INFINITESIMAL 1–PARAMETER SUBGROUPS AND COHOMOLOGY

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

This is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field k of characteristic p > 0. Whereas [SFB] is concerned with detection of cohomology classes, the present paper introduces the graded algebra k[V_{r}(G)] of functions on the scheme of infinitesimal 1-parameter subgroups of height ≤ r on an affine group scheme G and demonstrates that this algebra is essentially a retract of H^{ev}(G, k) provided that G is an infinitesimal group scheme of height ≤ r.

This work is a continuation of [F-S] in which the existence of certain universal extension classes was established, thereby enabling the proof of finite generation of H^{∗}(G, k) for any finite group scheme G over k. The role of the scheme of infinitesimal 1-parameter subgroups of G was foreshadowed in [F-P] where H^{∗}(G_{(1)}, k) was shown to be isomorphic to the coordinate algebra of the scheme of p-nilpotent elements of g = Lie(G) for G a smooth reductive group, G_{(1)} the first Frobenius kernel of G, and p = char(k) sufficiently large. Indeed, p-nilpotent elements of g correspond precisely to infinitesimal 1-parameter subgroups of G_{(1)}. Much of our effort in this present paper involves the analysis of the restriction of the universal extension classes to infinitesimal 1-parameter subgroups.

In section 1, we construct the affine scheme V_{r}(G) of homomorphisms from 𝔾_{a(r)} to G which we call the scheme of infinitesimal 1-parameter subgroups of height ≤ r in G. In the special case r = 1, this is the scheme of p-nilpotent elements of the p-restricted Lie algebra Lie(G); for various classical groups G, V_{r}(G) is the scheme of r-tuples of pnilpotent, pairwise commuting elements of Lie(G). More generally, an embedding G ⊂ GL_{n} determines a closed embedding of V_{r}(G) into the scheme of r-tuples of p-nilpotent, pairwise commuting elements of gl_{n} = Lie(GL_{n}). The relationship between k[V_{r}(G)], the coordinate algebra of V_{r}(G), and H^{∗}(G, k) is introduced in Theorem 1.14: a natural homomorphism of graded k-algebras

ψ : H^{ev}(G, k) → k[V_{r}(G)]

is constructed, a map which we show in [SFB] to induce a bijective map on associated schemes.

The universal classes e_{r} ∈ H^{2p^{r−1}}(GL_{n}, gl_{n}^{(r)}) constructed in [F-S] determine characteristic classes e_{r}(G, V) for any affine group scheme G provided with a finite dimensional representation V. Our central computation is the determination in Theorem 4.9 of the

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

characteristic classes associated to 1-parameter subgroups of GL_{n}. Since we apply this computation to the universal 1-parameter subgroup

exp\underline{α} : 𝔾_{a(r)} ⊗_{𝔽_{p}} A → GL_{n(r)} ⊗_{𝔽_{p}} A

where A is the coordinate algebra of V_{r}(GL_{n}), we consider characteristic classes of group schemes over a commutative (unital) ring. Section 2 presents a (straight-forward) extension of the theory of strict polynomial functors (used in [F-S] to construct the universal classes e_{r}) from the context of vector spaces over a field to projective modules over a commutative ring. In particular, we formulate the concept and basic properties of base change of such functors.

Since a homomorphism exp\underline{α} : 𝔾_{a(r)} ⊗_{𝔽_{p}} A → GL_{n} ⊗_{𝔽_{p}} A specified by the r-tuple \underline{α} = (α_{0}, …, α_{r−1}) is a product of homomorphisms exp_{α_{i}} ∘ F^{i} : 𝔾_{a(r)} ⊗_{𝔽_{p}} A → 𝔾_{a(r)} ⊗_{𝔽_{p}} A → GL_{n} ⊗_{𝔽_{p}} A (where F denotes the Frobenius map), we find it useful to introduce a coproduct structure on certain Ext-groups in the category of strict polynomial functors. This is done in section 3 by investigating Ext-algebras in the category P(2) of strict polynomial bifunctors. The complete determination of the Hopf algebra Ext_{P}^{∗}(I^{(r)}, I^{(r)}) is achieved in section 4.

The universal classes e_{i}^{(r−i)}, 1 ≤ i ≤ r, determine a natural map

ϕ : S^{∗}( ⨁_{1≤i≤r} gl_{n}^{(r)#}[2p^{i−1}]) → H^{ev}(GL_{n(r)}, k)

where the symmetric algebra can be viewed as the algebra of regular functions (suitably graded with appropriate Frobenius twists) on the affine scheme gl_{n}^{(r)×r}. The explicit form of the characteristic classes e_{r}(j)((𝔾_{a(r)})_{A}, V\underline{α}) given in Theorem 4.9 enables us to verify a very pleasing description of ψ ∘ ϕ. Indeed, Theorem 5.2 asserts that the composition of the maps of schemes

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

determined by ϕ and ψ is merely the r-th Frobenius twist of the natural embedding V_{r}(G) ⊂ gl_{n}^{×r}.

Finally, section 6 determines the homology algebra of the infinite general linear group over a finite field with coefficients in the adjoint representation. This multiplicative structure is easily deduced from our determination of the coproduct structure introduced in section 3 together with computations of [F-S].

The second author gratefully acknowledges the hospitality of the University of Heidelberg during the preparation of this paper.

§1. The scheme V_{r}(G).

As we establish in this section, the set of 1-parameter subgroups 𝔾_{a(r)} → G of height r into an affine group scheme G admits the structure of the set of rational points of an affine k-scheme V_{r}(G) whose coordinate algebra has the natural structure of a graded, connected k-algebra. Moreover, whenever G = GL_{n} or a classical group, then V_{r}(G) can be identified as the scheme of r-tuples of p-nilpotent, pair-wise commuting elements of g = Lie(G). A connection between the scheme V_{r}(G) and the cohomology of G is established in Theorem 1.14.

The following definition was first formulated by D. Gross [G]. As usual, k will denote a field of characteristic p > 0.

Definition 1.1. Let G be an affine group scheme/k and r > 0 be a positive integer. We define the functor

V_{r}(G) : (comm k-alg) → (sets)

by setting

V_{r}(G)(A) = Hom_{Gr/A}(𝔾_{a(r)} ⊗_{k} A, G ⊗_{k} A).

Proposition 1.2. For any commutative k-algebra A, we have a natural identification of V_{r}(GL_{n})(A) with

{(α_{0}, …, α_{r−1}) : α_{i} ∈ M_{n}(A), α_{i}^{p} = 0 = [α_{j}, α_{ℓ}] for all 0 ≤ i, j, ℓ < r}.

Proof. A homomorphism of group schemes over A

f : G_{A} → GL_{n} ⊗_{k} A

is equivalent to the data of a rational G_{A}-module structure on A^{n}. This data consists of an A-module homomorphism

Δ_{f} : A^{n} → A[G] ⊗_{A} A^{n}

making commutative the following two diagrams

\begin{matrix}A^{n} & ⟶^{Δ_{f}} & A[G] ⊗_{A} A^{n} \\ ^{=}↓ & & ↓^{ϵ⊗1} \\ A^{n} & ⟶^{1} & A^{n} \end{matrix}  (1.2.1)

\begin{matrix}A^{n} & ⟶^{Δ_{f}} & A[G] ⊗_{A} A^{n} \\ ^{Δ_{f}} ↓ & & ↓^{Δ⊗1} \\ A[G] ⊗_{A} A^{n} & ⟶^{1⊗Δ_{f}} & A[G] ⊗_{A} A[G] ⊗_{A} A^{n}. \end{matrix}  (1.2.2)

Let G denote 𝔾_{a(r)}, so that A[G] = A[T]/T^{p^{r}} is a free A-module of rank p^{r}. To give the A-module homomorphism Δ_{f} : A^{n} → A[G] ⊗_{A} A^{n} defining the rational G_{A}-module structure on A^{n} is equivalent to giving a sequence β_{0}, …, β_{p^{r}−1} ∈ M_{n}(A):

Δ_{f}(v) = ∑_{i=0}^{p^{r}−1} t^{i} ⊗ β_{i}(v).  (1.2.3)

The commutativity of (1.2.1) is equivalent to the condition that β_{0} = 1. The commutativity of (1.2.2) is equivalent to the conditions

β_{i}β_{j} = \binom{i + j}{j}β_{i+j} i + j < p^{r}

β_{i}β_{j} = 0 i + j ≥ p^{r}.  (1.2.4)

In particular, β_{i}^{p} = 0 = [β_{j}, β_{ℓ}] for all 0 ≤ i, j, ℓ < p^{r}.

Set α_{i} = β_{p^{i}}. For any j = ∑_{ℓ=0}^{r−1} j_{ℓ}p^{ℓ} with 0 ≤ j_{ℓ} < p, the equality

β_{j} = \frac{α_{0}^{j_{0}} ⋯ α_{r−1}^{j_{r−1}}}{(j_{0}!) ⋯ (j_{r−1}!)}  (1.2.5)

can be easily established using induction and the observation that

\binom{j + j′}{j} ≡ \binom{j_{0} + j′_{0}}{j_{0}} ⋯ \binom{j_{r−1} + j′_{r−1}}{j_{r−1}} mod(p)

(where j′ similarly equals ∑ j′_{ℓ}p^{ℓ} with 0 ≤ j′_{ℓ} < p).

Conversely, given an arbitrary r-tuple \underline{α} = (α_{0}, …, α_{r−1}) ∈ M_{n}(A)^{r} of commuting pnilpotent matrices, we define Δ_{f} using (1.2.3) by defining β_{j} according to (1.2.5). A direct verification shows that Δ_{f} so defined fits in the commutative squares (1.2.1) and (1.2.2).

Remark 1.3 The same argument as above may be used to show that for any commutative k-algebra A we have a natural identification (cf. [D-G])

Hom_{Gr/A}(𝔾_{a} ⊗_{k} A, GL_{n} ⊗_{k} A) = {(α_{0}, α_{1}, ... ) : α_{i} ∈ M_{n}(A), α_{i}^{p} = 0, [α_{i}, α_{j}] = 0 for all i, j; α_{i} = 0 for i big enough}.

The homomorphism exp\underline{α} : 𝔾_{a} ⊗_{k} A → GL_{n} ⊗_{k} A corresponding to a sequence \underline{α} = (α_{0}, α_{1}, ... ) of pairwise commuting p-nilpotent matrices is given by the formula (in which we assume that α_{i} = 0 for i ≥ r)

exp\underline{α}(s) = exp(sα_{0}) · exp(s^{p}α_{1}) · ... · exp(s^{p^{r−1}}α_{r−1}) ∈ GL_{n}(S)  (1.3.1)

for any A-algebra S and any s ∈ S, where for any p-nilpotent matrix α ∈ M_{n}(S) we set

exp(α) = 1 + α + \frac{α}{2} + ⋯ + \frac{α^{p−1}}{(p − 1)!} ∈ GL_{n}(S).

Whenever f : 𝔾_{a} ⊗_{k} A → G ⊗_{k} A is a group homomorphism (over A), we shall use the notation f^{(1)} for the composition homomorphism f ∘ F (where F denotes the Frobenius endomorphism of 𝔾_{a}). With this notation the formula (1.3.1) may be rewritten as follows

exp\underline{α} = exp_{α_{0}} · exp_{α_{1}}^{(1)} · ... · exp_{α_{r−1}}^{(r−1)}.  (1.3.2)

Note finally that the homomorphism exp\underline{α} : 𝔾_{a(r)}⊗_{k}A → GL_{n}⊗_{k}A defined by an r-tuple of pairwise commuting p-nilpotent matrices \underline{α} = (α_{0}, ..., α_{r−1}) coincides with the restriction of the corresponding homomorphism exp_{(\underline{α},0)} : 𝔾_{a} ⊗_{k} A → GL_{n} ⊗_{k} A to 𝔾_{a(r)} ⊗_{k} A.

As was noted in the proof of Proposition 1.2, to give a homomorphism 𝔾_{a(r)} → GL_{n} is the same as to make k^{n} into a comodule over the coalgebra k[𝔾_{a(r)}] and hence is the same as to make k^{n} into a module over the algebra k[𝔾_{a(r)}]^{#}. The following statement makes explicit this algebra.

Corollary 1.4. Let v_{0}, ..., v_{p^{r}−1} be the basis of k[𝔾_{a(r)}]^{#} = (k[T]/T^{p^{r}})^{#} dual to the standard basis of k[T]/T^{p^{r}}. Denote v_{p^{i}} by u_{i}. Then the algebra k[𝔾_{a(r)}]^{#} coincides with k[u_{0}, ..., u_{r−1}]/(u_{0}^{p}, ..., u_{r−1}^{p}). Moreover for any 0 ≤ j ≤ p^{r}−1 we have the following formula (in which j_{0}, ..., j_{r−1} are the p-adic digits of j, i.e. 0 ≤ j_{l} < p and j = ∑_{l=0}^{r−1} j_{l}p^{l})

v_{j} = \frac{u_{0}^{j_{0}} · ... · u_{r−1}^{j_{r−1}}}{j_{0}! · ... · j_{r−1}!} .

As we see in the proof of the following theorem, the representability of V_{r}(G) is an easy consequence of Proposition 1.2.

Theorem 1.5. For any affine group scheme G/k, the functor V_{r}(G) is represented by an affine scheme of finite type over k, which we’ll still denote V_{r}(G). Moreover G ↦ V_{r}(G) is a covariant functor from the category of affine group schemes/k to the category of affine schemes of finite type/k, which takes closed embeddings (of group schemes) to closed embeddings.

Proof. The equations α_{i}^{p} = 0 = [α_{j}, α_{ℓ}], 0 ≤ i, j, ℓ < p^{r} define the affine subscheme of 𝔸^{rn^{2}} which clearly represents the functor V_{r}(GL_{n}). For an arbitrary G choose an embedding G ↪ GL_{n} and denote by F_{1}, …, F_{M} ∈ k[GL_{n}] the defining equations of G, then V_{r}(G) is given by the additional Mp^{r} equations in the “rn^{2} variables” of \underline{α} ∈ 𝔸^{rn^{2}}

F_{i}(exp\underline{α}(t)) = 0 ∈ k[𝔾_{a(r)}] = k[t] = k[T]/T^{p^{r}}.

Whenever V/k is a finite dimensional vector space, we shall use the same notation V for the corresponding linear scheme Spec(S^{∗}(V^{#})) over k. Let g/k be a finite dimensional p-restricted Lie algebra. The p-th power map [p] : g → g is homogeneous polynomial (of degree p) and hence may be considered as a morphism of schemes. We shall denote by N_{p}(g) the p-nilpotent cone in g, i.e. the fiber over the point 0 ∈ g of the morphism [p] : g → g, so that for any k-algebra A the set of A-points of N_{p}(g) may be identified with {x ∈ g ⊗_{k} A : x^{[p]} = 0}.

Lemma 1.6. Let G/k be an affine group scheme over k and let g be the corresponding restricted Lie algebra. Then we have a natural identification V_{1}(G) = N_{p}(g).

Proof. Let A be a k-algebra. To give a homomorphism 𝔾_{a(1)} ⊗_{k} A → G ⊗_{k} A of group schemes over A is the same as to give a homomorphism 𝔾_{a(1)} ⊗_{k} A → G_{(1)} ⊗_{k} A (over A).

Since the category of height 1 group schemes is equivalent to the category of p-restricted Lie algebras (cf. [J;I.8.5,9.6]), this is equivalent to the data of a homomorphism of restricted Lie algebras g_{a} ⊗_{k} A → g ⊗_{k} A (over A), i.e. equivalent to the data of x ∈ g ⊗_{k} A with x^{[p]} = 0.

Since we shall demonstrate in [SFB] that the the spectrum of the cohomology algebra H^{∗}(G, k) is homeomorphic to V_{r}(G) whenever G is an infinitesimal group scheme of height ≤ r, it is of interest to obtain as explicit a description of V_{r}(G) as possible. As seen in Lemma 1.8, V_{r}(G) = V_{r}(G_{(r)}) has a description as r-tuples of p-nilpotent, pair-wise commuting elements of Lie(G) for classical groups.

Let G/k be an affine group scheme with Lie algebra g and let ϕ : G ↪ GL_{n} be a closed embedding with dϕ : g → gl_{n} the associated map of p-restricted Lie algebras. We say that ϕ is an embedding of exponential type (and ϕ(G) ⊂ GL_{n} is a subgroup of exponential type) if the following equivalent conditions are satisfied:

(1) For every k-algebra A and every p-nilpotent x ∈ g ⊗_{k} A, the homomorphism exp_{dϕ(x)} : 𝔾_{a} ⊗_{k} A → GL_{n} ⊗_{k} A factors via G ⊗_{k} A.

(2) The canonical morphism of schemes

N_{p}(g) ×_{k} 𝔾_{a} → N_{p}(gl_{n}) ×_{k} 𝔾_{a} ⟶^{exp} GL_{n}

factors via G.

Lemma 1.7. Let G be an affine group scheme with an embedding of exponential type G ↪ GL_{n}. Then

V_{r}(G)(A) = \{\underline{α} ∈ (Lie(G) ⊗_{k} A)^{r} : α_{i}^{[p]} = 0 = [α_{j}, α_{l}] for all 0 ≤ i, j, l < r\}. (1.7.1)

Proof. Let \underline{α} be an r-tuple in the right hand side of (1.7.1). By assumption, each exp_{α_{i}} : 𝔾_{a(r)} ⊗_{k} A → GL_{n} ⊗_{k} A factors via G ⊗_{k} A and hence so does each exp_{α_{i}}^{(i)}. Thus, the product exp\underline{α} does as well and so exp\underline{α} ∈ V_{r}(G).

Conversely, assume that exp\underline{α} : 𝔾_{a(r)} ⊗_{k} A → GL_{n} ⊗_{k} A factors via G⊗_{k} A. This implies immediately, that α_{0} = d(exp\underline{α})(1) ∈ (Lie(G) ⊗_{k} A) ∩ N_{p}(gl_{n})(A) = N_{p}(Lie(G))(A). The hypothesis of an embedding of exponential type implies that exp_{α_{0}} : 𝔾_{a(r)}⊗_{k}A → GL_{n}⊗_{k}A factors via G⊗_{k}A and hence (exp_{α_{1}} ·...·exp_{α_{r−1}}^{(r−2)})^{(1)} does so as well. ¿From this we conclude easily that exp_{(α_{1},...,α_{r−1})} = exp_{α_{1}} · ... · exp_{α_{r−1}}^{(r−2)} : 𝔾_{a(r−1)} ⊗_{k} A → GL_{n} ⊗_{k} A factors via G ⊗_{k} A so that induction on r concludes the proof.

The following lemma asserts that various classical subgroups of GL_{n} are of exponential type and hence satisfy the conclusion (1.7.1). We omit the proof which follows by inspection.

Lemma 1.8. 1) Any intersection of subgroups of exponential type of GL_{n} is again of exponential type.

2) The subgroups SL_{n} ⊂ GL_{n}, Sp_{2n} ⊂ GL_{2n}, B_{n} ⊂ GL_{n} (subgroup of upper triangular matrices), and U_{n} ⊂ GL_{n} (subgroup of strictly upper triangular matrices) are of exponential type.

3) Let ϕ be a nondegenerate symmetric bilinear form on the vector space k^{n}. Then the orthogonal group O(ϕ) ⊂ GL_{n} is a subgroup of exponential type.

Remark 1.9 Let G be an affine group scheme which admits an embedding G ↪ GL_{n} of exponential type. The choice of such an embedding gives us an identification of V_{r}(G) as above. We do not know whether this identification is canonical or depends on the choice of the embedding (of exponential type).

We next show that the scheme V_{r}(G) is a cone or, what amounts to the same thing, that the coordinate algebra k[V_{r}(G)] is graded connected. To do this, we first identify the scheme V_{r}(𝔾_{a(r)}).

Lemma 1.10. For any r > 0, V_{r}(𝔾_{a}) = V_{r}(𝔾_{a(r)}) = 𝔸^{r}.

Proof. A homomorphism 𝔾_{a(r)} ⊗_{k} A → 𝔾_{a(r)} ⊗_{k} A is equivalent to a polynomial P (T) ∈ A[T] of degree < p^{r} such that P (T + T′) = P (T) + P (T′). Such polynomials are those of the form P (T) = a_{0}T + a_{1}T^{p} + ... + a_{r−1}T^{p^{r−1}}.

Composition of morphisms defines for each affine group scheme G a natural morphism of schemes

V_{r}(G) × V_{r}(𝔾_{a(r)}) → V_{r}(G).

Taking G = 𝔾_{a(r)} we see that V_{r}(𝔾_{a(r)}) has a natural structure of a monoid scheme over k. Moreover for an arbitrary group scheme G the above morphism defines a right monoid action of V_{r}(𝔾_{a(r)}) on V_{r}(G). The monoid structure on V_{r}(𝔾_{a(r)}) corresponds to the composition of additive polynomials and so is relatively complicated. To simplify matters we restrict our attention to an abelian submonoid 𝔸^{1} ⊂ 𝔸^{r}, corresponding to linear polynomials. The monoid structure on this monoid 𝔸^{1} is just the usual multiplication.

Lemma 1.11. A right monoid action of 𝔸^{1} on an affine scheme X = Spec A over k is equivalent to the data of a (non-negative) grading on the ring A.

Proof. For any graded k-algebra A we have a canonical k-algebra homomorphism

A → A[T]  :  a = a_{0} + a_{1} + ... ↦ a_{0} + a_{1}T + a_{2}T^{2} + ...

which defines a morphism of schemes over k : X × 𝔸^{1} = Spec A[T] → Spec A = X. One readily verifies that this defines a monoid action of 𝔸^{1} on X.

Conversely, assume that we are given a right monoid action of 𝔸^{1} on X

f : X × 𝔸^{1} → X.

Consider the corresponding homomorphism of k-algebras f^{∗} : A = k[X] → k[X × 𝔸^{1}] = A[T] and set A_{i} = {a ∈ A : f^{∗}(a) ∈ A · T^{i}}. It’s easy to check that this defines a nonnegative grading on A and moreover the above two constructions are mutually inverse.

We sum up the results of the above discussion in the following lemma.

Lemma 1.12. For any affine group scheme G/k, the coordinate algebra k[V_{r}(G)] is a graded connected k-algebra. Moreover the homomorphism k[V_{r}(G′)] → k[V_{r}(G)] corresponding to a group scheme homomorphism G → G′ respects gradings.

Proof. The only thing left to prove is the connectedness of k[V_{r}(G)]. Consider first the case G = GL_{n}. In this case it’s straightforward to check that the canonical action

V_{r}(GL_{n}) × 𝔸^{1} → V_{r}(GL_{n})

corresponds to the following map of A-valued points (for any k-algebra A):

(α_{0}, α_{1}, ..., α_{r−1}) × a ↦ (aα_{0}, a^{p}α_{1}, ..., a^{p^{r−1}}α_{r−1}).

The closed embedding V_{r}(GL_{n}) ↪ (𝔸^{n^{2}})^{r} defines a surjective k-algebra homomorphism k[(𝔸^{n^{2}})^{r}] = k[{X_{ij}^{l}}_{l=0 i,j=1}^{r−1 n}] → k[V_{r}(GL_{n})] which is compatible with gradings provided that we define the grading on the left via the formula deg(X_{i,j}^{l}) = p^{l}. In any event, the graded algebra on the left is connected and hence k[V_{r}(G)] is connected as well. The general case now follows immediately, since the surjective homomorphism k[V_{r}(GL_{n})] → k[V_{r}(G)] corresponding to a closed embedding G ↪ GL_{n} is compatible with gradings.

We shall repeatedly use the structure of the cohomology algebra of 𝔾_{a} and 𝔾_{a(r)} as recalled in the following theorem.

Theorem 1.13[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 (and 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.

We can now introduce the homomorphism ψ which will play a key role in both this paper and [SFB].

Theorem 1.14. For any affine group scheme G/k, there is a natural homomorphism of graded commutative k-algebras

ψ : H^{ev}(G, k) → k[V_{r}(G)]

which multiplies degrees by \frac{p^{r}}{2} .

Proof. For any affine group scheme G over k, the canonical element

1 ∈ V_{r}(G)(k[V_{r}(G)]) = Hom_{k−alg}(k[V_{r}(G)], k[V_{r}(G)])

corresponds to a universal homomorphism of group schemes over k[V_{r}(G)]:

u : 𝔾_{a(r)} ⊗_{k} k[V_{r}(G)] → G ⊗_{k} k[V_{r}(G)].

This induces

u^{∗} : H^{∗}(G, k) → H^{∗}(G, k) ⊗_{k} k[V_{r}(G)] = H^{∗}(G ⊗_{k} k[V_{r}(G)], k[V_{r}(G)])
→ H^{∗}(𝔾_{a(r)} ⊗_{k} k[V_{r}(G)], k[V_{r}(G)]) = H^{∗}(𝔾_{a(r)}, k) ⊗_{k} k[V_{r}(G)].

For z ∈ H^{2n}(G, k), write

u^{∗}(z) = ∑_{j,i} λ^{j}x^{i} ⊗ f_{j,i}(z) ∈ H^{2n}(𝔾_{a(r)}, k) ⊗ k[V_{r}(G)]

where λ^{j}x^{i} = λ_{1}^{j_{1}} ⋯ λ_{r}^{j_{r}}x_{1}^{i_{1}} ⋯ x_{r}^{i_{r}} with j_{ℓ} ∈ {0, 1}. We define ψ(z) to be the coefficient of x_{r}^{n} in this expression. That ψ is a homomorphism of commutative k-algebras is obvious from the construction.

To verify that ψ multiplies degrees by \frac{p^{r}}{2} , it suffices to assume that k is algebraically closed by replacing k by its algebraic closure. The natural monoid action of 𝔸^{1} on 𝔾_{a(r)} is compatible with the action of 𝔸^{1} on V_{r}(G) given above in the sense that for any s ∈ k the following diagram commutes:

𝔾_{a(r)} × V_{r}(G) ⟶^{1×s} 𝔾_{a(r)} × V_{r}(G) ⟶^{u} G × V_{r}(G)
s×1↓  ↓^{pr_{1}}
𝔾_{a(r)} × V_{r}(G) ⟶^{u} G × V_{r}(G) ⟶^{pr_{1}}  G.

If we pull z ∈ H^{2n}(G, k) back via the lower horizontal and left vertical maps, then we obtain

∑_{j,i} s^{j_{1}+...+p^{r−1}j_{r}+pi_{1}+...+p^{r}i_{r}}λ^{j}x^{i} ⊗ f_{j,i}.

If we pull z ∈ H^{2n}(G, k) back via the right vertical and upper horizontal maps, then we obtain

∑_{j,i} λ^{j}x^{i} ⊗ s^{∗}(f_{j,i}).

Thus, each f_{j,i} is homogeneous of degree j_{1} + … + p^{r−1}j_{r} + pi_{1} + … + p^{r}i_{r}. In particular, ψ(z) has homogeneous degree p^{r}n.

In the special case p = 2, the previous theorem can be refined as follows to include cohomology classes of odd degree.

Theorem 1.14.1. Assume that p = 2. Then for any affine group scheme G/k, there is a natural homomorphism of graded commutative k-algebras

ψ : H^{∗}(G, k) → k[V_{r}(G)]

which multiplies degrees by 2^{r−1}.

Proof. This time we define ψ(z) (for z ∈ H^{n}(G, k)) to be the coefficient of λ_{r}^{n} in u^{∗}(z). The properties of this map ψ are verified in exactly the same way as above.

§2. Extension of scalars in polynomial functors.

The theory of strict polynomial functors developed in [F-S] is the basis for the construction of universal extension classes e_{r}. Since we need these universal classes in the context of group schemes over quite general commutative rings, we must extend the formulation of strict polynomial functors to this more general situation. We outline below this theory indicating the necessary (minor) changes. We then verify that the universal classes constructed in [F-S] (in the context of strict polynomial functors on finite dimensional vector spaces over a field) determine by base change universal classes in our more general context.

Let A be a commutative ring (always assumed to be unital). Denote by V = V_{A} the category of finitely generated projective A-modules and A-linear maps. For any V, W ∈ V_{A} we set Hom_{P ol}(V, W) = S^{∗}(V^{#})⊗_{A} W, where V^{#} = Hom_{A}(V, A) is the module dual to V (over A) and S^{∗} stands for the symmetric algebra (over A). The elements of Hom_{P ol}(V, W) are called polynomial maps from V to W. A polynomial map is said to be homogeneous of degree d if it belongs to S^{d}(V^{#})⊗_{A} W. The set of homogeneous polynomial maps of degree one coincides with Hom_{A}(V, W). Each bilinear map U × V → W defines a homogeneous polynomial map of degree two.

For any V ∈ V, we shall use the same notation V for the corresponding vector bundle Spec S^{∗}(V^{#}) over Spec A. With this notation, we have the following obvious formula

Hom_{P ol}(V, W) = Hom_{Sch/A}(V, W).

This formula enables us, in particular, to define the composition map

Hom_{P ol}(V, W) × Hom_{P ol}(U, V) → Hom_{P ol}(U, W).

Each polynomial map p : V → W defines a set map from V to W which we usually denote by the same letter p. Note also that we can extend scalars for polynomial maps, so that whenever p : V → W is a polynomial map (over A) and A′ is a commutative A-algebra we may consider a polynomial map p_{A′} : V_{A′} → W_{A′} over A′ and hence also the corresponding set map V_{A′} → W_{A′}.

Definition 2.1. A strict polynomial functor over a commutative ring A T : V_{A} → V_{A} consists of the data of T (V) ∈ V_{A} for each V ∈ V_{A} and a polynomial map T_{V,W} : Hom_{A}(V, W) → Hom_{A}(T (V), T (W)) for each pair V, W ∈ V_{A}, satisfying (2.1.1) T_{V,V} (1_{V}) = 1_{T (V)}. (2.1.2) For any U, V, W ∈ V_{A}, the following diagram of polynomial maps commute

Hom_{A}(V, W) × Hom_{A}(U, V)  ⟶  Hom_{A}(U, W)
T_{V,W} ×T_{U,V} ↓  ↓^{T_{U,W}}
Hom_{A}(T (V), T (W)) × Hom_{A}(T (U), T (V)) ⟶ Hom_{A}(T (U), T (W)).

For any V ∈ V_{A}, we denote by GL(V) the corresponding affine group scheme over A. As a scheme, GL(V) is the principal open subset in the vector bundle End_{A}(V) given by the non-vanishing of the determinant det ∈ S^{∗}(End_{A}(V)^{#}). If G is an affine group scheme of finite type over A, then to make V ∈ V_{A} into a rational G-module (i.e. into a comodule over the coalgebra A[G]) is equivalent to giving a group scheme homomorphism G → GL(V) over A.

Let T be a polynomial functor over A. The data of a polynomial functor provides, in particular, a morphism of schemes

T_{V,V} : End_{A}(V) → End_{A}(T (V)).

One checks easily that T_{V,V} takes the open subscheme GL(V) ⊂ End_{A}(V) to GL(T (V)) and the induced morphism of schemes T_{V,V} : GL(V) → GL(T (V)) is a group scheme homomorphism. Composing this homomorphism with the original homomorphism G → GL(V), we see that T (V) has a natural structure of a rational G-module.

Now the general theory developed in [F-S;§2] goes through practically without change. We briefly recall the essential definitions and results.

First of all, one can introduce the concept of a homogeneous strict polynomial functor of degree d (resp. of a strict polynomial functor of finite degree) by requiring that all polynomial maps T_{V,V} : End_{A}(V) → End_{A}(T (V)) are homogeneous of degree d (resp. degrees of these maps are bounded above). One checks easily that a strict polynomial functor of finite degree is a direct sum of (finitely many) homogeneous components.

One defines a natural transformation of strict polynomial functors exactly in the same way as was done in [F-S] for polynomial functors over a field. We denote by P = P_{A} the category of strict polynomial functors over A of finite degree and by P_{d} = P_{d,A} its full subcatagory consisting of homogeneous strict polynomial functors of degree d. As mentioned above, we have a canonical direct sum decomposition P_{A} = ⊕_{d≥0}P_{d,A}. In the present situation, the categories P_{A} and P_{d,A} are no longer abelian; however, they are exact categories in the sense of Quillen [Q] with admissible short exact sequences 0 → T′ → T → T′′ → 0 being those for which the sequence 0 → T′(V) → T (V) → T′′(V) → 0 is exact for any V ∈ V_{A}.

We recall the notation Γ^{i} for the i-th divided power, the dual (over A) of S^{i}.

Let T be a strict polynomial functor. For any V, W ∈ V_{A}, T defines a canonical A-linear homomorphism

θ : ⊕_{i≥0}Γ^{i}(Hom_{A}(V, W)) ⊗_{A} T (V) → T (W)

which is non-zero only on finitely many components (only on Γ^{d}(Hom_{A}(V, W)) ⊗_{A} T (V) if T is homogeneous of degree d). A functor T is said to be n-generated provided that the corresponding homomorphism

⊕_{i≥0}Γ^{i}(Hom_{A}(A^{n}, W)) ⊗_{A} T (A^{n}) → T (W)

is surjective for any W ∈ V_{A}. A functor T is said to be finitely generated if it is n-generated for some n ≥ 0. Arguing essentially in the same way as in [F-S], one can show that T is finitely generated if and only if it is of finite degree. In fact we have the following result.

Proposition 2.2 ([F-S] Proposition 2.9). A strict polynomial functor T is finitely generated if and only if it is of finite degree. More precisely, if T is n-generated, then deg(T) coincides with the degree of the polynomial map T_{A^{n},A^{n}} : End_{A}(A^{n}) → End_{A}(T (A^{n})); conversely if T has degree d, then it is d-generated.

The functor Γ^{d,n} ∈ P_{d} introduced in [F-S] (see also the proof of Proposition 3.6) is given by the formula Γ^{d,n}(V) = Γ^{d}(Hom_{A}(A^{n}, V)). This functor is relatively projective as one sees from the following result.

Theorem 2.3 ([F-S] Theorem 2.10). For any homogeneous strict polynomial functor of degree d and any n ≥ 0,

Hom_{P_{d}}(Γ^{d,n}, T) = T (A^{n}).

Thus, Γ^{d,n} is a relatively projective object of P_{d}. Moreover, for any homogeneous strict polynomial functor T of degree d, the associated natural transformation

Θ : Γ^{d,n} ⊗_{A} T (A^{n}) → T

is an admissible epimorphism if and only if T is n-generated.

Theorem 2.3 and Proposition 2.2 show that each T ∈ P_{d} has an admissible projective resolution. Thus we may develop the usual relative homological algebra in the category P_{d}. In particular, for any T, T′ ∈ P_{d} we may consider the corresponding Ext-groups Ext_{P}^{∗}(T, T′).

Denote by S(n, d) = S(n, d, A) the Schur algebra Γ^{d}(M_{n}(A)). For any T ∈ P_{d} and any n ≥ 0, the canonical homomorphism

θ : Γ^{d}(End_{A}(A^{n})) ⊗_{A} T (A^{n}) → T (A^{n})

makes T (A^{n}) into a finitely generated left S(n, d, A)-module, which is projective as an Amodule. Denote by Mod{S(n, d, A)} the exact category of all such modules. Associating to T ∈ P_{d} the S(n, d, A)-module T (A^{n}), we get an exact functor

Ψ : P_{d} → Mod{S(n, d, A)}.

Theorem 2.4([F-S] Theorem 3.1). Assume that n ≥ d. Then Ψ : P_{d,A} → Mod{S(n, d, A)} is an equivalence of categories with quasi-inverse defined by sending M ∈ Mod{S(n, d, A)} to Φ(M) ≡ Γ^{d,n} ⊗_{S(n,d,A)} M.

The proof is identical with that given in [F-S], the only additional problem being the verification that if M ∈ Mod{S(n, d, A)} and V ∈ V_{A} then Γ^{d,n}(V) ⊗_{S(n,d,A)} M is a projective A-module. To check this, it suffices to consider V = A^{m}. In this case, one can use the fact that Γ^{d,n}(A^{m}) = Γ^{d}(Hom_{A}(A^{n}, A^{m})) is a projective right S(n, d, A)-module.

Corollary 2.4.1. Let f : T → T′ be a homomorphism of homogenous strict polynomial functors of degree d. The following conditions are equivalent: (a.) f is an isomorphism. (b.) f_{A^{n}} : T (A^{n}) → T′(A^{n}) is an isomorphism for all n. (c.) f_{A^{n}} : T (A^{n}) → T′(A^{n}) is an isomorphism for some n ≥ d.

We now investigate the behavior of Ext-groups under base change. Let A′ be a commutative A-algebra and let T ∈ P_{A}, T′ ∈ P_{A′} be a pair of strict polynomial functors. We define a homomorphism f : T → T′ to be a collection of A′-linear maps

f_{V} : T (V) ⊗_{A} A′ → T′(V ⊗_{A} A′)

given for all V ∈ V_{A} and satisfying the following property: for any V, W ∈ V_{A}, the following diagram of polynomial maps commute

Hom_{A′}(V_{A′}, W_{A′}) = (Hom_{A}(V, W))_{A′} ⟶^{(T_{V,W})_{A′}} Hom_{A′}(T (V)_{A′}, T (W)_{A′})
T′_{V_{A′} ,W_{A′}} ↓  ↓^{f_{W} ∘}
Hom_{A′}(T′(V_{A′}), T′(W_{A′}))  ⟶^{∘f_{V}} Hom_{A′}(T (V)_{A′}, T′(W_{A′})).

We have used here and below the notation V_{A′} to denote V ⊗_{A} A′ for any V ∈ V_{A}.

Let V′ ∈ V_{A′} and consider the category V_{A}(V′), whose objects are pairs (V, ϕ), where V ∈ V_{A} and ϕ : V ⊗_{A} A′ → V′ is an A′-linear map, and morphisms from (V, ϕ) to (V_{1}, ϕ_{1}) are A′-linear maps ψ : V ⊗_{A} A′ → V_{1} ⊗_{A} A′ such that ϕ_{1}ψ = ϕ. Let T ∈ P_{A} be a strict polynomial functor. To each (V, ϕ) ∈ V_{A}(V′) we associate the A′-module T (V)_{A′} = T (V) ⊗_{A} A′, and to each ψ ∈ Hom((V, ϕ), (V_{1}, ϕ_{1})) we associate the A′-linear map

(T_{V,V_{1}})_{A′}(ψ) : T (V)_{A′} → T (V_{1})_{A′}.

In this way, we get a functor from the category V_{A}(V′) to the category of A′-modules. We define the A′-module T_{A′}(V′) via the formula

T_{A′}(V′) = lim_{⟶}_{V_{A}(V′)}T (V)_{A′}.

Observe that

T_{A′}(V_{A′}) = T (V)_{A′}, V ∈ V_{A},  (2.5.0)

since (V, 1_{V_{A′}}) is the final object of the category V_{A}(V_{A′}).

Proposition 2.5. Let T ∈ P_{A} and let A′ be a commutative A-algebra. Then the functor T_{A′} sending V′ ∈ V_{A′} to T_{A′}(V′) = lim_{⟶}_{V_{A}(V′)}T (V)_{A′} is a strict polynomial functor on V_{A′}.

Moreover, for any T′ ∈ P_{A′},

Hom(T, T′) = Hom_{P_{A′}} (T_{A′}, T′).

Proof. It’s clear that so defined T_{A′} is at least a functor in the usual sense from V_{A′} to the category of A′-modules. Furthermore, for any V′ ∈ P_{A′} we can find an embedding i : V′ ↪ A′^{n} and a projection p : A′^{n} → V′ for which p ∘ i = 1_{V′}. This implies that T_{A′}(p) ∘ T_{A′}(i) = 1_{T_{A′} (V′)} and hence T_{A′}(V′) is a direct summand in T_{A′}(A′^{n}) = T (A^{n})_{A′} and in particular T_{A′}(V′) ∈ P_{A′}.

To define polynomial maps

(T_{A′})_{V′,W′} : Hom_{A′}(V′, W′) → Hom_{A′}(T_{A′}(V′), T_{A′}(W′))

we proceed as follows. Choose embeddings i_{V′} : V′ ↪ A′^{n}, i_{W′} : W′ ↪ A′^{m} and projections p_{V′} : A′^{n} → V′, p_{W′} : A′^{m} → W′ such that p_{V′}i_{V′} = 1_{V′}, p_{W′}i_{W′} = 1_{W′} and define (T_{A′})_{V′,W′} as a composition

Hom_{A′}(V′, W′) ⟶ Hom_{A′}(A′^{n}, A′^{m}) ⟶^{(T_{A^{n},A^{m}})_{A′}} Hom_{A′}(T (A^{n})_{A′}, T (A^{m})_{A′}) =

= Hom_{A′}(T_{A′}(A′^{n}), T_{A′}(A′^{m})) → Hom_{A′}(T_{A′}(V′), T_{A′}(W′)).

A routine verification shows that the resulting polynomial map is well defined and that T_{A′} is a strict polynomial functor over A′ having the required universal mapping property.

Proposition 2.6. The extension of scalars functor ⊗_{A}A′ : P_{A} → P_{A′} sending T to T_{A′} is exact and preserves (relative) projectives. Furthermore,

Hom_{P_{A′}} (S_{A′}, T_{A′}) = Hom_{P_{A}}(S, T) ⊗_{A} A′, S, T ∈ P_{A}.

Proof. Let 0 → T^{1} → T → T^{2} → 0 be an exact sequence in P_{A}. Let further V′ be an object of V_{A′}. If V′ is of the form V′ = V_{A′}, then (2.5.0) implies that the sequence

0 → T_{A′}^{1}(V′) → T_{A′}(V′) → T_{A′}^{2}(V′) → 0  (2.6.1)

is isomorphic to

0 → T^{1}(V) ⊗_{A} A′ → T (V) ⊗_{A} A′ → T^{2}(V) ⊗_{A} A′ → 0

and hence is exact. In the general case we may choose a split embedding of V′ into A′^{n}. Then the sequence (2.6.1) is isomorphic to a direct summand in an exact sequence corresponding to A′^{n} and hence is again exact.

To show that ⊗_{A}A′ preserves projectives it suffices to establish the formula (Γ_{A}^{d,n}) ⊗_{A} A′ = Γ_{A′}^{d,n}, which is done by direct inspection.

In proving the last statement, we may clearly assume that S and T are homogeneous of degree d. If S = Γ^{d,n}, then our statement follows from Theorem 2.3 and (2.5.0). For a general S, we may consider a resolution of the form

(Γ^{d,d})^{m} → (Γ^{d,d})^{n} → S → 0

and use the right exactness of the functor Hom.

Proposition 2.6 has the following formal consequence.

Corollary 2.7. For any S, T ∈ P_{A}, we have a canonical homomorphism

Ext_{P_{A}}^{∗}(S, T) ⊗_{A} A′ → Ext_{P_{A′}}^{∗} (S_{A′}, T_{A′})

which is an isomorphism in case A′ is flat over A.

We shall be particularly interested in the base change of the functors I^{(r)} whose Extalgebra was determined in [F-S]. For any commutative 𝔽_{p}-algebra A, let f : A → A denote the Frobenius (p-th power) map sending a ∈ A to a^{p} ∈ A. For any V ∈ V_{A} and any r ≥ 0, set V^{(r)} = V ⊗_{f^{r}} A. The functor I^{(r)} = I_{A}^{(r)} : V ↦ V^{(r)} is an additive functor from the category V_{A} to itself satisfying the following properties:

(V ⊗_{A} W)^{(r)} = V^{(r)} ⊗_{A} W^{(r)}; (V^{#})^{(r)} = (V^{(r)})^{#} for any V, W ∈ V_{A}.

We’ll be using the notation v^{(r)} for the element v ⊗_{f^{r}} 1 ∈ V^{(r)}. It’s clear that if V is free with basis v_{1}, ..., v_{n}, then V^{(r)} is also free with basis v_{1}^{(r)}, ..., v_{n}^{(r)}. To make I^{(r)} into a polynomial functor, we note that for any V ∈ V_{A} we have a canonical A-linear homomorphism V^{(r)} → S^{p^{r}}(V) : v^{(r)} ↦ v^{p^{r}}. Applying this to V^{#} we get a canonical polynomial map V → V^{(r)} of degree p^{r} (the corresponding set map takes v to v^{(r)}). In particular, this gives us for any V, W ∈ V_{A} a canonical polynomial map

Hom_{A}(V, W) → Hom_{A}(V, W)^{(r)} = Hom_{A}(V^{(r)}, W^{(r)}).

A straightforward verification shows that with these maps I^{(r)} becomes a homogenous polynomial functor of degree p^{r} over A. We sum up the results of the previous discussion in the following proposition.

Proposition 2.8. Let A be a commutative 𝔽_{p}-algebra. Then I_{A}^{(r)} is a homogenous polynomial functor of degree p^{r}. Moreover I_{A}^{(r)} is additive and satisfies the following properties:

(V ⊗_{A} W)^{(r)} = V^{(r)} ⊗_{A} W^{(r)}; (V^{#})^{(r)} = (V^{(r)})^{#} for any V, W ∈ V_{A}.

Finally for any A as above we have a canonical isomorphism

I_{A}^{(r)} = I_{𝔽_{p}}^{(r)} ⊗_{𝔽_{p}} A.

Proof. Everything but the last statement has been established above. To prove the last statement we first mention that for any V_{0} ∈ V_{𝔽_{p}} we have a canonical isomorphism

V_{0}^{(r)} ⊗_{𝔽_{p}} A ⟶^{∼} (V_{0} ⊗_{𝔽_{p}} A)^{(r)}.

One checks easily that these isomorphisms define a homomorphism I_{𝔽_{p}}^{(r)} → I_{A}^{(r)}. The universal mapping property of Proposition 2.5 gives us further a canonical functor homomorphism I_{𝔽_{p}}^{(r)} ⊗_{𝔽_{p}} A → I_{A}^{(r)}. The formula (2.5.0) shows that the corresponding A-linear map

(I_{𝔽_{p}}^{(r)} ⊗_{𝔽_{p}} A)(V) → I_{A}^{(r)}(V)

is an isomorphism for modules of the form V = V_{0} ⊗_{𝔽_{p}} A (V_{0} ∈ P_{𝔽_{p}}), i.e. for free A-modules. To complete the proof, it suffices now to use Corollary 2.4.1.

§3. Characteristic classes and the Hopf Algebra E_{r}.

In [F-S], the determination of the (Yoneda) Ext-algebra E_{r} = Ext_{P_{𝔽p}}^{∗} (I^{(r)}, I^{(r)}) played an important role. In this section, we view elements of this algebra as universal characteristic classes for rational representations. We also introduce a natural coproduct on E_{r} which provides E_{r} with the structure of a commutative, cocommuative Hopf algebra (whose explicit form will be determined in the next section).

The following computation of E_{r} was achieved in [F-S].

Theorem 3.1([F-S] Theorem 4.10). Each E_{r} is a finite dimensional, commutative 𝔽_{p} algebra concentrated in even degrees. Frobenius twist determines an injective map E_{r−1} → E_{r} for each r > 0. Moreover, E_{0} = 𝔽_{p} (in degree 0) and E_{r} is obtained from E_{r−1} by adjoining a new generator

e_{r} ∈ Ext_{P_{𝔽p}}^{2p^{r−1}}(I^{(r)}, I^{(r)})

subject to the only relation e_{r}^{p} = 0. In particular, the dimension of E_{r} in degree n is 1 if n = 2j, 0 ≤ j < p^{r} and is 0 otherwise. If j = j_{0} + j_{1}p + … + j_{r−1}p^{r−1} with each j_{i} satisfying 0 ≤ j_{i} < p, then we define

e_{r}(j) = \frac{(e_{1}^{(r−1)})^{j_{0}}(e_{2}^{(r−2)})^{j_{1}}...e_{r}^{j_{r−1}}}{(j_{0}!)(j_{1}!) ⋯ (j_{r−1}!)} ∈ Ext_{P_{𝔽p}}^{2j} (I^{(r)}, I^{(r)}).

(Thus, e_{r} = e_{r}(p^{r−1})).

In the following proposition, we introduce characteristic classes for rational G-modules obtained from the universal classes e_{r} of Theorem 3.1.

Proposition 3.2. Let G be an affine group scheme over a commutative 𝔽_{p}-algebra A and let V ∈ V_{A} be a rational G-module with structure provided by ρ : G → GL(V). For each r > 0, this data determines the characteristic class

e_{r}(G, V) ∈ Ext_{G}^{2p^{r−1}}(V^{(r)}, V^{(r)})

(and characteristic classes e_{r}(j)(G, V) ∈ Ext_{G}^{2j}(V^{(r)}, V^{(r)}) for j < p^{r}).

Moreover,

a.) If ϕ : V → W is a homomorphism of rational G-modules, then

ϕ^{(r)} · e_{r}(G, V) = e_{r}(G, W) · ϕ^{(r)} ∈ Ext_{G}^{2p^{r−1}}(V^{(r)}, W^{(r)}).

b.) e_{r}(G, V_{1} ⊕ V_{2}) = e_{r}(G, V_{1}) ⊕ e_{r}(G, V_{2}).

c.) If A′ is a commutative A-algebra, then e_{r}(G_{A′}, V_{A′}) = e_{r}(G, V)_{A′}.

Proof. Sending F ∈ P_{𝔽_{p}} to F_{A}(V) ∈(rational GL(V)-modules) is an exact functor and thereby determines

Ext_{P_{𝔽p}}^{∗} (F, F′) ⟶^{⊗_{𝔽p} A} Ext_{P_{A}}^{∗}(F_{A}, F′_{A}) → Ext_{GL(V)}^{∗}(F_{A}(V), F′_{A}(V)) →
→ Ext_{G}^{∗}(F_{A}(V), F′_{A}(V)).

We define e_{r}(G, V) to be the image of e_{r} under this map (with F = F′ = I^{(r)}).

Let

0 → I^{(r)} → T_{1} → ⋯ → T_{2p^{r−1}} → I^{(r)} → 0

be an exact sequence in P_{𝔽_{p}} representing e_{r}. Then a.) and b.) are consequences of the following commutative diagrams of rational G-modules

0 → V^{(r)} ⟶ (T_{1})_{A}(V) ⟶ ⋯ → (T_{2p^{r−1}})_{A}(V) ⟶ V^{(r)} → 0
ϕ^{(r)}↓  ^{(T_{1})_{A}(ϕ)}↓  ^{(T_{2p^{r−1}} )_{A}(ϕ)}↓  ↓_{ϕ^{(r)}}
0 → W^{(r)} ⟶ (T_{1})_{A}(W) ⟶ ⋯ → (T_{2p^{r−1}})_{A}(W) ⟶ W^{(r)} → 0

0 → V_{1}^{(r)} ⊕ V_{2}^{(r)} ⟶ (T_{1})_{A}(V_{1}) ⊕ (T_{1})_{A}(V_{2}) ⟶ ⋯ → V_{1}^{(r)} ⊕ V_{2}^{(r)} → 0
(i_{1}^{(r)},i_{2}^{(r)})↓=^{∼}  ^{((T_{1})_{A}(i_{1}),(T_{1})_{A}(i_{2}))}↓  _{(i_{1}^{(r)},i_{2}^{(r)})}↓=^{∼}
0 → (V_{1} ⊕ V_{2})^{(r)} ⟶  (T_{1})_{A}(V_{1} ⊕ V_{2})  ⟶ ⋯ → (V_{1} ⊕ V_{2})^{(r)} → 0.

The last statement follows immediately from definitions and formula (2.5.0).

Remark 3.3. Let G/k be an infinitesimal group scheme of height ≤ r. Let further A be a commutative k-algebra and V ∈ V_{A} be a rational G_{A}-module. In this case the action of G_{A} on V^{(r)} is trivial and hence Ext_{G_{A}}^{∗}(V^{(r)}, V^{(r)}) = End_{A}(V^{(r)}) ⊗_{A} H^{∗}(G_{A}, A) = End_{A}(V)^{(r)} ⊗_{A} H^{∗}(G_{A}, A) = End_{A}(V)^{(r)} ⊗_{k} H^{∗}(G, k). Assume further that the module V is free and moreover we have a fixed basis of this module, so that V may be identified with A^{n} and End_{A}(V) may be identified with the matrix algebra M_{n}(A). The A-algebra M_{n}(A)^{(r)} may be further identified with M_{n}(A), this identification takes α^{(r)} (α ∈ M_{n}(A)) to a matrix (which we also denote α^{(r)}) whose entries are obtained from those of α by raising them to the power p^{r}. In the above situation, we get an isomorphism

Ext_{G_{A}}(V^{(r)}, V^{(r)}) = M_{n}(A) ⊗_{k} H^{∗}(G, k) = M_{n}(A ⊗_{k} H^{∗}(G, k)).

Moreover this isomorphism transforms the Yoneda pairing in Ext_{G_{A}}^{∗}(V^{(r)}, V^{(r)}) to the usual matrix multiplication in M_{n}(A ⊗_{k} H^{∗}(G, k)).

The isomorphism above enables us to interpret the characteristic clesses e_{r}(j)(G_{A}, V) as matrices with coefficients in the commutative ring A ⊗_{k} H^{ev}(G, k) = H^{ev}(G_{A}, A). We shall consistently use this point of view in the next section.

Convention 3.4. We adopt the following convention to fix the generators e_{r} (uniquely defined up to scalar multiple). This convention, possibly differing from that of [F-S], is chosen to insure the validity of Proposition 4.1.

Recall (cf. [F-P]) the May spectral sequence

E_{2}^{2s,t} = H^{t}(gl_{n}, k) ⊗ S^{s}(gl_{n}^{(1)#}) ⇒ H^{2s+t}(GL_{n(1)}, k),

where H^{∗}(gl_{n}, k) denotes Lie algebra cohomology of gl_{n} = Lie(GL_{n}). The composition of the natural map gl_{n}^{(r)#} → S^{p^{r−1}}(gl_{n}^{(1)#}) and the edge homomorphism of this spectral sequence determines a class in

Hom(gl_{n}^{(r)#}, H^{2p^{r−1}}(GL_{n(1)}, k)) ≃ H^{2p^{r−1}}(GL_{n(1)}, gl_{n}^{(r)}).

We define e_{r} (i.e., specify the scalar multiple) by requiring that e_{r}(GL_{n(1)}, k^{n}) equal the preceding class. To make this definition work we have to check that (with a definition of e_{r} used in [F-S]) the class e_{r}(GL_{n(1)}, k^{n}) ∈ H^{2p^{r−1}}(GL_{n(1)}, gl_{n}^{(r)}) differs from the preceding class by a non zero constant which is independent of n and k. This follows easily from properties of e_{r} proved in [F-S]; we leave the details to the reader.

The following definition is a natural extension of the definition of a strict polynomial functor. For simplicity, we work over a field k, although the following the definition can be immediately generalized to the case of an arbitrary commutative base ring. For the remainder of this section, we set P = P_{k}.

Definition 3.5. Let k be a field and V = V_{k}. A strict polynomial bifunctor

T : V × V → V

consists of the data of T (V, W) ∈ V for each pair (V, W) ∈ V^{2} and a polynomial map

T_{V_{1},W_{1};V_{2},W 2} : Hom_{k}(V_{1}, V_{2}) × Hom_{k}(W_{1}, W_{2}) → Hom_{k}(T (V_{1}, W_{1}), T (V_{2}, W_{2})) for all (V_{1}, W_{1}; V_{2}, W_{2}) ∈ V^{4} satisfying (3.5.1) T_{V,W;V,W} (1_{V} , 1_{W}) = 1_{T (V,W)}. (3.5.2) For any (V_{1}, W_{1}; V_{2}, W_{2}; V_{3}, W_{3}) ∈ V^{6}, the following diagram of polynomial maps commutes

[V_{2}, V_{3}] × [W_{2}, W_{3}] × [V_{1}, V_{2}] × [W_{1}, W_{2}]  ⟶ [V_{1}, V_{3}] × [W_{1}, W_{3}]
T_{V_{2},W_{2};V_{3},W_{3}} ×T_{V_{1},W_{1};V_{2},W_{2}} ↓  ↓^{T_{V_{1},W_{1};V_{3},W_{3}}}
[T (V_{2}, W_{2}), T (V_{3}, W_{3})] × [T (V_{1}, W_{1}), T (V_{2}, W_{2})] ⟶ [T (V_{1}, W_{1}), T (V_{3}, W_{3})]

where we have used [−, −] to denote Hom_{k}(−, −).

Such a strict polynomial bifunctor T is said to be homogeneous of bidegree (d, d′) provided that T_{V,W;V′,W′} is homogeneous of bidegree (d, d′) for all (V, W; V′, W′) ∈ V^{4}.

We denote by P(2) the abelian category whose objects are strict polynomial bifunctors of finite degree and whose maps are the evident analogues of natural transformations. Then P(2) is a direct sum of abelian categories P(2)_{d,d′} of bifunctors of bidegree (d, d′) each with enough projectives and injectives. (Indeed, projective generators in P(2)_{d,d′} are described in the proof of Proposition 3.6 below.)

We shall consider the following constructions of bifunctors from functors:

⊙ : P × P → P(2) , (F ⊙ G)(V, W) = F (V) ⊗ G(W)

Δ : P → P(2) , ΔF (V, W) = F (V ⊗ W).

Clearly, ⊙ is biexact and Δ is exact, so that these functors induce natural maps ⊙^{∗} : Ext_{P}^{s}(F, G) ⊗ Ext_{P}^{t}(F′, G′) → Ext_{P(2)}^{s+t}(F ⊙ F′, G ⊙ G′)

Δ^{∗} : Ext_{P}^{∗}(F, G) → Ext_{P(2)}^{∗}(Δ(F), Δ(G)).

Proposition 3.6. For any F, G ∈ P_{d}, F′, G′ ∈ P_{d′},

⊙^{∗} : Ext_{P}^{∗}(F, G) ⊗ Ext_{P}^{∗}(F′, G′) → Ext_{P(2)}^{∗}(F ⊙ F′, G ⊙ G′)

is an isomorphism of graded vector spaces.

Proof. We recall the strict polynomial functor Γ^{d,n} ∈ P_{d} defined to send V ∈ V to Γ^{d}(Hom_{k}(k^{n}, V)), where Γ^{d}(W) denotes the invariants under the symmetric group Σ_{d} of the d-fold tensor power W^{⊗d} of the vector space W. Then Hom_{P}(Γ^{d,n}, F) = F (k^{n}) for any F ∈ P_{d}, so that Γ^{d,n} is projective. Furthermore, as shown in [F-S;2.10], Γ^{d,n} is a projective generator of P_{d} provided that n ≥ d. A precisely parallel argument shows that

Hom_{P(2)}(Γ^{d,n} ⊙ Γ^{d′,n′}, T) = T (k^{n}, k^{n′})

for any T ∈ P(2), so that each Γ^{d,n} ⊙ Γ^{d′,n′} is projective; moreover, if n ≥ d, n′ ≥ d′, then Γ^{d,n} ⊙ Γ^{d′,n′} is a projective generator for P(2)_{d,d′}.

If F = Γ^{d,n}and F′ = Γ^{d′,n′}, then the natural map

G(k^{n}) ⊗ G(k^{n′}) = Hom_{P}(F, G) ⊗ Hom_{P}(F′, G′) →
→ Hom_{P(2)}(F ⊙ F′, G ⊙ G′) = G(k^{n}) ⊗ G′(k^{n′})

is the identity. Similarly, if F, F′ are sums of such projective objects, then this map is an isomorphism. More generally, we choose projective resolutions P_{•} → F, P′_{•} → F′ consisting of terms which are sums of such Γ^{d,n}. Then P_{•} ⊙ P′_{•} → F ⊙ F′ is a projective resolution in P(2), so that the preceding special case together with the Künneth isomorphsim implies that

Ext_{P}^{∗}(F, G) ⊗ Ext_{P}^{∗}(G′, G′) ≃ H^{∗}(Hom_{P}(P_{•}, G) ⊗ Hom_{P}(P′_{•}, G′))
→ H^{∗}(Hom_{P(2)}(P_{•} ⊙ P′_{•}, G ⊙ G′)) ≃ Ext_{P(2)}^{∗}(F ⊙ F′, G ⊙ G′)

is also an isomorphism.

Using the isomorphism of Proposition 3.6, we can now define a natural comultiplication on E_{r} which provides it with the structure of a Hopf algebra.

Proposition 3.7. The natural map induced by the exact functors ⊙ and Δ (and the natural isomorphism ΔI^{(r)} = I^{(r)} ⊙ I^{(r)})

∇ = (⊙^{∗})^{−1} ∘ Δ^{∗} : Ext_{P}^{∗}(I^{(r)}, I^{(r)}) → Ext_{P(2)}^{∗}(I^{(r)} ⊙ I^{(r)}, I^{(r)} ⊙ I^{(r)}) ⟶^{∼}

⟶^{∼} Ext_{P}^{∗}(I^{(r)}, I^{(r)}) ⊗ Ext_{P}^{∗}(I^{(r)}, I^{(r)})

is a k-algebra homomorphism which provides E_{r}(k) = Ext_{P}^{∗}(I^{(r)}, I^{(r)}) with the structure of a Hopf algebra (with counit) over k. Furthermore, the map E_{r−1}(k) → E_{r}(k) determined by Frobenius twist is an injective map of Hopf algebras. Finally the extension of scalars isomorphism E_{r} ⊗_{𝔽_{p}} k = E_{r}(k) is compatible with the Hopf algebra structure.

Proof. The isomorphism ⊙^{∗} of Proposition 3.6 is an isomorphism of algebras when F = G, F′ = G′ provided that the product structure in Ext_{P}^{∗}(F, F) ⊗ Ext_{P}^{∗}(F′, F′) is defined using the graded tensor product (i.e., (a ⊗ a′)(b ⊗ b′) = −1^{dega′degb}ab ⊗ a′b′ for a, a′, b, b′ homogeneous elements). Since E_{r}(k) is concentrated in even degrees, ∇ is a k-algebra homomorphism.

Coassociativity of ∇ is seen by identifying both functors (Δ, 1)∘Δ, (1, Δ)∘Δ : P → P(3) with the functor sending T to τ(T) ∈ P(3) defined by τ(T)(V_{1}, V_{2}, V_{3}) = T (V_{1} ⊗ V_{2} ⊗ V_{3}), where P(3) is the evident category of polynomial trifunctors.

The counit ϵ : E_{r}(k) → k is defined to be the projection onto degree 0:

ϵ : E_{r}(k) → Hom_{P}(I^{(r)}, I^{(r)}) = Hom_{k}(I^{(r)}(k), I^{(r)}(k)) = k.

Let ψ_{L}, ψ_{R} : P(2) → P be the exact functors defined by

ψ_{L}(T)(V) = T (k, V) , ψ_{R}(T)(V) = T (V, k).

Then

(ϵ ⊗ 1) ∘ ∇ = ψ_{L}^{∗} ∘ Δ^{∗} , (1 ⊗ ϵ) ∘ ∇ = ψ_{R}^{∗} ∘ Δ^{∗} : E_{r}(k) → E_{r}(k).

On the other hand, ψ_{L} ∘ Δ = 1 = ψ_{R} ∘ Δ : P → P, so that ϵ is a counit for ∇.

The fact that Frobenius twist determines a homomorphism (injective by Theorem 3.1) of Hopf algebras follows immediately from the observation that if we define (−)^{(1)} : P(2) → P(2) by setting T^{(1)}(V, W) = T (V^{(1)}, W^{(1)}) then ⊙ and Δ commute with Frobenius twists.

Compatibility of the extension of scalars isomorphism with comultiplication is obvious.

We introduce structure constants λ_{i,j} ∈ 𝔽_{p} for E_{r} defined by the following equation

∇(e_{r}(ℓ)) = ∑_{i+j=ℓ} λ_{i,j}e_{r}(i) ⊗ e_{r}(j) ∈ E_{r} ⊗ E_{r}.  (3.8.0)

Since the Hopf algebra map given by Frobenius twist, E_{r−1} → E_{r}, sends e_{r−1}(j) to e_{r}(j) whenever j < p^{r−1}, λ_{i,j} is well defined independent of r (provided that i + j < p^{r}). In Theorem 4.6, we show that each of these structure constants λ_{i,j} is equal to 1.

Proposition 3.8. Let G, G′ be affine group schemes over a field k. Let further ρ : G → GL(V), ρ′ : G′ → GL(V′) be two finite dimensional rational representations. Then we have the following formula relating characteristic classes

e_{r}(ℓ)(G × G′, V ⊗ V′) = ∑_{i+j=ℓ} λ_{i,j}e_{r}(i)(G, V) ⊗ e_{r}(j)(G′, V′) ∈

∈ H^{2ℓ}(G × G′, End(V ⊗ V′)^{(r)}) = ⨁_{i+j=2ℓ} H^{i}(G, End(V)^{(r)}) ⊗ H^{j}(G′, End(V′)^{(r)}). (3.8.1)

Proof. Equality (3.8.1) is a consequence of the compatibility of ⊙^{∗} and the Künneth isomorphism in group cohomology, in the sense of the commutativity of the following diagram

Ext_{P}^{∗}(I^{(r)}, I^{(r)})
Δ^{∗}↓
Ext_{P}^{∗}(I^{(r)}, I^{(r)}) ⊗ Ext_{P}^{∗}(I^{(r)}, I^{(r)}) ⟶^{⊙^{∗}} Ext_{P(2)}^{∗}(I^{(r)} ⊙ I^{(r)}, I^{(r)} ⊙ I^{(r)})
↓  ↓
H^{∗}(GL(V), E) ⊗ H^{∗}(GL(V′), E′) ⟶ H^{∗}(GL(V) × GL(V′), E ⊗ E′)
↓  ↓
H^{∗}(G, E) ⊗ H^{∗}(G′, E′)  ⟶  H^{∗}(G × G′, E ⊗ E′)

where E = End(V^{(r)}), E′ = End(V^{′(r)}). This commutativity is a consequence of the fact that each of the three horizontal isomorphisms arise from the Künneth theorem for the cohomology of the tensor product of chain complexes.

§4. Explicit computations

In this section, we explicitly determine the characteristic classes e_{r}((𝔾_{a(r)})_{A}, V\underline{α}) associated to a 1-parameter subgroup exp\underline{α} : 𝔾_{a(r)} ⊗_{𝔽_{p}} A → GL_{n} ⊗_{𝔽_{p}} A. This is done in several stages; for the special case r = 1, our convention establishes an agreeable answer; for the special case r ≤ 2, a weight argument enables us to extend this answer provided that our 1-parameter subgroup is given by a single p-nilpotent matrix. This is sufficient for us to determine explicitly the Hopf algebra structure on E_{r}; we use the fact that the product exp_{α} · exp_{β} can be rewritten as exp_{α+β} · exp_{ζ}^{(1)} (cf. (4.6.0)). Finally, the knowledge of this Hopf algebra structure facilitates an induction argument to complete the determination.

Let A be a commutative 𝔽_{p}-algebra and let \underline{α} = (α_{0}, ..., α_{l−1}) ∈ M_{n}(A)^{×l} be an l-tuple of pairwise commuting p-nilpotent n × n matrices. For every r > 0, this data defines a representation

𝔾_{a(r)}^{×l} ⊗_{𝔽_{p}} A → GL_{n} ⊗_{𝔽_{p}} A : (t_{0}, ..., t_{l−1}) ↦ exp_{α_{0}}(t_{0}) · ... · exp_{α_{l−1}}(t_{l−1}).

This representation makes A^{n} into a rational (𝔾_{a(r)}^{×l})_{A}-module, which we denote W\underline{α}. Furthermore, for every r ≥ l we may consider the corresponding representation

exp\underline{α} : 𝔾_{a(r)} ⊗_{𝔽_{p}} A → GL_{n} ⊗_{𝔽_{p}} A : t ↦ exp_{α_{0}}(t) · ... · exp_{α_{l−1}}(t^{p^{l−1}}).

We denote by V\underline{α} the free A-module A^{n} made into a rational 𝔾_{a(r)} ⊗_{𝔽_{p}} A-module via this representation. Clearly V\underline{α} is a pull-back of the rational module W\underline{α} under the homomor-

phism 𝔾_{a(r)} ⟶^{1×F ×...×F^{l−1}} 𝔾_{a(r)}^{×l}.

Throughout this section k denotes a field of characteristic p > 0 and A denotes an arbitrary commutative 𝔽_{p}-algebra.

Proposition 4.1. Consider the representation exp_{α} : 𝔾_{a(1)} → GL_{n} determined by the p-nilpotent matrix α ∈ M_{n}(k) and let V_{α} denote the associated rational 𝔾_{a(1)}-module. Then

e_{r}(𝔾_{a(1)}, V_{α}) = x_{1}^{p^{r−1}} ⊗ α^{(r)} ∈ H^{2p^{r−1}}(𝔾_{a(1)}, M_{n}(k)^{(r)}) =
= H^{2p^{r−1}}(𝔾_{a(1)}, k) ⊗_{k} M_{n}(k)^{(r)} = H^{2p^{r−1}}(𝔾_{a(1)}, k) ⊗_{k} M_{n}(k),

where x_{1} ∈ H^{2}(𝔾_{a(1)}, k) is the canonical generator of H^{∗}(𝔾_{a(1)}, k) (see Theorem 1.13).

Proof. The construction of the May spectral sequence implies that the element x_{1} ∈ H^{2}(𝔾_{a(1)}, k) coincides with the image of t^{(1)} ∈ g_{a}^{(1)#} under the edge homomorphism of the May spectral sequence, where k[𝔾_{a(1)}] = k[T]/T^{p} and t denotes the class of T in Lie(𝔾_{a})^{#} = (T)/(T^{2}). Let {E^{i,j}} denote the basis of M_{n}(k)^{#} dual to the usual basis {E_{i,j}} of M_{n}(k). The naturality of the May spectral sequence implies the commutativity of the following diagram

M_{n}(k)^{(r)#} ⟶ S^{p^{r−1}}(M_{n}(k)^{(1)#}) ⟶ H^{2p^{r−1}}(GL_{n(1)}, k)
α^{(r)#}↓  _{S^{p^{r−1}} (α^{(1)#})}↓  ↓^{exp_{α}^{∗}}
g_{a}^{(r)#} ⟶ S^{p^{r−1}}(g_{a}^{(1)#}) ⟶ H^{2p^{r−1}}(𝔾_{a(1)}, k).

The top row coincides (according to the Convention 3.4) with the map gl_{n}^{(r)#} → H^{2p^{r−1}}(GL_{n(1)}, k)

corresponding to e_{r}(GL_{n(1)}, k^{n}). The bottom row sends t^{(r)} to x_{1}^{p^{r−1}} by the preceding remark. Thus the map M_{n}(k)^{(r)#} → H^{2p^{r−1}}(𝔾_{a(1)}, k) corresponding to e_{r}(𝔾_{a(1)}, V_{α}) sends

(E^{i,j})^{(r)} to (α_{ij})^{p^{r}}x_{1}^{p^{r−1}}.

The following lemma establishes the effect of Frobenius on our characteristic classes.

Lemma 4.2. Let F^{∗} : H^{∗}(𝔾_{a(ℓ−1)}, k) → H^{∗}(𝔾_{a(ℓ)}, k) denote the map induced by Frobenius. For any 0 < ℓ ≤ r and any ℓ-tuple \underline{α} = (α_{0}, ..., α_{ℓ−1}) ∈ M_{n}(k)^{×ℓ} of commuting p-nilpotent matrices,

a.) e_{r}(j)(𝔾_{a(ℓ)}, V_{(0,α_{1},...α_{ℓ−1})}) = F^{∗}(e_{r}(j)(𝔾_{a(ℓ−1)}, V_{(α_{1},...,α_{ℓ−1})})).

b.) If j < p^{r−1}, then

e_{r}(j)(𝔾_{a(ℓ)}, V_{(α_{0},...,α_{ℓ−1})}) = F^{∗}(e_{r−1}(j)(𝔾_{a(ℓ−1)}, V_{(α_{0}^{(1)},...,α_{ℓ−2}^{(1)})})).

Proof. Assertion (a.) follows from the definition of e_{r}(j)(G, V) and the fact that exp_{(0,α_{1},...,α_{ℓ−1})} = exp_{(α_{1},...,α_{ℓ−1})} ∘ F : 𝔾_{a(ℓ)} → GL_{n}. To prove the assertion (b.), it suffices to establish the commutativity of the following diagram:

M_{n}(k)^{#} ⟶^{e_{r−1}(j)} H^{2j}(GL_{n(r−1)}, k) ⟶^{exp_{(α{(1) \atop 0},...,α{(1) \atop ℓ−2})}^{∗}} H^{2j}(𝔾_{a(ℓ−1)}, k)
↓^{=}  ↓_{F^{∗}}  ↓_{F^{∗}}
M_{n}(k)^{#} ⟶^{e_{r}(j)} H^{2j}(GL_{n(r)}, k) ⟶^{exp_{(α_{0},...,α_{ℓ−1})}^{∗}} H^{2p^{r−1}}(𝔾_{a(ℓ)}, k).

Commutativity of the right square follows from the obvious relation exp_{(α_{0}^{(1)},...,α_{ℓ−2}^{(1)})} ∘ F = F ∘ exp_{(α_{0},...,α_{ℓ−1})} : 𝔾_{a(ℓ)} → GL_{n}. To check the commutativity

of the left square note first of all that e_{r}(j) = e_{r−1}(j)^{(1)} (for j < p^{r−1}). Let

0 → I^{(r−1)} ⟶^{d} T_{1} ⟶^{d} ... ⟶^{d} T_{2j} ⟶^{d} I^{(r−1)} → 0

be an extension of polynomial functors representing e_{r−1}(j). The extension class e_{r}(j) = e_{r−1}(j)^{(1)} is represented by the extension

0 → I^{(r)} ⟶^{d^{(1)}} T_{1}^{(1)} ⟶^{d^{(1)}} ... ⟶^{d^{(1)}} T_{2j}^{(1)} ⟶^{d^{(1)}} I^{(r)} → 0

(where T^{(1)} = T ∘ I^{(1)} for any T ∈ P) and hence e_{r}(j)(GL_{n(r)}, k^{n}) is represented by the extension of rational GL_{n(r)}-modules

0 → (k^{n})^{(r)} ⟶^{d(k^{n(1)})} T_{1}(k^{n(1)}) ⟶^{d(k^{n(1)})} ... ⟶^{d(k^{n(1)})} T_{2j}(k^{n(1)}) ⟶^{d(k^{n(1)})} (k^{n})^{(r)} → 0.

For any polynomial functor T ∈ P, the vector space T (k^{n(1)}) may be identified with T (k^{n}) and the representation of GL_{n} in T (k^{n(1)}) coincides with the composition of the representation of GL_{n} in T (k^{n}) and the Frobenius endomorphism of GL_{n}. This shows that the above extension of rational GL_{n(r)}-modules is the pull-back via the Frobenius

homomorphism GL_{n(r)} ⟶^{F} GL_{n(r−1)} of the extension representing e_{r−1}(j)(GL_{n(r−1)}, k^{n}).

The following lemma reduces our computations to the special case A = 𝔽_{p} and the very special representation given as the external tensor product of p × p Jordan blocks. This is particularly useful in that Proposition 3.8 applies to analyze e_{r}(𝔾_{a(ℓ)}^{×i} ⊗_{𝔽_{p}} A, W_{(α_{1},...,α_{i})}).

Lemma 4.3. Let P (X_{1}, ..., X_{i}) be a polynomial in i variables with coefficients in the cohomology ring H^{ev}(𝔾_{a(ℓ)}^{×i}, 𝔽_{p}) . Let further r ≥ ℓ be an integer. Assume that the formula

e_{r}(𝔾_{a(ℓ)}^{×i} ⊗_{𝔽_{p}} A, W_{(α_{1},...,α_{i})}) = P (α_{1}^{(r)}, ..., α_{i}^{(r)})  (4.3.1)

holds in the following special case: A = 𝔽_{p};

α_{1} = α ⊗ 1 ⊗ ... ⊗ 1 , …, α_{i} = 1 ⊗ 1 ⊗ ... ⊗ α,

where α is a p × p Jordan block. Then the above formula holds for any A and any i-tuple of commuting p-nilpotent matrices.

Proof. To give an i tuple of commuting p-nilpotent matrices α_{1}, ..., α_{i} ∈ M_{n}(A) is equivalent to making A^{n} into a module over the A-algebra

B = A[Y_{1}, ..., Y_{i}]/(Y_{1}^{p}, ..., Y_{i}^{p}) = A[Y_{1}]/Y_{1}^{p} ⊗_{A} ... ⊗_{A} A[Y_{i}]/Y_{i}^{p}.

The set of matrices corresponding to the free rank one B-module B is precisely the above special set {α_{1}, …, α_{i}}. Thus our condition and Proposition 3.2 c.) show that the formula in question holds for a free B-module of rank one. Proposition 3.2 b) implies that the formula holds for any free B-module of finite rank. Finally we observe that any finitely generated B-module is a quotient of a free B-module of finite rank. So to end the proof, it suffices to establish the following fact 4.3.2. Let α_{1}, ..., α_{i} ∈ M_{n}(A) and β_{1}, ..., β_{i} ∈ M_{m}(A) be two i-tuples of commuting pnilpotent matrices. Assume further that the equality (4.3.1) holds for β and that there exists a surjective A-linear map ϕ : A^{m} → A^{n} such that α_{j}ϕ = ϕβ_{j} for all j = 1, ..., i. Then the formula (4.3.1) holds for α as well.

Proof. To simplify the notation, denote the cohomology ring H^{ev}(𝔾_{a(ℓ)}^{×i}, 𝔽_{p}) by H. Since ϕ is surjective, the induced map

End_{A}((A^{n})^{(r)}) ⊗_{𝔽_{p}} H ⟶^{(ϕ^{(r)})^{∗}⊗1} Hom_{A}((A^{m})^{(r)}, (A^{n})^{(r)}) ⊗_{𝔽_{p}} H

is injective. Thus it suffices to check that both sides of the formula (4.3.1) give the same result being composed with ϕ^{(r)} on the right. However according to Proposition 3.2 a) we have:

e_{r}(𝔾_{a(ℓ)}^{×i} ⊗_{𝔽_{p}} A, W\underline{α}) · ϕ^{(r)} = ϕ^{(r)} · e_{r}(𝔾_{a(ℓ)}^{×i} ⊗_{𝔽_{p}} A, W\underline{β}) = ϕ^{(r)} · P (β_{1}^{(r)}, ..., β_{i}^{(r)}) =

= P (α_{1}^{(r)}, ...α_{i}^{(r)}) · ϕ^{(r)}.

The next proposition is the extension of Proposition 4.1 to 𝔾_{a(2)}.

Proposition 4.4. Consider the representation exp_{α} : 𝔾_{a(2)} → GL_{n} determined by the p-nilpotent matrix α ∈ M_{n}(k) and let V_{α} denote the associated rational 𝔾_{a(2)}-module. Then for any r ≥ 2

e_{r}(𝔾_{a(2)}, V_{α}) = x_{1}^{p^{r−1}} ⊗ α^{(r)} ∈ H^{2p^{r−1}}(𝔾_{a(2)}, k) ⊗_{k} M_{n}(k)^{(r)}.

Proof. By Lemma 4.3 it suffices to consider α to be a p × p Jordan block. We may also assume (extending the scalars if necessary) that the field k is infinite.

Let T ⊂ GL_{p} be the rank-1 split torus whose k-rational points are matrices of the form d_{s} ≡ diag(1, s, …, s^{p−1}) (s ∈ k^{∗}). So defined, T stabilizes exp_{α} since d_{s}αd_{s}^{−1} = s^{−1}α and hence d_{s}exp_{α}d_{s}^{−1} = exp_{α} ∘ s^{−1}. Since the cohomology class e_{r} ∈ H^{2p^{r−1}}(GL_{p}, M_{p}(k)^{(r)}) is invariant under the action of GL_{p} by conjugation on GL_{p} and M_{p}(k), we conclude that the class e_{r}(𝔾_{a(2)}, V_{α}) is invariant under the action of T (where d_{s} acts on M_{p}(k)^{(r)} by conjugation and on 𝔾_{a(2)} by multiplication by s^{−1}).

We write

e_{r}(𝔾_{a(2)}, V_{α}) = ∑_{i,j=1}^{p} x_{i,j} ⊗ E_{i,j}^{(r)} ∈ H^{2p^{r−1}}(𝔾_{a(2)}, M_{n}(k)^{(r)}) =

= H^{2p^{r−1}}(𝔾_{a(2)}, k) ⊗_{k} M_{p}(k)^{(r)}

for some elements x_{i,j} ∈ H^{2p^{r−1}}(𝔾_{a(2)}, k). As d_{s}E_{i,j}^{(r)}d_{s}^{−1} = s^{p^{r}(i−j)}E_{i,j}^{(r)}, we must have (d_{s^{−1}})^{∗}(x_{i,j}) = s^{−p^{r}(i−j)}x_{i,j} = s^{p^{r}(j−i)}x_{i,j} for all i, j. In other words, x_{i,j} ∈ H^{2p^{r−1}}(𝔾_{a(2)}, k) must have weight p^{r}(j − i) with respect to the standard action of 𝔾_{m} on 𝔾_{a(2)}.

Sublemma 4.4.1. Let r ≥ ℓ > 0 be a pair of integers. The only monomial of weight p^{r} in H^{2p^{r−1}}(𝔾_{a(ℓ)}, k) (with respect to the standard action of 𝔾_{m} on 𝔾_{a(ℓ)}) is x_{1}^{p^{r−1}}. If ℓ ≤ 2 then there are no monomials in H^{2p^{r−1}}(𝔾_{a(ℓ)}, k) of weight p^{r}m with 1 < m < p.

Proof. Consider an arbitrary monomial x = x_{1}^{n_{1}} ·...·x_{ℓ}^{n_{ℓ}} ·λ_{1}^{ϵ_{1}} ·...·λ_{ℓ}^{ϵ_{ℓ}} (0 ≤ ϵ_{j} ≤ 1). Its degree equals 2(n_{1}+...+n_{ℓ})+(ϵ_{1}+...+ϵ_{ℓ}) and its weight equals n_{1}p+...+n_{ℓ}p^{ℓ}+ϵ_{1}+ϵ_{2}p+...+ϵ_{ℓ}p^{ℓ−1}. Since w(x) has to be divisible by p we conclude that ϵ_{1} = 0. Now it’s straightforward to check that w(x) ≥ \frac{p}{2}deg(x) and the equality holds only if ϵ_{2} = ... = ϵ_{ℓ} = n_{2} = ... = n_{ℓ} = 0. This proves the first statement. The second statement is trivial for ℓ = 1. Assume that ℓ = 2 and x = x_{1}^{n_{1}}x_{2}^{n_{2}}λ_{1}^{ϵ_{1}}λ_{2}^{ϵ_{2}} has degree 2p^{r−1} and weight p^{r−1}m with 1 < m < p. As before ϵ_{1} = 0 and since deg(x) has to be even we conclude that ϵ_{2} = 0. Thus we have a system of equations n_{1} + n_{2} = p^{r−1}, n_{1}p + n_{2}p^{2} = p^{r}m. From which we conclude that n_{2}(p − 1) = p^{r−1}(m − 1) and thus m − 1 ≡ 0 mod p − 1 which gives us a contradiction.

Lemma 4.4.1 shows that x_{i,j} = 0 unless j = i + 1 and x_{i,i+1} = c_{i}x_{1}^{p^{r−1}} for some c_{i} ∈ k. Restricting the class e_{r}(𝔾_{a(2)}, V_{α}) to 𝔾_{a(1)} and applying Proposition 4.1 we conclude that

each c_{i} = 1 and so e_{r}(𝔾_{a(2)}, V_{α}) = ∑_{i=1}^{p−1} x_{1}^{p^{r−1}} ⊗ E_{i,i+1}^{(r)} = x_{1}^{p^{r−1}} ⊗ α^{(r)}.

We can now apply Lemma 4.3 to get an almost complete understanding of charactersitic classes for 𝔾_{a(2)}.

Corollary 4.5. a.) Let α ∈ M_{n}(k) be a p-nilpotent matrix. Then for any r ≥ 2 and any 0 ≤ j < p^{r}

e_{r}(j)(𝔾_{a(2)}, V_{α}) = \frac{x_{r}^{j_{0}} · ... · x_{1}^{j_{r−1}p^{r−1}}}{j_{0}! · ... · j_{r−1}!} ⊗ (α^{(r)})^{s(j)}

where j = j_{0} + j_{1}p + ... + j_{r−1}p^{r−1} (0 ≤ j_{i} < p) is the p-adic expansion of j and s(j) = j_{0} + ... + j_{r−1} is the sum of p-adic digits of j. In particular, e_{r}(j)(𝔾_{a(2)}, V_{α}) = 0 unless j ≡ 0 mod p^{r−2} since x_{i} = 0 ∈ H^{∗}(G_{a(2)}, k) whenever i > 2. b.) Let α, β ∈ M_{n}(k) be a pair of commuting p-nilpotent matrices. Then for any r ≥ 2

e_{r}(𝔾_{a(2)}^{×2}, W_{(α,β)}) = (x_{1}^{p^{r−1}} ⊗ 1) ⊗ α^{(r)} + (1 ⊗ x_{1}^{p^{r−1}}) ⊗ β^{(r)}+

+ ∑_{i+j=p,0≠i≠p} λ(ip^{r−2}, jp^{r−2})\frac{x_{2}^{ip^{r−2}} ⊗ x_{2}^{jp^{r−2}}}{i! · j!} ⊗ (α^{(r)})^{i}(β^{(r)})^{j}.

Proof. According to the definition of e_{r}(j), we have

e_{r}(j)(𝔾_{a(2)}, V_{α}) = \frac{e_{1}^{(r−1)}(𝔾_{a(2)}, V_{α})^{j_{0}} · ... · e_{r}(𝔾_{a(2)}, V_{α})^{j_{r−1}}}{j_{0}! · . . . · j_{r−1}!} .

Proposition 4.4, Proposition 4.1, and Lemma 4.2 b) show that

e_{i}^{(r−i)}(𝔾_{a(2)}, V_{α}) = x_{r+1−i}^{p^{i−1}} ⊗ α^{(r)} (= 0 if i < r − 1).

This proves the first part of our statement. Lemma 4.3 shows that in proving the second part we may assume that α = γ ⊗ 1, β = 1 ⊗ γ, where γ is the p × p Jordan block. In this case W_{(α,β)} = V_{γ} ⊗ V_{γ}, so that our statement follows from a.) and Proposition 3.8.

If α and β are two commuting p-nilpotent matrices in M_{n}(k), then for any s we have the equality

exp(s · α) · exp(s · β) = exp(s · (α + β)) · exp(s^{p} · ξ)

where ξ = ∑_{i+j=p,0≠i≠p} \frac{α^{i}β^{j}}{i!j!} . Thus, we conclude

exp_{α} · exp_{β} = exp_{α+β} · exp_{ξ}^{(1)} : 𝔾_{a(r)} → GL_{n}.  (4.6.0)

This simple observation in conjunction with Corollary 4.5 enables us to determine the structure constants λ_{i,j} of the Hopf algebra E_{r}.

Theorem 4.6. E_{r} is isomorphic as a Hopf algebra to the dual (𝔽_{p}[X]/X^{p^{r}})^{#} of the primitively generated truncated polynomial algebra 𝔽_{p}[X]/X^{p^{r}}. Furthermore, the structure constants λ_{i,j} of (3.8.0) all equal 1; in particular,

∇(e_{r}) = ∑_{i+j=p^{r−1}} e_{r}(i) ⊗ e_{r}(j).

Proof. Recall that (𝔽_{p}[X]/X^{p^{r}})^{#} = 𝔽_{p}[u_{0}, ..., u_{r−1}]/(u_{0}^{p}, ..., u_{r−1}^{p}) (see Corollary 1.4). Theorem 3.1 shows that sending u_{i} to e_{i+1}^{(r−i−1)} defines an isomorphism of algebras ϕ_{r} :

(𝔽_{p}[X]/X^{p^{r}})^{#} ⟶^{∼} E_{r}. This isomorphism takes v_{j} (as in Corollary 1.4) to e_{r}(j) for all 0 ≤ j < p^{r}. To show that ϕ_{r} is an isomorphism of Hopf algebras, it suffices (since ∇(v_{ℓ}) = ∑_{i+j=ℓ} v_{i} ⊗ v_{j}) to show that all structure constants λ_{i,j} are equal to 1. To prove this, we proceed by induction on r. For r = 0 there’s nothing to prove. Assume now that the statement is true for r − 1 and consider the case of E_{r}. Our induction hypothesis and the fact that λ_{i,j} is independent of r (provided that i + j < p^{r}) show that λ_{i,j} = 1 provided that i + j < p^{r−1}. Let α, β ∈ M_{n}(𝔽_{p}) be a pair of commuting p-nilpotent matrices. Consider the representation exp_{α} · exp_{β} : 𝔾_{a(2)} → GL_{n}. Denote by U the vector space 𝔽_{p}^{n} made into a rational 𝔾_{a(2)}-module via this representation. Note that U is the pull-back of the representation W_{(α,β)} under the diagonal embedding Δ : 𝔾_{a(2)} → 𝔾_{a(2)}^{×2}. On the other hand, (4.6.0) shows that U is a pull-back of the representation W_{(α+β,ξ)} under the

homomorphism 𝔾_{a(2)} ⟶^{(1,F)} 𝔾_{a(2)}^{×2}. These remarks and Corollary 4.5 give us two ways to compute e_{r}(𝔾_{a(2)}, U). Comparing the results we get the following relation

x_{1}^{p^{r−1}} ⊗ (α^{(r)} + β^{(r)}) + x_{2}^{p^{r−1}} ⊗ ∑_{i+j=p,0≠i≠p} \frac{λ(ip^{r−2}, jp^{r−2})}{i! · j!} · (α^{(r)})^{i}(β^{(r)})^{j} =

= x_{1}^{p^{r−1}} ⊗ (α + β)^{(r)} + x_{2}^{p^{r−1}} ⊗ ξ^{(r)}.

Since x_{2}^{p^{r−1}} ≠ 0, we conclude from the previous formula that

∑_{i+j=p,0≠i≠p} \frac{λ(ip^{r−2}, jp^{r−2}) − 1}{i! · j!}α^{i}β^{j} = 0.  (4.6.1)

Taking α and β to be multiplication by x and y in the ring B = 𝔽_{p}[x, y]/(x^{p}, y^{p}) (in which case the monomials α^{i}β^{j} with i, j < p are linearly independent over 𝔽_{p}) we conclude immediately that all coefficients in (4.6.1) are zero, i.e. λ(ip^{r−2}, jp^{r−2}) = 1 whenever i + j = p, 0 ≠ i ≠ p.

Next we show that λ(i, j) = 1 whenever i + j = p^{r−1}. To do so, we note that coassociativity of E_{r} gives the following relation on λ :

λ(a, b) · λ(a + b, c) = λ(a, b + c) · λ(b, c) (0 ≤ a, b, c; a + b + c < p^{r})

Assume first that i ≥ p^{r−2}. Taking in the above relation a = p^{r−2}, b = i − p^{r−2}, c = j and taking into account that λ(p^{r−2}, (p − 1)p^{r−2}) = 1 and also λ(a, b) = λ(b, c) = 1 since a + b < p^{r−1}, b + c < p^{r−1} we conclude that λ(i, j) = 1. If i < p^{r−2}, then j ≥ p^{r−2} and a similar argument works.

Consider the diagram

(𝔽_{p}[u_{0}, ..., u_{r−1}]/(u_{0}^{p}, ..., u_{r−1}^{p}) ⟶^{∇} (𝔽_{p}[u_{0}, ...u_{r−1}]/(u_{0}^{p}, ..., u_{r−1}^{p}))^{⊗2}
=^{∼}↓^{ϕ_{r}}  =^{∼}↓^{ϕ_{r}^{⊗2}}
E_{r}  ⟶^{∇}  E_{r}^{⊗2}.

The above computation shows that the two images of u_{r−1} in E_{r}^{⊗2} coincide. The same being true for u_{0}, ..., u_{r−2} in view of the induction hypothesis we conclude that ϕ_{r} is an isomorphism of Hopf algebras and λ_{i,j} = 1 for all i, j such that i + j < p^{r}.

Using Theorem 4.6, we can now complete our determination of the characteristic classes e_{r}(j)(𝔾_{a(ℓ)}, V_{α}) associated to a 1-parameter subgroup given by a single p-nilpotent matrix α.

Theorem 4.7. For any 0 < ℓ ≤ r, any 0 ≤ j < p^{r} and any p-nilpotent matrix α ∈ M_{n}(k),

e_{r}(j)(𝔾_{a(ℓ)}, V_{α}) = \frac{x_{r}^{j_{0}} ⋯ x_{1}^{p^{r−1}j_{r−1}}}{(j_{0}!) ⋯ (j_{r−1}!)} ⊗ (α^{(r)})^{s(j)}

where j = j_{0} + j_{1}p + ⋯ + j_{r−1}p^{r−1} (0 ≤ j_{i} < p) is the p-adic expansion of j and where s(j) = ∑ j_{i} is the sum of p-adic digits of j.

Proof. The cases ℓ = 1, 2 have been verified in Propositions 4.1 and 4.4. In dealing with the general case, we may assume, using induction on ℓ, that the theorem holds for ℓ − 1.

Assume first that j < p^{r−1} (so that j_{r−1} = 0). In this case induction and Lemma 4.2 b) show

e_{r}(j)(𝔾_{a(ℓ)}, V_{α}) = F^{∗}(e_{r−1}(j)(𝔾_{a(ℓ−1)}, V_{α^{(1)}})) = F^{∗}(\frac{x_{r−1}^{j_{0}} · ... · x_{1}^{j_{r−2}p^{r−2}}}{j_{0}! · ... · j_{r−2}!} )⊗

⊗ (α^{(r)})^{s(j)} = \frac{x_{r}^{j_{0}} · ... · x_{1}^{j_{r−1}p^{r−1}}}{j_{0}! · ... · j_{r−1}!} ⊗ (α^{(r)})^{s(j)}.

To settle the general case, it suffices now to prove the formula

e_{r}(𝔾_{a(ℓ)}, V_{α}) = x_{1}^{p^{r−1}} ⊗ α^{(r)}.

Moreover Lemma 4.3 shows that we may assume that α is a p × p Jordan block. As in the proof of Proposition 4.4, we may write

e_{r}(𝔾_{a(ℓ)}, V_{α}) = ∑_{1≤i<j<p} x_{i,j} ⊗ E_{i,j}^{(r)} ∈ H^{2p^{r−1}}(𝔾_{a(ℓ)}, k) ⊗_{k} M_{p}(k)^{(r)}  (4.7.1)

for some x_{i,j} ∈ H^{2p^{r−1}}(𝔾_{a(ℓ)}, 𝔽_{p}) of weight p^{r}(j − i) with respect to the standard action of 𝔾_{m} on 𝔾_{a(ℓ)}. The same reasoning as in the proof of the Proposition 4.4 shows that

x_{i,i+1} = x_{1}^{p^{r−1}}. For ℓ > 2, one cannot simply argue by weights that x_{i,j} = 0 for j > i + 1 so we proceed in a different way.

Since the cohomology class e_{r} ∈ H^{2p^{r−1}}(GL_{p}, M_{p}(k)^{(r)}) is invariant under the action of GL_{p} by conjugation on GL_{p} and M_{p}(k)^{(r)}, we conclude that the class e_{r}(𝔾_{a(ℓ)}, V_{α}) is invariant under the action of 𝔾_{a} (where the action of 𝔾_{a} on M_{p}(k) is the composition of the standard action by conjugation of GL_{p} and the homomorphism exp_{α} : 𝔾_{a} → GL_{p} and the action of 𝔾_{a} on 𝔾_{a(ℓ)} is trivial). This remark implies immediately that the matrix

(x_{i,j})_{1≤i,j≤p} = e_{r}(𝔾_{a(ℓ)}, V_{α}) ∈ H^{2p^{r−1}}(𝔾_{a(ℓ)}, k) ⊗ M_{p}(k)^{(r)} ⊂
⊂ M_{p}(H^{ev}(𝔾_{a(ℓ)}, k))

commutes with α^{(r)} = α, i.e. x_{i,j} = x_{i′,j′} whenever j − i = j′ − i′. Thus we may rewrite (4.7.1) as follows:

e_{r}(𝔾_{a(ℓ)}, V_{α}) = ∑_{m=1}^{p−1} y_{m} ⊗ (α^{(r)})^{m}  (4.7.2)

where y_{m} ∈ H^{2p^{r−1}}(𝔾_{a(ℓ)}, 𝔽_{p}) has weight p^{r}m and y_{1} = x_{1}^{p^{r−1}}. Note that the formula (4.7.2), established above in case α is a p×p Jordan block actually holds for any p-nilpotent matrix α in view of Lemma 4.3.

Let α ∈ M_{n}(k), β ∈ M_{m}(k) be a pair of p-nilpotent matrices. Proposition 3.8 and Theorem 4.6 give the following formula:

e_{r}(𝔾_{a(ℓ)}^{×2}, V_{α} ⊗ V_{β}) = e_{r}(𝔾_{a(ℓ)}, V_{α}) ⊗ 1 + 1 ⊗ e_{r}(𝔾_{a(ℓ)}, V_{β})+

+ ∑_{i+j=p^{r−1},i≠0≠j} e_{r}(i)(𝔾_{a(ℓ)}, V_{α}) ⊗ e_{r}(j)(𝔾_{a(ℓ)}, V_{β}).

The terms in the above formula corresponding to i ≠ 0 ≠ j are already known; moreover the corresponding term can be non zero only if s(i) < p, s(j) < p. Finally it’s easy to see that if i + j = p^{r−1} and s(i) < p, s(j) < p then i, j ≡ 0 mod p^{r−2}. This remark, Lemma 4.3 and formula (4.7.2) give (4.7.3) Let α, β ∈ M_{n}(k) be a pair of commuting p-nilpotent matrices. Then the following formula holds

e_{r}(𝔾_{a(ℓ)}^{×2}, W_{(α,β)}) = ∑_{m=1}^{p−1}(y_{m} ⊗ 1) ⊗ (α^{(r)})^{m} + ∑_{m=1}^{p−1}(1 ⊗ y_{m}) ⊗ (β^{(r)})^{m}+

+ ∑_{i+j=p,i≠0≠j} \frac{x_{2}^{ip^{r−2}} ⊗ x_{2}^{jp^{r−2}}}{i! · j!} ⊗ (α^{(r)})^{i} · (β^{(r)})^{j}.

Let U be a rational 𝔾_{a(ℓ)}-module corresponding to the representation exp_{α} · exp_{β} = exp_{α+β} · exp_{ξ}^{(1)} (see (4.6.0)). Proceeding in the same way as in the proof of Theorem 4.6, we get two answers for e_{r}(𝔾_{a(ℓ)}, U):

e_{r}(𝔾_{a(ℓ)}, U) = ∑_{m=1}^{p−1} y_{m} ⊗ ((α^{(r)})^{m} + (β^{(r)})^{m}) + ∑_{i+j=p,0≠i≠p} \frac{x_{2}^{p^{r−1}}}{i! · j!} ⊗ (α^{(r)})^{i} · (β^{(r)})^{j}

e_{r}(𝔾_{a(ℓ)}, U) = ∑_{m=1}^{p−1} y_{m} ⊗ ((α + β)^{(r)})^{m} + ∑_{m=1}^{p−1} F^{∗}(y_{m}) ⊗ (ξ^{(r)})^{m}+

+ ∑_{i+j=p,0≠i≠p} \frac{x_{2}^{ip^{r−2}} · x_{3}^{jp^{r−2}}}{i! · j!} ⊗ ((α + β)^{i} · ξ^{j})^{(r)}.

Note that (α + β)^{i} · ξ^{j} is a homogeneous polynomial in α, β of degree i + pj = (i + j) + (p − 1)j ≥ 2p − 1 and hence equals zero. The same reasoning shows that ξ^{2} = 0. Thus the last formula simplifies to

e_{r}(𝔾_{a(ℓ)}, U) = ∑_{m=0}^{p−1} y_{m} ⊗ ((α + β)^{(r)})^{m} + x_{2}^{p^{r−1}} ⊗ ξ^{(r)}.

Comparing the two answers, we get

∑_{m=2}^{p−1} y_{m} ⊗ ((α + β)^{m} − α^{m} − β^{m})^{(r)} = 0.

Taking now α (resp. β) to be multiplication by x (resp. y) in the ring B = k[x, y]/(x^{p}, y^{p}) (in which case the monomials α^{i} · β^{j} with 0 ≤ i < p, 0 ≤ j < p are linearly independent over k) we conclude immediately that y_{m} = 0 for m > 1.

Corollary 4.8. Let \underline{α} = (α_{0}, …, α_{ℓ−1}) ∈ M_{n}(A)^{×ℓ} be an ℓ-tuple of p-nilpotent, pairwise commuting n × n matrices over a commutative 𝔽_{p}-algebra A. Then for any 0 ≤ j < p^{r}

e_{r}(j)((𝔾_{a(r)}^{×ℓ})_{A}, W\underline{α}) = ∑_{j_{0}+...+j_{ℓ−1}=j_{0≤j_{0},...,j_{ℓ−1}}} \frac{x_{r}^{j_{0,0}} · ... · x_{1}^{p^{r−1}j_{0,r−1}}}{j_{0,0}! · ... · j_{0,r−1}!} ⊗ ...⊗

⊗ \frac{x_{r}^{j_{ℓ−1,0}} · ... · x_{1}^{p^{r−1}j_{ℓ−1,r−1}}}{j_{ℓ−1,0}! · ... · j_{ℓ−1,r−1}!} ⊗ (α_{0}^{(r)})^{s(j_{0})} · ... · (α_{ℓ−1}^{(r)})^{s(j_{ℓ−1})} (4.8.0)

where j_{i,0}, ..., j_{i,r−1} are p-adic digits of j_{i} and s(j_{i}) = j_{i,0} + ... + j_{i,r−1} is their sum.

Proof. Lemma 4.3 shows that it suffices to consider the case when A = 𝔽_{p} and

α_{0} = α ⊗ ... ⊗ 1 , …, α_{ℓ−1} = 1 ⊗ ... ⊗ α

where α is a p × p Jordan block. In this case, our statement follows from Proposition 3.8, the determination that λ_{i,j} = 1 in Theorem 4.6, and Theorem 4.7.

We now present our explicit computation of the characteristic classes e_{r}(j)((𝔾_{a(r)})_{A}, V\underline{α}).

Theorem 4.9. Let \underline{α} = (α_{0}, ..., α_{r−1}) ∈ M_{n}(A)^{×r} be an r-tuple of pairwise commuting p-nilpotent matrices over a commutative 𝔽_{p}-algebra A. Then for any 0 ≤ j < p^{r}

e_{r}(j)((𝔾_{a(r)})_{A}, V\underline{α}) = ∑_{j_{0}+pj_{1}+...+p^{r−1}j_{r−1}=j_{0≤j_{0},...,j_{r−1}}} \frac{x_{r}^{j_{0,0}} · ... · x_{1}^{p^{r−1}j_{0,r−1}}}{j_{0,0}! · ... · j_{0,r−1}!} ·

· \frac{x_{r}^{pj_{1,0}} · ... · x_{2}^{p^{r−1}j_{1,r−2}}}{j_{1,0}! · ... · j_{1,r−2}!} · ... · \frac{x_{r}^{p^{r−1}j_{r−1,0}}}{j_{r−1,0}!} ⊗ (α_{0}^{(r)})^{s(j_{0})} · ... · (α_{r−1}^{(r)})^{s(j_{r−1})}

where j_{i,o}, ..., j_{i,r−1−i} are p-adic digits of j_{i} (note that j_{i} < p^{r−i}) and s(j_{i}) = j_{i,o} + ... + j_{i,r−1−i} is their sum.

Proof. The characteristic class e_{r}(j)((𝔾_{a(r)})_{A}, V\underline{α}) coincides with the pull-back of e_{r}(j)((𝔾_{a(r)})_{A}^{×r}, W\underline{α}) under the morphism

𝔾_{a(r)} ⟶^{1×F ×...×F^{r−1}} 𝔾_{a(r)}^{×r}.

Thus, our statement follows from Corollary 4.8 once we observe that the pull-back of the summand corresponding to j_{0}, ..., j_{r−1} in (4.8.0) is zero unless j_{i} ≡ 0 mod p^{i} for all i.

Corollary 4.10. In conditions and notations of Theorem 4.9, the coefficient with which x_{r}^{j}

appears  in  e_{r}(j)((𝔾_{a(r)})_{A}, V\underline{α})  is  equal  to  \frac{(α_{0}^{(r)})^{j_{0}} ·...·(α_{r−1}^{(r)})^{j_{r−1}}}{j_{0}!·...·j_{r−1}!}

where j_{0}, ..., j_{r−1} are p-adic digits of j. In particular the coefficient with which x_{r}^{p^{ℓ−1}} appears in e_{ℓ}^{(r−ℓ)}((𝔾_{a(r)})_{A}, V\underline{α}) is equal to α_{r−ℓ}^{(r)}.

Proof. We get powers of x_{r} in the formula of Theorem 4.9 only when 0 ≤ j_{i} < p for all i , i.e. when j_{1}, ..., j_{r−1} are p-adic digits of j.

§5 The morphism Θ : V_{r}(G) → M_{n}(k)^{×r}.

In Theorem 5.2, we identify the map ψ : H^{ev}(GL_{n(r)}, k) → k[V_{r}(GL_{n})] on the subalgebra of H^{ev}(GL_{n(r)}, k) determined by the universal classes e_{ℓ}^{(r−ℓ)}. This is done by using the explicit computation of e_{r}(j)(𝔾_{a(r)}, V\underline{α}) given in Theorem 4.9 and Corollary 4.10. As we see in several corollaries, this identification gives considerable information concerning the cohomology of infinitesimal group schemes. We then proceed to discuss some plausible refinements of these results.

Let n, r > 0 be a pair of integers. For each 0 ≤ ℓ ≤ r − 1, the characteristic class e_{r−ℓ}^{(ℓ)}(GL_{n(r)}, k^{n}) ∈ H^{2p^{r−ℓ−1}}(GL_{n(r)}, k) ⊗ M_{n}(k)^{(r)} defines a k-linear map gl_{n}^{(r)#} → H^{2p^{r−ℓ−1}}(GL_{n(r)}, k) and hence a homomorphism of commutative k-algebras

S^{∗}(gl_{n}^{(r)#}) → H^{ev}(GL_{n(r)}, k).

Taking the product of these homomorphisms over all ℓ (0 ≤ ℓ ≤ r − 1) we get a homomorphism

ϕ : S^{∗}(⊕_{ℓ=0}^{r−1}gl_{n}^{(r)#}) → H^{ev}(GL_{n(r)}, k)

of commutative k-algebras and the associated morphism of schemes

Φ : Spec H^{ev}(GL_{n(r)}, k) → (gl_{n}^{(r)})^{×r} = gl_{n}^{×r}.

The main result of [F-S] shows that the morphism Φ is finite. The following proposition is a partial description of its image.

Proposition 5.1. The image of Φ : Spec H^{ev}(GL_{n(r)}, k) → gl_{n}^{×r} is contained in V_{r}(GL_{n}) ⊂ gl_{n}^{×r}.

Proof. Let X^{i,j}(ℓ) (1 ≤ i, j ≤ n, 0 ≤ ℓ ≤ r − 1) denote the coordinate function on gl_{n}^{×r} which sends (α_{0}, ..., α_{r−1}) ∈ M_{n}(k)^{×r} to the (i, j)-th entry of α_{ℓ}. The ideal defining the closed subscheme V_{r}(GL_{n(r)}) ⊂ gl_{n}^{×r} is generated by functions of the form (see Theorem 1.5)

∑_{t} X^{i,t}(ℓ) · X^{t,j}(ℓ′) − X^{i,t}(ℓ′) · X^{t,j}(ℓ)

∑_{t_{1},...,t_{p−1}} X^{i,t_{1}}(ℓ) · X^{t_{1},t_{2}}(ℓ) · ... · X^{t_{p−1},j}(ℓ).

Thus it suffices to check that ϕ takes the above generators to zero. The image of X^{i,j}(ℓ) under ϕ is equal to the (i, j)-th entry of the matrix e_{r−ℓ}^{(ℓ)}(GL_{n(r)}, k^{n}). Thus the image under ϕ of the first of the above generators is the (i, j)-th entry of the matrix

e_{r−ℓ}^{(ℓ)}(GL_{n(r)}, k^{n}) · e_{r−ℓ′}^{(ℓ′)}(GL_{n(r)}, k^{n}) − e_{r−ℓ′}^{(ℓ′)}(GL_{n(r)}, k^{n}) · e_{r−ℓ}^{(ℓ)}(GL_{n(r)}, k^{n})

whereas the image under ϕ of the second generator is the (i, j)-th entry of the matrix (e_{r−ℓ}^{(ℓ)}(GL_{n(r)}, k^{n}))^{p}. But the above matrices are zero according to the Theorem 3.1.

Proposition 5.1 shows that the homomorphism ϕ defines a homomorphism of commutative k-algebras

\overline{ϕ} : k[V_{r}(GL_{n})] → H^{ev}(GL_{n(r)}, k).

Let Φ, Ψ denote the maps of affine schemes associated to the maps \overline{ϕ}, ψ of commutative k-algebras.

Theorem 5.2. The composition

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

equals the r-th Frobenius twist morphism corresponding to the (defined over 𝔽_{p}) scheme V_{r}(GL_{n(r)}) = V_{r}(GL_{n}).

Proof. It suffices to show that for k = 𝔽_{p} the ring homomorphism

θ : 𝔽_{p}[V_{r}(GL_{n(r)})] ⟶^{\overline{ϕ}} H^{ev}(GL_{n(r)}, 𝔽_{p}) ⟶^{ψ} 𝔽_{p}[V_{r}(GL_{n(r)})]

coincides with the p^{r}-th power map. Denote the 𝔽_{p}-algebra 𝔽_{p}[V_{r}(GL_{n(r)})] by A. The universal group homomorphism

exp\underline{α} : 𝔾_{a(r)} ⊗_{𝔽_{p}} A → GL_{n(r)} ⊗_{𝔽_{p}} A

is determined by the universal r-tuple of pairwise commuting p-nilpotent matrices \underline{α} = (α_{0}, ..., α_{r−1}) ∈ M_{n}(A)^{×r}, where α_{ℓ} = (X^{i,j}(ℓ))_{i,j=1}^{n}. The definition of ϕ implies that ϕ(X^{i,j}(ℓ)) coincides with the (i, j)-th entry of the matrix e_{r−ℓ}^{(ℓ)}(GL_{n(r)}, 𝔽_{p}^{n}) and hence ψϕ(X^{i,j}(ℓ) coincides with the coefficient of x_{r}^{p^{r−ℓ−1}} in the (i, j)-th entry of the matrix e_{r−ℓ}^{(ℓ)}(𝔾_{a(r)}, V\underline{α}). Corollary 4.10 shows that this coefficient equals X^{i,j}(ℓ)^{p^{r}}.

As we make explicit in the following (immediate) corollary of Theorem 5.2, the theorem provides a reasonably explicit subalgebra of H^{ev}(GL_{n(r)}, k).

Corollary 5.3. If f ∈ Ker \overline{ϕ}, then f^{p^{r}} = 0.

Remark 5.4 Consider the special case r = 1. The scheme

V_{1}(GL_{n}) = N_{p}(gl_{n}) ⊂ gl_{n} = M_{n}(k)

is defined by setting all entries of the matrix α^{p} (where α = (X^{i,j})_{i,j=1}^{n}) to be zero. Since the trace of α lies in Ker \overline{ϕ} [F-S;1.4], this kernel is always non-trivial. According to a theorem of Friedlander and Parshall [F-P] (with bound on p improved in [A-J]), for p > n the homomorphism

ϕ : S^{∗}(M_{n}(k)^{#}) = k[X^{i,j}]_{i,j=1}^{n} → H^{ev}(GL_{n(1)}, k)

is surjective and its kernel is precisely the ideal generated by the coefficients of the characteristic polynomial of α. Apparently the same is true for n = p. However for p < n the situation appears more complicated as can be seen in examples computed in [A-J].

Corollary 5.5. Let G/k be an infinitesimal group scheme of height ≤ r. The image of the canonical homomorphism

ψ : H^{ev}(G, k) → k[V_{r}(G)]

contains k[V_{r}(G)]^{p^{r}}.

Proof. Choose a closed embedding G ↪ GL_{n(r)}. Since the homomorphism k[V_{r}(GL_{n(r)})] → k[V_{r}(G)] is surjective, it suffices to consider the case G = GL_{n(r)}. In this case, our statement follows immediately from Theorem 5.2.

The following question is motivated by the observation that Theorem 5.2 implies that Φ and Ψ are bijections and by the verification in Theorem 5.10 of one special case.

Question 5.6. With notation as in Theorem 5.2, does the composition

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

also equal the r-th Frobenius twist morphism?

An affirmative answer to this question would imply in particular that z^{p^{r}} = 0 for any element z ∈ Ker(ψ : H^{ev}(GL_{n(r)}, k) → k[V_{r}(GL_{n(r)})]). We show in [SFB] that each element in Ker ψ is nilpotent, which gives additional evidence in favor of an affirmative answer.

Question 5.7. Let G/k be an infinitesimal group scheme of height ≤ r. The choice of a closed embedding i : G ↪ GL_{n(r)} defines a morphism of schemes

Φ_{G} : Spec H^{ev}(G, k) ⟶^{i_{∗}} Spec H^{ev}(GL_{n(r)}, k) ⟶^{Φ} V_{r}(GL_{n(r)}) = V_{r}(GL_{n(r)})^{(r)}.

Is it the case that Φ_{G}(Spec H^{ev}(G, k)) ⊂ V_{r}(G)^{(r)} and that the resulting morphism Φ_{G} : Spec H^{ev}(G, k) → V_{r}(G)^{(r)} is independent of the choice of the embedding G ↪ GL_{n(r)}?

The answer to the above question is affirmative for r = 1. If g denotes the Lie algebra of G, then the morphism

Φ_{G} : Spec H^{ev}(G, k) → V_{1}(G)^{(1)} = N_{p}(g)^{(1)}

corresponds to the homomorpism g^{(1)} → H^{2}(G, k) arising from the May spectral sequence.

Remark 5.8. If the answer to Question 5.7 is affirmative in general, then Theorem 5.2 admits the following generalization: For any infinitesimal group scheme G/k of height ≤ r, the composition

V_{r}(G) ⟶^{Ψ} Spec H^{ev}(G, k) ⟶^{Φ_{G}} V_{r}(G)^{(r)}

equals the r-th Frobenius twist morphism.

Question 5.9. If the answer to Question 5.7 is affirmative, then one can go further and ask the following: for any infinitesimal group scheme G/k of height ≤ r, does the composition

Spec H^{ev}(G, k) ⟶^{Φ_{G}} V_{r}(G)^{(r)} ⟶^{Ψ^{(r)}} (Spec H^{ev}(G, k))^{(r)}

equal the r-th Frobenius twist morphism?

The following theorem provides an affirmative answer to Question 5.9 in the very special case for which r = 1, p = 2.

Theorem 5.10. Let G be an infinitesimal group scheme of height ≤ 1 over a field k of characteristic 2. Then the composition

Spec H^{∗}(G, k) ⟶^{Φ_{G}} V_{1}(G)^{(1)} ⟶^{Ψ^{(1)}} (Spec H^{∗}(G, k))^{(1)}

equals the Frobenius twist morphism.

Proof. The proof is based on the use of the following invariant version of May’s resolution [M]. Let g be a finite dimensional restricted Lie algebra over a field k of characteristic 2. Denote by U = U(g) the universal restricted enveloping algebra of g. Consider the following complex

0 ← k ⟵^{ϵ} U ⟵^{d} U ⊗ g ⟵^{d} U ⊗ Γ^{2}(g) ⟵^{d} · · · .  (5.10.0)

The differential d : U ⊗ Γ^{n}(g) → U ⊗ Γ^{n−1}(g) is a sum of two maps. The first one is the U-linear extension of the canonical embedding

Γ^{n}(g) ↪ g ⊗ Γ^{n−1}(g) ↪ U ⊗ Γ^{n−1}(g).

The second is defined as follows. The squaring map g → g (x ↦ x^{[2]}) is homogeneous polynomial of degree 2 and hence defines a k-linear map ξ : Γ^{2}(g) → g. Consider the k-linear homomorphism

ξ_{n} : Γ^{n}(g) ⟶^{comult} ⨁_{i=0}^{n−2} Γ^{i}(g) ⊗ Γ^{2}(g) ⊗ Γ^{n−i−2}(g) ⟶^{⊕_{i}1⊗ξ⊗1}

⟶^{⊕_{i}1⊗ξ⊗1} ⨁_{i=0}^{n−2} Γ^{i}(g) ⊗ g ⊗ Γ^{n−i−2}(g) ⟶^{can} g^{⊗(n−1)}.

Extending scalars if necessary, we may assume that the field k is infinite. In this case the vector space Γ^{n}(g) is generated by tensors of the form x^{⊗n} (x ∈ g). Since ξ_{n}(x^{⊗n}) = x^{[2]} ⊗ ... ⊗ x + ... + x ⊗ ... ⊗ x^{[2]} ∈ Γ^{n−1}(g), we conclude that ξ_{n} is a k-linear homomorphism from Γ^{n}(g) to Γ^{n−1}(g). Now the second summand in d is defined to be the U-linear extension of ξ_{n}. A straightforward verification shows that d is really a differential; moreover using the results of May [M] or arguing directly one shows that (5.10.0) is a U-projective resolution of k. We may use this resolution to compute the (co)homology of g. Thus we see in particular that H^{∗}(G, k) coincides with homology of the complex (cf. [F-P])

0 → g^{#} ⟶^{ξ^{#}} S^{2}(g^{#}) ⟶^{ξ_{3}^{#}} S^{3}(g^{#}) → · · · .  (5.10.1)

The complex (5.10.1) is a commutative differential graded k-algebra and multiplication in cohomology is induced by multiplication in this complex.

Denote the coordinate algebra k[V_{1}(G)] = k[N_{p}(g)] by A. Thus A is a factor algebra of S^{∗}(g^{#}) modulo the ideal generated by ξ^{#}(g^{#}) ⊂ S^{2}(g^{#}). The universal homomorphism g_{a} ⊗ A → g ⊗ A is defined by the element 1_{g} ∈ g ⊗ g^{#} ⊂ g ⊗ A. The dual map g^{#} ⊗ A → g_{a}^{#} ⊗ A = A is an A-linear extension of the canonical embedding λ : g^{#} ↪ A. Our definitions imply that the homomorphism ψ : H^{∗}(G, k) → A is given by the following homomorphism of complexes

g^{#} ⟶^{ξ^{#}} S^{2}(g^{#}) ⟶^{ξ_{3}^{#}} ... ⟶^{ξ_{n−1}^{#}} S^{n}(g^{#}) ⟶^{ξ_{n}^{#}} ...
λ↓  ^{λ_{2}}↓  ↓  ^{λ_{n}}↓
A ⟶^{0} A ⟶^{0} ... ⟶^{0} A ⟶^{0} ...

where λ_{n} : S^{n}(g^{#}) → A is the n-th component of the canonical projection S^{∗}(g^{#}) → A. In other words, the canonical projection S^{∗}(g^{#}) → A is a homomorphism of differential graded algebras (A is equipped with zero differential) and the induced map in cohomology is our homomorphism ψ. On the other hand, the homomorphism ϕ_{G} : A^{(1)} → H^{∗}(G, k) is induced by the homomorphism S^{∗}(g^{#(1)}) → H^{∗}(G, k) arising from the k-linear map g^{#(1)} ↪ S^{2}(g^{#}) : x^{(1)} ↦ x^{2}. A straightforward computation completes the proof.

§6. An application to discrete group homology.

In [F-S], information about the rational cohomology groups H^{∗}(GL_{n}, gl_{n}^{(r)}) was used to compute homology of the discrete group GL(𝔽_{q}) with coefficients in the adjoint representation. In this section, we show that our present knowledge of comultiplication in H^{∗}(GL_{n}, gl_{n}^{(r)}) enables us to determine the multiplicative structure of H_{∗}(GL(𝔽_{q}), M(𝔽_{q})).

For a field k, we denote by GL_{n}(k) the discrete group of invertible n × n matrices over k. The group GL_{n}(k) acts by conjugation on M_{n}(k). We embed GL_{n}(k) into GL_{n+1}(k) (and M_{n}(k) into M_{n+1}(k)) in a standard way and we set GL(k) = lim_{→} GL_{n}(k), M(k) = lim_{→} M_{n}(k). Since group homology commutes with direct limits, we conclude that

H_{i}(GL(k), M(k)) = lim_{→} H_{i}(GL_{n}(k), M_{n}(k)).

By a stability theorem of W. Dwyer[D], H_{i}(GL(k), M(k)) = H_{i}(GL_{n}(k), M_{n}(k)) for all n big enough (depending on i).

Choosing an ordering for the basis of k^{n} ⊗ k^{m}, we get a homomorphism in the category of pairs (group,module)

(GL_{n}(k) × GL_{m}(k), M_{n}(k) ⊗ M_{m}(k)) → (GL_{nm}(k), M_{nm}(k))

which induces a homomorphism in homology

H_{∗}(GL_{n}(k),M_{n}(k)) ⊗ H_{∗}(GL_{m}(k), M_{m}(k)) = H_{∗}(GL_{n}(k) × GL_{m}(k),
M_{n}(k) ⊗ M_{m}(k)) → H_{∗}(GL_{nm}(k), M_{nm}(k)) → H_{∗}(GL(k), M(k)).

This homomorphism is compatible with stabilization, thereby inducing a pairing

H_{∗}(GL(k), M(k)) ⊗ H_{∗}(GL(k), M(k)) → H_{∗}(GL(k), M(k))

which makes H_{∗}(GL(k), M(k)) into an associative graded k-algebra. We proceed to determine this algebra in case k = 𝔽_{q} is a finite field.

For an affine group scheme G/k and a rational G-module M, viewing a regular morphism as a set map on k-valued points determines a natural homomorphism H^{∗}(G, M) → H^{∗}(G(k), M). When G and M are defined over 𝔽_{p}, the Frobenius twist M^{(r)} may be identified as a G-module with M endowed with a new action of G via the r-th power of the Frobenius endomorphism F : G → G. Assume now that k is a finite field. Then the action of F on the group of rational points G(k) is an isomorphism and hence H^{∗}(G(k), M^{(r)}) = H^{∗}(G(k), M). Thus, for any r ≥ 0, we get a homomorphism H^{∗}(G, M^{(r)}) → H^{∗}(G(k), M^{(r)}) = H^{∗}(G(k), M) and each of the diagrams

\begin{matrix}H^{∗}(G, M^{(r−1)}) & ⟶^{z↦z^{(1)}} & H^{∗}(G, M^{(r)}) \\ ↓ & & ↓ \\ H^{∗}(G(k), M) & ⟶^{=} & H^{∗}(G(k), M) \end{matrix}  (6.0)

commute. Applying the previous construction to G = GL_{n} and M = M_{n}(k), we obtain canonical homomorphisms (cf. [CPSvdK])

H^{∗}(GL_{n}, M_{n}(k)^{(r)}) → H^{∗}(GL_{n}(k), M_{n}(k)).

Let ℓ ≥ 0 be an integer. Choose r > 0 such that ℓ < p^{r} and denote by e(ℓ) the image of e_{r}(ℓ)(GL_{n}, k^{n}) ∈ H^{2ℓ}(GL_{n}, M_{n}(k)^{(r)}) in H^{2ℓ}(GL_{n}(k), M_{n}(k)) = = Hom_{k}(H_{2ℓ}(GL_{n}(k), M_{n}(k)), k). Commutativity of the diagram (6.0) implies that the resulting cohomology class is independent of the choice of r. Moreover one checks easily that the diagram

H_{2ℓ}(GL_{n}(k), M_{n}(k)) ⟶^{e(ℓ)} k
can↓  ^{=}↓
H_{2ℓ}(GL_{n+1}(k), M_{n+1}(k)) ⟶^{e(ℓ)} k

commutes. Passing to a limit in n we get a homomorphism

e(ℓ) : H_{2ℓ}(GL(k), M(k)) → k.

The following theorem recalls the determination of the additive structure of H_{∗}(GL(k), M(k)) as presented in [F-S].

Theorem 6.1 ([F-S] Theorem 7.6). Let k = 𝔽_{q} be a finite field. Then H_{m}(GL(k), M(k)) = 0 if m is odd. On the other hand, if m = 2ℓ is even, then the homomorphism e(ℓ) : H_{m}(GL(k), M(k)) → k is an isomorphism.

Denote by x(ℓ) ∈ H_{2ℓ}(GL(k), M(k)) the element for which e(ℓ)(x(ℓ)) = 1. The multiplicative structure of H_{∗}(GL(k), M(k)) is given by the following theorem.

Theorem 6.2. Let k = 𝔽_{q} be a finite field. Then x(ℓ) · x(ℓ′) = x(ℓ + ℓ′). In other words, H_{∗}(GL(k), M(k)) is a polynomial algebra over k in one generator x(1) ∈ H_{2}(GL(k), M(k)).

Proof. Proposition 3.8 and Theorem 4.6 show that (for r big enough)

e_{r}(ℓ + ℓ′)(GL_{n} × GL_{n′}, k^{n} ⊗ k^{n′}) = ∑_{i+j=ℓ+ℓ′} e_{r}(i)(GL_{n}, k^{n}) ⊗ e_{r}(j)(GL_{n′}, k^{n′}).

This formula implies the commutativity of the following diagram

H_{2ℓ}(GL_{n}(k), M_{n}(k)) ⊗ H_{2ℓ′}(GL_{n′}(k), M_{n′}(k)) ⟶^{e(ℓ)⊗e(ℓ′)} k
mult↓  ^{=}↓
H_{2(ℓ+ℓ′)}(GL_{nn′}(k), M_{nn′}(k))  ⟶^{e(ℓ+ℓ′)} k.

Now we conclude immediately that e(ℓ + ℓ′)(x(ℓ)x(ℓ′)) = e(ℓ)(x(ℓ)) · e(ℓ′)(x(ℓ′)) = 1, i.e. x(ℓ)x(ℓ′) = x(ℓ + ℓ′).

References

[A-J] H. Andersen and J. Jantzen, Cohomology of induced representations for algebraic groups, Math.

Annalen 269 (1985), 487–525.

[CPSvdK] E. Cline, B. Parhall, L. Scott, and W. van der Kallen, Rational and generic cohomology, Inventiones Math. 39 (1977), 143–163.

[D] W. Dwyer, Twisted homological stability for general linear groups, Annals of Math. 111 (1980), 239-251.

[D-G] M. Demazure and P. Gabriel, Groupes algébriques I (Masson/North-Holland, ed.), 1970. [D-M] B Dondas and R. McCarthy, Stable K-theory and Hochschild homology, Annals of Math. 142

(1994), 685-701.

[F-P] E. Friedlander and B. Parshall, Cohomology of Lie algebras and algebraic groups, Amer. J.

Math. 108 (1986), 235-253.

[F-S] E. Friedlander and A. Suslin, Cohomology of finite group schemes over a field, Inventiones

Math. (in press).

[G] D. Gross, Higher nullcones and commuting varieties, Comm. in Algebra 21 (1993), 1427–1455.

[J] J. Jantzen, Representations of Algebraic Groups (Academic Press, ed.), 1987.

[M] J.P. May, The cohomology of restricted Lie algebras and of Hopf algebras, J. Algebra 3 (1966), 123–146.

[Q] D. Quillen, Higher algebraic K-theory I, Algebraic K-Theory I (Springer-Verlag, Lecture Notes in Math 341, ed.), 1973, pp. 85-147.

[SFB] A. Suslin, E. Friedlander, and C. Bendel, Support varieties for infinitesimal group schemes.