RATIONAL COHOMOLOGY AND COHOMOLOGICAL STABILITY IN GENERIC REPRESENTATION THEORY

NICHOLAS J. KUHN

May 27, 1997

Abstract. With F_{q} a finite field of characteristic p, let F(q) be the category whose objects are functors from finite dimensional F_{q}–vector spaces to F̄_{p}– vector spaces. Extension groups in F(q) can be interpreted as MacLane (or Topological Hochschild) cohomology with twisted coefficients. Furthermore, evaluation on an m dimensional vector space V_{m} induces a homorphism from Ext_{F(q)}^{∗}(F, G) to finite group cohomology Ext_{GL_{m}(F_{q})}^{∗}(F (V_{m}), G(V_{m})).

E. Friedlander and A. Suslin have introduced a category P of “strict polynomial functors” which has the same relationship to the category of rational GL_{m}–modules that F(q) has to the category of GL_{m}(F_{q})–modules. Our main theorem says that, for all finite objects F, G ∈ P, and all s, the natural restriction map

Ext_{P}^{s}(F_{(k)}, G_{(k)}) ⟶ Ext_{F(q)}^{s}(F, G)

is an isomophism for all large enough k and q. Here F_{(k)} denotes F twisted by the Frobenious k times.

This theorem is an analogue of an old stability theorem of E. Cline, B. Parshall, L. Scott, and W. van der Kallen relating rational GL_{m}–modules to GL_{m}(F_{q})–modules. These two theorems then combine with an observation of Friedlander and Suslin to show that, for all finite F, G ∈ P, and all s, the natural map

Ext_{F(q)}^{s}(F, G) ⟶ Ext_{GL_{m}(F_{q})}^{s}(F (V_{m}), G(V_{m}))

is an isomorphism for all large enough m and q. Thus group cohomology of the finite general linear groups has often been identified with (the more computable) MacLane cohomology.

1. Introduction

Let p be a fixed prime, F_{q} the characteristic p field with q elements, and F̄_{p} the algebraic closure of F_{p}.

Given two rational modules M and N for the algebraic group GL_{m} (defined over F_{p}), one can consider the rational cohomology Ext_{GL_{m}}^{∗}(M, N). The inclusion of the finite general linear group GL_{m}(F_{q}) ⟶ GL_{m} induces restriction maps from rational cohomology to finite group cohomology

Ext_{GL_{m}}^{∗}(M, N) ⟶ Ext_{GL_{m}(F_{q})}^{∗}(M, N)

1991 Mathematics Subject Classification. Primary 19D55; Secondary 18G60, 20G10, 20J06. Partially supported by the N.S.F.

The Frobenious map Φ : GL_{m} ⟶ GL_{m} induces homomorphisms

Ext_{GL_{m}}^{∗}(M_{(k)}, N_{(k)}) ⟶ Ext_{GL_{m}}^{∗}(M_{(k+1)}, N_{(k+1)}),

and compatible isomorphisms

Ext_{GL_{m}(F_{q})}^{∗}(M_{(k)}, N_{(k)}) ≃ Ext_{GL_{m}(F_{q})}^{∗}(M_{(k+1)}, N_{(k+1)}),

where M_{(k)} denotes the module M twisted by the Frobenious k times. Thus one gets maps

Ext_{GL_{m}}^{s}(M_{(k)}, N_{(k)}) ⟶ Ext_{GL_{m}(F_{q})}^{s}(M, N).

When specialized to the algebraic group GL_{m}, the main result of [CPSvK] says that, if M and N are finite dimensional F̄_{p}–vector spaces, this last map is an isomorphism for all large enough k and q (where “large enough” is a function of s and the weights of M and N).

The papers [K:I, K:II, K:III] study certain categories of functors that I view as the categories of ‘generic’ representations of the general linear groups over finite fields. More precisely, there are abelian categories F(q) and compatible exact ‘restriction’ maps

F(q^{r}) ⟶ GL_{m}(F_{q^{r}})–modules
↓  ↓
F(q) ⟶ GL_{m}(F_{q})–modules,

for all q, m, and r. Many aspects of GL_{m}(F_{q})–modules seem best understood by the consideration of the analogous aspect of the category F(q); for example, the simple GL_{m}(F_{q})–modules are precisely the objects that are the restriction of simple functors in F(q). Moreover, F(q) is of K-theoretic interest: Ext–groups in F(q) can be interpreted as MacLane (or Topological Hochschild) cohomology groups with twisted coefficients.

E. Friedlander and A. Suslin [FS] have recently introduced a parallel category of “strict polynomial” functors which has the same relationship to the category of rational GL_{m}–modules that F(q) has to the category of GL_{m}(F_{q})–modules. One has an abelian category P and exact restriction maps:

P ⟶ GL_{m}–modules
↓  ↓
F(q) ⟶ GL_{m}(F_{q})–modules,

which induce commutative diagrams

Ext_{P}^{∗}(F_{(k)}, G_{(k)}) ⟶ Ext_{GL_{m}}^{∗}(F_{(k)}, G_{(k)})
↓  ↓
Ext_{F(q)}^{∗}(F, G) ⟶ Ext_{GL_{m}(F_{q})}^{∗}(F, G),

for all F, G ∈ P. As before, F_{(k)} denotes F twisted by the Frobenious k times.

The analogue in P or F(q) of a finite dimensional GL_{m}–module is a finite functor: an object with only a finite number of simple composition factors. Our main theorem is the following analogue of the [CPSvK] stability result.

Theorem 1.1. If F, G ∈ P are finite, the natural map

Ext_{P}^{s}(F_{(k)}, G_{(k)}) ⟶ Ext_{F(q)}^{s}(F, G)

is an isomophism for all large enough k and q.

As will be specified more precisely later, for k, “large enough” will be a function of s and the weights of F and G, while stability occurs very quickly for q, and is only a function of the weights of F and G.

Combining this theorem with [CPSvK] and a result from [FS] yields the following consequence, which proves many cases of [K:II, Conjecture 3.18].

Theorem 1.2. If F, G ∈ P are finite, the natural map

Ext_{F(q)}^{s}(F, G) ⟶ Ext_{GL_{m}(F_{q})}^{s}(F, G)

is an isomorphism for all large enough m and q.

The computational point of this theorem (and the general conjecture, which says that the “large q” hypothesis here should not be needed) is that there are powerful theorems and general techniques available for computing Ext_{F(q)}^{s}(F, G) for many F and G (as in [K:III, FLS]). Note also that included here is the statement that the groups Ext_{GL_{m}(F_{q})}^{s}(F, G) are independent of m, if m is large enough, which is a special case of the main result of [D]; however our proof is independent of Dwyer’s theorem. (These cohomology groups have a K–theoretic interpretation as stable K–groups as in [DM].)

The proofs of Theorem 1.1 and the main theorem of [CPSvK] have in common the heavy use of vanishing theorems and spectral sequence techniques. However, there the similarities seem to end: the two proofs are very different in both structure and detail. In the remainder of this introduction, we explain the organization of the rest of the paper as we outline our argument.

Firstly, we need to give definitions of our various functor categories allowing for convenient comparison. This involves slight modification of the categories denoted by F(q) and P in [K:I, K:II, K:III] and [FS] respectively.

Definition 1.3. Let q be a power of the prime p.

(1) Let V(q) denote the F_{q}–linear category whose objects are the finite dimensional F_{p}–vector spaces V, and with Hom_{V(q)}(V, W) = Hom_{F_{q}}(F_{q} ⊗_{F_{p}} V, F_{q} ⊗_{F_{p}} W).

(2) Let F(q) denote the category whose objects are functors

F : V(q) ⟶ F̄_{p} − vector spaces,

and whose morphisms are the natural transformations between such functors.

(3) For n ∈ N (or even n ∈ N[1/p]), let F(q)_{n} denote the full subcategory of F(q) with objects F such that F (λ) = λ^{n} : F (V) ⟶ F (V), for all V ∈ V(q) and λ ∈ F_{q}.

(4) Let F_{ω}(q) be the full subcategory of F(q) whose objects are locally finite.

We recall that an object in an abelian category is locally finite if it is the union of its finite subobjects.

F(q) is evidently equivalent to the category denoted by F̄_{p} ⊗_{F_{q}} F(q) in [K:II, §5.1]. As observed in [K:I, §3.3], there is a decomposition of abelian categories

F(q) ≃ F(q)_{0} × F(q)_{1} × ⋯ × F(q)_{q−1},

and it is enlightening to regard F(q) as graded by the semigroup {0} ∪ Z/(q − 1). If F ∈ F(q)_{n}, we will say that F has scalar degree n.

Definition 1.4. Let S^{n}, Φ^{−1}S^{n}, I_{q}, and I_{q,n} be the following functors from V(p) to F_{p}-vector spaces. Firstly, S^{n}(V) = (V^{⊗n})/Σ_{n}, and I_{q}(V) = S^{∗}(V)/(x^{q} −x | x ∈

S^{∗}(V)). Then Φ^{−1}S^{n} = colim {S^{n} ⟶^{Φ} S^{pn} ⟶^{Φ} S^{p^{2}n} ⟶ … }, where Φ : S^{n} ⟶ S^{pn}

is the p^{th} power map. Finally, I_{q,0} = S^{0}, and, for n > 0, I_{q,n} is the image in I_{q} of ⨁ S^{m} where the sum is over m > 0 such that m ≡ n mod (q − 1).

One can consider a small F_{p}–linear category A as an “F_{p}–algebra with many objects”, and an F_{p}–linear functor from A to the category of F̄_{p}–vector spaces as an “A–module”. As will be elaborated on in §2, there are the obvious dual notions of an F_{p}–coalgebra with many objects, and its category of comodules.

The dual of V(p) will be an example of such a multiobject coalgebra, and will be denoted by C(p). The functors S^{n}, Φ^{−1}S^{n}, I_{q}, and I_{q,n} all have the property that applying them to a multiobject coalgebra canonically yields another multiobject coalgebra, and we observe:

Lemma 1.5. F(q) is the category of I_{q}(C(p))–comodules, and F(q)_{n} is the category of I_{q,n}(C(p))–comodules.

Definition 1.6.

(1) Let P^{n} be the category of S^{n}(C(p))–comodules, and P the N–graded category of S^{∗}(C(p))–comodules.

(2) Let Φ^{−1}P^{n} be the category of Φ^{−1}S^{n}(C(p))–comodules, and Φ^{−1}P the N[1/p]– graded category of Φ^{−1}S^{∗}(C(p))–comodules.

(3) Let Φ^{−1}P_{ω} be the full subcategory of Φ^{−1}P whose objects are locally finite.

It is easy to see that P, so defined, is the category of “strict polynomial functors” of [FS], modified by including arbitrary colimits of objects and extending the scalars to F̄_{p}. If F ∈ P^{n}, we will say that F has scalar degree n.

The p^{th} power map Φ : S^{n} ⟶ S^{pn} induces the Frobenious twists Φ : P^{n} ⟶ P^{pn}, Φ : Φ^{−1}P^{n} ≃ Φ^{−1}P^{pn}, and Φ : F(q)_{n} ≃ F(q)_{pn}. The projection I_{q^{r}} ⟶ I_{q} induces the restriction map F(q^{r}) ⟶ F(q). Furthermore, the projection S^{∗} ⟶ I_{q} factors canonically as S^{∗} ⟶ Φ^{−1}S^{∗} ⟶ I_{q}, inducing exact functors P^{n} ⟶ Φ^{−1}P_{ω}^{n} ⟶ F(q)_{n} which are compatible with both Φ and F(q^{r}) ⟶ F(q).

If n has p–adic expansion n = ∑_{j=0}^{l} n_{j}p^{j}, 0 ≤ n_{j} ≤ p − 1, let α_{p}(n) = ∑_{j=0}^{l} n_{j}, and let ν_{p}(n) be the smallest j such that n_{j} ≠ 0 (i.e. the p–exponent of n).

Theorem 1.1 follows from the following four propositions.

Proposition 1.7. For all finite F, G ∈ P, the natural map

colim_{k} Ext_{P}^{∗}(F_{(k)}, G_{(k)}) ⟶ Ext_{Φ^{−1}P_{ω}}^{∗}(F, G)

is an isomorphism.

Proposition 1.8. For all F, G ∈ P^{n}, and all s, the natural map

Φ : Ext_{P}^{s}(F_{(k)}, G_{(k)}) ⟶ Ext_{P}^{s}(F_{(k+1)}, G_{(k+1)})

is an isomorphism for all k > (n − α_{p}(n))/(p − 1) − ν_{p}(n) + log_{p}(s).

Proposition 1.9. For all finite F, G ∈ P, the natural map

Ext_{Φ^{−1}P_{ω}}^{∗}(F, G) ⟶ lim_{q} Ext_{F(q)}^{∗}(F, G)

is an isomophism.

Proposition 1.10. For all F, G ∈ F(q)_{n}, and r ≥ 0, the natural map

Ext_{F(q^{r})}^{∗}(F, G) ⟶ Ext_{F(q)}^{∗}(F, G)

is an isomorphism if q ≥ n.

As will be observed in §3, Proposition 1.7 is a formal consequence of the general considerations about categories of comodules over multiobject coalgebras developed in §2. A weak form of Proposition 1.10, strong enough to prove Theorem 1.1, follows from an easy restriction/induction argument as in [K:II, §5.2], and a quantitative examination of the argument yields the proposition as stated. This will be shown in §4. In §5, Proposition 1.9 follows quickly from the vanishing theorem of [K:III]: Φ^{−1}S^{n} (with scalars extended) is an injective object in the category F_{ω}(q). Finally, Proposition 1.8 will be proved in §6. Like key parts of [FS], the argument here is a transferal of the ingenious arguments of [FLS] and [F] to our situation.

The deduction of Theorem 1.2 from Theorem 1.1 is in §7.

Remark 1.11. The results in this paper were presented by me at the March 1997 Northwestern University conference on algebraic topology. Just before the conference, I found out that Theorem 1.1 and Theorem 1.2 have been independently proved by the foursome V. Franjou, Friedlander, Skorichenko, and Suslin. Since then, they have proved [K:II, Conjecture 3.18] in complete generality.

2. multiobject coalgebras

2.1. comodules over multiobject coalgebras. If F is a field, a small F–linear category A has the following structure:

• a set of objects ob(A)

• an F–vector space A(V, W), for all V, W ∈ ob(A)

• unit maps η_{V} : F ⟶ A(V, V), for all V ∈ ob(A)

• product maps μ_{U,V,W} : A(V, W) ⊗_{F} A(U, V) ⟶ A(U, W), for all U, V, W ∈ ob(A).

These satisfy the usual axioms reflecting the unital and associative properties. If the set of objects of A has only one element 0, then the structure of A is encoded in the statement that A(0, 0) is an F–algebra. Thus an F–linear category can be thought of a multiobject F–algebra. (This is the point of view in the treatise [M].)

There is an obvious, but less known, dual notion. We will say that C is a multiobject coalgebra over F if C has the following structure:

• a set of objects ob(C)

• an F–vector space C(V, W), for all V, W ∈ ob(C)

• counit maps ϵ_{V} : C(V, V) ⟶ F, for all V ∈ ob(C)

• coproduct maps Ψ_{U,V,W} : C(U, W) ⟶ C(V, W)⊗_{F} C(U, V), for all U, V, W ∈ ob(C).

These satisfy the obvious axioms reflecting the counital and coassociative properties. The usual notion of a coalgebra will be a coalgebra in our new sense having just one object.

We will say that a multiobject algebra A (or coalgebra C) is of finite type if each A(V, W) (or C(V, W)) is finite dimensional. Taking F–duals of all the structure for an algebra A of finite type yields a coalgebra C of finite type, and conversely: C(V, W) = A(V, W)^{#}.

Let A be a multiobject algebra over F. Letting F̄ denote the algebraic closure of F, we call an F–linear functor F : A ⟶ F̄–vector spaces a (left) A–module. Unraveling the definitions, we see that an A–module F has the following stucture:

• an F̄–vector space F (V), for all V ∈ ob(A)

• multiplication maps μ_{V,W}^{F} : A(V, W) ⊗_{F} F (V) ⟶ F (W), such that

A(V, W) ⊗ A(U, V) ⊗ F (U) ⟶^{1⊗μ^{F}} A(V, W) ⊗ F (V)
↓^{μ⊗1}  ↓_{μ^{F}}
A(U, W) ⊗ F (U)  ⟶^{μ^{F}}  F (W)

commutes for all U, V, W, and also each composite

F ⊗ F (V) ⟶^{μ⊗1} A(V, V) ⊗ F (V) ⟶^{μ^{F}} F (V)

is the canonical isomorphism.

There is an obvious notion of a morphism α : F ⟶ G between two A–modules: α consists of a set of linear maps {α_{V} : F (V) ⟶ G(V) | V ∈ ob(A)}, commuting with the multiplication maps. Furthermore, the resulting category of A–modules is an abelian category: α : F ⟶ G is monic (respectively, epic) if all the component maps α_{V} : F (V) ⟶ G(V) are monic (respectively, epic).

Dually, if C is a multiobject coalgebra over F, a (right) C–comodule F has the following structure:

• an F̄–vector space F (V), for all V ∈ ob(C)

• multiplication maps Ψ_{V,W}^{F} : F (V) ⟶ F (W) ⊗_{F} C(V, W), such that

F (U)  ⟶^{Ψ^{F}}  F (W) ⊗ C(U, W)
↓_{Ψ^{F}}  ↓^{1⊗Ψ}
F (V) ⊗ C(U, V) ⟶^{Ψ^{F} ⊗1} F (W) ⊗ C(V, W) ⊗ C(U, V)

commutes for all U, V, W, and also each composite

F (V) ⟶^{Ψ^{F}} F (V) ⊗ C(V, V) ⟶^{1⊗Ψ} F (V) ⊗ F

is the canonical isomorphism.

As before, there is an obvious notion of a morphism between C–comodules, and the resulting category of C–comodules is abelian. This is an AB5 category, i.e. an abelian category with direct sums and exact direct limits, but will not have arbitrary products unless C is of finite type.

Note that if A is an algebra of finite type with dual coalgebra C, then the stucture maps

A(V, W) ⊗ F (V) ⟶ F (W),

defining an A–module F, are equivalent to maps

F (V) ⟶ F (W) ⊗ C(V, W)

defining a C–comodule. Thus the categories of A–modules and C–comodules are isomorphic abelian categories.

2.2. projectives and injectives. Let A be a multiobject F–algebra. Given U ∈ ob(A), let F_{U} be the A–module defined by letting F_{U}(V) = F̄ ⊗_{F} A(U, V) with multiplication maps induced from the product maps in A. By Yoneda’s Lemma, given any A–module F, these satisfy:

Hom_{A}(F_{U}, F) ≃ F (U), (2.1)

thus the modules F_{U} form a set of projective generators for the category of A– modules.

Dually, let C be a multiobject F–coalgebra. Given W ∈ ob(C), let F_{W} be the C–comodule defined by letting F_{W} (V) = F̄ ⊗_{F} C(V, W) with comultiplication maps induced from the coproduct maps in C. By a Yoneda’s Lemma type of argument, given any C–comodule F, these satisfy:

Hom_{C}(F, F_{W}) ≃ F (W)^{#}, (2.2)

thus the comodules F_{W} form a set of injective objects in the category of C– comodules.

2.3. locally finite comodules. In any abelian category, an object is called simple if it has no nontrivial proper subobjects, finite if it has a finite filtration with simple subquotients, and locally finite if it is the union of its finite subobjects.

If C is a multiobject coalgebra, and W ∈ ob(C), let (F_{W})_{ω} ⊆ F_{W} be the sum of the finite objects in the injective C–comodule F_{W}. Clearly (F_{W})_{ω} is an injective in the category of locally finite C–comodules, and we have

Proposition 2.1. In the category of locally finite C–comodules, every object F has an injective resolution of the form

F ⟶ I(0) ⟶ I(1) ⟶ I(2) ⟶ …

in which each I(j) is a direct sum of summands of (F_{W})_{ω}’s.

Proof. The general theory of locally finite AB5 categories (as in [Po]) implies that F will have such a resolution in which each I(j) is the direct sum of injective hulls of simple comodules. But by (2.2), every simple comodule admits an embedding in some F_{W}, and thus its injective hull will be a summand of F_{W}.

If F and G are locally finite C–comodules, we let Ext_{C_{ω}}^{∗}(F, G) denote Ext groups in the category of locally finite C–comodules.

Proposition 2.2. Let C be the union of subcoalgebras C(0) ⊂ C(1) ⊂ C(2) ⊂ …, all having the same set of objects.

(1) Every finite C–comodule is a C(k)–comodule, for some k.

(2) If F and G are finite C(j)–comodules, then the natural map

colim_{k≥j} Ext_{C(k)_{ω}}^{∗}(F, G) ⟶ Ext_{C_{ω}}^{∗}(F, G)

is an isomophism.

Proof. We first check (1). For each V ∈ ob(C), the injective C–comodule F_{W} will be the union of the corresponding injective C(k)–comodules F (k)_{W}. Since a finite C–comodule F is Artinian, (2.2) implies that F embeds in a finite product of F_{W} ’s, and then, since F is also Noetherian, F will embed in the corresponding product of F (k)_{W} ’s, for large enough k, verifying (1).

Given (1), it is easy to see that, if F and G are finite C(j)–comodules, then the natural map

colim_{k≥j} Ext_{C(k)_{f}}^{∗} (F, G) ⟶ Ext_{C_{f}}^{∗} (F, G)

is an isomophism, where Ext_{C_{f}}^{∗} (F, G) denotes Yoneda Ext groups using only finite objects. But, as observed in [K:I, Proposition B.1(2)], in any locally finite AB5 category B,

Ext_{B_{f}}^{∗} (F, G) ≃ Ext_{B}^{∗}(F, G),

for any two finite objects F, G ∈ B. Thus (2) is true.

2.4. new coalgebras from old. Suppose we are given

• a functor H from F–vector spaces to F–vector spaces

• a natural transformation Ψ : H(A ⊗ B) ⟶ H(A) ⊗ H(B), and

• a linear map ϵ : H(F) ⟶ F.

We will call the triple (H, Ψ, ϵ) coherent if each diagram

[figure: H((A ⊗ B) ⊗ C); H(A ⊗ (B ⊗ C)); H(A ⊗ B) ⊗ H(C); H(A) ⊗ H(B ⊗ C); (H(A) ⊗ H(B)) ⊗ H(C); H(A) ⊗ (H(B) ⊗ H(C)); Ψ; Ψ; Ψ⊗1; 1⊗Ψ]

commutes, where the horizontal maps are the canonical isomorphisms, and, furthermore, the composites

H(A) ≃ H(A ⊗ F) ⟶^{Ψ} H(A) ⊗ H(F) ⟶^{ϵ} H(A) ⊗ F

and

H(A) ≃ H(F ⊗ A) ⟶^{Ψ} H(F) ⊗ H(A) ⟶^{ϵ} F ⊗ H(A)

are the canonical isomorphisms.

A straightforward check of the definitions proves

Proposition 2.3. If (H, Ψ, ϵ) is coherent and C is a multiobject coalgebra, then so is H(C), where H(C) has the same objects as C, H(C)(V, W) = H(C(V, W)), and the structure maps are the composites

H(C(U, W)) ⟶^{H(Ψ_{U,V,W})} H(C(V, W) ⊗ C(U, V)) ⟶^{Ψ} H(C(V, W)) ⊗ H(C(U, V)),

H(C(V, V)) ⟶^{H(ϵ_{V} )} H(F) ⟶^{ϵ} F.

Note that this construction is natural in both C and (H, Ψ, ϵ).

3. the categories P, Φ^{−1}P, and F(q)

In this section we specialize the general theory of the last section to the categories of interest. Recall that C(p) is the multiobject coalgebra dual to the category V(p) of finite dimensional F_{p}–vector spaces.

3.1. The category P. Recall that S^{n}(V) = (V^{⊗n})/Σ_{n}, i.e. degree n polynomials in V. The evident maps

Ψ : S^{n}(V ⊗ W) ⟶ S^{n}(V) ⊗ S^{n}(W)

and

ϵ : S^{n}(F_{p}) ≃ F_{p}

define a coherent triple (S^{n}, Ψ, ϵ) in the terminology of §2.4. S^{n}(C(p)) is thus a coalgebra, and we define P^{n} to be the category of S^{n}(C(p))–comodules. Note that S^{n}(C(p)) is of finite type, and it is easy to see that P^{n} can also be viewed as the category of Γ^{n}(V(p))–modules, where Γ^{n}(V) = (V^{⊗})^{Σ_{n}}.

The general theory immediately implies that P^{n} has a set of projective generators given by Γ_{U}^{n} defined by Γ_{U}^{n}(V) = F̄_{p} ⊗_{F_{p}} Γ^{n}(Hom(U, V)), and a set of injective cogenerators S_{W}^{n} defined by S_{W}^{n} (V) = F̄_{p} ⊗_{F_{p}} S^{n}(Hom(V, W)^{#}). (Compare with [FS, §2].)

In subsection 3.4 below, we will check that each Γ_{U}^{n} is finite, thus every object in P^{n} is locally finite.

The p^{th} power map Φ : S^{n} ⟶ S^{pn} induces a map of associated coherent triples, thus an inclusion of coalgebras Φ : S^{n}(C(p)) ⟶ S^{pn}(C(p)), and finally the exact functor “Frobenious twist” Φ : P^{n} ⟶ P^{pn}.

3.2. The category Φ^{−1}P. The coalgebra Φ^{−1}S^{n}(C(p)) is defined to be the union of the coalgebras

S^{n}(C(p)) ⟶^{Φ} S^{pn}(C(p)) ⟶^{Φ} S^{p^{2}n}(C(p)) ⟶ ….

We let Φ^{−1}P^{n} be its category of comodules, and Φ^{−1}P_{ω}^{n} ⊂ Φ^{−1}P^{n} the full subcategory of locally finite objects.

Once again the general theory immediately tells us that Φ^{−1}P^{n} has set of injective objects Φ^{−1}S_{W}^{n} defined by Φ^{−1}S_{W}^{n} (V) = F̄_{p} ⊗_{F_{p}} Φ^{−1}S^{n}(Hom(V, W)^{#}).

Furthermore, as Φ^{−1}S_{W}^{n} is evidently the union of the finite subcomodules S_{W}^{p^{k}n}, we can apply Proposition 2.1 to conclude

Proposition 3.1. Every F ∈ Φ^{−1}P_{ω}^{n} has an injective resolution

F ⟶ I(0) ⟶ I(1) ⟶ I(2) ⟶ …

in which each I(j) is a direct sum of summands of Φ^{−1}S_{W}^{n} ’s.

We can apply Proposition 2.2 to conclude

Proposition 3.2. If F and G are finite objects in P^{n}, then the natural map

colim_{k} Ext_{P^{p^{k}n}}^{∗}(F_{(k)}, G_{(k)}) ⟶ Ext_{Φ^{−1}P_{ω}^{n}}^{∗}(F, G)

is an isomophism.

This is Proposition 1.7 of the introduction.

3.3. The category F(q). Let L(V(q)) be the F_{q}–linear extension of the category V(q). Explicitly, L(V(q)) is the multiobject F_{q}–algebra with objects finite dimensional F_{p}–vector spaces, and with L(V, W) = F_{q}[Hom_{F_{q}}(F_{q} ⊗_{F_{p}} V, F_{q} ⊗_{F_{p}} W)]. Then F(q) is the category of L(V(q))–modules.

Since L(V(q)) is of finite type, F(q) is also the category of comodules over the dual F_{q}–coalgebra DL. Examining the definitions reveals that

DL(V, W) = F_{q}^{Hom_{Fq} (F_{q}⊗_{Fp} V,F_{q}⊗_{Fp} W)} = F_{q} ⊗_{F_{p}} I_{q}(Hom_{F_{p}}(V, W)^{#}),

where the second equality is [K:I, Lemma 4.12]. Thus we conclude that F(q) is the category of I_{q}(C(p))–comodules, as asserted in Lemma 1.5

Φ : S^{∗} ⟶ S^{∗} induces an isomophism Φ : I_{q} ⟶ I_{q} (of order [F_{q} : F_{p}]). Thus S^{∗} ⟶ I_{q} uniquely extends to a map Φ^{−1}S^{∗} ⟶ I_{q}, inducing exact functors P_{n} ⟶ Φ^{−1}P_{n} ⟶ F(q)_{n} which are compatible with both Φ and F(q^{r}) ⟶ F(q). Each map

S^{n}(C(p)) ⟶^{Φ} S^{pn}(C(p)) ⟶^{Φ} … ⟶ Φ^{−1}S^{n}(C(p)) ⟶ lim_{q} I_{q,n}(C(p))

is monic, and so we deduce

Lemma 3.3.

(1) For all F, G ∈ P, each of the natural maps

Hom_{P}(F, G) ⟶ Hom_{P}(F_{(k)}, G_{(k)}) ⟶ Hom_{Φ^{−1}P}(F, G)

are isomorphisms.

(2) For all F, G ∈ Φ^{−1}P, the natural map

Hom_{Φ^{−1}P}(F, G) ⟶ lim_{q} Hom_{F(q)}(F, G)

is an isomorphism.

3.4. finite objects and polynomial functor degree. Let F be an object in one of the categories P, Φ^{−1}P, and F(q). If F (V) is finite dimensional for all V, we can define a growth function d_{F} : N ⟶ N by d_{F} (m) = dim F ((F_{p})^{m}).

Proposition 3.4. Let F be an object in one of the categories P, Φ^{−1}P, and F(q). The following are equivalent.

(1) F is a finite object.

(2) The restriction of F to F(p) is finite.

(3) F takes finite dimensional values, and d_{F} (m) is a polynomial function of m.

(4) F takes finite dimensional values, and the restriction of F to F(p) is a polynomial functor in the sense of Eilenberg and MacLane [EM]. Furthermore, if these conditions hold, the degree of d_{F} equals the degree of the polynomial functor.

Proof. For F ∈ F(q), the equivalence of all four conditions was shown in [K:I]. That condition (2) implies condition (1) in general is clear. In the categories P and Φ^{−1}P, we know that finite subobjects must be subobjects of finite sums of S_{W}^{n} ’s. But S_{W}^{n} remains finite when restricted to F(p), thus (1) implies (2).

Given F in one of our categories, we write deg F = d if the restriction of F to F(p) has polynomial degree d.

There is a relationship between polynomial degree and scalar degree.

Proposition 3.5. Every P ∈ P^{n} has finite polynomial degree. If d = deg F, then n and d satisfy

(1) d ≡ n mod (p − 1),

(2) α_{p}(n) ≤ d ≤ n.

This proposition follows immediately from the next proposition, which will describe a small bit of the structure of the simple objects in P.

We need some definitions. Let Λ^{+} be the set of nonincreasing sequences of positive integers λ = (λ_{1}, λ_{2}, …) that are eventually 0. It is useful to view λ ∈ Λ^{+} as describing the rows of a Young diagram. Let n(λ) = ∑_{i} λ_{i}, and let d(λ) = ∑_{k} n(λ(k)), where λ has p–adic expansion λ = ∑_{k} p^{k}λ(k). λ ∈ Λ^{+} is called q– regular if λ_{i} −λ_{i+1} < q for all i, and we let Λ(q) ⊂ Λ^{+} be set of q–regular sequences.

Proposition 3.6. There are simple objects F^{λ} ∈ P, with λ ∈ Λ^{+} such that

(1) {F^{λ} | λ ∈ Λ^{+}} is a complete set of simples in P.

(2) {F^{λ} | λ ∈ Λ(q)} is a complete set of simples in F(q).

(3) F^{λ} has scalar degree n(λ) and polynomial degree d(λ).

This proposition can be gleaned from [FS, §3] and [K:II, §6]. Following the algebraic groups literature, one would say that F^{λ} is the simple object with highest weight λ.

We end this section by noting that, though all simple objects in F(q) arise as the restriction of simple objects in P, this is not the case for more general finite objects. The following is perhaps the simplest example of this phenomenon.

Example 3.7. Let F ∈ F(2) be the functor that arises as a nonsplit extension

0 ⟶ F^{(1,1)} ⟶ F ⟶ F^{(1,1,1)} ⟶ 0.

(It is a quotient of S^{3}.) It is easy to check that F is not the restriction of any functor in F(4).

4. Scalar decomposition and stability as q varies

In [K:II, §5.2], we used an induction/restriction argument to show that the restriction map Ext_{F(q)}^{∗}(S^{1}, S^{1}) ⟶ Ext_{F(p)}^{∗}(S^{1}, S^{1}) is an isomorphism for all q. Here, roughly the same ideas yield a proof of Proposition 1.10.

Fix q and r. As in [K:II, §5.2], the restriction Res : F(q^{r}) ⟶ F(q) has an exact left adjoint Ind : F(q) ⟶ F(q^{r}). Thus, for all F, G ∈ F(q^{r}), the restriction map

Res : Ext_{F(q^{r})}^{∗}(F, G) ⟶ Ext_{F(q)}^{∗}(F, G)

can be identified with

η^{∗} : Ext_{F(q^{r})}^{∗}(F, G) ⟶ Ext_{F(q^{r})}^{∗}(Ind(Res(F)), G),

where η : F ⟶ Ind(Res(F)) is the unit of the adjunction.

Lemma 4.1. For all F ∈ F(q^{r}), η : F ⟶ Ind(Res(F)) is naturally split monic.

Proof. Explicitly, Ind(Res(F)(V) = F (V ⊕V_{(1)}⊕⋯⊕V_{(r−1)}), where (in this section only) V_{(k)} denotes a F_{q^{r}}–vector space V with scalars twisted by the q^{k} power map, and η is the inclusion induced by V ⊂ (V ⊕ ⋯ ⊕ V_{(r−1)}). Since this inclusion has an evident natural left inverse given by projection, η is naturally split monic.

Corollary 4.2. For all F, G ∈ F(q^{r}), Res : Ext_{F(q^{r})}^{∗}(F, G) ⟶ Ext_{F(q)}^{∗}(F, G) is naturally split monic

The following finiteness result was proved by Schwartz in [S, Theorem 5.3.8]

Theorem 4.3. If F, G ∈ F(q) are finite objects, then Ext_{F(q)}^{s}(F, G) is finite for all s.

This theorem, combined with the previous corollary yields the following weak version of Proposition 1.10, which suffices for proving Theorem 1.1.

Corollary 4.4. If F, G ∈ P are finite, and s is fixed, then for all large enough q,

Res : Ext_{F(q^{r})}^{s}(F, G) ⟶ Ext_{F(q)}^{s}(F, G)

is an isomorphism for all r.

To get the stronger stability result, we need to use the scalar decomposition: F ∈ F(q^{r}) naturally decomposes

F ≃ F_{0} ⊕ ⋯ ⊕ F_{q^{r}−1},

with F_{n} ∈ F(q^{r})_{n}. Furthermore, Ext_{F(q^{r})}^{∗}(F_{n}, G_{m}) = 0 unless m = n. Given our remarks above, Proposition 1.10 thus follows from

Theorem 4.5. If n ≤ q, then, for all F ∈ F(q^{r})_{n},

η : F = F_{n} ⟶ Ind(Res(F))_{n}

is an isomorphism.

Proof. We need to begin by generalizing the scalar decomposition to multivariable functors. Suppose H(V_{0}, …, V_{r−1}) is an r variable functor from F_{q^{r}}–vector spaces to F̄_{p}–vector spaces. Then H will have a natural decomposition

H ≃ ⨁_{I} H_{I},

where I ranges over sequences I = (i_{0}, …, i_{r−1}) with 0 ≤ i_{j} ≤ q^{r} − 1, and H_{I} is defined by

H_{I}((V_{0}, …, V_{r−1}) =

{x ∈ H(V_{0}, …, V_{r−1}) | H(λ_{0}, …, λ_{r−1})(x) = λ_{0}^{i_{0}} … λ_{r−1}^{i_{r−1}}x for all (λ_{0}, …, λ_{r−1}) ∈ (F_{q^{r}})^{r}}.

We apply this to the multivariable functor F (V_{0} ⊕ ⋯ ⊕ V_{r−1}), yielding a decomposition

F (V_{0} ⊕ ⋯ ⊕ V_{r−1}) ≃ ⨁_{I} F_{I}(V_{0}, …, V_{r−1}),

where we observe that, since F has scalar degree n, F_{I} = 0 unless |I| = i_{0} + i_{1} + ⋯ + i_{r−1} = n. Thus we have a decomposition

Ind(Res(F))(V) ≃ ⨁_{I} F_{I}(V, V_{(1)}, …, V_{(r−1)}),

with I running over this same sum. As a functor of V, F_{I}(V, V_{(1)}, …, V_{(r−1)}) clearly has scalar degree n_{q}(I) where n_{q}(I) = i_{0} + i_{1}q + ⋯ + i_{r−1}q^{r−1}. Thus, restricting to the components of scalar degree n, we have

Ind(Res(F))_{n}(V) ≃ ⨁_{I} F_{I}(V, V_{(1)}, …, V_{(r−1)}),

with the sum over I such that both |I| = n and n_{q}(I) ≡ n mod (q^{r} − 1). The summand F (V) corresponds to the sequence I = (n, 0, …, 0). The next lemma shows that, under the hypothesis that n ≤ q, this is the only summand, proving the theorem.

Lemma 4.6. If |I| ≤ q and |I| ≡ n_{q}(I) mod (q^{r} − 1), then I = (|I|, 0, …, 0).

Proof. Since |I| ≤ q, it follows that each i_{j} ≤ q, and if any i_{j} = q then it is the only nonzero term in I. In this latter case, it is easy to check that I = (q, 0, …, 0) is the only possibility making true |I| ≡ n_{q}(I) mod (q^{r} − 1). So we can assume i_{j} ≤ q − 1. But then n_{q}(I) ≤ (q^{r} − 1) and we easily deduce that |I| ≡ n_{q}(I) mod (q^{r} − 1) implies that |I| = n_{q}(I), as needed.

5. Proof of Proposition 1.9

Recall that F_{ω}(q) is the category of locally finite objects in F(q). In our context, the “Vanishing Theorem” of [K:III] reads

Theorem 5.1. For all n and W, Φ^{−1}S_{W}^{n} is an injective object in F_{ω}(q).

Combined with Proposition 3.1, this implies

Corollary 5.2. Every G ∈ Φ^{−1}P_{ω}^{n} has an injective resolution

0 ⟶ G ⟶ I(0) ⟶ I(1) ⟶ I(2) ⟶ …

that remains injective when restricted to F_{ω}(q).

Proof of Proposition 1.9. Let F and G be any two objects in Φ^{−1}P_{ω}^{n}. With I(∗) an injective resolution of G as in the last corollary, Ext_{Φ^{−1}P_{ω}^{n}}^{∗}(F, G) is the homology of the complex Hom_{Φ^{−1}P_{ω}^{n}}(F, I(∗)), and Ext_{F_{ω}(q)}^{∗}(F, G) is the homology of the complex Hom_{F_{ω}(q)}(F, I(∗)). By Lemma 3.3, there are isomorphisms of complexes

Hom_{Φ^{−1}P_{ω}^{n}}(F, I(∗)) ≃ lim_{q} Hom_{F_{ω}(q)}(F, I(∗)).

Thus there are short exact sequences

0 ⟶ lim_{q}^{1} Ext_{F_{ω}(q)}^{s−1}(F, G) ⟶ Ext_{Φ^{−1}P_{ω}^{n}}^{s}(F, G) ⟶ lim_{q} Ext_{F_{ω}(q)}^{s}(F, G) ⟶ 0.

The proof is completed by noting that Schwartz’s finiteness theorem, Theorem 4.3, implies that if F and G are finite, then the lim^{1} term vanishes, and also Ext_{F_{ω}(q)}^{∗}(F, G) ≃ Ext_{F(q)}^{∗}(F, G) (see [K:III, Prop.4.2]).

6. Proof of Proposition 1.8

In this section we prove Proposition 1.8 by using De Rham complexes as was done in [FLS], [FS], and particularly [F].

6.1. Hypercohomology spectral sequences. We introduce notation for hypercohomology spectral sequences as in [FLS]. See [W] for more about the homological algebra.

Let A be an abelian category with enough injectives. Given a pair (A, C^{∗}), with A ∈ A, and C^{∗} a complex C^{0} ⟶ C^{1} ⟶ C^{2} ⟶ … of objects in A, one gets two convergent spectral sequences, {I_{r}} and {II_{r}}, satisfying

I_{1}^{s,t} = Ext_{A}^{t}(A, C^{s}), and
II_{2}^{s,t} = Ext_{A}^{s}(A, H^{t}(C^{∗})),

with common abutment “Ext^{s+t}(A, C^{∗})” (a Hom–group in the derived category of A).

In either spectral sequence, the differential d_{r} has bidegree (r, 1 − r). The composite of the edge homomorphisms

Ext_{A}^{s}(A, H^{0}(C^{∗})) = II_{2}^{s,0} ⟶ Ext^{s}(A, C^{∗}) ⟶ I_{1}^{0,s} = Ext_{A}^{s}(A, C^{0})

is the map induced by the inclusion H^{0}(C^{∗}) ↪ C^{0}.

Finally, we note that these constructions are natural in all of the data.

6.2. The De Rham complex. Given an F_{p}–vector space W, let Λ_{W}^{n} ∈ P^{n} be defined by Λ_{W}^{n} (V) = F̄_{p} ⊗_{F_{p}} Λ^{n}(Hom(V, W)^{#}).

Define a complex Ω_{n,W}^{∗} ∈ P^{n} by letting Ω_{n,W}^{i} = S_{W}^{n−i} ⊗ Λ_{W}^{i}, and by letting d : Ω_{n,W}^{i} ⟶ Ω_{n,W}^{i+1} be the composite

S_{W}^{n−i} ⊗ Λ_{W}^{i} ⟶ S_{W}^{n−i−1} ⊗ S_{W}^{1} ⊗ Λ_{W}^{i} ⟶ S_{W}^{n−i−1} ⊗ Λ_{W}^{i+1}.

Lemma 6.1. There is an isomorphism (Ω_{n,W}^{∗})_{(1)} ≃ H^{∗}(Ω_{pn,W}^{∗}; d). Under this isomorphism, Φ : (S_{W}^{n})_{(1)} ⟶ S_{W}^{pn} corresponds to the inclusion H^{0}(Ω_{pn,W}^{∗}) ↪ Ω_{pn,W}^{0} .

See [C] or [FLS].

6.3. The polynomial degree vanishing theorem. The purpose of this subsection is to establish the following theorem and corollary.

Theorem 6.2. Suppose F ∈ P^{n} has deg F ≤ d. If n = n_{0} + ⋯ + n_{d} with each n_{j} > 0, then given any G_{0}, …, G_{d} with G_{j} ∈ P^{n_{j}}, Ext_{P}^{∗}(F, G_{0} ⊗ ⋯ ⊗ G_{d}) = 0.

Corollary 6.3. Suppose F ∈ P^{n} has deg F ≤ d. If α_{p}(n − i) + α_{p}(i) > d, then Ext_{P}(F_{(k)}, (Ω_{n,W}^{i})_{(k)}) = 0 for all k.

The corollary follows from the theorem by noting that, by simple transfer arguments, if α_{p}(n − i) + α_{p}(i) > d, then (Ω_{n,W}^{i})_{(k)} will be a direct summand of a functor of the form G_{0} ⊗ ⋯ ⊗ G_{d} as in the theorem.

Proof of Theorem 6.2. We need to use P⟨d + 1⟩, the d + 1 variable version of the category P. P⟨d+1⟩ is the sum of categories P⟨d+1⟩^{n}, where n = (n_{0}, …, n_{d}) and P⟨d + 1⟩^{n} is the category of comodules over the multiobject coalgebra S^{n_{0}}(C(p)) ⊗ ⋯ ⊗ S^{n_{d}}(C(d)). Explicitly, an object G ∈ P⟨d + 1⟩^{n} will consist of an F̄_{p}–vector space G(V_{0}, …, V_{d}) for every d + 1 tuple of finite dimensional F_{p}–vector spaces (V_{0}, …, V_{d}), together with stucture maps

G(V_{0}, …, V_{d}) ⟶ G(W_{0}, …, W_{d})⊗S^{n_{0}}(Hom(V_{0}, W_{0})^{#})⊗⋯⊗S^{n_{0}}(Hom(V_{d}, W_{d})^{#})

making appropriate diagrams commute.

Using the natural isomorphims S^{∗}(V ⊕ W) ≃ S^{∗}(V) ⊗ S^{∗}(W), it is straightforward to check that there are adjoint functors

∇ : P ⟶ P⟨d + 1⟩ and Δ : P⟨d + 1⟩ ⟶ P

with (∇F)(V_{0}, …, V_{d}) = F (V_{0} ⊕ ⋯ ⊕ V_{d}) and (ΔG)(V) = G(V, …, V). These are both exact, so ones gets

Ext_{P⟨d+1⟩}^{∗}(∇F, G) ≃ Ext_{P}^{∗}(F, ΔG)

for F ∈ P, G ∈ P⟨d + 1⟩.

Using the multivariable scalar degree (defined as in the proof of Theorem 4.5), this refines as follows. If n = (n_{0}, …, n_{d}) and n = n_{0} + ⋯ + n_{d}, ones gets isomorphisms

Ext_{P⟨d+1⟩^{n}}^{∗}((∇F)_{n}, G) ≃ Ext_{P^{n}}^{∗}(F, ΔG)

for F ∈ P^{n}, G ∈ P⟨d+1⟩^{n}, where (∇F)_{n} denotes the component of ∇F in P⟨d+1⟩^{n}.

Now let G_{0}□ × ⋯ □ ×G_{d} ∈ P⟨d + 1⟩^{n} be defined by (G_{0}□ × ⋯ □ ×G_{d})(V_{0}, …, V_{d}) = G_{0}(V_{0}) ⊗ ⋯ ⊗ G_{d}(V_{d}). Since Δ(G_{0}□ × ⋯ □ ×G_{d}) = G_{0} ⊗ ⋯ ⊗ G_{d}, there is an isomorphism

Ext_{P^{n}}^{∗}(F, G_{0} ⊗ ⋯ ⊗ G_{d}) ≃ Ext_{P⟨d+1⟩^{n}}^{∗}((∇F)_{n}, G_{0}□ × ⋯ □ ×G_{d}).

The proof of the theorem is now completed by observing that under our hypotheses, (∇F)_{n} = 0: the assumption that deg F ≤ d means that all (d + 1)–fold cross effects are zero [EM], thus (∇F)_{n} ≠ 0 implies that at most d of the n_{j} can be nonzero, contradicting our hypothesis that all d + 1 of the n_{j} are positive.

Remark 6.4. Theorems like this, with objects in F(q), have been noted before by many people, with the idea (misreferenced in [FS]) going back to T. Pirashvili’s 1988 article [Pi]). Closely related, and also sufficient for proving Corollary 6.3, are “Kunneth theorems” for calculating Ext–groups of tensor products in these sorts of categories (as in [K:III, §4.4]). Here the idea, as far as the author knows, first appeared in [K:I, §7], which was widely circulated in preprint form in 1990.

6.4. More vanishing Ext groups. We take our first steps towards proving Proposition 1.8, closely following [F, §IV].

The following combinatorial lemma was essentially proved in [F, proof of Prop.4.1.2]

Lemma 6.5. If 0 < t ≤ p^{ν_{p}(n)} then

α_{p}(n − t) + α_{p}(t) = α_{p}(n) + (p − 1)(ν_{p}(n) − ν_{p}(t)).

Proof. Write n = n_{0}p^{ν_{p}(n)} and t = t_{0}p^{ν_{p}(t)}. Then

α_{p}(n − t) = α_{p}((n_{0} − 1)p^{ν_{p}(n)}) + α_{p}(p^{ν_{p}(n)} − t)
= α_{p}(n_{0} − 1) + α_{p}(p^{(ν_{p}(n)−ν_{p}(t))} − t_{0})
= α_{p}(n_{0}) − 1 + (p − 1)(ν_{p}(n) − ν_{p}(t)) − α_{p}(t_{0}) + 1
= α_{p}(n) + (p − 1)(ν_{p}(n) − ν_{p}(t)) − α_{p}(t),

where the first equality holds because t ≤ p^{ν_{p}(n)}.

Lemma 6.6. If ν_{p}(n) > (d − α_{p}(n))/(p − 1) + log_{p}(s), then the natural map

Ext_{P}^{t}(F_{(k)}, (S_{W}^{n})_{(k)}) ⟶ Ext_{P}^{t}(F_{(k)}, (S_{W}^{pn})_{(k−1)})

is an isomorphism, for all W, t ≤ s, k ≥ 1, and F ∈ P^{n} with deg F ≤ d.

Proof. Consider the hypercohomology sequences associated to the pair (F_{(k)}, (Ω_{pn,W}^{∗})_{(k−1)}). We will have

I_{1}^{t_{1},t_{2}} = Ext_{P}^{t_{2}}(F_{(k)}, (S_{W}^{pn−t_{1}} ⊗ Λ_{W}^{t_{1}} )_{(k−1)}), and
II_{2}^{t_{1},t_{2}} = Ext_{P}^{t_{1}}(F_{(k)}, (S_{W}^{n−t_{2}} ⊗ Λ_{W}^{t_{2}} )_{(k)}).

RATIONAL COHOMOLOGY AND GENERIC REPRESENTATION THEORY 17

Thus, by Corollary 6.3,

I_{1}^{t,∗} = 0 if α_{p}(pn − t) + α_{p}(t) > d, and
II_{2}^{∗,t} = 0 if α_{p}(n − t) + α_{p}(t) > d.

The numerical hypothesis ensures that the previous lemma can be used, and we learn that

I_{1}^{t,∗} = 0 for 0 < t ≤ s + (p − 1), and
II_{2}^{∗,t} = 0 for 0 < t ≤ s.

We deduce that, for t ≤ s, the composite of the edge homomorphisms

II_{1}^{t,0} ⟶ I_{1}^{0,t}

is an isomorphism, which is precisely the conclusion of the lemma. Proposition 6.7. If h > (d − α_{p}(n))/(p − 1) + log_{p}(s) − ν_{p}(n), then

Ext_{P}^{t}(F_{(k)}, (S_{W}^{p^{h}n})_{(k)}) = 0

for all W, 0 < t ≤ s, k ≥ 0, and F ∈ P^{p^{h}n} with deg F ≤ d. Proof. The numerical hypothesis can be rewritten as

ν_{p}(p^{h}n) > (d − α_{p}(p^{h}n))/(p − 1) + log_{p}(s),

so that the previous lemma applies. Using the lemma k times implies that

Ext_{P}^{t}(F_{(k)}, (S_{W}^{p^{h}n})_{(k)}) ≃ Ext_{P}^{t}(F_{(k)}, (S_{W}^{p^{h+k}n})).

But this latter group is 0 for 0 < t ≤ s, as S_{W}^{p^{h+k}n} is injective in P. Remark 6.8. Comparison with [F, Prop.4.1.2] suggests that s might be replaced by s/2 in the numerical hypothesis of the proposition (which would double the range of vanishing Ext groups), but the author has been unable to follow Franjou’s argument on this point. 6.5. Stabilization with respect to Frobenious twisting. The following theorem is a strengthened version of Proposition 1.8. Theorem 6.9. If k > (d − α_{p}(n))/(p − 1) + log_{p}(s) − ν_{p}(n), then

Φ : Ext_{P}^{t}(F_{(k)}, G_{(k)}) ⟶ Ext_{P}^{t}(F_{(k+1)}, G_{(k+1)})

is an isomorphism for all t ≤ s, and F, G ∈ P^{n} with deg F ≤ d.

The structure of the proof is as follows. Fixing n and d, we will define two families of statements, S(s, h) and S′(s, h), for s, h ≥ 0. Statement S(s, 0) will be the statement of the theorem. Thus the next four lemmas will prove the theorem by upwards induction on s, followed by downwards induction on h. Lemma 6.10. S(0, h) is true for all h. Lemma 6.11. S′(s, h) is true for all h > (d − α_{p}(n))/(p − 1) + log_{p}(s) − ν_{p}(n). Lemma 6.12. S′(s, h) implies S(s, h).

Lemma 6.13. S(s − 1, h) and S(s, h + 1) imply S′(s, h).

Statement S(s, h): If k > (d − α_{p}(n))/(p − 1) + log_{p}(s) − ν_{p}(n) − h, then

Φ : Ext_{P}^{t}(F_{(k)}, G_{(k)}) ⟶ Ext_{P}^{t}(F_{(k+1)}, G_{(k+1)})

is an isomorphism for all t ≤ s, and F, G ∈ P^{p^{h}n} with deg F ≤ d.

Statement S′(s, h): If k > (d − α_{p}(n))/(p − 1) + log_{p}(s) − ν_{p}(n) − h, then

Φ : Ext_{P}^{t}(F_{(k)}, (S_{W}^{p^{h}n})_{(k)}) ⟶ Ext_{P}^{t}(F_{(k+1)}, (S_{W}^{p^{h}n})_{(k+1)})

is an isomorphism for all W, t ≤ s, and F ∈ P^{p^{h}n} with deg F ≤ d.

The careful reader will note that the implication S(s, h) ⇒ S′(s, h) is obvious.

Proof of Lemma 6.10. This is a weak form of Lemma 3.3.

Proof of Lemma 6.11. By Proposition 6.7, if 0 < t ≤ s, both Ext groups will be 0.

Proof of Lemma 6.12. Any G ∈ P^{p^{h}n} will have a resolution of the form

0 ⟶ G ⟶ C^{0} ⟶ C^{1} ⟶ …

in which each C^{j} is a sum of summands of S_{W}^{p^{h}n}’s. Let {I_{r}(k)} be the first hypercohomology spectral sequence associated to the pair (F_{(k)}, C_{(k)}^{∗}). This satisfies

Ext_{P}^{t_{2}}(F_{(k)}, C_{(k)}^{t_{1}}) = I_{1}^{t_{1},t_{2}}(k) ⇒ Ext_{P}^{t_{1}+t_{2}}(F_{(k)}, G_{(k)}).

By naturality, Φ induces a map of spectral sequences

Φ : {I_{r}(k)} ⟶ {I_{r}(k + 1)}.

Statement S′(s, h) implies that

Φ : I_{1}^{∗,t}(k) ⟶ I_{1}^{∗,t}(k + 1)

is an isomophism for t ≤ s. Thus

Φ : I_{∞}^{t_{1},t_{2}}(k) ⟶ I_{∞}^{t_{1},t_{2}}(k + 1)

will be an isomophism for t_{1} + t_{2} ≤ s, and S(s, h) follows.

Proof of Lemma 6.13. We can assume that k > 0, since if k = 0, then both Ext groups will be 0, by Proposition 6.7. Let {I_{r}(k)} and {II_{r}(k)} denote the hypercohomology spectral sequences associated to the pair (F_{(k)}, (Ω_{p^{h+1}n,W}^{∗})_{(k−1)}). We will have

I_{1}^{t_{1},t_{2}} = Ext_{P}^{t_{2}}(F_{(k)}, (S_{W}^{p^{h+1}n−t_{1}} ⊗ Λ_{W}^{t_{1}} )_{(k−1)}), and
II_{2}^{t_{1},t_{2}} = Ext_{P}^{t_{1}}(F_{(k)}, (S_{W}^{p^{h}n−t_{2}} ⊗ Λ_{W}^{t_{2}} )_{(k)}).

Our numerical assumption rewrites as

k − 1 > (d − α_{p}(n))/(p − 1) + log_{p}(s) − ν_{p}(n) − (h + 1).

Thus statement S(s, h + 1) implies that

Φ : I_{1}^{∗,t}(k) ⟶ I_{1}^{∗,t}(k + 1)

is an isomophism for t ≤ s. Meanwhile, statement S(s − 1, h) implies that

Φ : II_{2}^{t,∗}(k) ⟶ II_{2}^{t,∗}(k + 1)

is an isomophism for t ≤ s − 1. We deduce that

Φ : II_{2}^{s,0}(k) ⟶ I_{2}^{s,0}(k + 1)

is also an isomophism, and we have proved S′(s, h).

7. Proof of Theorem 1.2

The main theorem of [CPSvK] implies

Theorem 7.1. For a fixed n and s, there exist numbers K(n, s) and Q(n, s) so that, for all F, G ∈ P^{n}, and for all m,

Ext_{GL_{m}}^{s}(F_{(k)}, G_{(k)}) ⟶ Ext_{GL_{m}(F_{q})}^{s}(F, G)

will be an isomorphism if k > K(n, s) and q > Q(n, s).

Remark 7.2. The important thing here is that stability is independent of m. (Friedlander and Suslin also suggest in [FS, Thm.7.3] that K(n, s) can be taken to be independent of n, but I have been unable to read this from [CPSvK].)

Using “good filtrations” of GL_{m}–modules (as in [Do1, Do2]), Friedlander and Suslin [FS, Cor. 3.13] show

Theorem 7.3. For all F, G ∈ P^{n},

Ext_{P}^{∗}(F, G) ⟶ Ext_{GL_{m}}^{∗}(F, G)

will be an isomorphism if m ≥ n.

Theorem 1.2 follows easily from these two theorems and Theorem 1.1.

Proof of Theorem 1.2. Combining Theorem 1.1 and Theorem 7.1, we learn that given F, G, and s, there exists k and Q such that

Ext_{P}^{s}(F_{(k)}, G_{(k)}) ⟶ Ext_{F(F_{q})}^{s}(F, G) and
Ext_{GL_{m}}^{s}(F_{(k)}, G_{(k)}) ⟶ Ext_{GL_{m}(F_{q})}^{s}(F, G)

are isomorphisms for all q > Q, and for all m.

We can clearly assume that F, G ∈ P^{n} for some n, and so Theorem 7.3 implies that

Ext_{P}^{s}(F_{(k)}, G_{(k)}) ⟶ Ext_{GL_{m}}^{s}(F_{(k)}, G_{(k)})

will be an isomorphism if m ≥ p^{k}n.

Thus, in the commutative diagram

Ext_{P}^{s}(F_{(k)}, G_{(k)}) ⟶ Ext_{GL_{m}}^{s}(F_{(k)}, G_{(k)})
↓  ↓
Ext_{F(q)}^{s}(F, G) ⟶ Ext_{GL_{m}(F_{q})}^{s}(F, G),

we know that the top and side maps are isomorphisms, for all large enough m and q, thus so is the bottom map.

We end the paper by pointing out that there is still an unproven statement of the same sort as Theorem 1.1 and [CPSvK].

Question 7.4. If M, N are modules over the algebraic semigroup M_{m} (defined over F_{p}, will the map

Ext_{M_{m}}^{s}(M_{(k)}, N_{(k)}) ⟶ Ext_{M_{m}(F_{q})}^{s}(M, N)

be an isomorphism for large k and q?

Theorem 7.1 and results in [FS] (basically Theorem 7.3 above) combine to say this map is monic for large k and q.

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Department of Mathematics, University of Virginia, Charlottesville, VA 22903