G_{(r)}}^{G}(M)).

By Proposition 3.1, the abutment may be identified with H^{p+q}(G, Ind_{G_{(r)}}^{G}(M)). Since G/G_{(r)} = G^{(r)} is affine, this can be identified with H^{p+q}(G_{(r)}, M) (cf. [J;I.5.12]). Proposition 3.4 implies that the E_{2}^{p,q} term can be identified as

H^{p}(B^{(r)}, H^{q}(B_{(r)}, Ind_{G_{(r)}}^{G}(M))) = H^{p}(X^{(r)}, H^{q}(B_{(r)}, M)).

Remark 3.6.1 If M in (3.6) is the restriction of a rational G-module (this is the case considered in [A-J]), then the tensor identity (see [J]) shows that the quasi-coherent sheaf H^{q}(B_{(r)}, M) is canonically isomorphic to the locally free sheaf L(H^{q}(B_{(r)}, M)).

§4. The detection theorem for infinitesimal group schemes

In this section, we extend Theorem 2.5 to an arbitrary infinitesimal group scheme. Our strategy is to use the spectral sequence of the previous section, valid for Frobenius kernels of smooth group schemes, to reduce the detection theorem for Frobenius kernels to the unipotent case. We then extend the validity of the detection theorem to arbitrary infinitesimal group schemes G by embedding G in some Frobenius kernel GL_{n(r)} and identifying H^{∗}(G, Λ) with H^{∗}(GL_{n(r)}, Ind_{G}^{GL_{n(r)}}(Λ)).

We begin with the detection theorem for Frobenius kernels.

Theorem 4.1. Let G denote a connected, smooth affine group scheme over k and assume that there exists a Borel subgroup B ⊂ G. Let Λ be an associative unital rational G_{(r)}- algebra. If z ∈ H^{n}(G_{(r)}, Λ) satisfies the property that for any field extension K/k and any group scheme homomorphism ν : 𝔾_{a(r)} ⊗_{k} K → G ⊗_{k} K (over K) the cohomology class ν^{∗}(z_{K}) ∈ H^{n}(𝔾_{a(r)} ⊗_{k} K, Λ_{K}) is nilpotent, then z is itself nilpotent.

Proof. As in the proof of Proposition 3.1, it suffices to assume that k = \overline{k}. (Since G has a k-rational point - namely, the identity - G ⊗_{k} \overline{k} is connected.) Consider the spectral sequence of Theorem 3.6:

H^{p}(X^{(r)}, H^{q}(B_{(r)}, Λ)) ⟹ H^{p+q}(G_{(r)}, Λ).

This spectral sequence appears as the Hochschild-Serre spectral sequence corresponding to the group extension

1 → B_{(r)} → B → B^{(r)} → 1.

By Lemma 2.2,

F^{i}(H^{n}(G_{(r)}, Λ)) · F^{j}(H^{m}(G_{(r)}, Λ)) ⊂ F^{i+j}(H^{n+m}(G_{(r)}, Λ)).

On the other hand F^{i}(H^{∗}(G_{(r)}, Λ)) = 0 for i > dim X. This shows that the ideal F^{1}(H^{∗}(G_{(r)}, Λ)) is nilpotent and hence it suffices to show that the image of z in H^{0}(X^{(r)}, H^{n}(B_{(r)}, Λ)) is nilpotent. The k-algebra H^{0}(X^{(r)}, H^{∗}(B_{(r)}, Λ)) = H^{0}(B^{(r)}, H^{∗}(B_{(r)}, Ind Λ)) is a subalgebra of the k[G^{(r)}]-algebra H^{∗}(B_{(r)}, Ind Λ) (here and in the sequel we abbreviate the notation Ind_{G_{(r)}}^{G} to Ind). Thus it suffices to show that the image z^{∼} of z in H^{∗}(B_{(r)}, Ind Λ) is nilpotent.

Denote by I^{•} the standard G_{(r)}-injective resolution of Λ: I^{m} = Λ ⊗ k[G_{(r)}]^{⊗(m+1)} (here G_{(r)} acts on I^{m} via the right regular representation on the last tensor factor). Note that I^{•} is naturally a differential graded k-algebra. The differential graded k[G^{(r)}]-algebra J^{•} = (Ind I^{•})^{B_{(r)}} satisfies H^{∗}(J^{•}) = H^{∗}(B_{(r)}, Ind Λ). Let g be any point of G and let g^{(r)} be the image of g in G^{(r)}. Consider the differential graded k(g^{(r)})-algebra J^{•}⊗_{k[G^{(r)}]}k(g^{(r)}) and tensor it over k(g^{(r)}) with k(g). To understand the structure of the resulting differential graded k(g)-algebra assume first that the point g is k-rational.

For any rational G_{(r)}-module M, there is a canonical G_{(r)}-equivariant homomorphism ϵ_{M} : Ind M → M: f ⊗ m ↦ f(1) · m. This homomorphism is clearly k[G^{(r)}]-linear if we make M into a k[G^{(r)}]-module via the evaluation at 1 ∈ G^{(r)}(k) map k[G^{(r)}] → k. Consider further the composition

Ind M ⟶^{g^{−1}} Ind M ⟶^{ϵ_{M}} M.

This homomorphism takes (Ind M)^{B_{(r)}} ⊂ Ind M to M^{g^{−1}B_{(r)}g} ⊂ M. Furthermore, this homomorphism is k[G^{(r)}]-linear if we make M into a k[G^{(r)}]-module via the evaluation at g^{(r)} ∈ G^{(r)}(k) map. Thus we get a natural homomorphism

θ_{M} : ((Ind M)^{B_{(r)}} ⊗_{k[G^{(r)}]} k(g^{(r)})) ⊗_{k(g^{(r)})} k(g) → M^{g^{−1}B_{(r)}g}

The homomorphism θ_{k[G_{(r)}]} is an isomorphism as one sees from the Cartesian square

\begin{matrix}g^{−1}B_{(r)}g\G_{(r)} & ⟶ & Spec k \\ ^{x↦gx}↓ & & _{g^{(r)}}↓ \\ B_{(r)}\G & ⟶ & G_{(r)}\G = G^{(r)} \end{matrix}  (4.1.1)

(To check that (4.1.1) is Cartesian, one realizes this square as the sheafification in the fppf topology of a pull-back square of presheaves of sets of the same form in which the indicated quotients are quotients as presheaves). Hence, θ_{M} is an isomorphism for any injective G_{(r)}-module M. This implies that

(J^{•} ⊗_{k[G^{(r)}]} k(g^{(r)})) ⊗_{k(g^{(r)})} k(g) = (I^{•})^{g^{−1}B_{(r)}g}.

Using extension of scalars, one sees easily that the same kind of an answer holds for any point g ∈ G. More precisely, we have the following general formula:

(J^{•} ⊗_{k[G^{(r)}]} k(g^{(r)})) ⊗_{k(g^{(r)})} k(g) = (I^{•} ⊗_{k} k(g))^{g^{−1}(B_{(r)}⊗_{k}k(g))g}.

In this formula, we use the same notation g for the k(g)-rational point determined by the point g ∈ G. This formula implies, in particular, that

H^{∗}(J^{•} ⊗_{k[G^{(r)}]} k(g)) =H^{∗}(g^{−1}(B_{(r)} ⊗_{k} k(g))g, Λ ⊗_{k} k(g)) ⊂
⊂ H^{∗}(g^{−1}(U_{(r)} ⊗_{k} k(g))g, Λ ⊗_{k} k(g)).

A direct computation shows that the image of z^{∼} in this cohomology algebra coincides with the restriction of z_{k(g)} ∈ H^{n}(G_{(r)} ⊗_{k} k(g), Λ_{k(g)}) to g^{−1}(U_{(r)} ⊗_{k} k(g))g. Thus, we get a cohomology class of the unipotent group g^{−1}(U_{(r)} ⊗_{k} k(g))g with coefficients in the associative unital rational algebra Λ ⊗_{k} k(g) which restricts nilpotently (according to our assumptions on z) to all one parameter subgroups of g^{−1}(U_{(r)} ⊗_{k} k(g))g. Theorem 2.5 shows that this cohomology class is nilpotent. Since extension of scalars from k(g^{(r)}) to k(g) gives an injective map in cohomology, we conclude that the image of z^{∼} in the cohomology of J^{•} ⊗_{k[G^{(r)}]} k(x) is nilpotent for each point x ∈ G^{(r)}. Finally we can use the following result.

Proposition 4.2. Let A be a commutative noetherian ring of finite Krull dimension and let J^{•} be an associative unital differential graded A-algebra. Assume that z ∈ H^{n}(J^{•}) is a cohomology class such that for each prime ideal μ ∈ Spec A the image of z in H^{∗}(J^{•} ⊗_{A} k(μ)) is nilpotent. Then z is nilpotent.

Proof. We proceed by induction on d = dim A. Assume first that d = 0 (i.e., A is an artinian ring). Since every artinian ring is (canonically) a product of local artinian rings : A = ∏ A_{i} and similarly J^{•} = ∏ J_{i}^{•}, where J_{i}^{•} is a differential graded A_{i}-algebra we may further assume that A is local. In this case, we use an additional induction on the length of the maximal ideal 𝔐(A) of A. If this length is equal to zero, then A is a field and there is nothing to prove. Otherwise, choose any nonzero element t ∈ 𝔐(A). Applying the induction hypothesis to A/tA and a differential graded A/tA-algebra J^{•}/tJ^{•}, we see that there exists N > 0 such that the image of z^{N} in J^{•}/tJ^{•} is a boundary. Replacing z by z^{N} we my assume that z is a cohomology class of a cycle of the form tu (u ∈ J^{n}). Since t is nilpotent the statement is clear in this case.

Assume now that d > 0 and the statement has been proved already for differential graded algebras over commutative rings of dimension < d. Consider the following chain of ideals in A

Ann_{A}z ⊂ Ann_{A}z^{2} ⊂ Ann_{A}z^{3} ⊂ ...

Since A is noetherian this chain eventually stabilizes. Replacing z by its appropriate power we may assume in the sequel that

Ann_{A}z = Ann_{A}z^{2} = ...

Denote this ideal by I. Let μ ∈ Spec A be a prime ideal of height < d. Applying the induction hypothesis to A_{μ} and J_{μ}^{•} we conclude that the image of z in H^{∗}(J_{μ}^{•}) = H^{∗}(J^{•})_{μ} is nilpotent which implies readily that I ⊄ μ. This shows that the only prime ideals which could possibly contain I are maximal ideals of height d. Since d > 0 we can find t ∈ I such that dim A/tA < d. Applying the induction hypothesis to A/tA and J^{•}/tJ^{•}, we conclude that (after replacing z by an appropriate power of z) we may assume that z is a cohomology class of a cycle of the form tu with u ∈ J^{n}. Note that t · du = 0 and hence t · d(u^{2}) = t · (u ∪ du + du ∪ u) = 0. Thus tu^{2} is a cycle. Denoting the cohomology class of this cycle by y we obtain the equality z^{2} = ty and hence the vanishing z^{3} = z ∪ z^{2} = z ∪ ty = tz ∪ y = 0.

Remark 4.2.1 The proof of Theorem 4.1 can be much simplified provided that one assumes that Λ is a rational G-algebra. In this case it suffices to show that the image of z ∈ H^{n}(G_{(r)}, Λ) in H^{0}(X^{(r)}, L(H^{n}(B_{(r)}, Λ))) is nilpotent. One checks readily that the fiber of the vector bundle L(H^{n}(B_{(r)}, Λ)) at a rational point x^{(r)} = g^{(r)}B^{(r)} ∈ X^{(r)} (g ∈ G(k)) coincides with H^{n}((B^{g})_{(r)}, Λ). Furthermore the image of z ∈ H^{n}(G_{(r)}, Λ) under the composition

H^{n}(G_{(r)}, Λ) ⟶^{ρ} H^{0}(X^{(r)}, L(H^{n}(B_{(r)}, Λ))) → H^{n}((B^{g})_{(r)}, Λ)

coincides with z_{|(B^{g})_{(r)}}. Now the proof is easily concluded using Theorem 2.5 and induction on the dimension of the support of the section

ρ(z) ∈ H^{0}(X^{(r)}, L(H^{n}(B_{(r)}, Λ).

In [S-F-B], we introduced for any affine group scheme G/k an affine scheme V_{r}(G) of finite type over k, whose A-points for any commutative k-algebra A are in one to one correspondence with group scheme homomorphisms (over A) 𝔾_{a(r)} ⊗_{k} A → G ⊗_{k} A. Let s ∈ V_{r}(G) be a point. This point defines a canonical k(s)-point of V_{r}(G) and hence an associated group scheme homomorphism over k(s) which we denote

ν_{s} : 𝔾_{a(r)} ⊗_{k} k(s) → G ⊗_{k} k(s).

Note that if K/k is a field extension and ν : 𝔾_{a(r)} ⊗_{k} K → G ⊗_{k} K is a group scheme homomorphism, then this data defines a point s ∈ V_{r}(G) and a field embedding k(s) ↪ K such that ν is obtained from ν_{s} by extending scalars from k(s) to K.

Theorem 4.3. Let G be an infinitesimal group scheme of height ≤ r over k and let Λ be an associative unital rational G-algebra. Then the following conditions on a cohomology class z ∈ H^{n}(G, Λ) are equivalent:

(a.) z is nilpotent. (b.) For every field extension K/k and every group scheme homomorphism over K ν : 𝔾_{a(r)} ⊗_{k} K → G ⊗_{k} K, the cohomology class ν^{∗}(z_{K}) ∈ H^{n}(𝔾_{a(r)} ⊗_{k} K, Λ_{K}) is nilpotent. (c.) For every point s ∈ V_{r}(G), the cohomology class ν_{s}^{∗}(z_{k(s)}) ∈ H^{n}(𝔾_{a(r)} ⊗_{k} k(s), Λ_{k(s)}) is nilpotent.

Proof. The equivalence of conditions (b.) and (c.) is obvious from the discussion above. Clearly (a.) implies (b.). We now proceed to prove the converse.

Embed G in some G′ = GL_{n(r)} and set Λ′ = Ind_{G}^{G′}(Λ). Thus, Λ′ is an associative, unital rational G′-algebra. Moreover, H^{∗}(G, Λ) = H^{∗}(G′, Λ′) as graded algebras. By Theorem 4.1, it suffices to show that the element z′ ∈ H^{n}(G′, Λ′) which corresponds to z under the above identification satisfies the condition that ν′^{∗}(z′) ∈ H^{n}(𝔾_{a(r)} ⊗_{k} K, Λ′_{K}) is nilpotent for every field extension K/k and every group scheme homomorphism ν′ : 𝔾_{a(r)} ⊗_{k} K → G′ ⊗_{k} K. We consider the cartesian square

\begin{matrix}𝔾_{a(s)} ⊗_{k} K & ⟶^{ν} & G ⊗_{k} K \\ ↓ & & ↓ \\ 𝔾_{a(r)} ⊗_{k} K & ⟶^{ν′} & G′ ⊗_{k} K. \end{matrix}  (4.3.1)

where the upper left corner is of the indicated form (with s ≤ r) because it is a K-subgroup of 𝔾_{a(r)} ⊗_{k} K. For notational convenience, we re-write (4.3.1) as

H ⟶^{ν} G_{K}
↓  ↓
H′ ⟶^{ν′} G′_{K}.

Consider the homomorphism of rational H′-modules

θ : Λ′_{K} = Ind_{G_{K}}^{G′_{K}} (Λ_{K}) → Ind_{H}^{H′}(Λ_{K})

determined by the adjointness of Ind and Res. This homomorphism is surjective. To prove this consider the following commutative diagram of schemes

H′ ⟶^{ν′} G′_{K}
p_{H} ↓  ^{p_{G}}↓
H′/H ⟶^{\overline{ν′}} G′_{K}/G_{K}.

According to our definitions, Ind_{G_{K}}^{G′_{K}} (Λ_{K}) = Γ(G′_{K}/G_{K}, L(Λ_{K})) and Ind_{H}^{H′}(Λ_{K}) = Γ(H′/H, L(Λ_{K}))

(cf.  (3.3) and (3.4)).  Observe that (p_{H})^{∗}(\overline{ν′}^{∗}(L(Λ_{K})))  =

= ν′^{∗}(p_{G}^{∗}(L(Λ_{K}))) = ν′^{∗}({ \atop K[G′] ⊗_{K} Λ_{K}}) = { \atop K[H′] ⊗_{K} Λ_{K}} = (p_{H})^{∗}(L(Λ_{K})). This together with Proposition 3.2 shows that \overline{ν′}^{∗}(L(Λ_{K})) = L(Λ_{K}). Now, it suffices to observe that \overline{ν′} is a closed embedding.

We rewrite θ (as a map of K-algebras) as

θ : (K[G′] ⊗_{K} Λ_{K})^{G_{K}} = Ind_{G_{K}}^{G′_{K}} (Λ_{K}) → Ind_{H}^{H′}(Λ_{K}) = (K[H′] ⊗_{K} Λ_{K})^{H}.

Since Ker{K[G′] → K[H′]} is contained in the maximal ideal of the local artinian ring K[G′], it is nilpotent. From this we conclude immediately that I = Ker θ is also nilpotent. Now consider the following portion of the exact cohomology sequence

H^{n}(H′, I) → H^{n}(H′, Λ′_{K}) → H^{n}(H′, Ind_{H}^{H′}(Λ_{K})) = H^{n}(H, Λ_{K}).

To prove that ν′^{∗}(z′_{K}) ∈ H^{n}(H′, Λ′_{K}) is nilpotent, we first observe that its image in H^{n}(H, Λ_{K}) is nilpotent by our original hypothesis on z and Lemma 2.4, for this image equals ν^{∗}(z_{K}). Thus, replacing z by its appropriate power, we may assume that ν′^{∗}(z′_{K}) ∈ H^{n}(H′, I) . Since I is nilpotent, the ring without unit H^{∗}(H′, I) is also nilpotent and thus ν′^{∗}(z′_{K}) is nilpotent.

Since Ext_{G}^{∗}(M, M) = H^{∗}(G, End_{k}(M)) for a rational G-module M and and as End_{k}(M)_{K} = End_{K}(M_{K}) if M is finite dimensional, Theorem 4.3 has the following immediate corollary.

Corollary 4.4. Let G be an infinitesimal group scheme of height ≤ r over k and let M be a finite dimensional rational G-module. Then the following conditions on an extension class e ∈ Ext_{G}^{n}(M, M) are equivalent: a. e is nilpotent. b. For every field extension K/k and every group scheme homomorphism over K ν :

𝔾_{a(r)}⊗_{k} K → G⊗_{k} K, the extension class ν^{∗}(e_{K}) ∈ Ext_{𝔾_{a(r)}⊗_{k}K}^{n}(M_{K}, M_{K}) is nilpotent. c. The extension class ν_{s}^{∗}(e_{k(s)}) ∈ Ext_{𝔾_{a(r)}⊗_{k}k(s)}^{n}(M_{k(s)}, M_{k(s)}) is nilpotent, for every point

s ∈ V_{r}(G).

§5. Support schemes

In [S-F-B], we introduced and analyzed a natural ring homomorphism ψ : H^{ev}(G, k) → k[V_{r}(G)] and the associated morphism of schemes Ψ : V_{r}(G) → Spec H^{ev}(G, k). In the special case G = GL_{n(r)}, we identified the composition

V_{r}(GL_{n(r)}) ⟶^{Ψ} Spec H^{ev}(GL_{n(r)}, k) ⟶^{Φ} V_{r}(GL_{n(r)})

as the r-th Frobenius twist morphism (cf. [S-F-B;5.2] by identifying ψ applied to the universal classes e_{i}^{(r−i)} ∈ H^{2p^{r−i}}(GL_{n(r)}, gl_{n}^{(r)}) introduced in [F-S]. This result implies immediately that for any infinitesimal group scheme G of height ≤ r the morphism Ψ is finite and universally injective. As we shall see in this section, our detection theorem together with this result enables us to demonstrate that

Ψ : V_{r}(G) → |G| ≡ Spec H^{ev}(G, k)

is a (universal) homeomorphism.

Throughout this section, r will denote a fixed positive integer and G will denote an infinitesmal group scheme over k of height ≤ r.

The following proposition is a somewhat sharper version of Theorem 4.3 in case of the cohomology algebra with trivial coefficients.

Proposition 5.1. For any z ∈ H^{2n}(G, k), z is nilpotent whenever ψ(z) ∈ k[V_{r}(G)] is nilpotent.

Proof. Consider some z ∈ H^{2n}(G, k) with ψ(z) nilpotent. By Theorem 4.3, it suffices to prove for every field extension K/k and every group scheme homomorphism over K, ν : 𝔾_{a(r)} ⊗_{k} K → G ⊗_{k} K that ν^{∗}(z_{K}) ∈ H^{2n}(𝔾_{a(r)} ⊗_{k} K, K) is nilpotent. Furthermore, it suffices to restrict attention to fields K which are algebraically closed. Write

\overline{ν^{∗}(z_{K})} = ∑_{i_{1}+...+i_{r}=n} a_{i_{1},... ,i_{r}}x_{1}^{i_{1}} · ... · x_{r}^{i_{r}} ∈ K[x_{1}, ..., x_{r}] = H^{∗}(𝔾_{a(r)} ⊗_{k} K, K)_{red}.

According to definitions, (see [S-F-B]) the coefficient a_{0,...,0.n} coincides with ψ(z)(s), where s ∈ V_{r}(G) is the point defined by ν. Since ψ(z) is nilpotent, its value at each point is trivial and hence a_{0,... ,0,n} = 0.

Let γ\underline{c} : 𝔾_{a(r)} ⊗_{k} K → 𝔾_{a(r)} ⊗_{k} K denote the composition

𝔾_{a(r)} ⊗_{k} K ⟶^{Δ} (𝔾_{a(r)} ⊗_{k} K)^{×r} ⟶^{c_{1},c_{2}∘F,... ,c_{r}∘F^{r−1}} (𝔾_{a(r)} ⊗_{k} K)^{×r} ⟶^{m} 𝔾_{a(r)} ⊗_{k} K

where m denotes the group operation (addition) of 𝔾_{a(r)} ⊗_{k} K. We easily verify that

γ∗ ; \underline{c}(x_{i}) = c_{1}^{p^{i}}x_{i} + c_{2}^{p^{i}}x_{i+1} + ⋯ + c_{r−i+1}^{p^{i}}x_{r}

using (1.3.2), (1.3.3) and (1.4). Let ν′ = ν ∘ γ\underline{c} : 𝔾_{a(r)} ⊗_{k} K → G ⊗_{k} K. The coefficient of x_{r}^{n} in ν′^{∗}(z) ∈ H^{2n}(𝔾_{a(r)} ⊗_{k} K, K) equals

∑_{i_{1}+...+i_{r}=n} a_{i_{1},... ,i_{r}}(c_{r}^{p})^{i_{1}} · ... · (c_{1}^{p^{r}})^{i_{r}}

which equals 0 by our assumption on z. Since \underline{c} is an arbitrary r-tuple in the algebraically closed field K, we conclude that each a_{i_{1},... ,i_{r}} equals 0 and thus that \overline{ν^{∗}(z)} = 0 as required.

As shown in [S-F-B;5.2], the image of the homomorphism ψ : H^{ev}(G, k) → k[V_{r}(G)] contains the p^{r}-th power of each element of k[V_{r}(G)]. This, together with Proposition 5.1, immediately implies the following theorem.

Theorem 5.2. Let G be an infinitesimal group scheme of height ≤ r. Then the kernel of the canonical homomorphism

ψ : H^{ev}(G, k) → k[V_{r}(G)]

is nilpotent and its image contains the p^{r}-th power of each element of k[V_{r}(G)]. Consequently, the associated morphism of schemes

Ψ : V_{r}(G) → |G| = Spec H^{ev}(G, k)

is a finite universal homeomorphism.

Proposition 5.3. Let G be a closed subgroup of GL_{n(r)}. Then the homomorphism

Res ∘ ϕ : S^{∗}(⊕_{i=1}^{r}gl_{n}^{#(r)}[2p^{i−1}]) → H^{ev}(G, k)_{red}

induced by the universal cohomology classes e_{1}^{(r−1)}, …, e_{r} ∈ H^{∗}(GL_{n(r)}, gl_{n}^{(r)}) has image containing all p^{r}-th powers. Consequently the associated morphism of schemes

|G| → |GL_{n(r)}| ⟶^{Φ} V_{r}(GL_{n(r)}) ⊂ gl_{n}^{×r}

is finite and (universally) injective.

Proof. Recall that the closed embedding G ↪ GL_{n(r)} defines a closed embedding V_{r}(G) ↪ V_{r}(GL_{n}) (cf. [S-F-B;1.5]), so that the induced map on coordinate algebras k[V_{r}(GL_{n(r)})] → k[V_{r}(G)] is surjective. Consider the commutative diagram

k[V_{r}(GL_{n(r)})] ⟶^{\overline{ϕ}} H^{ev}(GL_{n(r)}, k) ⟶^{ψ} k[V_{r}(GL_{n(r)})]
=↓  ^{Res}↓  ↓
k[V_{r}(GL_{n(r)})] ⟶^{Res∘\overline{ϕ}} H^{ev}(G, k) ⟶^{ψ} k[V_{r}(G)].

For any cohomology class z ∈ H^{ev}(G, k), consider ψ(z) ∈ k[V_{r}(G)] and lift it to a (homogenous) element y′ ∈ k[V_{r}(GL_{n(r)})] = k ⊗_{𝔽_{p}} 𝔽_{p}[V_{r}(GL_{n(r)})]. Applying to k-coefficients of y′ the r-th power of the Frobenius endomorphism, we get an element y ∈ k[V_{r}(GL_{n(r)})] whose r-th Frobenius twist is y′^{p^{r}}. Commutativity of the above diagram and the fact (discussed above) that the composition of the homomorphisms of the top row is the r-th Frobenius twist immediately implies that the image of z^{p^{r}} − Res(\overline{ϕ}(y)) in k[V_{r}(G)] is trivial and hence (by Proposition 5.1) z^{p^{r}} − Res(\overline{ϕ}(y)) is nilpotent.

Since the map Res ∘ ϕ of Proposition 5.3 factors through H^{ev}(GL_{n(r)}, k), we conclude that the restriction map

H^{ev}(GL_{n(r)}, k)_{red} → H^{ev}(G, k)_{red}

has image containing all p^{r}-th powers. This immediately implies the following.

Corollary 5.4. Let G be an infinitesimal group scheme of height ≤ r and let H be a closed subgroup scheme of G. Then the homomorphism

H^{ev}(G, k)_{red} ⟶^{Res} H^{ev}(H, k)_{red}

has image containing all p^{r}-th powers. Consequently, for each z ∈ H^{ev}(H, k) there exists N ≥ 0 such that z^{p^{N}} may be lifted to H^{ev}(G, k).

Corollary 5.4.1. In conditions and notations of Corollary 5.4, the associated morphism of support schemes |H| → |G| is finite and universally injective.

Remark 5.5 As seen above, the morphism Φ : |GL_{n(r)}| → V_{r}(GL_{n(r)}) induced by \overline{ϕ} is a universal homeomorphism, the topological inverse to Ψ. Taken in conjunction with Theorem 5.2, this implies that Φ restricts to a topological homeomorphism from |G| to V_{r}(G)^{(r)}. In [S-F-B], we asked whether Φ(|G|) ⊂ V_{r}(GL_{n(r)}) is contained scheme-theoretically in V_{r}(G)^{(r)} and whether the morphism Φ_{G} : |G| → V_{r}(G)^{(r)} which would then be determined is independent of the embedding G ↪ GL_{n(r)}. We see that these questions have an affirmative answer if weakened to set-theoretic statements.

§6. Support varieties for modules

The purpose of this section is to give a non-cohomological, geometric interpretation of the cohomological support variety |G|_{M} of a finite dimesnional, rational G-module M for an infinitesimal group scheme G . Indeed, Theorem 6.7 presents the appropriate analogue of Proposition 5.1 with k replaced by M; this easily implies in Corollary 6.8 a homeomorphism between |G|_{M} and the non-cohomological V_{r}(G)_{M} ⊂ V_{r}(G). The key ingredient in our analysis is the detection result of Theorem 4.3 for H^{∗}(G, Λ), where Λ = End_{k}(M). This, however, is not sufficient, for we must carefully analyze the special case G = 𝔾_{a(r)}.

Throughout this section, G will denote an infinitesimal group scheme of height ≤ r and M a finite dimensional rational G-module. Moreover, Λ will denote End_{k}(M) = M ⊗M^{#}, a finite dimensional, associative, unital rational G-algebra.

We recall that the (cohomological) support variety of M (denoted |G|_{M}) is defined as the

Zariski closed subset of |G| = Spec H^{ev}(G, k) defined by the ideal I_{M} = Ker(H^{ev}(G, k) ⟶^{ρ_{Λ}} H^{ev}(G, Λ)). (In particular, |G|_{M} may be considered as a reduced k-scheme.)

We begin by introducing a closed subset V_{r}(G)_{M} ⊂ V_{r}(G) which we shall show maps homeomorphically onto |G|_{M}.

Proposition 6.1. Let G be an infinitesimal group scheme over k and M a finite dimensional rational G-module. Then

V_{r}(G)_{M} = {s ∈ V_{r}(G) : M ⊗_{k} k(s) is not projective over the subalgebra
k(s)[u_{r−1}]/u_{r−1}^{p} ⊂ k(s)[u_{0}, ..., u_{r−1}]/(u_{0}^{p}, ..., u_{r−1}^{p}) = k(s)[𝔾_{a(r)}]^{#}}

is a Zariski closed conical subset in V_{r}(G).

Proof. Let A denote the coordinate algebra k[V_{r}(G)] and consider the canonical morphism of group schemes over A 𝔾_{a(r)} ⊗_{k} A → G ⊗_{k} A. Using this homomorphism, we make M_{A} into a rational module over 𝔾_{a(r)} ⊗_{k} A, i.e. into a module over A[𝔾_{a(r)}]^{#} = A[u_{0}, ..., u_{r−1}]/(u_{0}^{p}, ..., u_{r−1}^{p}). Observe that M_{A} is evidently a free A-module. Using this fact, one checks easily that if M ⊗_{k} k(s) is projective (=free) over k(s)[u_{r−1}]/u_{r−1}^{p} ⊂ k(s)[𝔾_{a(r)}]^{#} for some point s ∈ V_{r}(G), then M ⊗_{k} A_{s} is a free A_{s}[u_{r−1}]/u_{r−1}^{p}-module. Here A_{s} denotes the localization of A at a prime ideal μ_{s} corresponding to s. Since the rank of M ⊗_{k} A is finite, this implies immediately the existence of a ∈ A \ μ_{s} such that M ⊗_{k} A_{a} is a free A_{a}[u_{r−1}]/u_{r−1}^{p}-module. Hence, for all points s′ of the principal open set Spec A_{a} ⊂ Spec A, the module M ⊗_{k} k(s′) is free over k(s′)[u_{r−1}]/u_{r−1}^{p}.

Let V_{r}(G)_{M} also denote the closed reduced subscheme of V_{r}(G) associated to this closed subset. Then field valued points Spec K → V_{r}(G)_{M} are in one to one correspondence with group scheme homomorphisms ν : 𝔾_{a(r)} ⊗_{k} K → G ⊗_{k} K for which M_{K} is not projective over K[u_{r−1}]/u_{r−1}^{p} ⊂ K[𝔾_{a(r)}]^{#}. To prove that V_{r}(G)_{M} is conical, it suffices to show that if s ∈ V_{r}(G)_{M}(K) is a field valued point and c ∈ K, then c · s ∈ V_{r}(G)_{M}(K). For c = 0, this is evident since the point 0 · s ∈ V_{r}(G)(K) corresponds to the trivial group scheme homomorphism 𝔾_{a(r)} ⊗_{k} K → G ⊗_{k} K. Assume next that c ≠ 0. The morphism

ν_{c·s} : 𝔾_{a(r)} ⊗_{k} K → G ⊗_{k} K is given by the composition 𝔾_{a(r)} ⊗_{k} K ⟶^{c} 𝔾_{a(r)} ⊗_{k} K ⟶^{ν_{s}} G.

One checks immediately that the homomorphism c_{∗} : K[𝔾_{a(r)}]^{#} → K[𝔾_{a(r)}]^{#} takes u_{i} to c^{p^{i}}u_{i}. Thus a K[u_{r−1}]/u_{r−1}^{p}-module structure on M, corresponding to the point c · s, is obtained from the module structure corresponding to s via the ring homomorphism

K[u_{r−1}]/u_{r−1}^{p} ⟶^{u_{r−1}↦c^{p^{r−1}} u_{r−1}} K[u_{r−1}]/u_{r−1}^{p}.

Since this ring homomorphism is clearly an isomorphism, the statement follows.

In the unipotent case, one can give a slightly different description of the support of a module using the following lemma.

Lemma 6.2. Let M be a finite dimensional rational module over an infinitesimal group scheme G. Denote by J_{M} the annihilator of the left H^{ev}(G, k)-module H^{∗}(G, M). Then a. I_{M} ⊂ J_{M}. b. If G is unipotent then \sqrt{I_{M}} = \sqrt{J_{M}}.

Proof. We have a natural left action of H^{ev}(G, Λ) on H^{∗}(G, M) and the action of H^{ev}(G, k) on H^{∗}(G, M) is obtained from this one via the homomorphism ρ_{Λ} : H^{ev}(G, k) → H^{ev}(G, Λ). This makes the first statement obvious.

Assume now that G is unipotent. In this case M^{#} admits a filtration of length d = dim_{k}M with trivial one-dimensional factors. Assuming now that z ∈ H^{2n}(G, k) annihilates H^{∗}(G, M) we conclude easily that z^{d} annihilates H^{∗}(G, M ⊗ M^{#}) = H^{∗}(G, Λ), i.e. z^{d} ∈ I_{M}.

The following proposition is a special case (namely, G is assumed to have height 1) of our main Theorem 6.7.

Proposition 6.3. Let G be an infinitesimal group scheme of height 1 and let M be a finite dimensional rational G-module. The following conditions on a cohomology class z ∈ H^{2n}(G, k) are equivalent: (a.) z ∈ \sqrt{I_{M}} (b.) The function ψ(z) ∈ k[V_{1}(G)] takes zero value at all points s ∈ V_{1}(G)_{M}.

Proof. The statement is trivial for zero dimensional classes, so we assume that n > 0. By definition z ∈ \sqrt{I_{M}} if and only if ρ(z) ∈ H^{2n}(G, Λ) is nilpotent. By Theorem 4.3, ρ(z) is nilpotent if and only if ν_{s}^{∗}(ρ(z)_{k(s)}) = ρ(ν_{s}^{∗}(z_{k(s)})) ∈ H^{2n}(𝔾_{a(1)} ⊗_{k} k(s), Λ_{k(s)}) is nilpotent for every point s ∈ V_{1}(G). Lemma 6.2 shows further that this last condition is equivalent to the condition that for every s ∈ V_{1}(G) a sufficiently high power of ν_{s}^{∗}(z_{k(s)}) annihilates H^{∗}(𝔾_{a(1)} ⊗_{k} k(s), M_{k(s)}). By definition, ν_{s}^{∗}(z_{k(s)}) = ψ(z)(s) · x_{1}^{n}. Thus the previous condition means that either ψ(z)(s) = 0 or a sufficiently high power of x_{1} ∈ H^{2}(𝔾_{a(1)} ⊗_{k} k(s), k(s)) annihilates H^{∗}(𝔾_{a(1)} ⊗_{k} k(s), M_{k(s)}). Since multiplication by x_{1} defines periodicity in the cohomology of any rational 𝔾_{a(1)} ⊗_{k} k(s)-module, either ψ(z)(s) = 0 or H^{i}(𝔾_{a(1)} ⊗_{k} k(s), M_{k(s)}) = 0 for i ≥ 2 so that M_{k(s)} is a projective k(s)[𝔾_{a(1)}]^{#} = k(s)[u_{1}]/u_{1}^{p}-module.

Proposition 6.3 admits the following formulation in terms of support varieties.

Corollary 6.3.1. Let M be a finite dimensional rational module over an infinitesimal group scheme G of height one. Then

Ψ^{−1}(|G|_{M}) = V_{1}(G)_{M}.

Proof. Let k[V_{1}(G)_{M}] denote the coordinate algebra of the reduced closed subscheme of V_{1}(G) associated to the closed subset V_{1}(G)_{M} ⊂ V_{1}(G) and let I_{M} denote the kernel of the surjective homomorphism k[V_{1}(G)] → k[V_{1}(G)_{M}]. Proposition 6.3 asserts that ψ^{−1}(I_{M}) = \sqrt{I_{M}}. Since Ψ : V_{1}(G) → |G| is the map induced by ψ, we conclude that Ψ^{−1}(|G|_{M}) = V_{1}(G)_{M}.

To compute the support varieties for modules over 𝔾_{a(r)} we need the following lemma. A very similar result has been proved by J. Carlson in the context of elementary abelian groups (cf. [C1;6.4]).

Lemma 6.4. Let M be a finite dimensional vector space over k equipped with an r-tuple of commuting p-nilpotent endomorphisms α_{1}, ..., α_{r}. Furthermore, let f ∈ k[X_{1}, ..., X_{r}] be a polynomial without constant and linear terms. Set α^{∼}_{1} = α_{1} + f(α_{1}, ..., α_{r}). If M is a projective k[u]/u^{p}-module with the action of u being given by α_{1}, then it’s also projective over k[u]/u^{p} with the action of u being given by α^{∼}_{1} (and vice versa).

Proof. The general case is immediately reduced to the case α^{∼}_{1} = α_{1} + α_{2}α_{3} and if p ≠ 2 is reduced further to the case α^{∼}_{1} = α_{1} + α_{2}^{2}. We shall assume that p ≠ 2, the case p = 2 may be treated similarly. We use the notation H^{i}(α_{1}, M) for the cohomology of k[u]/u^{p} with coefficients in M, where u acts on M via α_{1}. To verify that M is projective over k[α_{1} + α_{2}^{2}] it suffices to check that Ker(α_{1} + α_{2}^{2})/Im(α_{1} + α_{2}^{2})^{p−1} = H^{2}(α_{1} + α_{2}^{2}, M) is a trivial vector space. Furthermore, since the action of α_{2} on H^{2}(α_{1} + α_{2}^{2}, M) is nilpotent it suffices to show that the homomorphism

α_{2} : H^{2}(α_{1} + α_{2}^{2}, M) → H^{2}(α_{1} + α_{2}^{2}, M)

is injective. So let x ∈ Ker(α_{1} + α_{2}^{2}) be an element such that α_{2}(x) = (α_{1} + α_{2}^{2})^{p−1}(y) for some y ∈ M. Note the formula

(X + Y)^{p−1} = \frac{X^{p} + Y^{p}}{X + Y} = X^{p−1} − X^{p−2}Y + ... − XY^{p−2} + Y^{p−1}.  (6.4.1)

Using this formula, we get

(α_{1} + α_{2}^{2})^{p−1} = ∑_{i=[ \frac{p}{2} ]}^{p−1}(−1)^{i}α_{1}^{i}α_{2}^{2(p−1−i)}

and further

α_{1}(x) = −α_{2}^{2}(x) = −α_{2}( ∑_{i=[ \frac{p}{2} ]}^{p−1}(−1)^{i}α_{1}^{i}α_{2}^{2(p−1−i)}y) =

= α_{1}( ∑_{i=[ \frac{p}{2} ]}^{p−1}(−1)^{i−1}α_{1}^{i−1}α_{2}^{2(p−1−i)+1}y).

Thus x − ∑_{i=[ \frac{p}{2} ]}^{p−1}(−1)^{i−1}α_{1}^{i−1}α_{2}^{2(p−1−i)+1}y ∈ Ker α_{1} = Im α_{1}^{p−1} and hence

x = ∑_{i=[ \frac{p}{2} ]}^{p−1}(−1)^{i−1}α_{1}^{i−1}α_{2}^{2(p−1−i)+1}y + α_{1}^{p−1}z

for some z ∈ M. Applying α_{2} to this formula and taking into account that α_{2}(x) = (α_{1} + α_{2}^{2})^{p−1}(y), we conclude

∑_{i=[ \frac{p}{2} ]}^{p−1}(−1)^{i}α_{1}^{i}α_{2}^{2(p−1−i)}y = ∑_{i=[ \frac{p}{2} ]−1}^{p−2}(−1)^{i}α_{1}^{i}α_{2}^{2(p−1−i)}y + α_{2}α_{1}^{p−1}z.

The term corresponding to i = [\frac{p}{2}]−1 on the right vanishes and the above formula simplifies to α_{1}^{p−1}y = α_{1}^{p−1}α_{2}z. Hence y −α_{2}(z) ∈ Ker α_{1}^{p−1} = Im α_{1}; in other words, y = α_{2}z +α_{1}t for some t ∈ M. Now, one checks easily that x = (α_{1} + α_{2}^{2})^{p−1}(z − α_{2}t) and hence \overline{x} = 0 ∈ H^{2}(α_{1} + α_{2}^{2}, M).

Proposition 6.5. Let M be a finite dimensional rational 𝔾_{a(r)}-module. Then

Ψ^{−1}(|𝔾_{a(r)}|_{M}) = V_{r}(𝔾_{a(r)})_{M}.

Proof. Since k[𝔾_{a(r)}]^{#} = k[u_{0}, ..., u_{r−1}]/(u_{0}^{p}, ..., u_{r−1}^{p}) = k[𝔾_{a(1)}^{×r}]^{#}, the category of rational 𝔾_{a(r)}-modules is equivalent to the category of rational 𝔾_{a(1)}^{×r}-modules. For any

rational 𝔾_{a(r)}-module M, we shall denote by M^{∼} the same module considered as a rational 𝔾_{a(1)}^{×r}-module. Note that for any M we have a natural isomorphism H^{∗}(𝔾_{a(r)}, M) =

H^{∗}(𝔾_{a(1)}^{×r}, M^{∼}). Moreover the isomorphism H^{∗}(𝔾_{a(r)}, k) = H^{∗}(𝔾_{a(1)}^{×r}, k) is an isomorphism

of k-algebras and the isomorphism H^{∗}(𝔾_{a(r)}, M) = H^{∗}(𝔾_{a(1)}^{×r}, M^{∼}) is compatible with the action of H^{∗}(𝔾_{a(r)}, k) = H^{∗}(𝔾_{a(1)}^{×r}, k) on both sides. Thus we get a natural isomorphism of support schemes |𝔾_{a(r)}| →^{∼} |𝔾_{a(1)}^{×r}| and this isomorphism takes |𝔾_{a(r)}|_{M} onto |𝔾_{a(1)}^{×r}|_{M^{∼}} for each M.

In [S-F-B;§1], we exhibited an identification of the scheme V_{r}(𝔾_{a(r)}) with the affine space 𝔸^{r}. This identification takes a point with coordinates (a_{0}, ..., a_{r−1}) to the endomorphism ν_{a} of 𝔾_{a(r)} given by the additive polynomial a_{0}T + a_{1}T^{p} + ... + a_{r−1}T^{p^{r−1}}. With this identification,

ν_{a}^{∗}(x_{i}) = a_{0}x_{i} + … + a_{r−1}^{p^{r−i}}x_{r} ∈ H^{2}(𝔾_{a(r)}, k).

We can also identify 𝔸^{r} with V_{1}(𝔾_{a(1)}^{×r}), this time identifying a point (b_{0}, ..., b_{r−1}) with a morphism 𝔾_{a(1)} → 𝔾_{a(1)}^{×r} (t ↦ (b_{0}t, ..., b_{r−1}t)). The map h : V_{r}(𝔾_{a(r)}) → V_{1}(𝔾_{a(1)}^{×r}) making the following square commute

𝔸^{r} = V_{r}(𝔾_{a(r)}) ⟶^{Ψ_{𝔾_{a(r)}}} |𝔾_{a(r)}|
h↓  =^{∼}↓

𝔸^{r} = V_{1}(𝔾_{a(1)}^{×r}) ⟶^{Ψ_{𝔾{×r \atop a(1)}}} |𝔾_{a(1)}^{×r}|

is given by the formula h(a_{0}, …, a_{r−1}) = (a_{r−1}, ..., a_{1}^{p^{r−2}}, a_{0}^{p^{r−1}}), where |𝔾_{a(r)}| =^{∼} |𝔾_{a(1)}^{×r}|

is the isomorphism described above. The discussion above and Corollary 6.3.1 show that

Ψ_{𝔾_{a(r)}}^{−1}(|𝔾_{a(r)}|_{M}) = h^{−1}(V_{1}(𝔾_{a(1)}^{×r})_{M^{∼}}) =

= {(a_{0}, ..., a_{r−1}) : M is not projective over k[u]/u^{p}, with the action of u on M

being given by the formula u · m = (a_{r−1}u_{0} + a_{r−2}^{p}u_{1} + ... + a_{0}^{p^{r−1}}u_{r−1}) · m}.

We want to show that the set we get on the right of the above formula equals V_{r}(𝔾_{a(r)})_{M}. Consider the action of k[u]/u^{p} = k[u_{r−1}]/u_{r−1}^{p} on M, corresponding to the morphism ν_{a} : 𝔾_{a(r)} → 𝔾_{a(r)}. A straightforward computation shows that

(ν_{a})_{∗}(u_{r−1}) = ∑_{i=0}^{p^{r}−1} ∑ ; \begin{matrix}i_{0} + i_{1} + .. & . + i_{r−1} = i \\ i_{0} + pi_{1} + .. & . + p^{r−1}i_{r−1} = p^{r−1} \end{matrix}(i_{0}, ..., i_{r−1})(a_{0}^{i_{0}} · ... · a_{r−1}^{i_{r−1}}) · v_{i} (6.5.1)

where (i_{0}, …, i_{r−1}) = \frac{(i_{1}+···+i_{r−1})!}{i_{0}!···i_{r−1}!} and v_{0}, ..., v_{p^{r}−1} is the standard basis of k[𝔾_{a(r)}]^{#}. We proved in [S-F-B] the following formula for v_{i}:

v_{i} = \frac{u_{0}^{i^{(0)}} · ... · u_{r−1}^{i^{(r−1)}}}{i^{(0)}! · ... · i^{(r−1)}!}  (6.5.2)

where i = i^{(0)} + i^{(1)}p + ... + i^{(r−1)}p^{r−1} (0 ≤ i^{(j)} ≤ p − 1) is the p-adic expansion of i. This formula shows that each v_{i} is a monomial in u_{0}, ..., u_{r−1} of degree s(i) = sum of digits of i. It is clear that the coefficient at v_{0} = 1 in (6.5.1) is equal to zero. Next we look at coefficients at u_{0} = v_{1}, u_{1} = v_{p}, ..., u_{r−1} = v_{p^{r−1}}. Assume that i_{0} + ... + i_{r−1} = p^{l}, i_{0} + pi_{1} + ... + p^{r−1}i_{r−1} = p^{r−1}. Consider the p-adic expansions of i_{0}, ..., i_{r−1}:

i_{0} = i_{0}^{(0)} + i_{0}^{(1)}p + ... + i_{0}^{(r−1)}p^{r−1}, …, i_{r−1} = i_{r−1}^{(0)} + i_{r−1}^{(1)}p + ... + i_{r−1}^{(r−1)}p^{r−1}.

One checks easily that (i_{0}, ..., i_{r−1}) ≢ 0 mod p ⟺ i_{0}^{(j)} + ... + i_{r−1}^{(j)} ≤ p − 1 for all 0 ≤ j ≤ r − 1. Thus, non-zero terms in the coefficient at u_{l} correspond to those i_{0}, ..., i_{r−1} for which

p^{l} = i_{0} + ... + i_{r−1} = (i_{0}^{(0)} + ... + i_{r−1}^{(0)}) + ... + (i_{0}^{(r−1)} + ... + i_{r−1}^{(r−1)})p^{r−1}

is the p-adic expansion of p^{l}. This implies readily that all i_{j} exept one has to be zero and the only non zero one equals p^{l}. Looking finally at the equation i_{0} +i_{1}p+...+i_{r−1}p^{r−1} = p^{r−1} we conclude that i_{r−l−1} = p^{l}, i_{j} = 0 (j ≠ r − l − 1). All in all the coefficient at u_{l} equals a_{r−l−1}^{p^{l}}. Thus, the linear term in (6.5.1) equals

a_{r−1}u_{0} + a_{r−2}^{p}u_{1} + ... + a_{0}^{p^{r−1}}u_{r−1}.

It is easy to check that for r = 2 there are no nonlinear terms in (6.5.1). However for r ≥ 3 nonlinear terms really appear in (6.5.1). For example for r = 3 the formula looks as follows:

(ν_{a})_{∗}(u_{2}) = a_{2}u_{0} + a_{1}^{p}u_{1} + a_{0}^{p^{2}}u_{2} + ∑_{i=1}^{p−1} a_{0}^{p(p−i)}a_{1}^{i} · u_{0}^{i}u_{1}^{p−i}/(i!(p − i)!).

Nevertheless the proposition follows using Lemma 6.4.

The following lemma follows immediately from the definitions of |G|_{M} and V_{r}(G)_{M}.

Lemma 6.6. Let f : H → G be a homomorphism of infinitesimal group schemes of height ≤ r and let M be a finite dimensional, rational G-module (considered also as a rational H-module via f). Denote by f_{∗} : V_{r}(H) → V_{r}(G) and f_{∗} : |H| → |G| the associated morphisms of schemes. Then

|H|_{M} ⊂ f_{∗}^{−1}|G|_{M}, V_{r}(H)_{M} = f_{∗}^{−1}(V_{r}(G)_{M}).

Theorem 6.7. Let G be an infinitesimal group scheme of height ≤ r and let M be a finite dimensional rational G-module. Let further z ∈ H^{2n}(G, k) be a cohomology class. The following conditions are equivalent:

a. z ∈ \sqrt{I_{M}}. b. The function ψ(z) ∈ k[V_{r}(G)] takes zero value at all points s ∈ V_{r}(G)_{M}.

Proof. Assume first that z ∈ \sqrt{I_{M}} and let s be any point of V_{r}(G)_{M}. As usual we’ll use the notation ν_{s} for the group scheme homomorphism 𝔾_{a(r)} ⊗_{k} k(s) → G ⊗_{k} k(s) corresponding to s. The cohomology class ν_{s}^{∗}(z_{k(s)}) obviously belongs to \sqrt{I_{M,s}}, where I_{M,s} = Ker{H^{∗}(𝔾_{a(r)} ⊗_{k} k(s), k(s)) → H^{∗}(𝔾_{a(r)} ⊗_{k} k(s), Λ_{k(s)})}. Proposition 6.5 implies now that the function ν_{s}^{∗}(ψ(z)_{k(s)}) = ψ(ν_{s}^{∗}(z_{k(s)})) ∈ k(s)[V_{r}(G)] takes zero value at all points of V_{r}(𝔾_{a(r)} ⊗_{k} k(s))_{M_{k(s)}} and, in particular, at the unit point 1 ∈ V_{r}(𝔾_{a(r)} ⊗_{k} k(s)) of the monoid scheme V_{r}(𝔾_{a(r)} ⊗_{k} k(s)). However ν_{s}^{∗}(ψ(z)_{k(s)})(1) = ψ(z)(s) ∈ k(s).

Assume now that ψ(z) takes zero value at all points of V_{r}(G)_{M}. Let s be any point of V_{r}(G). Using Lemma 6.6 we conclude immediately that the function ψ(ν_{s}^{∗}(z_{k(s)})) = ν_{s}^{∗}(ψ(z)_{k(s)}) vanishes at all points of V_{r}(𝔾_{a(r)} ⊗_{k} k(s))_{M_{k(s)}}. Proposition 6.5 implies that ν_{s}^{∗}(z_{k(s)}) ∈ \sqrt{I_{M,s}}. In other words, the cohomology class ν_{s}^{∗}(ρ(z)_{k(s)}) ∈ H^{2n}(𝔾_{a(r)} ⊗_{k} k(s), Λ_{k(s)}) is nilpotent. Since this holds for all points s ∈ V_{r}(G) we conclude from Theorem 4.3 that ρ(z) is nilpotent, i.e. z ∈ \sqrt{I_{M}}.

The following corollary follows from Theorem 6.7 exactly as Corollary 6.3.1 follows from Proposition 6.3.

Corollary 6.8. Assuming the conditions and notations of Theorem 6.7,

Ψ^{−1}(|G|_{M}) = V_{r}(G)_{M}.

In other words, Ψ induces a homeomorphism Ψ : V_{r}(G)_{M} ⟶^{∼} |G|_{M}.

Example 6.9. Let G be an infinitesimal algebraic group scheme and let M be a finite dimensional rational G-module. If M has dimension not divisible by p, then |G|_{M} = |G|. This can be seen immediately using Corollary 6.8 and the observation that a necessary condition for a finite dimesional module over k[u]/u^{p} to be projective is that the dimension of the module must be divisible by p.

We conclude this section by considering in some detail the example of induced and irreducible modules for SL_{2(r)}. Let

H^{0}(λ) ≡ Ind_{B_{2}}^{SL_{2}}(k_{λ}) = (k[SL_{2}] ⊗_{k} k_{λ})^{B_{2}}

denote the induced module, where B_{2} ⊂ SL_{2} is the subgroup of lower triangualr matrices, k_{λ} is the 1-dimensional B_{2}-module with character the dominant weight λ (with respect to the diagonal matrices of determinant 1) and let L(λ) ⊂ H^{0}(λ) denote the simple socle. Recall that for SL_{2} the dominant weights are of the form λ = nρ for n ∈ ℕ and ρ = \frac{1}{2}α, where α is the lone simple root.

Proposition 6.10. Let N denote the p-nilpotent matrices in sl_{2} ≡ Lie(SL_{2}). Using Theorem 5.2, identify |SL_{2(r)}| with V_{r}(SL_{2(r)}) which can be identified with {(α_{0}, …, α_{r−1}) ∈ N^{r} : [α_{i}, α_{j}] = 0 ∀ i, j} (cf. [S-F-B;1.7]). Let λ ∈ X(T) be dominant and decompose λ as λ_{0} + pλ_{1} + ⋯ + p^{q}λ_{q} with λ_{i} = n_{i}ρ where 0 ≤ n_{i} < p. Then, for all r ≥ 1, a. |SL_{2(r)}|_{L(λ)} = {(α_{0}, …, α_{r−1}) ∈ N^{r} : [α_{i}, α_{j}] = 0 ∀ i, j and α_{r−i−1} = 0 if n_{i} = p − 1}

and b. |SL_{2(r)}|_{H^{0}(λ)} = {(α_{0}, α_{1}, …, α_{r−s−1}, 0, …, 0) ∈ N^{r} : [α_{i}, α_{j}] = 0 ∀ i, j and s ≥

0 is the least integer such that n_{s} ≠ p − 1}.

To verify Proposition 6.10, we require the following sublemma.

Sublemma 6.11. Let St_{s} denote the Steinberg module H^{0}((p^{s} − 1)ρ) ≃ S^{p^{s}−1}(V) for each s ≥ 1 (where V is the natural two-dimensional module on which SL_{2} acts). a. For all 1 ≤ s ≤ r, |SL_{2(r)}|_{St_{s}} = {(α_{0}, …, α_{r−s−1}, 0, …, 0) ∈ N^{r} : [α_{i}, α_{j}] = 0}. b. For all 0 ≤ i ≤ r − 1, |SL_{2(r)}|_{St_{1}^{(i)}} = {(α_{0}, …, α_{r−i−2}, 0, α_{r−i}, …, α_{r−1}) ∈ N^{r} :

[α_{i}, α_{j}] = 0}.

Proof. (of the sublemma) Let M be an arbitrary SL_{2}-module and consider the embedding of 𝔾_{a} into SL_{2} as the sugroup of upper triangular 2 × 2 matrices. The action of SL_{2} on H^{∗}(SL_{2(r)}, k) induces an action of SL_{2} on |SL_{2(r)}| such that (via the action on H^{∗}(G, M)) |SL_{2(r)}|_{M} ⊂ |SL_{2(r)}| is closed under this action. After the identification of |SL_{2(r)}| with V_{r}(SL_{2(r)}), one observes that the action of g ∈ SL_{2} is simply given by componentwise conjugation on a tuple \underline{α} ∈ |SL_{2(r)}|. Hence, \underline{α} ∈ |SL_{2(r)}|_{M} if and only if g · \underline{α} ∈ |SL_{2(r)}|_{M} for any g ∈ SL_{2}. Given \underline{α} ∈ |SL_{2(r)}|, there exists (by the Kolchin Fixed Point Theorem) g ∈ SL_{2} such that g·\underline{α} ∈ |𝔾_{a(r)}| ⊂ |SL_{2(r)}| (i.e. each component of g·\underline{α} is upper triangular) and so it suffices to compute |𝔾_{a(r)}|_{M}.

Let M = St_{s} and \underline{α} = (a_{0}, …, a_{r−1}) ∈ |𝔾_{a(r)}| ≃ 𝔸^{r}. As in the proof of Proposition 6.5, to determine whether \underline{α} is in |𝔾_{a(r)}|_{M} we must consider the action of u\underline{α} = a_{r−1}u_{0} + ⋯ +

a_{0}^{p^{r−1}}u_{r−1} on St_{s}. By direct calculation, one sees that u_{s}, u_{s+1}, …, u_{r−1} all act trivially on St_{s}, so that we are reduced to the action of a_{r−1}u_{0} + ⋯ + a_{r−s}^{p^{s−1}}u_{s−1}. On the other hand, St_{s} is projective over SL_{2(s)}, and so St_{s} ⊗_{k} k(\underline{α}) is projective over k(\underline{α})[u\underline{α}]/up ; \underline{α} if and only if at least one of a_{r−s}, …, a_{r−1} is nonzero. In other words, |𝔾_{a(r)}|_{St_{s}} = {(a_{0}, …, a_{r−s−1}, 0, …, 0) ∈ 𝔸^{r}}, from which (6.11.a) follows. The proof of (6.11.b) is analogous, once one observes that only a_{r−i−1}^{p^{i}}u_{i} acts nontrivially on St_{1}^{(i)}.

Proof. (of Proposition 6.10) By Steinberg’s twisted tensor product theorem, L(λ) ≃ L(λ_{0})⊗ L(λ_{1})^{(1)} ⊗ ⋯ ⊗ L(λ_{q})^{(q)}, and so by Proposition 7.2,

|SL_{2(r)}|_{L(λ)} = ⋂_{i=1}^{q} |SL_{2(r)}|_{L(λ_{i})^{(i)}}.

If n_{i} ≠ p−1, dimL(λ_{i})^{(i)} = n_{i}+1 < p and hence |SL_{2(r)}|_{L(λ_{i})^{(i)}} = |SL_{2(r)}| by Example 6.9. On the other hand, if n_{i} = p − 1, then L(λ_{i})^{(i)} = St_{1}^{(i)}. Applying (6.11.b) and combining the two cases, we conclude (6.10.a).

Note that if λ_{0} ≠ p−1, then the dimension of H^{0}(λ) (= n+1 for λ = nρ) is not divisible by p. Let s be the least integer such that n_{s} ≠ p − 1. Then

λ = (p^{s} − 1)ρ + p^{s}n_{s}ρ + ⋯ + p^{q}n_{q}ρ ≡ (p^{s} − 1)ρ + p^{s}λ′

and so by [J;II.3.19], H^{0}(λ) = H^{0}((p^{s} − 1)ρ + (λ′)^{(s)}) ≃ St_{s} ⊗ H^{0}(λ′)^{(s)}. By assumption on s, H^{0}(λ′)^{(s)} has dimension not divisible by p and so applying Example 6.9 to H^{0}(λ′)^{(s)}, (6.11.a) to St_{s}, and Proposition 7.2, we conclude (6.11.b).

§7. Applications

In this section, we provide various applications of Theorems 5.2 and 6.7 closely related to analogous results for finite groups proved by J. Carlson (cf. [C2]). These applications are based upon the fact that our theorems provide a non-cohomological, geometric description of cohomological support varieties.

Proposition 7.1. Let f : H → G be a homomorphism of infinitesimal group schemes and let M be a finite dimensional rational G-module (considered also as a rational H-module via f). Then

f_{∗}^{−1}(|G|_{M}) = |H|_{M}.

Proof. Assume that both H and G are of height ≤ r. The statement follows from the following commutative square, whose vertical arrows are homeomorphisms

V_{r}(H) ⟶^{f_{∗}} V_{r}(G)
Ψ_{G}↓  ^{Ψ_{H}} ↓
|H| ⟶^{f_{∗}} |G|

and Lemma 6.6.

Theorem 7.2. Let G be an infinitesimal group scheme, and let M, N be finite dimensional rational G-modules. Then

|G|_{M⊗N} = |G|_{M} ∩ |G|_{N}.

Proof. Using the Kunneth Theorem, one sees easily that if G and H are infinitesimal group schemes then the canonical morphism ((pr_{1})_{∗}, (pr_{2})_{∗}) : |G × H| → |G| × |H| is a finite universal homeomorphism. Moreover if M (resp. N) is a rational G-module (resp. H-module), then

|G × H|_{M ˜⊗N} = ((pr_{1})_{∗}, (pr_{2})_{∗})^{−1}(|G|_{M} × |H|_{N})

where M ˜⊗N = pr_{1}^{∗}M ⊗ pr_{2}^{∗}N is the external tensor product. Applying this remark in the case G = H and applying Proposition 7.1 to Δ : G → G × G, we conclude

|G|_{M⊗N} = Δ_{∗}^{−1}(|G × G|_{M ˜⊗N}) = Δ_{∗}^{−1}((pr_{1})_{∗}^{−1}(|G|_{M}) ∩ (pr_{2})_{∗}^{−1}(|G|_{N})) =
= |G|_{M} ∩ |G|_{N}.

For our next application, we require the following two lemmas.

Lemma 7.3. Let G be an infinitesimal group scheme of height ≤ r and let ν : 𝔾_{a(r)} → G be a non-trivial group scheme homomorphism. Consider the associated homomorphism of k-algebras

k[u_{r−1}]/u_{r−1}^{p} ↪ k[u_{0}, ..., u_{r−1}]/(u_{0}^{p}, ..., u_{r−1}^{p}) = k[𝔾_{a(r)}]^{#} ⟶^{ν_{∗}} k[G]^{#}.

This homomorphism makes k[G]^{#} into a projective (left) k[u_{r−1}]/u_{r−1}^{p}-module.

Proof. The homomorphism ν may be decomposed as

𝔾_{a(r)} ⟶^{p} 𝔾_{a(s)} ↪ G

where 1 ≤ s ≤ r, the first arrow is the standard projection and the second arrow is the closed embedding. Since the quotient scheme G/𝔾_{a(s)} is affine we conclude (cf. [J]) that k[G] is an injective rational 𝔾_{a(s)}-module. The functor (−)^{#} sending a k-vector space to its k-linear dual defines an anti-equivalence between the category of finite dimensional rational 𝔾_{a(s)}-modules and the category of finite dimensional k[𝔾_{a(s)}]^{#}-modules, which shows that k[G]^{#} is a projective k[𝔾_{a(s)}]^{#}-module. Consider finally the composition

k[u_{r−1}]/u_{r−1}^{p} ↪ k[𝔾_{a(r)}]^{#} ⟶^{p_{∗}} k[𝔾_{a(s)}]^{#}.

Since p_{∗}(u_{r−1}) = u_{s−1} we conclude that k[𝔾_{a(s)}]^{#} = k[u_{0}, ..., u_{s−1}]/(u_{0}^{p}, ..., u_{s−1}^{p}) is a projective k[u_{r−1}]/u_{r−1}^{p}-module, which concludes the proof.

The following lemma is closely related to lemmas proven by J. Carlson in [C2].

Lemma 7.4. Let x ∈ H^{2n}(k[u]/u^{p}, k) be a cohomology class. Consider a k[u]/u^{p}- projective resolution

0 ⟵ k ⟵ P_{0} ⟵^{d} P_{1} ⟵ …

of k and choose a representing cocycle f : P_{2n} → k for x. Finally set \overline{P}_{2n} = P_{2n}/d(P_{2n+1}), M = Ker\{f : \overline{P}_{2n} → k\}. The following conditions are equivalent: (a.) x ≠ 0 (b.) M is a projective k[u]/u^{p}-module.

Proof. Observe that we have a long exact sequence with all terms except the first and the last one projective:

0 ⟵ k ⟵ P_{0} ⟵^{d} … ⟵^{d} P_{2n−1} ⟵^{\overline{d}} \overline{P}_{2n} ⟵ 0.

Applying the same construction to the standard periodic resolution Q_{•} of k, we get another long exact sequence of the same kind

0 ⟵ k ⟵ Q_{0} ⟵ … ⟵ Q_{2n−1} ⟵ \overline{Q}_{2n} = k ⟵ 0.

By Schanuel’s Lemma (cf. [B]),

\overline{P}_{2n} ⊕ ⨁_{i=0}^{n−1} P_{2i} ⊕ ⨁_{i=0}^{n−1} Q_{2i+1} =^{∼} \overline{Q}_{2n} ⊕ ⨁_{i=0}^{n−1} Q_{2i} ⊕ ⨁_{i=0}^{n−1} P_{2i+1}.

Using now the Krull-Schmidt Theorem, we conclude that \overline{P}_{2n} is of the form

\overline{P}_{2n} = k ⊕ P

where P is a projective (=free) k[u]/u^{p}-module. If f_{|k} ≠ 0, then f : \overline{P}_{2n} → k splits and hence M =^{∼} P is projective. On the other hand, if f_{|k} = 0, then the trivial k[u]/u^{p}-module k is a direct summand in M = Ker\{\overline{P}_{2n} → k\} and hence M is not projective. Thus, condition (b.) is equivalent to the condition f_{|k} ≠ 0 (which we designate as condition (c.) ). Observe that each k[u]/u^{p}-linear homomorphism P_{2n−1} → k restricts trivially to the maximal trivial submodule Ker{u : P_{2n−1} → P_{2n−1}} of P_{2n−1}. Hence if f_{|k} ≠ 0 then f can not be extended to a k[u]/u^{p}-linear homomorphism P_{2n−1} → k, i.e. x ≠ 0. Assume, on the other hand, that f_{|k} = 0. Since every projective k[u]/u^{p}-module is also injective, a projective submodule P ⊂ \overline{P}_{2n} ⊂ P_{2n−1} is a direct summand and hence we may extend f_{|P} to a k[u]/u^{p}-linear homomorphism g : P_{2n−1} → k. Since f_{|k} = 0 = g_{|k} we conclude that f = g_{|\overline{P}_{2n}} and hence x = 0. Thus the condition (a.) is also equivalent to (c.).

Theorem 7.5. Let G be an infinitesimal group scheme of height ≤ r. Then W ⊂ |G| is the support |G|_{M} of some finite dimensional rational G-module M if and only if W is a closed, conical subset of |G|.

Proof. The necessity of the conditions that W be closed and conical is obvious (since the ideal I_{M} is homogenous).

To prove the converse, it suffices in view of Theorem 7.2 to show that the zero locus Z(x) ⊂ |G| of any cohomology class x ∈ H^{2n}(G, k) may be realized in the form |G|_{M} for an appropriate M. We proceed as in [C2]. Let

0 ← k ⟵ P_{0} ⟵^{d} P_{1} ⟵^{d} …

be a resolution of k by finitely generated projective k[G]^{#}-modules. Choose a representing cocycle f : P_{2n} → k for x and set, \overline{P}_{2n} = P_{2n}/d(P_{2n+1}), M = Ker\{f : \overline{P}_{2n} → k\}. Modifying the resolution P_{•} if necessary, we may assume that M ≠ 0. In this case, we claim that

|G|_{M} = Z(x).

In view of Theorem 6.7, it suffices to check the equality

V_{r}(G)_{M} = Ψ^{−1}(Z(x)) = Z(ψ(x)).  (7.5.1)

Considering both sides of (7.5.1) as closed reduced subschemes of V_{r}(G), it suffices to establish that they have the same field valued points. This amounts to showing that if K/k is a field extension and ν : 𝔾_{a(r)} ⊗_{k} K → G ⊗_{k} K is a group scheme homomorphism, then the following conditions are equivalent: (a.) M_{K} is not a projective K[u_{r−1}]/u_{r−1}^{p}-module. (b.) The pullback of x_{K} ∈ H^{2n}(G ⊗_{k} K, K) via the K-algebra homomorphism

K[u_{r−1}]/u_{r−1}^{p} ↪ K[𝔾_{a(r)}]^{#} ⟶^{ν_{∗}} K[G]^{#}

is trivial in H^{2n}(K[u_{r−1}]/u_{r−1}^{p}, K). If ν is the trivial homomorphism, then both conditions are satisfied (here we need the assumption M ≠ 0). If, on the other hand, ν is non trivial, then Lemma 7.3 shows that P_{•} ⊗_{k} K is a K[u_{r−1}]/u_{r−1}^{p}-projective resolution of K, so that the equivalence of (a.) and (b.) follows from Lemma 7.4.

Our next application provides retrospective motivation for the detection theorems we have presented. Recall that if G is a simple smooth group scheme with Borel subgroup B ⊂ G and maximal torus T ⊂ B, then the group scheme T G_{(r)} is defined to fit in the following cartesian square:

T G_{(r)} ⟶ T
↓  ↓
G ⟶^{F^{r}} G

The “Main Theorem” of [CPS] asserts for a simple smooth group scheme G over an algebraically closed field k that a finite dimensional rational T G_{(r)}-module M is injective if and only if the restrictions of M to root subgroups U_{α(r)} ⊂ G_{(r)} are injective.

Proposition 7.6. Let G be an infinitesimal group scheme of height ≤ r over an algebraically closed field k. Let further M be a finite dimensional rational G-module. Assume that whenever H ⊂ G is a subgroup scheme isomorphic to 𝔾_{a(s)} (with s ≤ r) the restriction of M to H is an injective rational H-module. Then M is injective as a rational G-module.

Proof. Let x ∈ V_{r}(G) be a k-rational point different from the origin and let ν_{x} : 𝔾_{a(r)} → G be the corresponding (non-trivial) group scheme homomorphism (over k). Denote the image of ν_{x} by H. Clearly H =^{∼} 𝔾_{a(s)} for some s ≤ r. Our assumption implies that M is an injective (=projective) k[H]^{#}-module. Using Lemma 7.3 we conclude immediately that M is also projective over k[u_{r−1}]/u_{r−1}^{p}, i.e. x ∉ V_{r}(G)_{M}. Since rational points are dense in any k-scheme of finite type, we conclude that either V_{r}(G)_{M} is empty (in which case M = 0) or consists of the origin only. In the latter case we conclude from Theorem 6.7 that \sqrt{I_{M}} coincides with the augmentation ideal of H^{ev}(G, k). The H^{ev}(G, k)-module Ext_{G}^{∗}(M, M) = H^{∗}(G, End_{k}(M)) is finitely generated according to [FS] and is killed by a sufficiently high power of the augmentation ideal. This shows that there exists an integer N > 0 such that Ext_{G}^{i}(M, M) = 0 for i ≥ N. All we have to do now is to use the following result.

Lemma 7.6.1. Let M be a finite dimensional rational module over an infinitesimal group scheme G. Assume that there exists an integer N such t hat Ext_{G}^{i}(M, M) = 0 for i ≥ N. Then M is injective (=projective).

Proof. Denote by L_{j}(j = 1, ..., n) the simple G-modules. For each j the Ext-group Ext_{G}^{∗}(L_{j}, M) has a natural structure of a left module over the Yoneda algebra Ext_{G}^{∗}(M, M). Moreover the action of H^{ev}(G, k) on Ext_{G}^{∗}(L_{j}, M) is the composition of the above action and the homomorphism ρ : H^{ev}(G, k) → Ext_{G}^{∗}(M, M). Our condition implies now that the finitely generated H^{ev}(G, k)-module Ext_{G}^{∗}(L_{j}, M) = H^{∗}(G, L_{j}^{#} ⊗ M) is killed by a sufficiently high power of the augmentation ideal of H^{ev}(G, k) and hence Ext_{G}^{i}(L_{j}, M) = 0 for all sufficiently high i. Thus, increasing N if necessary, we may assume that Ext_{G}^{i}(L_{j}, M) = 0 for all j and all i ≥ N. An obvious induction on the length shows further that Ext_{G}^{i}(L, M) = 0 for i ≥ N and all finite dimensional modules L. Consider finally an injective resolution of M

0 → M → I^{0} → I^{1} → ... → I^{N} → ...

consisting of finite dimensional modules and denote by Z^{i} the submodule of i-cocycles. We prove that all Z^{i} are injective (=projective) by decreasing induction on i. Assume first that i ≥ N. The embedding Z^{i} ↪ I^{i} defines an element in Ext_{G}^{i}(Z^{i}, M) which has to be

trivial since i ≥ N. The triviality of this element means that the projection I^{i−1} ⟶^{d} Z^{i} splits and hence Z^{i} is injective. To make the induction step we use the crucial fact that injectives and projectives are the same in our situation. Thus injectivity of Z^{i} implies that the exact sequence

0 → Z^{i−1} → I^{i−1} ⟶^{d} Z^{i} → 0

splits and hence Z^{i−1} is injective as well.

The proof of the following theorem is merely a repetition of the module-theoretic arguments given by Carlson in [C2].

Theorem 7.7. (cf. [C2;Thm1′]) Let G be an infinitesimal group scheme of height ≤ r and let M be a finite dimensional rational G-module. If V_{r}(G)_{M} can be written as a union of closed subsets, V_{r}(G)_{M} = V_{1} ∪ V_{2}, with V_{1} ∩ V_{2} = {0} and dimV_{1}, dimV_{2} ≥ 1, then M can be written as M_{1} ⊕ M_{2} with V_{r}(G)_{M_{i}} = V_{i}.

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Department of Mathematics, Northwestern University, Evanston, IL 60208 E-mail: suslin@math.nwu.edu, eric@math.nwu.edu, bendel@math.nwu.edu