GENERAL LINEAR AND FUNCTOR

COHOMOLOGY OVER FINITE FIELDS

Vincent Franjou^{†}, Eric M. Friedlander^{∗}, Alexander Scorichenko, and Andrei Suslin^{∗∗},

Introduction

In recent years, there has been considerable success in computing Ext-groups of modular representations associated to the general linear group by relating this problem to one of computing Ext-groups in functor categories [F-L-S], [F-S]. In this paper, we extend our ability to make such Ext-group calculations by establishing several fundamental results. Throughout this paper, we work over fields of positive characteristic p.

We consider two different categories of functors, the category F(𝔽_{q}) of all functors from finite dimensional 𝔽_{q}-vector spaces to 𝔽_{q}-vector spaces, where 𝔽_{q} is the finite field of cardinality q, and the category P(𝔽_{q}) of strict polynomial functors of finite degree as defined in [F-S]. The category P(𝔽_{q}) presents several advantages over the category F(𝔽_{q}) from the point of view of computing Ext-groups. These are the accessibility of injectives and projectives, the existence of a base change, and an even easier access to Ext-groups of tensor products. This explains the usefulness of our comparison in Theorem 3.9 of Ext-groups in the category P(𝔽_{q}) with Ext-groups in the category F(𝔽_{q}). Weaker forms of this theorem have been known to us since 1995 and to T. Pirashvili (unpublished) independently. This early work apparently inspired the paper of N. Kuhn [K2] as well as the present paper.

The calculation (for an arbitrary field k of positive characteristic) of the Ext_{P(k)}-groups from a Frobenius-twisted divided power functor to a Frobeniustwisted symmetric power functor is presented in Theorem 4.5. This calculation is extended in section 5 to various other calculations of Ext_{P(k)}-groups between divided powers, exterior powers, and symmetric powers. This leads to similar Ext_{F(𝔽_{q})}-group calculations in Theorem 6.3. The results are given with their structure as tri-graded Hopf algebras. A result, stated in this form, has been obtained by N. Kuhn for natural transformations (Hom_{F(𝔽_{q})}-case) from a divided power functor

Typeset by AMS-TEX

†Partially supported by the CNRS, UMR 6629

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

∗∗Partially supported by the N.S.F. grant DMS - 9510242

to a symmetric power functor [K3, §5]. Computing Ext-groups between symmetric and exterior powers is the topic in [F], which contains partial results for the category F(𝔽_{p}).

The final result, proved by the last-named author in the appendix, is the proof of equality of the Ext_{F(𝔽_{q})}-groups and Ext_{GL_{∞}(𝔽_{q})}-groups associated to strict polynomial functors of finite degree. This has an history of its own that will be briefly recalled at the beginning of the appendix.

These results complement the important work of E. Cline, B. Parshall, L. Scott, and W. van der Kallen in [CPSvdK]. The results in that paper apply to general reductive groups defined and split over the prime field and to general (finite dimensional) rational modules, but lack the computational applicability of the present paper. As observed in [CPSvdK], Ext-groups of rational G-modules are isomorphic to Ext-groups of associated Chevalley groups, provided that Frobenius twist is applied sufficiently many times to the rational modules, and provided that the finite field is sufficiently large. One consequence of our work is a strong stability result for the effect of iterating Frobenius twists (Corollary 4.10); it applies to the Extgroups of rational modules arising from strict polynomial functors of finite degree. A second consequence is an equally precise lower bound for the order of the finite field 𝔽_{q} required to compare these “stably twisted” rational Ext-groups with the Ext-groups computed for the infinite general linear group GL_{∞}(𝔽_{q}). For explicit calculations of Ext-groups for GL_{n}(𝔽_{q}) for various fundamental GL_{n}(𝔽_{q})-modules, one then can combine results of this paper with explicit stability results of W. van der Kallen [vdK].

What follows is a brief sketch of the contents of this paper. Section 1 recalls the category P(k) of strict polynomial functors of finite degree on finite dimensional k-vector spaces and further recalls the relationship of P to the category of rational representations of the general linear group. The investigation of the forgetful functor P(k) → F(k) is begun by observing that Hom_{P(k)}(P, Q) = Hom_{F(k)}(P, Q) for strict polynomial functors of degree d provided that cardinality of the field k is at least d. Theorem 1.7 presents the key property for Ext-groups involving functors of exponential type; this property enables our Ext-group calculations and appears to have no evident analogue for GL_{n} (with n finite).

In section 2, we employ earlier work of N. Kuhn and the first named author to provide a first comparison of Ext_{P(𝔽_{q})}^{n}(P, Q) and Ext_{F(𝔽_{q})}^{n}(P, Q). The weakness of this comparison is that the lower bound on q depends upon the Ext-degree n as well as the degrees of P and Q. Section 3 remedies this weakness, providing in Theorem 3.10 a comparison of Ext-groups for which the lower bound for q depends only upon the degrees of P and Q. Our proof relies heavily upon an analysis of base change, i.e. the effect of an extension E/F of finite fields on Ext_{F}(P, Q)-groups when P and Q are strict polynomial functors.

For every integer d, we compute in Theorem 4.5, Ext_{P(k)}(Γ^{d(r)}, S^{dp^{r−j}(j)}) and Ext_{P(k)}(Γ^{d(r)}, Λ^{dp^{r−j}(j)}), where Γ^{d}, S^{d}, Λ^{d} denote the d-fold divided power, symmetric power, and exterior power functor respectively. These computations are fundamental, for Γ^{d} (respectively its dual S^{d}) is projective (resp. injective) in P(k) and tensor products of Γ^{i} (resp. S^{i}) of total degree d constitute a family of projective generators (resp. injective cogenerators) for P_{d}(k), the full subcategory of P(k) consisting of functors homogeneous of degree d. For example, Theorem 4.5 leads to the strong stability result (with respect to Frobenius twist) of Corollary 4.10 which is applicable to arbitrary strict polynomial functors of finite degree. The proof of Theorem 4.5 is an intricate nested triple induction argument. Readers of [F-L-S] or [F-S] will recognize here a new ingredient: the computation of the differentials in hypercohomology spectral sequences as Koszul differentials.

In section 5, we extend the computation of Theorem 4.5 to other fundamental pairs of strict polynomial functors. Finally, in section 6, we combine these computations of Ext_{P(k)}-groups with the strong comparison theorem of section 3 and our understanding of base change for Ext_{F(𝔽_{q})}-groups. The result is various complete calculations of Ext_{F(𝔽_{q})}^{∗}(−, −), as tri-graded Hopf algebras.

The appendix, written by the last-named author, demonstrates a natural isomorphism Ext_{F}^{∗}(A, B) ⟶^{∼} Ext_{GL(𝔽_{q})}^{∗}(A(𝔽_{q}^{∞}), B(𝔽_{q}^{∞})) for finite functors A, B in F.

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

§1. Recollections of Functor Categories

The purpose of this expository section is to recall the definitions and basic properties of the category P of strict polynomial functors of finite degree (on k-vector spaces) introduced in [F-S] and to be used in subsequent sections. We also contrast the category P with the category F of all functors from finite dimensional k-vector spaces to k-vector spaces. Strict polynomial functors were introduced in order to study rational cohomology of the general linear groups GL_{n} over k (i.e., cohomology of comodules for the Hopf algebra k[GL_{n}], the coordinate algebra of the algebraic group GL_{n}). Although one can consider strict polynomial functors over arbitrary commutative rings (as was necessarily done in [S-F-B]), we shall restrict attention throughout this paper to such functors defined on vector spaces over a field k of characteristic p > 0. Indeed, in subsequent sections we specialize further to the case in which k is a finite field.

We let V = V_{k} denote the category of k-vector spaces and k-linear homomorphisms and we denote by V^{f} the full subcategory of finite dimensional kvector spaces. A polynomial map T : V → W between finite dimensional vector spaces is defined to be a morphism of the corresponding affine schemes over k: Spec(S^{∗}(V^{#})) → Spec(S^{∗}(W^{#})) (where Spec(S^{∗}(V^{#})) is the affine scheme associated to the symmetric algebra over k of the k-linear dual of V). Equivalently, such a polynomial map is an element of S^{∗}(V^{#}) ⊗ W. The polynomial map T : V → W is said to be homogeneous of degree d if T ∈ S^{d}(V^{#}) ⊗ W. A polynomial map between finite dimensional vector spaces is uniquely determined by its associated set-theoretic function from V to W provided that k is infinite; this is more readily understood as the observation that a polynomial (in any number of variables) is uniquely determined by its values at k-rational points provided that the base field is infinite.

Strict polynomial functors

We recall [F-S, 2.1] that a strict polynomial functor P : V^{f} → V^{f} is the following collection of data: for any V ∈ V^{f} a vector space P (V) ∈ V^{f}; for any V, W in V^{f}, a polynomial map P_{V,W} : Hom_{k}(V, W) → Hom_{k}(P (V), P (W)). These polynomial maps should satisfy appropriate compatibility conditions similar to the ones used in the usual definition of a functor. A strict polynomial functor is said to be homogeneous of degree d if P_{V,W} : Hom_{k}(V, W) → Hom_{k}(P (V), P (W)) is homogeneous of degree d for each pair V, W in V^{f}. A strict polynomial functor is said to be of finite degree provided that the degrees of the polynomial maps P_{V,W} are bounded independent of V, W ∈ V^{f}.

Replacing the polynomial maps P_{V,W} : Hom_{k}(V, W) → Hom_{k}(P (V), P (W)) by the associated set-theoretic functions, we associate to each strict polynomial functor P a functor in the usual sense V^{f} → V^{f} (which we usually denote by the same letter P). The above remarks about polynomial maps imply that over infinite fields strict polynomial functors may be viewed as functors (in the usual sense) which satisfy an appropriate additional property. However, over finite fields (and a fortiori over more general base rings), this is no longer the case: the concept of a strict polynomial functor incorporates more data than that of a functor in the usual sense.

We denote by P (or by P(k) if there is need to specify the base field k) the abelian category of strict polynomial functors of finite degree. One easily verifies that P splits as a direct sum ⊕_{d}P_{d}, where P_{d} denotes the full subcategory of strict polynomial functors homogeneous of degree d.

Typical examples of strict polynomial functors homogeneous of degree d are the d-fold tensor power functor ⊗^{d}, the d-fold exterior power functor Λ^{d}, the d-fold symmetric power functor S^{d} (defined as the Σ_{d}-coinvariants of ⊗^{d}), and the d-fold divided power functor Γ^{d} (defined as the Σ_{d}-invariants of ⊗^{d}). The functor S^{d} is injective, and Γ^{d} is projective in P_{d}.

Let ϕ : k → k denote the p^{th}-power map. The Frobenius twist functor I^{(1)} sends a vector space V to the base change of V via the map ϕ (which we denote by V^{(1)}). So defined, I^{(1)} is a strict polynomial functor homogeneous of degree p; I_{V,W}^{(1)} is the p-th power map

Hom_{k}(V^{(1)}, W^{(1)})^{#} = (Hom_{k}(V, W)^{#})^{(1)} → S^{p}(Hom_{k}(V, W)^{#})

viewed as an element of S^{p}(Hom_{k}(V, W)^{#}) ⊗ Hom_{k}(V^{(1)}, W^{(1)}). For any functor G : V^{f} → V, we define G^{(1)} as G ∘ I^{(1)}.

As observed in [S-F-B, §2], there is a natural construction of base change of a strict polynomial functor. Namely, if k → K is a field extension and V is a k-vector space, let V_{K} denote K ⊗_{k} V with the evident K vector space structure. If P is a strict polynomial functor over k, then the base change P_{K} is defined by setting P_{K}(V_{K}) = P (V)_{K} and setting the polynomial map (P_{K})_{V_{K},W_{K}} : Hom_{K}(V_{K}, W_{K}) → Hom_{K}(P (V_{K}), P (W_{K})) to be the base change of P_{V,W} as a morphism of affine schemes. As observed in [S-F-B, 2.6], this base change is both exact and preserves projectives. Consequently, we have the following elementary base change property.

Proposition 1.1. [S-F-B, 2.7] Let P, Q be strict polynomial functors of finite degree over a field k. For any field extension k → K, there is a natural isomorphism of K-vector spaces

Ext_{P(K)}^{∗}(P_{K}, Q_{K}) ≅ Ext_{P(k)}^{∗}(P, Q) ⊗_{k} K .

If P is a strict polynomial functor, then for any n > 0 the vector space P (k^{n}) inherits a natural structure of a rational GL_{n}-module. (In fact, functoriality of P implies that P (k^{n}) inherits a natural structure of a comodule for the coalgebra k[M_{n}] whose restriction to k[GL_{n}] ⊂ k[M_{n}] provides the rational GL_{n}-structure). We recall the following relationship between Ext-groups in the category P and in the category of rational GL_{n}-modules.

Theorem 1.2. [F-S, 3.13] Let P, Q be strict polynomial functors homogeneous of degree d and let n ≥ d. Then there is a natural isomorphism

Ext_{P}^{∗}(P, Q) ⟶^{≃} Ext_{GL_{n}}^{∗}(P (k^{n}), Q(k^{n}))

induced by the exact functor sending a strict polynomial functor to its value on k^{n}.

By a theorem of H. Andersen (cf. [J, II, 10.14]), the Frobenius twist induces an injection on rational Ext-groups: for any two finite dimensional rational GL_{n}- modules M and N, the natural map induced by the Frobenius twist (which we view as an exact functor on the category of rational GL_{n}-modules)

Ext_{GL_{n}}^{∗}(M, N) → Ext_{GL_{n}}^{∗}(M^{(1)}, N^{(1)})

is injective. Thus, Theorem 1.2 gives us the following useful corollary.

Corollary 1.3. Let P, Q be strict polynomial functors homogeneous of degree d. The Frobenius twist is an exact functor on P which induces an injective map on Ext-groups:

Ext_{P}^{∗}(P, Q) → Ext_{P}^{∗}(P^{(1)}, Q^{(1)}).

Consider now the abelian category F of functors V^{f} → V. If we need to explicitly indicate the base field k, we shall denote this category by F(k). The forgetful functor P → F is clearly exact, thereby inducing a natural map on Ext-groups

Ext_{P}^{∗}(P, Q) → Ext_{F}^{∗}(P, Q)

where we have abused notation by using P, Q to denote strict polynomial functors and their images in F.

The following elementary proposition provides a key to understanding the forgetful functor P → F.

Proposition 1.4. Assume that k has at least d elements. Then for any P, Q in P_{d}, the natural inclusion Hom_{P}(P, Q) ↪ Hom_{F}(P, Q) is an isomorphism.

Proof. Let f ∈ Hom_{F}(P, Q) be a homomorphism of functors. To check that f is a homomorphism of strict polynomial functors, we have to verify that for any V, W in V^{f} the following diagram of polynomial maps commutes

Hom_{k}(V, W)  ⟶^{Q_{V,W}} Hom_{k}(Q(V), Q(W))
P_{V,W} ↓  ↓^{∘f_{V}}
Hom_{k}(P (V), P (W)) ⟶^{f_{W} ∘} Hom_{k}(P (V), Q(W))

To do so, we observe that both compositions are homogeneous polynomial maps of degree d from Hom_{k}(V, W) to Hom_{k}(P (V), Q(W)) whose values at all rational points coincide. Finally we observe that if a field F contains at least d elements then a homogeneous polynomial of degree d (in any number of variables) which takes zero values at all rational points is necessarily zero (as a polynomial).

Polynomial functors

We recall polynomial functors in the category F. For a functor F in F, define its difference functor Δ(F) by

Δ(F)(V) = Ker{F (V ⊕ k) → F (V)}.

The functor F is said to be polynomial if the r-th difference functor Δ^{r}(F) vanishes for r sufficiently large. The Eilenberg-MacLane degree of a polynomial functor F is the least integer d such that Δ^{d+1}(F) = 0. The same functors (or rather their images under the forgetful functor P → F) used as examples of strict polynomial functors homogeneous of degree d provide examples of polynomial functors of Eilenberg-MacLane degree d: the d-fold tensor power functor ⊗^{d}, the d-fold divided power functor Γ^{d}, etc. More generally one checks immediately that for any strict polynomial functor P in P_{d} its image under the forgetful functor P → F is polynomial of Eilenberg-MacLane degree less than or equal to d.

Following N. Kuhn [K1] we say that a functor F in F is finite if it is of finite Eilenberg-MacLane degree and takes values in V^{f}. The previous remarks imply that the image in F of any strict polynomial functor P ∈ P is finite.

For any vector space V ∈ V^{f} define a functor P_{V} ∈ F by the formula P_{V} (W) = k[Hom_{k}(V, W)]. The Yoneda Lemma shows that for any Q in F we have a natural isomorphism Hom_{F}(P_{V} , Q) = Q(V). This implies immediately that the functor P_{V} is projective in F. A functor P ∈ F is said to be of finite type if it admits an epimorphism from a finite direct sum of functors of the form P_{V}. To say that a functor Q ∈ F admits a projective resolution of finite type is clearly equivalent to saying that Q admits a resolution each term of which is isomorphic to a finite direct sum of functors of the form P_{V}. On several occasions we’ll need the following useful fact.

Proposition 1.5. [Sc],[F-L-S, 10.1] Assume that the field k is finite. Then every finite functor Q in F admits a projective resolution of finite type.

A clear difference between the two categories of functors F and P appears when considering the Frobenius twist. If k is perfect, then the Frobenius map ϕ : k → k is an isomorphism so that the Frobenius twist (−)^{(1)} becomes invertible when viewed in F. Indeed, if k is the prime field 𝔽_{p}, then I^{(1)} = I in F; note that even for k = 𝔽_{p}, (−)^{(1)} is not invertible in P.

Observe that if k is perfect, then for any P, Q in F we have a natural isomorphism Ext_{F}^{∗}(P, Q) ⟶^{∼} Ext_{F}^{∗}(P^{(1)}, Q^{(1)}). Hence for any strict polynomial functors P, Q in P we get a natural map

\underrightarrow{lim}_{r} Ext_{P}^{∗}(P^{(r)}, Q^{(r)}) → \underrightarrow{lim}_{r} Ext_{F}^{∗}(P^{(r)}, Q^{(r)}) ≅ Ext_{F}^{∗}(P, Q) .

Theorem 3.10 gives conditions for this map to be an isomorphism.

Exponential functors

An exponential functor is a graded functor A^{∗} = (A^{0}, A^{1}, ..., A^{n}, ...) from V^{f} to V^{f} together with natural isomorphisms

A^{0}(V) ≅ k , A^{n}(V ⊕ W) ≅ ⨁_{m=0}^{n} A^{m}(V) ⊗ A^{n−m}(W) , n > 0 .

Lemma 1.6. Let A^{∗} be an exponential functor.

(1) The functors A^{1}, A^{2}, ... are without constant term, i.e. A^{i}(0) = 0 for i > 0.

(2) The natural maps

A^{n}(V) = A^{n}(V) ⊗ A^{0}(W) ↪ ⨁_{m=0}^{n} A^{m}(V) ⊗ A^{n−m}(W) = A^{n}(V ⊕ W)

A^{n}(V ⊕ W) = ⨁_{m=0}^{n} A^{m}(V) ⊗ A^{n−m}(W) ↠ A^{n}(V) ⊗ A^{0}(W) = A^{n}(V)

coincide (up to an automorphism of the functor A^{n}) with the map induced by the inclusion of the first factor A^{n}(i_{1}) and the map induced by the projection onto the first factor A^{n}(p_{1}) respectively.

(3) The Eilenberg-MacLane degree of the functor A^{n} is at most n and is equal to n provided that A^{1} ≠ 0. In particular the functors A^{n} are finite.

Proof. (1) The exponential condition shows that for n > 0 we have an isomorphism A^{n}(0) = A^{n}(0 ⊕ 0) ≅ A^{0}(0) ⊗ A^{n}(0) ⊕ ... ⊕ A^{n}(0) ⊗ A^{0}(0) = A^{n}(0) ⊕ ... ⊕ A^{n}(0). Thus dim A^{n}(0) ≥ 2dim A^{n}(0) and hence dim A^{n}(0) = 0.

(2) Denote the homomorphisms in question by i_{V,W}^{n} and p_{V,W}^{n} respectively. The functoriality of all the maps involved implies the commutativity of the following diagram

A^{n}(V) ⟶^{i_{V,0}^{n}} A^{n}(V ⊕ 0) = A^{n}(V)
=↓  ↓^{A^{n}(i_{1})}
A^{n}(V) ⟶^{i_{V,W}^{n}}  A^{n}(V ⊕ W)

Now it suffices to note that (according to (1) ) i_{−,0}^{n} is an automorphism of the functor A^{n}. The same reasoning applies to p_{V,W}^{n}.

(3) The exponential condition and (2) show that the functor Δ(A^{n}) is isomorphic to the direct sum A^{1}(k) ⊗ A^{n−1} ⊕ ... ⊕ A^{n}(k). Immediate induction on n now concludes the proof.

Typical examples of exponential functors are given by the symmetric algebra S^{∗} = (S^{0}, S^{1}, ...S^{n}, ...), the exterior algebra Λ^{∗} = (Λ^{0}, Λ^{1}, ..., Λ^{n}, ...) and the divided power algebra Γ^{∗} = (Γ^{0}, Γ^{1}, ..., Γ^{n}, ...). Another exponential functor L^{∗} = (L^{0}, L^{1}, ..., L^{n}, ...) is obtained as the quotient of the symmetric power algebra by the ideal of p-th powers; it coincides with the exterior algebra when p = 2.

Note also that if we define the tensor product of graded functors via the usual formula

(A^{∗} ⊗ B^{∗})^{n} = ⨁_{m=0}^{n} A^{m} ⊗ B^{n−m},

then the tensor product of two exponential functors is again exponential. Clearly, the Frobenius twist of an exponential functor is again exponential.

The previous definition generalizes immediately to the case of strict polynomial functors. We skip the obvious details.

The following theorem, in the case of the category F, was used in [F]. It generalizes a result due to Pirashvili [P] (much used in [F-S] and [F-L-S]) which asserts that if A is an additive functor and B, C are functors without constant term, then all Ext-groups from A to B ⊗ C are 0. The isomorphism of Theorem 1.7, which fails in the context of Ext-groups over GL_{n}, provides an important tool enabling computations of functor cohomology. From now on we assume (if not specified otherwise) that the base field k is finite.

Theorem 1.7. Let A^{∗} be an exponential functor. For any B, C in F, we have natural isomorphisms

Ext_{F}^{∗}(A^{n}, B ⊗ C) = ⨁_{m=0}^{n} Ext_{F}^{∗}(A^{m}, B) ⊗ Ext_{F}^{∗}(A^{n−m}, C).

Furthermore, if B and C take values in the category V^{f} then we also have natural isomorphisms

Ext_{F}^{∗}(B ⊗ C, A^{n}) = ⨁_{m=0}^{n} Ext_{F}^{∗}(B, A^{m}) ⊗ Ext_{F}^{∗}(C, A^{n−m}).

Similarly, if A^{∗} is an exponential strict polynomial functor (of finite degree), then for any B, C in P we have natural isomorphisms

Ext_{P}^{∗}(A^{n}, B ⊗ C) = ⨁_{m=0}^{n} Ext_{P}^{∗}(A^{m}, B) ⊗ Ext_{P}^{∗}(A^{n−m}, C)

Ext_{P}^{∗}(B ⊗ C, A^{n}) = ⨁_{m=0}^{n} Ext_{P}^{∗}(B, A^{m}) ⊗ Ext_{P}^{∗}(C, A^{n−m}).

Proof. In case of the category F the first part is proved in [F, 1.4.2] (we recall the proof below). The same argument gives the second part. Alternatively, the second part follows from the first by the duality isomorphism 1.12. The proof for the category P is identical, using the theory of strict polynomial bifunctors of finite degree as developed in [S-F-B]. It should be noted also that in case of the category P the above theorem holds over arbitrary (not necessarily finite) fields.

Let bi–F denote the abelian category of bifunctors V^{f} × V^{f} → V. Consider a pair of adjoint (on both sides) functors

V^{f} ⟶^{D} ; ⟵^{Π}V^{f} × V^{f}.

Here Π is the direct sum functor Π(V, W) = V ⊕ W and D is the diagonal functor D(V) = (V, V). Taking compositions on the right with these functors we get a pair of adjoint on both sides functors between F and bi–F.

F ⟶^{P ↦P ∘Π} ; ⟵^{Q∘D←Q}bi–F.

Since both functors are exact they also preserve projectives and injectives and we get the usual adjunction isomorphisms (1.7.1) For any P in F and any Q in bi–F we have natural isomorphisms

Ext_{F}^{∗}(P, Q ∘ D) =Ext_{bi–F}^{∗}(P ∘ Π, Q)
Ext_{F}^{∗}(Q ∘ D, P) =Ext_{bi–F}^{∗}(Q, P ∘ Π).

For any functors B, C in F we define their external tensor product B ⊠ C ∈ bi–F via the formula B ⊠C(V, W) = B(V)⊗C(W). Exactness of tensor products and an obvious formula for the external tensor product of projective generators: P_{V} ⊠P_{W} = P_{(V,W)}, give us the following Künneth-type formula: (1.7.2) Assume that A_{1}, A_{2} are functors in F which admit projective resolutions of finite type, then for any B, C in F we have natural isomorphisms

Ext_{bi–F}^{∗}(A_{1} ⊠ A_{2}, B ⊠ C) = Ext_{F}^{∗}(A_{1}, B) ⊗ Ext_{F}^{∗}(A_{2}, C).

Using (1.7.1) and the exponential property of A^{∗} we get isomorphisms:

Ext_{F}^{∗}(A^{n}, B ⊗ C) = Ext_{F}^{∗}(A^{n}, (B ⊠ C) ∘ D) = Ext_{bi–F}^{∗}(A^{n} ∘ Π, B ⊠ C) =

= Ext_{bi–F}^{∗}(⨁_{m=0}^{n} A^{m} ⊠ A^{n−m}, B ⊠ C) = ⨁_{m=0}^{n} Ext_{bi–F}^{∗}(A^{m} ⊠ A^{n−m}, B ⊠ C).

Finally we observe that all functors A^{i} are finite according to Lemma 1.6 and hence admit projective resolutions of finite type (see Proposition 1.5), so that we may use (1.7.2) to conclude the proof.

Observe that for homogeneous strict polynomial functors A^{n}, B and C, the sum in Theorem 1.7 cannot have more than one non-zero term. Thus, in the case of the category P, Theorem 1.7 generalizes [F-S, Proposition 5.2]. This, and the injectivity of the symmetric powers in P, makes computing Ext-groups in the category P easier than in the category F.

Applying Theorem 1.7 to the exponential functors \underbrace{A^{∗} ⊗ ... ⊗ A^{∗}}_{n} and

\underbrace{B^{∗} ⊗ ... ⊗ B^{∗}}_{m} we obtain the following corollary.

Corollary 1.8. Let A^{∗}, B^{∗} be exponential functors. For any non negative integers k_{1}, ..., k_{n}; l_{1}, ..., l_{m} we have natural isomorphisms

Ext_{F}^{∗}(A^{k_{1}} ⊗ ... ⊗ A^{k_{n}}, B^{l_{1}} ⊗ ... ⊗ B^{l_{m}}) =
= _{⨁_{k_{1,1}+...+k_{1,m}=k_{1} ... k_{n,1}+...+k_{n,m}=k_{n}} ; l_{1,1}+...+l_{1,n}=l_{1} ... l_{m,1}+...+l_{m,n}=l_{m}}  ⨂_{s=1, t=1}^{n m} Ext_{F}^{∗}(A^{k_{s,t}}, B^{l_{t,s}}) .

Similarly, if A^{∗} and B^{∗} are exponential strict polynomial functors of finite degree, then we have corresponding natural isomorphisms obtained by replacing Ext_{F}^{∗} by Ext_{P}^{∗}.

All our examples of exponential functors are bi-algebras. Indeed, for any exponential functor A^{∗}, the natural maps

A^{i}(V) ⊗ A^{j}(V) ↪ A^{i+j}(V ⊕ V) ⟶^{A^{i+j}(Σ)} A^{i+j}(V)

A^{i+j}(V) ⟶^{A^{i+j}(Δ)} A^{i+j}(V ⊕ V) ↠ A^{i}(V) ⊗ A^{j}(V)

define natural product and coproduct operations A^{i} ⊗A^{j} → A^{i+j}, A^{i+j} → A^{i} ⊗A^{j}. Here, Σ is the sum map (x, y) ↦ x + y and Δ is the diagonal map x ↦ (x, x).

Definition 1.9. We say that A^{∗} is an Hopf exponential functor provided that for any V in V^{f} the above product operation makes A^{∗}(V) into a (graded) associative k-algebra with unit 1 ∈ A^{0}(V) = k.

The name Hopf functor is justified by the following Lemma.

Lemma 1.10. Assume that A^{∗} is an Hopf exponential functor. Then for any V in V^{f} the above operations make A^{∗}(V) into a (graded) Hopf algebra with counit ϵ : A^{∗}(V) → A^{0}(V) = k.

Proof. Note first that the (right) multiplication by 1 ∈ A^{0}(V) coincides with the natural isomorphism

i_{V,0}^{n} : A^{n}(V) = A^{n}(V) ⊗ A^{0}(0) ⟶^{∼} A^{n}(V ⊕ 0) = A^{n}(V)

(cf. the proof of Lemma 1.6 (2)). In the same way the right comultiplication by ϵ coincides with the natural isomorphism

p_{V,0}^{n} : A^{n}(V) = A^{n}(V ⊕ 0) ⟶^{∼} A^{n}(V) ⊗ A^{0}(0) = A^{n}(V).

Since p_{V,0}^{n} = (i_{V,0}^{n})^{−1} we conclude that 1 ∈ A^{0}(V) is a unit iff ϵ is a counit of A^{∗}(V).

Observe next that the natural homomorphism A^{m}(V)⊗A^{n−m}(W) → A^{n}(V ⊕W) in the definition of the exponential functor may be expressed in terms of the product operation as the composition

A^{m}(V)⊗A^{n−m}(W) ⟶^{A^{m}(i_{1})⊗A^{n−m}(i_{2})} A^{m}(V ⊕W)⊗A^{n−m}(V ⊕W) ⟶^{mult} A^{n}(V ⊕W).

This remark implies immediately that the associativity of A^{∗}(V) (for all V) is equivalent to the fact that the two possible identifications of graded tri-functors A^{∗}(U) ⊗ A^{∗}(V) ⊗ A^{∗}(W) and A^{∗}(U ⊕ V ⊕ W) coincide. Now the verification of the fact that A^{∗} is a Hopf algebra becomes a straightforward computation. For example to check the coassociativity we have to verify the commutativity of the following diagram

A^{∗}(V)  ⟶^{comult}  A^{∗}(V) ⊗ A^{∗}(V)
comult↓  ^{comult⊗1}↓
A^{∗}(V) ⊗ A^{∗}(V) ⟶^{1⊗comult} A^{∗}(V) ⊗ A^{∗}(V) ⊗ A^{∗}(V).

However one checks easily that both compositions coincide with the homomorphism

A^{∗}(V) ⟶^{A^{∗}(Δ)} A^{∗}(V ⊕V ⊕V) = A^{∗}(V)⊗A^{∗}(V)⊗A^{∗}(V), where Δ : V → V ⊕V ⊕V

is the diagonal map.

We say that the exponential functor A^{∗} is commutative (respectively, skewcommutative) if for all non-negative integers i, j and every V in V^{f} the map τ : V ⊕ V → V ⊕ V, (x, y) ↦ (y, x) gives rise to a commutative diagram (resp. a diagram commutative up to a sign (−1)^{ij}):

A^{i}(V) ⊗ A^{j}(V) ⟶ A^{i+j}(V ⊕ V)
T ↓  ^{A(τ)}↓
A^{j}(V) ⊗ A^{i}(V) ⟶ A^{i+j}(V ⊕ V)

where T is the twist map. For a commutative exponential functor A^{∗} we set ϵ(A^{∗}) = 1 and for a skew commutative one we set ϵ(A^{∗}) = −1. We say in these cases that A^{∗} is ϵ(A^{∗})-commutative. When an exponential functor is commutative (resp. skew-commutative), the product in the algebra A^{∗}(V) is (skew-) commutative and the coproduct is (skew-) cocommutative. We readily verify that the functors Γ^{∗}, Λ^{∗}, S^{∗}, L^{∗} and their Frobenius twists are Hopf exponential functors and that ϵ(Γ^{∗}) = ϵ(S^{∗}) = ϵ(L^{∗}) = +1, ϵ(Λ^{∗}) = −1.

Assume that A^{∗} and B^{∗} are exponential functors (respectively exponential strict polynomial functors of finite degree). In this case the tri-graded vector space Ext^{∗}(A^{∗}, B^{∗}) (a notation to denote Ext_{F}^{∗}(A^{∗}, B^{∗}) as well as Ext_{P}^{∗}(A^{∗}, B^{∗})) acquires a natural structure of (tri-graded) bi-algebra. The product operation is defined as the composition

Ext^{∗}(A^{∗}, B^{∗}) ⊗ Ext^{∗}(A^{∗}, B^{∗}) ⟶^{∼} Ext^{∗}(A^{∗} ⊠ A^{∗}, B^{∗} ⊠ B^{∗}) = Ext^{∗}(A^{∗} ∘ Π, B^{∗} ∘ Π)

→ Ext^{∗}(A^{∗} ∘ Π ∘ D, B^{∗} ∘ Π ∘ D) → Ext^{∗}(A^{∗}, B^{∗}).

Here the second arrow is induced by the exact functor bi–F ⟶^{Q↦Q∘D} F and the last arrow is defined by the adjunction homomorphisms Δ : I → Π∘D and Σ : Π∘D → I. Equivalently the product of the classes e ∈ Ext^{i}(A^{n}, B^{m}), e′ ∈ Ext^{i′}(A^{n′}, B^{m′}) may be described as the image of the tensor product e ⊗ e′ ∈ Ext^{i+i′}(A^{n} ⊗ A^{n′}, B^{m} ⊗ B^{m′}) under the homomorphism Ext^{∗}(A^{n} ⊗A^{n′}, B^{m} ⊗B^{m′}) → Ext^{∗}(A^{n+n′}, B^{m+m′}) induced by the product operation B^{m}⊗B^{m′} → B^{m+m′} and the coproduct operation A^{n+n′} → A^{n} ⊗ A^{n′}. Finally the product operation may be also described (using the isomorphisms of Theorem 1.7) as any of the following two compositions

Ext^{∗}(A^{∗}, B^{∗}) ⊗ Ext^{∗}(A^{∗}, B^{∗}) ⟶^{∼} Ext^{∗}(A^{∗} ⊗ A^{∗}, B^{∗}) ⟶ Ext^{∗}(A^{∗}, B^{∗})

Ext^{∗}(A^{∗}, B^{∗}) ⊗ Ext^{∗}(A^{∗}, B^{∗}) ⟶^{∼} Ext^{∗}(A^{∗}, B^{∗} ⊗ B^{∗}) ⟶ Ext^{∗}(A^{∗}, B^{∗}).

In a similar fashion the coproduct is defined as the composition

Ext^{∗}(A^{∗}, B^{∗}) → Ext^{∗}(A^{∗} ∘ Π, B^{∗} ∘ Π) = Ext^{∗}(A^{∗} ⊠ A^{∗}, B^{∗} ⊠ B^{∗}) =
= Ext^{∗}(A^{∗}, B^{∗}) ⊗ Ext^{∗}(A^{∗}, B^{∗}).

The coproduct operation may be also described (using the isomorphisms of Theorem 1.7) as any one of the following compositions

Ext^{∗}(A^{∗}, B^{∗}) ⟶ Ext^{∗}(A^{∗} ⊗ A^{∗}, B^{∗}) ⟶^{∼} Ext^{∗}(A^{∗}, B^{∗}) ⊗ Ext^{∗}(A^{∗}, B^{∗})

Ext^{∗}(A^{∗}, B^{∗}) ⟶ Ext^{∗}(A^{∗}, B^{∗} ⊗ B^{∗}) ⟶^{∼} Ext^{∗}(A^{∗}, B^{∗}) ⊗ Ext^{∗}(A^{∗}, B^{∗})

The proof of the following lemma is straightforward (but tiresome).

Lemma 1.11. Let A^{∗} and B^{∗} be Hopf exponential functors (resp. Hopf exponential strict polynomial functors of finite degree). Then the tri-graded vector space Ext^{∗}(A^{∗}, B^{∗}) has a natural structure of a (tri-graded) Hopf algebra. Moreover if A^{∗} is ϵ(A^{∗})-commutative and B^{∗} is ϵ(B^{∗})-commutative, then the following diagrams commute up to a sign (−1)^{st+ \frac{ϵ(A)−1}{2} ·ik+ \frac{ϵ(B)−1}{2} ·jl}

Ext^{s}(A^{i}, B^{j}) ⊗ Ext^{t}(A^{k}, B^{l}) ⟶^{mult} Ext^{s+t}(A^{i+k}, B^{j+l})
=^{∼}↓  ^{=}↓
Ext^{t}(A^{k}, B^{l}) ⊗ Ext^{s}(A^{i}, B^{j}) ⟶^{mult} Ext^{s+t}(A^{i+k}, B^{j+l})
Ext^{s+t}(A^{i+k}, B^{j+l}) ⟶^{comult} Ext^{s}(A^{i}, B^{j}) ⊗ Ext^{t}(A^{k}, B^{l})
=↓  =^{∼}↓
Ext^{s+t}(A^{i+k}, B^{j+l}) ⟶^{comult} Ext^{t}(A^{k}, B^{l}) ⊗ Ext^{s}(A^{i}, B^{j}) .

Dual Functors

For a functor P : V^{f} → V, we define its dual P^{#} : V^{f} → V by P^{#}(V) = P (V^{#})^{#}. Two examples of this duality which will be important in what follows are

(S^{d})^{#} = Γ^{d} , (Λ^{d})^{#} = Λ^{d}.

The contravariant functor ^{#} : F → F is clearly exact and hence for any P, Q in F we get a natural duality homomorphism ^{#} : Ext_{F}^{∗}(P, Q) → Ext_{F}^{∗}(P^{#}, Q^{#}). The same construction obviously applies to the category P as well.

Lemma 1.12. Assume that P, Q ∈ F take values in the category V^{f}. Then the duality homomorphism

# : Ext_{F}^{∗}(P, Q) → Ext_{F}^{∗}(Q^{#}, P^{#})

is an isomorphism. Similarly for any P, Q ∈ P the duality homomorphism

# : Ext_{P}^{∗}(P, Q) → Ext_{P}^{∗}(Q^{#}, P^{#})

is an isomorphism.

Proof. For any F in F we have a natural homomorphism F → F^{##}, which is an isomorphism provided that F takes values in V^{f}. In particular P^{##} = P, Q^{##} = Q. Now it suffices to show that the composition

Ext_{F}^{∗}(P, Q) ⟶^{#} Ext_{F}^{∗}(Q^{#}, P^{#}) ⟶^{#} Ext_{F}^{∗}(P^{##}, Q^{##}) = Ext_{F}^{∗}(P, Q)

is the identity map. Let e ∈ Ext_{F}^{n}(P, Q) be represented by an extension

0 → Q → P_{n} → ... → P_{1} → P → 0.

Then e^{##} is represented by the extension

0 → Q^{##} → P_{n}^{##} → ... → P_{1}^{##} → P^{##} → 0

and our statement follows from the commutativity of the diagram

0 ⟶ Q ⟶ P_{n} ⟶ ... ⟶ P_{n} ⟶ P ⟶ 0
≅↓  ↓  ↓  ≅↓
0 ⟶ Q^{##} ⟶ P_{n}^{##} ⟶ ... ⟶ P_{n}^{##} ⟶ P^{##} ⟶ 0

The case of the category P is trivial since in this case ^{#} : P → P is an antiequivalence.

Clearly, the dual of an exponential functor is again exponential. The following result (to be used in §5) is straightforward from the definitions.

Lemma 1.13. Let A^{∗}, B^{∗} be Hopf exponential functors (resp. Hopf exponential strict polynomial functors of finite degree). The natural isomorphism of Lemma 1.12

Ext^{∗}(A^{∗}, B^{∗}) ⟶^{∼} Ext^{∗}(B^{∗}, A^{∗})

is an anti-isomorphism of graded Hopf algebras (i.e. (xy)^{#} = y^{#}x^{#} etc.).

§2. Weak Comparison Theorem.

In this section, we investigate the map on Ext-groups induced by the forgetful functor P → F. Throughout this section, and in much of §3, we shall restrict our attention to the case in which the base field k is a finite field k = 𝔽_{q} of characteristic p, with q = p^{N}. We show for strict polynomial functors A, B of degree d that the natural map

Ext_{P}^{i}(A^{(m)}, B^{(m)}) → Ext_{F}^{i}(A^{(m)}, B^{(m)}) = Ext_{F}^{i}(A, B)

is an isomorphism provided that m and q are sufficiently large compared to d and i. In the following section, we shall show that the condition on q can be weakened to the simple condition that q ≥ d.

We begin with the following vanishing theorem which complements the fact that symmetric power functors S^{d} are injective in P_{d}. The proof of this theorem uses the finiteness of the base field k in an essential way.

Theorem 2.1 [F, §4]. Let A : V^{f} → V^{f} be a finite functor of Eilenberg-MacLane degree ≤ d. For any i > 0,

Ext_{F}^{i}(A, S^{p^{h}}) = 0

provided that h ≥ log_{p} \frac{i+2}{2} + [\frac{d−1}{p−1}].

Proof. By [F, 4.1.2], the natural embedding S^{p^{h}(l+1)} ↪ S^{p^{h+1}(l)} induces an isomorphism

Ext_{F}^{i}(A, S^{p^{h}(l+1)}) ⟶^{∼} Ext_{F}^{i}(A, S^{p^{h+1}(l)})

provided that h ≥ log_{p} \frac{i+2}{2} + [\frac{d−1}{p−1}]. Applying this remark N times we see that (for the same values of h) the natural embedding

S^{p^{h}} = S^{p^{h}(N)} ↪ S^{p^{h+1}(N−1)} ↪ ... ↪ S^{p^{h+N}}

induces an isomorphism

Ext_{F}^{i}(A, S^{p^{h}}) ⟶^{∼} Ext_{F}^{i}(A, S^{p^{h+N}}).

On the other hand, the vanishing theorem of N. Kuhn [K3, Theorem 1.1] shows that

\underrightarrow{lim}_{k≥0} Ext_{F}^{i}(A, S^{p^{h+kN}}) = 0

for any functor A admitting a projective resolution of finite type. Since each finite functor A admits such a resolution (see Proposition 1.5), this completes the proof.

The following classical lemma was already used in [F] and thus implicitly in the proof of Theorem 2.1. In Proposition 2.3, it enables us to extend the vanishing statement of Theorem 2.1 to tensor products of symmetric power functors.

Lemma 2.2. Let n be an integer with p-adic expansion

n = n_{0} + n_{1}p + ... + n_{k}p^{k} (0 ≤ n_{i} < p) .

Then the functor S^{n} is canonically a direct summand in (S^{1})^{⊗n_{0}} ⊗ ... ⊗ (S^{p^{k}})^{⊗n_{k}}.

Proof. For any integer m with 0 ≤ m ≤ n, the multiplication and comultiplication of the exponential functor S^{∗} yield natural homomorphisms

S^{n} ⟶^{comult} S^{m} ⊗ S^{n−m} ⟶^{mult} S^{n} .

Explicitly, S^{n} ⟶^{comult} S^{m} ⊗ S^{n−m} sends an n-fold product x_{1} ⋯ x_{n} ∈ S^{n}(V) to the sum indexed by Σ_{n}/Σ_{m} × Σ_{n−m} of tensors of the form x_{i_{1}} ⋯ x_{i_{m}} ⊗ x_{i_{m+1}} ⋯ x_{i_{n}}. Composition of these homomorphisms is multiplication by \binom{n}{m}, so S^{n} is a direct summand in S^{m} ⊗ S^{n−m} provided that \binom{n}{m} is prime to p. One easily verifies that \binom{n}{m} is prime to p if and only if all p-adic digits of m are less or equal to the corresponding digits of n. The lemma now follows by an induction on the sum of p-adic digits of n.

Proposition 2.3. Let A^{∗} ∈ F be an exponential functor. Assume that

h ≥ log_{p} \frac{s + 2}{2} + [\frac{d − 1}{p − 1}] , and n_{1} + ... + n_{k} ≡ 0 mod p^{h}.

Then for all 0 < i ≤ s

Ext_{F}^{i}(A^{d}, S^{n_{1}} ⊗ ... ⊗ S^{n_{k}}) = 0 .

Proof. Using Lemma 2.2, we are easily reduced to the case that n_{i} = p^{m_{i}} is a power of p for all i. By Theorem 1.7,

Ext_{F}^{∗}(A^{d}, S^{p^{m_{1}}} ⊗ ... ⊗ S^{p^{m_{k}}}) = ⨁_{d_{1}+···+d_{k}=d} ⨂_{j=1}^{k} Ext_{F}^{∗}(A^{d_{j}}, S^{p^{m_{j}}}).

Since the functor A^{0} is constant, all its Ext-groups to functors without constant term are trivial; thus the summand corresponding to the k-tuple (d_{1}, ..., d_{k}) is trivial provided that d_{j} = 0 for some j. Consider now a summand corresponding to a ktuple (d_{1}, ..., d_{k}) with all d_{j} > 0. In this case we have inequalities d_{j} ≤ d − (k − 1). On the other hand, an easy induction on k establishes that if p^{m_{1}} + ... + p^{m_{k}} ≡ 0 mod p^{h}, then m_{j} ≥ h − [\frac{k−1}{p−1}] for all j. Consequently,

m_{j} ≥ h − [\frac{k − 1}{p − 1}] ≥ log_{p} \frac{s + 2}{2} + [\frac{d − 1}{p − 1}] − [\frac{k − 1}{p − 1}] ≥ log_{p} \frac{s + 2}{2} + [\frac{d_{j} − 1}{p − 1}] .

Theorem 2.1 shows that Ext_{F}^{i}(A^{d_{j}}, S^{p^{m_{j}}}) = 0 for 0 < i ≤ s, thus concluding the proof.

Corollary 2.4. Assume that h ≥ log_{p} \frac{s+2}{2} + [\frac{d−1}{p−1}]. Let d_{1}, ..., d_{k}; n_{1}, ..., n_{l} be integers such that d_{1} + ... + d_{k} = d and n_{1} + ... + n_{l} ≡ 0 mod p^{h}. Then

Ext_{F}^{i}((Γ^{d_{1}} ⊗ ... ⊗ Γ^{d_{k}})^{(m)}, S^{n_{1}} ⊗ ... ⊗ S^{n_{l}}) = 0

for all m and all 0 < i ≤ s.

Proof. This follows immediately from Proposition 2.3 since (Γ^{d_{1}} ⊗ ... ⊗ Γ^{d_{k}})^{(m)} is a direct summand in A^{d}, where A^{∗} is the exponential functor (\underbrace{Γ^{∗} ⊗ ... ⊗ Γ^{∗})}_{k}^{(m)}.

Corollary 2.4 easily implies our first, and weakest, comparison of the Ext-groups Ext_{P}^{∗} and Ext_{F}^{∗}. A first form of this proposition was proved by E. Friedlander [1995, unpublished] and T. Pirashvili [B-P2] (and discussion with the first author).

Proposition 2.5. Let A and B be strict polynomial functors homogeneous of degree d, over a field with q elements. Assume that

m ≥ log_{p} \frac{s + 2}{2} + [\frac{d − 1}{p − 1}] , q ≥ dp^{m}.

Then the canonical homomorphisms

Ext_{P}^{i}(A^{(m)}, B^{(m)}) → Ext_{F}^{i}(A^{(m)}, B^{(m)}) ≅ Ext_{F}^{i}(A, B)

are isomorphisms for all 0 ≤ i ≤ s.

Proof. Consider first the special case A = Γ^{d_{1}} ⊗ ... ⊗ Γ^{d_{k}}, d_{1} + ... + d_{k} = d. The functor B^{(m)} admits an injective resolution in P_{dp^{m}}

0 → B^{(m)} → I^{•}

in which every I^{j} is a direct sum of functors of the form S^{n_{1}} ⊗ ... ⊗ S^{n_{l}} with n_{1} + ... + n_{l} = dp^{m} [F-S, §2]. We employ the hypercohomology spectral sequence

E_{1}^{ij} = Ext_{F}^{i}(A^{(m)}, I^{j}) ⟹ Ext_{F}^{i+j}(A^{(m)}, B^{(m)}).

Corollary 2.4 shows that E_{1}^{ij} = 0 for all 0 < i ≤ s. This shows that for 0 ≤ i ≤ s the group Ext_{F}^{i}(A^{(m)}, B^{(m)}) coincides with the i-th homology group of the complex Hom_{F}(A^{(m)}, I^{•}), which equals Hom_{P}(A^{(m)}, I^{•}) by Proposition 1.4. The homology of this latter complex equals Ext_{P}^{∗}(A^{(m)}, B^{(m)}).

In the general case we consider a projective resolution

0 ⟵ A ⟵ P_{•}

in which each P_{i} is a direct sum of functors of the form Γ^{d_{1}}⊗...⊗Γ^{d_{k}}, d_{1}+...+d_{k} = d. This gives us two hypercohomology spectral sequences

PE_{1}^{ij} = Ext_{P}^{i}(P_{j}^{(m)}, B^{(m)}) ⟹ Ext_{P}^{i+j}(A^{(m)}, B^{(m)})
FE_{1}^{ij} = Ext_{F}^{i}(P_{j}^{(m)}, B^{(m)}) ⟹ Ext_{F}^{i+j}(A^{(m)}, B^{(m)})

and a homomorphism of spectral sequences _{P}E → _{F}E. According to the special case considered above, the homomorphisms _{P}E_{1}^{ij} → _{F}E_{1}^{ij} on E_{1}^{ij}-terms are isomorphisms for all j and all 0 ≤ i ≤ s. The standard comparison theorem for spectral sequences thus implies that Ext_{P}^{i}(A^{(m)}, B^{(m)}) → Ext_{F}^{i}(A^{(m)}, B^{(m)}) = Ext_{F}^{i}(A, B) is an isomorphism for 0 ≤ i ≤ s.

Combining Proposition 2.5 with the injectivity of Frobenius twists established in Corollary 1.3, we obtain the following theorem asserting that Ext_{P}-groups stabilize with respect to repeated Frobenius twists. The lower bound of the number of twists required to stabilize Ext_{P}^{s}(A, B) depends upon both the cohomological degree s and the degree d of the functors A and B; in Corollary 4.10, we shall eliminate the dependence of this bound on d.

Theorem 2.6 (Weak Twist Stability). Let A and B be homogeneous strict polynomial functors of degree d. The Frobenius twist homomorphism

Ext_{P}^{s}(A^{(m)}, B^{(m)}) → Ext_{P}^{s}(A^{(m+1)}, B^{(m+1)})

is injective for all m and is an isomorphism for m ≥ log_{p} \frac{s+2}{2} + [\frac{d−1}{p−1}].

Proof. Injectivity is given by Corollary 1.3. To prove the asserted surjectivity, we pick a finite extension E/k with more that dp^{m+1} elements. Consider the following commutative diagram

Ext_{P(E)}^{s}(A_{E}^{(m)}, B_{E}^{(m)}) ⟶ Ext_{P(E)}^{s}(A_{E}^{(m+1)}, B_{E}^{(m+1)})
=^{∼}↓  =^{∼}↓
Ext_{F(E)}^{s}(A_{E}^{(m)}, B_{E}^{(m)}) ⟶^{=^{∼}} Ext_{F(E)}^{s}(A_{E}^{(m+1)}, B_{E}^{(m+1)}).

By Proposition 2.5, the vertical arrows in this diagram are isomorphisms, whereas the lower horizontal arrow is an isomorphism because Frobenius twist is an invertible functor in F(E). Thus, the top horizontal arrow is also an isomorphism. The proof is completed by applying Proposition 1.1: the upper horizontal arrow of this commutative square is the extension of scalars of Ext_{P(k)}^{s}(A^{(m)}, B^{(m)}) → Ext_{P(k)}^{s}(A^{(m+1)}, B^{(m+1)}).

Using Theorem 2.6, we significantly strengthen Proposition 2.5 by having the bound on q (the cardinality of our base field) depend only upon the Ext-degree and the degree of the homogeneous polynomial and not also upon the the number of Frobenius twists. Theorem 2.7 follows immediately from Proposition 2.5 and Theorem 2.6.

Theorem 2.7 (Weak Comparison Theorem). Let A and B be strict polynomial functors homogeneous of degree d, over a field with q elements. Let m_{0} be the least integer ≥ log_{p} \frac{s+2}{2} + [\frac{d−1}{p−1}]. Assume that q ≥ dp^{m_{0}}. Then for any m ≥ m_{0} the canonical homomorphisms

Ext_{P}^{i}(A^{(m)}, B^{(m)}) → Ext_{F}^{i}(A^{(m)}, B^{(m)}) = Ext_{F}^{i}(A, B)

are isomorphisms for all 0 ≤ i ≤ s.

The following is an immediate corollary of Theorem 2.7.

Corollary 2.8. In conditions and notations of Theorem 2.7, the canonical homomorphism

\underrightarrow{lim}_{m} Ext_{P}^{i}(A^{(m)}, B^{(m)}) → Ext_{F}^{i}(A, B)

is an isomorphism for 0 ≤ i ≤ s , provided that q ≥ dp^{m_{0}}.

§3. Extension of Scalars and Ext-groups.

The aim of this section is to understand the effect on Ext_{F}-groups of base change from P(k) to P(K), where k → K is an extension of finite fields. In Theorem 3.9, we see that this effect is simply one of base change provided that the cardinality of the base field k is greater than or equal to the degree of the strict polynomial functors involved. This will enable us to provide in Theorem 3.10 a considerably stronger version of the comparison of Theorem 2.7: the condition on cardinality of the (finite) base field is weakened to the condition that the cardinality be greater or equal to the degree of the strict polynomial functors involved.

For any finite extension k → K of fields, we have pairs of functors

V_{K}^{f} ⟵^{t} ; ⟶_{τ} V_{k}^{f} V_{K} ⟵^{t} ; ⟶_{τ} V_{k}

where τ is the forgetful (or restriction) functor and t is the extension of scalars (or induction) functor. Clearly, both t and τ are exact and t is left adjoint to τ.

The following proposition is less evident and presumably less well known. One can informally describe this result as saying that induction equals coinduction. We include a sketch of proof for the sake of completeness.

Proposition 3.1. Let k → K be a finite extension of fields. A choice of a non-zero k-linear homomorphism T : K → k determines adjunction transformations

θ : τ ∘ t → I_{V_{k}} , η : I_{V_{K}} → t ∘ τ

which establish that t is right adjoint to τ (as functors on either V or V^{f}).

Proof. The choice of T determines a non-degenerate symmetric k-bilinear form

K × K ⟶ k
(a, b) ↦ T (ab) .

We define

x = ∑_{i=1}^{n} e_{i} ⊗ e_{i}^{#} ∈ K ⊗_{k} K ,

where e_{1}, …, e_{n} is a basis for K over k and e_{1}^{#}, …, e_{n}^{#} is the dual basis with respect to the above bilinear form. For any V in V_{k}, we define

θ_{V} = T ⊗_{k} 1_{V} : τ ∘ t(V) = K ⊗_{k} V → V ;

for any W in V_{K}, we define

η_{W} = x · (−) : W → t ∘ τ(W) = K ⊗_{k} W

(i.e., multiplication by x ∈ K ⊗_{k} K). One easily check that x is independent of the choice of the basis e_{1}, …, e_{n}. This, in turn, easily implies that

x · (a ⊗ 1 − 1 ⊗ a) = 0 ∈ K ⊗_{k} K ∀a ∈ K,  (3.1.1)

so that η_{W} is K-linear. Similarly, one checks that

(T ⊗ 1_{K})(x) = (1_{K} ⊗ T)(x) = 1 ∈ K  (3.1.2)

which easily implies that

Hom_{K}(W, K ⊗_{k} V) ⟶ Hom_{k}(W, V)  Hom_{k}(W, V) ⟶ Hom_{K}(W, K ⊗_{k} V)
ϕ ↦ θ_{V} ϕ  ψ ↦ (1_{K} ⊗_{k} ψ)η_{W}

are mutually inverse natural isomorphisms.

Remark 3.1.3. If K/k is separable, then we shall without special mention always choose T : K → k to be the trace map.

The functors τ and t determine adjoint (on both sides) pairs

F(k) = Funct(V_{k}^{f}, V_{k}) ⟶^{t_{∗}} ; ⟵_{τ_{∗}}F(K/k) = Funct(V_{k}^{f}, V_{K}) ⟶^{τ^{∗}} ; ⟵_{t^{∗}}F(K) = Funct(V_{K}^{f}, V_{K}) .

Here t_{∗} and τ_{∗} are given by taking the composition with t and τ on the left, whereas t^{∗} and τ^{∗} are given by taking composition with t and τ on the right. Note that all functors in the above diagram are exact. Since these functors have exact adjoints, we conclude further that they take injectives to injectives and projectives to projectives.

These observations immediately imply the following proposition.

Proposition 3.2. Let k → K be a finite extension of fields. For any functors A in F(K), B in F(K/k), C in F(k), we have natural isomorphisms of graded K vector-spaces:

Ext_{F(K/k)}^{∗}(B, t^{∗}A) = Ext_{F(K)}^{∗}(τ^{∗}B, A) Ext_{F(K/k)}^{∗}(B, t_{∗}C) = Ext_{F(k)}^{∗}(τ_{∗}B, C)
Ext_{F(K/k)}^{∗}(t^{∗}A, B) = Ext_{F(K)}^{∗}(A, τ^{∗}B) Ext_{F(K/k)}^{∗}(t_{∗}C, B) = Ext_{F(k)}^{∗}(C, τ_{∗}B) .

Corollary 3.3. Let k → K be a finite extension of fields. Let A ∈ F(K), A_{0} ∈ F(k) be functors such that t^{∗}A = t_{∗}A_{0} ∈ F(K/k). For any C in F(k), we have natural isomorphisms of graded K-vector spaces

K ⊗_{k} Ext_{F(k)}^{∗}(A_{0}, C) = Ext_{F(K)}^{∗}(A, τ^{∗}t_{∗}C)
K ⊗_{k} Ext_{F(k)}^{∗}(C, A_{0}) = Ext_{F(K)}^{∗}(τ^{∗}t_{∗}C, A) .

Proof. According to Proposition 3.2 we have natural isomorphisms

Ext_{F(K)}^{∗}(A, τ^{∗}t_{∗}C) = Ext_{F(K/k)}^{∗}(t^{∗}A, t_{∗}C) = Ext_{F(K/k)}^{∗}(t_{∗}A_{0}, t_{∗}C) =

= Ext_{F(k)}^{∗}(A_{0}, τ_{∗}t_{∗}C).

It now suffices to observe that the functor τ_{∗}t_{∗}C is given by the formula V ↦ K ⊗_{k} C(V). If we forget about the action of K, then this is just a direct sum of finitely many copies of C, so that the natural homomorphism

K ⊗_{k} Ext_{F(k)}^{∗}(A_{0}, C) → Ext_{F(k)}^{∗}(A_{0}, τ_{∗}t_{∗}C)

is an isomorphism of graded K-vector spaces.

Using Corollary 3.3, we obtain the following understanding of the effect of base change on Ext_{F}-groups, an understanding which we later compare to the corresponding property of Ext_{P}-groups, as given by the Proposition 1.1.

Theorem 3.4. For any strict polynomial functors P, Q in P(k) we have natural isomorphisms of graded K-vector spaces

K ⊗_{k} Ext_{F(k)}^{∗}(P, Q) = Ext_{F(K)}^{∗}(P_{K}, τ^{∗}t^{∗}Q_{K}) = Ext_{F(K)}^{∗}(τ^{∗}t^{∗}P_{K}, Q_{K}) .

Proof. This follows immediately from Corollary 3.3, taking into account the obvious formula

t^{∗}P_{K} = t_{∗}P ∈ F(K/k).

Remark 3.4.1 It’s easy to see from the discussion above that the isomorphisms

K ⊗_{k} Ext_{F(k)}^{∗}(P, Q) = Ext_{F(K)}^{∗}(P_{K}, τ^{∗}t^{∗}Q_{K}) = Ext_{F(K)}^{∗}(P_{K}, Q_{K} ∘ (t ∘ τ))
K ⊗_{k} Ext_{F(k)}^{∗}(P, Q) = Ext_{F(K)}^{∗}(τ^{∗}t^{∗}P_{K}, Q_{K}) = Ext_{F(K)}^{∗}(P_{K} ∘ (t ∘ τ), Q_{K})

are obtained as K-linear extensions of the canonical k-linear maps

Ext_{F(k)}^{∗}(P, Q) → Ext_{F(K)}^{∗}(P_{K}, Q_{K} ∘ (t ∘ τ))
Ext_{F(k)}^{∗}(P, Q) → Ext_{F(K)}^{∗}(P_{K} ∘ (t ∘ τ), Q_{K})

which may be described as follows. The exact functor τ^{∗}t_{∗} : F(k) → F(K) defines a canonical homomorphism τ^{∗}t_{∗} : Ext_{F(k)}^{∗}(P, Q) → Ext_{F(K)}^{∗}(τ^{∗}t_{∗}P, τ^{∗}t_{∗}Q) = Ext_{F(K)}^{∗}(τ^{∗}t^{∗}P_{K}, τ^{∗}t^{∗}Q_{K}) = Ext_{F(K)}^{∗}(P_{K} ∘ (t ∘ τ), Q_{K} ∘ (t ∘ τ)). Furthermore left adjointness of t to τ defines a functorial homomorphism γ : t ∘ τ → I, whereas right adjointness of t to τ defines a functorial homomorphism η : I → t ∘ τ. The homomorphisms in question are the compositions of the homomorphism induced by τ^{∗}t_{∗} with homomorphisms

Ext_{F(K)}^{∗}(P_{K} ∘ (t ∘ τ), Q_{K} ∘ (t ∘ τ)) ⟶^{X↦X·P_{K}(η)} Ext_{F(K)}^{∗}(P_{K}, Q_{K} ∘ (t ∘ τ))
Ext_{F(K)}^{∗}(P_{K} ∘ (t ∘ τ), Q_{K} ∘ (t ∘ τ)) ⟶^{X↦Q_{K}(γ)·X} Ext_{F(K)}^{∗}(P_{K} ∘ (t ∘ τ), Q_{K})

defined by the functor homomorphisms P_{K}(η) : P_{K} → P_{K} ∘ (t ∘ τ) and Q_{K}(γ) : Q_{K} ∘ (t ∘ τ) → Q_{K} respectively.

The following theorem relates the effects of the base change on Ext-groups in the categories P and F.

Theorem 3.5. Let k → K be a finite extension of fields and let P, Q be in P(k). The following diagram commutes

K ⊗_{k} Ext_{P(k)}^{∗}(P, Q) ⟶^{∼}  Ext_{P(K)}^{∗}(P_{K}, Q_{K})
↓  ↓
K ⊗_{k} Ext_{F(k)}^{∗}(P, Q) ⟶^{∼} Ext_{F(K)}^{∗}(P_{K}, Q_{K} ∘ (t ∘ τ)) .

(3.5.1)

Here the top horizontal arrow is the isomorphism of Proposition 1.1, the bottom horizontal arrow is the isomorphism of Theorem 3.4, the left vertical arrow is the standard homomorphism, and the right vertical arrow is the composition of the standard homomorphism Ext_{P(K)}^{∗}(P_{K}, Q_{K}) → Ext_{F(K)}^{∗}(P_{K}, Q_{K}) and the homomorphism on Ext-groups induced by the functor homomorphism Q_{K}(η) : Q_{K} ↦ Q_{K} ∘ (t ∘ τ).

Similarly, we have a commutative diagram

K ⊗_{k} Ext_{P(k)}^{∗}(P, Q) ⟶^{∼}  Ext_{P(K)}^{∗}(P_{K}, Q_{K})
↓  ↓
K ⊗_{k} Ext_{F(k)}^{∗}(P, Q) ⟶^{∼} Ext_{F(K)}^{∗}(P_{K} ∘ (t ∘ τ), Q_{K}) .

(3.5.2)

Proof. Let x in Ext_{P(k)}^{m}(P, Q) be represented by an extension of strict polynomial functors

x : 0 → Q ⟶ X_{m} ⟶ ⋯ → X_{1} ⟶ P ⟶ 0.

The extension τ^{∗}t_{∗}(x) coincides with τ^{∗}t^{∗}(x_{K}). Note further that we have a canonical homomorphism of extensions

τ^{∗}t^{∗}(x_{K}) : 0 → Q_{K} ∘ (t ∘ τ) → … → P_{K} ∘ (t ∘ τ) → 0
Q_{K}(η)↑  ^{P_{K}(η)}↑
x_{K}  : 0 →  Q_{K}  → … →  P_{K}  → 0 .

The existence of such a homomorphism gives us the desired equality

τ^{∗}t_{∗}(x) · P_{K}(η) = τ^{∗}t^{∗}(x_{K}) · P_{K}(η) = Q_{K}(η) · x_{K}.

Essentially the same proof gives (3.5.2); alternatively, (3.5.2) follows by applying the duality isomorphism of (1.12) to (3.5.1).

The following lemma will enable us to reinterpret in the special case of an extension of finite fields the Ext-groups Ext_{F(K)}^{∗}(P_{K}, Q_{K} ∘ (t ∘ τ)) occurring in the lower right corner of the commutative square (3.5.1).

Lemma 3.6. Let k be a finite field with q = p^{N} elements and let k → K be a finite fields extension of degree n. The corresponding functor t ∘ τ : V_{K} → V_{K} coincides with I ⊕ I^{(N)} ⊕ ... ⊕ I^{((n−1)N)}. Under this identification, the adjunction homomorphism η : I → t∘τ (respectively, γ : t∘τ → I) corresponds to the canonical embedding of I into the above direct sum (resp. the canonical projection of the above direct sum onto I). In particular, the composition I ⟶^{η} t ∘ τ ⟶^{γ} I is the identity endomorphism.

Proof. The functor in question is given by the formula W ↦ K ⊗_{k} W = = (K ⊗_{k} K) ⊗_{K} W. The K − K bimodule K ⊗_{k} K is canonically isomorphic to the direct sum ⊕_{σ∈Gal(K/k)}^{σ}K, where ^{σ}K is K with the standard right K-module structure and the left K-module structure being defined via σ. In case σ is the lN-th power of the Frobenius automorphism, the K-vector space^{σ}K⊗_{K}W coincides with W^{(lN)}. These remarks allow us to identify t ∘ τ with I ⊕ I^{(N)} ⊕ ... ⊕ I^{((n−1)N)}. The homomorphism γ_{W} : K ⊗_{k} W → W is given by the formula a ⊗ w ↦ aw and hence corresponds to the multiplication homomorphism K ⊗_{k} K → K, i.e. to the projection of K ⊗_{k} K onto the summand corresponding to σ = Id_{K}. Finally the homomorphism η_{W} : W → K ⊗_{k} W is given by multiplication by x ∈ K ⊗_{k} K. The property (3.1.1) shows that x is killed by the kernel of the projection K ⊗_{k} K → K and hence all σ-components of x with σ ≠ Id_{K} are trivial. Finally the property (3.1.2) implies easily that Id_{K}-component of x is equal to 1.

As we now show, injectivity of Ext_{P}^{∗} → Ext_{F}^{∗} is an easy consequence of Theorems 2.7 and 3.5 provided that we work over a finite base field. (The hypothesis of finiteness of the base field is required in the proof of Theorem 2.7.)

Corollary 3.7. Let k be a finite field and consider strict polynomial functors P and Q of finite degree. The canonical homomorphism

Ext_{P(k)}^{s}(P, Q) → Ext_{F(k)}^{s}(P, Q)

is injective for all s.

Proof. According to Theorem 2.7, we may find an extension K/k and an integer m ≥ 0 such that the homomorphism Ext_{P(K)}^{s}(P_{K}^{(m)}, Q_{K}^{(m)}) → Ext_{F(K)}^{s}(P_{K}, Q_{K}) is an isomorphism. Applying Theorem 3.5 to the functors P^{(m)} and Q^{(m)} and noting that the right vertical arrow of (3.5.1) is the composition of an isomorphism Ext_{P(K)}^{s}(P_{K}^{(m)}, Q_{K}^{(m)}) → Ext_{F(K)}^{s}(P_{K}^{(m)}, Q_{K}^{(m)}) and a split monomorphism

Ext_{F(K)}^{s}(P_{K}^{(m)}, Q_{K}^{(m)}) ⟶^{X↦X·Q_{K}^{(m)}(η)} Ext_{F(K)}^{s}(P_{K}^{(m)}, Q_{K}^{(m)} ∘ (t ∘ τ)) , we conclude

that the map Ext_{P(k)}^{s}(P^{(m)}, Q^{(m)}) → Ext_{F(k)}^{s}(P^{(m)}, Q^{(m)}) = Ext_{F(k)}^{s}(P, Q) is injective. Injectivity of the map in question now follows from the injectivity of the twist map (given in Corollary 1.3).

We construct now an extension of scalars homomorphism on Ext_{F}^{∗}-groups applied to strict polynomial functors.

Proposition 3.8. Let k → K be a finite extension of fields. For P and Q in P(k), there is a natural “extensions of scalars” k-homomorphism

Ext_{F(k)}^{∗}(P, Q) → Ext_{F(K)}^{∗}(P_{K}, Q_{K})

which fits in a commutative square of K-linear maps

K ⊗_{k} Ext_{P(k)}^{∗}(P, Q) ⟶^{∼} Ext_{P(K)}^{∗}(P_{K}, Q_{K})
↓  ↓
K ⊗_{k} Ext_{F(k)}^{∗}(P, Q) ⟶ Ext_{F(K)}^{∗}(P_{K}, Q_{K})

(3.8.1)

whose upper horizontal arrow is the isomorphism of Proposition 1.1 and whose vertical maps are the standard homomorphisms.

Proof. Let γ : t ∘ τ → I be the adjunction homomorphism establishing the left adjointness of t to τ. The extension of scalars homomorphism Ext_{F(k)}^{∗}(P, Q) → Ext_{F(K)}^{∗}(P_{K}, Q_{K}) is defined to be the composition of the homomorphism Ext_{F(k)}^{∗}(P, Q) → Ext_{F(K)}^{∗}(P_{K}, Q_{K} ∘ (t ∘ τ)) of Theorem 3.4 with the map

Ext_{F(K)}^{∗}(P_{K}, Q_{K} ∘ (t ∘ τ)) ⟶^{X↦Q_{K}(γ)·X} Ext_{F(K)}^{∗}(P_{K}, Q_{K}).

Theorem 3.5 and the fact that Q_{K}(γ) ∘ Q_{K}(η) = Q_{K}(γ ∘ η) = Id implies the commutativity of (3.8.1).

Using a weight argument together with Lemma 3.6, we now verify that the extension of scalars homomorphism induces an isomorphism

K ⊗_{k} Ext_{F(k)}^{∗}(P, Q) ⟶^{∼} Ext_{F(K)}^{∗}(P_{K}, Q_{K})

provided that P and Q are homogeneous strict polynomial functors of degree d and q = |k| ≥ d.

Theorem 3.9. Let k be a finite field with q = p^{N} elements and let k → K be an extension of finite fields. Consider strict polynomial functors P, Q homogeneous of degree d. If q ≥ d, then the extension of scalars homomorphism

K ⊗_{k} Ext_{F(k)}^{∗}(P, Q) → Ext_{F(K)}^{∗}(P_{K}, Q_{K})

is an isomorphism.

Proof. Theorem 3.4 shows that the left hand side coincides with Ext_{F(K)}^{∗}(P_{K}, Q_{K} ∘ (t ∘ τ)). After this identification our homomorphism coincides with the map

induced by the functor homomorphism Q_{K} ∘ (t ∘ τ) ⟶^{Q_{K}(γ)} Q_{K}. This functor homomorphism is split by Q_{K}(η) : Q_{K} → Q_{K} ∘ (t ∘ τ), which gives us a direct sum decomposition Q_{K} ∘ (t ∘ τ) = Q_{K} ⊕ Q_{K}′. To prove the theorem, we must show that Ext_{F(K)}^{∗}(P_{K}, Q_{K}′) = 0.

Assume first that there exists an exponential strict polynomial functor Q^{∗} such that each Q^{i} is homogeneous of degree i and Q = Q^{d}. By Lemma 3.6 and the exponential property of Q^{∗}, we have a natural direct sum decomposition

Q_{K} ∘(t∘τ) = Q_{K} ∘(I ⊕⋯⊕I^{((n−1)N)}) = ⨁_{d_{0}+···+d_{n−1}=d} Q_{K}^{d_{0}} ⊗⋯⊗(Q_{K}^{d_{n−1}})^{((n−1)N)} .

The summand corresponding to the n-tuple (d, 0, …, 0) is Q_{K} and hence Q_{K}′ coincides with the direct sum over n-tuples (d_{0}, …, d_{n−1}) ≠ (d, 0, …, 0). The summand in the above direct sum decomposition indexed by (d_{0}, …, d_{n−1}) is homogeneous of degree d_{0} +d_{1}q+⋯+d_{n−1}q^{n−1} = d+d_{1}(q−1)+⋯+d_{n−1}(q^{n−1} −1). On the other hand P_{K} is homogeneous of degree d. Since all Ext-groups in the category F_{K} between two homogeneous functors are zero unless their degrees are congruent modulo q^{n} − 1 (see e. g. [K1, 3.3]), it suffices to note now that for any n-tuple ≠ (d, …, 0) we have inequalities

0 < d_{1}(q − 1) + ⋯ + d_{n−1}(q^{n−1} − 1) ≤ d(q^{n−1} − 1) ≤ q(q^{n−1} − 1) < q^{n} − 1 .

In the general case, we have an injective resolution 0 ⟶ Q ⟶ Q_{•} in which each functor Q_{i} is a direct sum of functors of the form S^{l_{1}} ⊗ ⋯ ⊗ S^{l_{d}}, l_{1} + … + l_{d} = d. Since the functor P(k) → F(K), Q ↦ Q_{K}′ is obviously exact, we get a resolution

0 → Q′_{K} → (Q_{•})_{K}′ .

The above special case shows that Ext_{F(K)}^{∗}(P_{K}, (Q_{i})_{K}′) = 0, so that considering the hypercohomology spectral sequence we easily conclude the proof.

We now have all the ingredients in place to easily conclude our strong comparison theorem.

Theorem 3.10 (Strong Comparison Theorem). Let k be a finite field with q elements Let further P and Q be homogeneous strict polynomial functors of degree d. If q ≥ d, then the canonical homomorphism

\underrightarrow{lim}_{m} Ext_{P(k)}^{∗}(P^{(m)}, Q^{(m)}) → Ext_{F(k)}^{∗}(P, Q)

is an isomorphism (in all degrees).

Proof. For any s ≥ 0 we can find, according to Corollary 2.8, a finite extension K/k such that the standard homomorphism \underrightarrow{lim}_{m} Ext_{P(K)}^{∗}(P_{K}^{(m)}, Q_{K}^{(m)}) → Ext_{F(K)}^{∗}(P_{K}, Q_{K}) is an isomorphism for all 0 ≤ ∗ ≤ s. According to Proposition 3.8 and Theorem 3.9 we have a commutative diagram

K ⊗_{k} \underrightarrow{lim}_{m} Ext_{P(k)}^{∗}(P^{(m)}, Q^{(m)}) ⟶^{∼} \underrightarrow{lim}_{m} Ext_{P(K)}^{∗}(P_{K}^{(m)}, Q_{K}^{(m)})
↓  ≅↓
K ⊗_{k} Ext_{F(k)}^{∗}(P, Q)  ⟶^{∼}  Ext_{F(K)}^{∗}(P_{K}, Q_{K})

in which all arrows except possibly the left vertical one are isomorphisms (for ∗ ≤ s). This implies immediately that the left vertical arrow is an isomorphism as well.

Theorem 3.10 is complemented by the following elementary observation.

Lemma 3.11. Let k be a finite field with q elements. Assume that P, Q ∈ P(k) are homogeneous strict polynomial functors with 0 ≤ deg P ≠ deg Q < q. Then

\underrightarrow{lim}_{m} Ext_{P(k)}^{∗}(P^{(m)}, Q^{(m)}) = Ext_{F(k)}^{∗}(P, Q) = 0

Proof. Vanishing of the left hand side is obvious and vanishing of the right hand side follows from the fact that deg P ≢ deg Q mod (q − 1) - cf. [K1, 3.3]

§4. Computation of Ext_{P}^{∗}(Γ^{d(r)}, S^{dp^{r−j}(j)}) and Ext_{P}^{∗}(Γ^{d(r)}, Λ^{dp^{r−j}(j)}).

In Theorem 4.5, we compute the Ext-groups of the title (for all d ≥ 0 and all 0 ≤ j ≤ r). Throughout this section, P will denote the category of strict polynomial functors of finite degree on the category of finite dimensional vector spaces over an arbitrary (but fixed) field k. The starting point is the determination of the Extgroups

V_{j} = V_{r,j} = Ext_{P}^{∗}(I^{(r)}, S^{p^{r−j}(j)}) and W_{j} = W_{r,j} = Ext_{P}^{∗}(I^{(r)}, Λ^{p^{r−j}(j)}).

This was achieved in [F-S, 4.5, 4.5.1] by using the hypercohomology spectral sequences for the Koszul and De Rham complexes, as initiated by [F-L-S, §3, 5]. Again, our computation relies heavily on the same technique, using the complexes:

Ω_{dp^{r−j+1}}^{•(j−1)} : 0 → S^{dp^{r−j+1}(j−1)} → S^{dp^{r−j−1}−1(j−1)} ⊗ Λ^{1(j−1)} ⋯ → Λ^{dp^{r−j+1}(j−1)} → 0
Kz_{dp^{r−j}}^{•(j)} : 0 → Λ^{dp^{r−j}(j)} → Λ^{dp^{r−j}−1(j)} ⊗ S^{1(j)} ⋯ → S^{dp^{r−j}(j)} → 0 .

Since a theorem of Cartier tells us that the cohomology of the De Rham complex has a form similar to the De Rham itself but with an additional Frobenius twist, the De Rham complex provides the structure for an inductive argument on the number j of Frobenius twists. This requires, however, that we consider S^{dp^{r−j}(j)} and Λ^{dp^{r−j}(j)} simultaneously, and the Koszul complex serves to tightly link Extgroups for symmetric powers with those for exterior powers.

Since all but the outer terms of the complexes involve the tensor product of a symmetric and exterior functor, Theorem 1.7 suggests an inductive argument on the integer d.

Finally, we compute by an induction on the cohomological degree the hypercohomology spectral sequences for the two complexes. When studying the second hypercohomology spectral sequences for the De Rham complex, a new feature appears, the use of the generalized Koszul complex associated to a map. Since the cohomology of such a complex is readily computed, it is useful to view the (only) non-trivial differential in our hypercohomology spectral sequence as fitting into a generalized Koszul complex. The map involved is nothing but the non-trivial differential for the case d = 1, which relates the graded vector spaces V_{j} and W_{j}. The same feature reappears when computing the first hypercohomology spectral sequence for the Koszul complex: this has only one non-zero differential, which we identify as the differential for the generalized Koszul complex associated once again to the graded map from W_{j} to V_{j} arising in the case d = 1.

Hypercohomology spectral sequences

In a familiar manner, there are two spectral sequences associated to applying RHom_{P}(Γ^{d(r)}, −) to a complex of functors F^{•} in P. The first has E_{1}^{s,t}-term Ext_{P}^{t}(Γ^{d(r)}, F^{s}), whereas the second has E_{2}^{s,t}-term Ext_{P}^{s}(Γ^{d(r)}, H^{t}(F^{•})). Since the Koszul complex is exact, the second spectral sequence associated to Kz_{dp^{r−j}}^{•(j)} is triv-

ial (and hence the abutment of the first one is zero). We shall use the notation E^{∼} for the first spectral sequence associated to the Koszul complex Kz_{dp^{r−j}}^{•(j)}:

E^{∼}_{1}^{s,t} = Ext_{P}^{t}(Γ^{d(r)}, S^{s(j)} ⊗ Λ^{dp^{r−j}−s(j)}) ⟹ 0

Furthermore, we shall use the notation I (resp. II) for the first (resp. second) spectral sequence associated to the De Rham complex Ω_{dp^{r−j+1}}^{•(j−1)}:

I_{1}^{s,t} = Ext_{P}^{t}(Γ^{d(r)}, S^{dp^{r−j+1}−s(j−1)} ⊗ Λ^{s(j−1)}) ⟹ Ext_{P}^{s+t}(Γ^{d(r)}, Ω_{dp^{r−j+1}}^{•(j−1)})
II_{2}^{s,t} = Ext_{P}^{s}(Γ^{d(r)}, S^{dp^{r−j}−t(j)} ⊗ Λ^{t(j)}) ⟹ Ext_{P}^{s+t}(Γ^{d(r)}, Ω_{dp^{r−j+1}}^{•(j−1)}).

To compute the differentials in the spectral sequences E^{∼}, I, II, we shall use the following proposition.

Proposition 4.1. Let T_{1}^{•}, T_{2}^{•} be bounded below complexes of homogeneous strict polynomial functors of degrees d_{1}p^{r} and d_{2}p^{r} respectively. Set d = d_{1} + d_{2}, then the first (respectively, the second) spectral sequence associated to applying the functor RHom_{P}(Γ^{d(r)}, −) to the complex T_{1}^{•} ⊗ T_{2}^{•} is naturally isomorphic to the tensor product of the first (resp., second) spectral sequences associated to applying the functors RHom(Γ^{d_{i}(r)}, −) to T_{i}^{•}.

Proof. Let T_{1}^{•} → I_{1}^{••}, T_{2}^{•} → I_{2}^{••} be the Cartan-Eilenberg resolutions of T_{1}^{•} and T_{2}^{•} respectively. An immediate verification shows that Tot(T_{1}^{•} ⊗ T_{2}^{•}) → I_{1}^{••} ⊗ I_{2}^{••} is the Cartan-Eilenberg resolution of Tot(T_{1}^{•} ⊗T_{2}^{•}) (here I_{1}^{••} ⊗I_{2}^{••} should be considered as a bicomplex with (I_{1} ⊗ I_{2})^{s,t} = ⨁_{s_{1}+s_{2}=s,t_{1}+t_{2}=t} I_{1}^{s_{1},t_{1}} ⊗ I_{2}^{s_{2},t_{2}}). The two spectral sequences in question are the two spectral sequences of the bicomplex

Hom_{P}(Γ^{d(r)}, I_{1}^{••} ⊗ I_{2}^{••})

which can be identified using Theorem 1.7 with

Hom_{P}(Γ^{d_{1}(r)}, I_{1}^{••}) ⊗ Hom_{P}(Γ^{d_{2}(r)}, I_{2}^{••}) .

The result now follows immediately [M, 11, §3, Ex. 6].

The following corollary makes explicit the determination of the differentials in the tensor product of spectral sequences as considered in Proposition 4.1.

Corollary 4.2. All differentials of the first hypercohomology spectral sequence associated to applying the functor RHom_{P}(Γ^{d(r)}, −) to the complex (Ω_{p^{r−j+1}}^{•(j−1)})^{⊗d} are trivial.

In the second hypercohomology spectral sequence associated to applying the functor RHom_{P}(Γ^{d(r)}, −) to (Ω_{p^{r−j+1}}^{•(j−1)})^{⊗d} (respectively, the first hypercohomology spec-

tral sequence associated to applying the functor RHom_{P}(Γ^{d(r)}, −) to (Kz_{p^{r−j}}^{•(j)})^{⊗d}), all differentials except for d_{p^{r−j}+1} (resp. except for d_{p^{r−j}}) are trivial. The only non-trivial differential sends a decomposable, homogeneous element v_{1} ⊗ ... ⊗ v_{d} to:

∑_{i=1}^{d}(−1)^{σ(i)}v_{1} ⊗ ... ⊗ ∂(v_{i}) ⊗ ... ⊗ v_{d}

where ∂ is the only non-zero differential in the second hypercohomology spectral sequence associated to applying the functor RHom_{P}(I^{(r)}, −) to Ω_{p^{r−j+1}}^{•(j−1)} (resp. the first hypercohomology spectral sequence associated to applying RHom_{P}(I^{(r)}, −) to Kz_{p^{r−j}}^{•(j)}) and σ(i) denotes the number of terms of odd total degree among v_{1}, ..., v_{i−1}.

Proof. This follows immediately from Proposition 4.1 and the known information about differentials in the spectral sequences associated to applying the functor RHom_{P}(I^{(r)}, −) to Ω_{p^{r−j+1}}^{•(j−1)} and Kz_{p^{r−j}}^{•(j)} -see [F-S, §4].

Generalized Koszul complexes

The graded vector space V_{j} is concentrated in even degrees. It is one dimensional in degrees s ≡ 0 mod 2p^{r−j}, 0 ≤ s < 2p^{r}, and zero otherwise [F-S, 4.5]. The graded vector space W_{j} is related to V_{j} by two homomorphisms. The first one, which we denote by θ, is the (only non-zero) differential

θ := d_{p^{r−j}} : Ext_{P}^{∗}(I^{(r)}, Λ^{p^{r−j}(j)}) → Ext_{P}^{∗−(p^{r−j}−1)}(I^{(r)}, S^{p^{r−j}(j)})  (4.2.1)

in the first hypercohomology spectral sequence associated to applying the functor RHom(I^{(r)}, −) to the Koszul complex Kz_{p^{r−j}}^{•(j)}.

This homomorphism is an isomorphism, shifting degrees by p^{r−j} − 1, and gives an inverse to the Yoneda multiplication by the Koszul complex. This implies, in particular, that W_{j} is concentrated in even degrees for p ≠ 2 and in odd degrees for p = 2, j < r.

The second homomorphism from W_{j} to V_{j}, which we denote by ∂, is the (only non-trivial) differential

∂ := d_{p^{r−j}+1} : Ext_{P}^{∗}(I^{(r)}, Λ^{p^{r−j}(j)}) → Ext_{P}^{∗+p^{r−j}+1}(I^{(r)}, S^{p^{r−j}(j)})  (4.2.2)

in the second hypercohomology spectral sequence associated to applying the functor RHom(I^{(r)}, −) to the De Rham complex Ω_{p^{r−j+1}}^{•(j−1)}. This homomorphism fits into an exact sequence

0 → W_{j−1} ⟶^{α} W_{j} ⟶^{∂} V_{j} → V_{j−1} → 0

which can be interpreted in terms of edge homomorphisms of the corresponding spectral sequence (see [F-S, §4]). Here the homomorphism V_{j} → V_{j−1} is the natural map of degree zero induced by the embedding S^{p^{r−j}(j)} ↪ S^{p^{r−j+1}(j−1)}, whereas the homomorphism α is the Yoneda multiplication by a class in Ext_{P}^{p^{r−j+1}−p^{r−j}}(Λ^{p^{r−j+1}(j−1)}, Λ^{p^{r−j}(j)}) obtained by truncating the De Rham complex. In particular, α increases degrees by p^{r−j+1} − p^{r−j}.

We shall use the notation K_{j} (resp. C_{j}) for the kernel (resp. cokernel) of ∂, so that the previous exact sequence gives natural isomorphisms of graded vector spaces

W_{j−1}[p^{r−j} − p^{r−j+1}] ⟶^{∼} K_{j} = Ker ∂ , Coker ∂ = C_{j} ⟶^{∼} V_{j−1} .

In order to analyze the spectral sequences Ẽ and II, we shall observe that iterations of θ and ∂ form “generalized Koszul complexes” in the following sense. Let f : W → V be a homomorphism of finite dimensional k-vector spaces. Set

Q_{d}^{i}(f) = S^{d−i}(V) ⊗ Λ^{i}(W) , Q_{∗}^{∗}(f) = ⨁_{0≤i≤d} Q_{d}^{i}(f).

We refer to d as the total degree and to i as the cohomological degree.

Note that Q_{∗}^{∗}(f) has a natural structure of a bigraded, graded commutative (with respect to the cohomological degree) algebra. We endow the algebra Q(f) with a Koszul differential κ (of cohomological degree -1), defined in terms of f and product and coproduct operations as follows:

κ : Q_{d}^{i}(f) = S^{d−i}(V) ⊗ Λ^{i}(W) ⟶^{1_{S^{d−i}(V )}⊗comult} S^{d−i}(V) ⊗ W ⊗ Λ^{i−1}(W) ⟶^{1⊗f⊗1}

⟶^{1⊗f⊗1} S^{d−i}(V) ⊗ V ⊗ Λ^{i−1}(W) ⟶^{mult⊗1_{Λ^{i−1}(W)}} S^{d−i+1}(V) ⊗ Λ^{i−1}(W) = Q_{d}^{i−1}(f)

We refer to the resulting complex Q_{d}^{•}(f) as the generalized Koszul complex for f.

One checks easily that κ is a (graded) derivation and hence H^{∗}(Q(f)) has a natural structure of a bigraded graded commutative (with respect to the cohomological degree) algebra. Moreover, we have canonical identifications

H_{1}^{1}(Q(f)) = Ker f H_{1}^{0}(Q(f)) = Coker f .

The usefulness of this formalism lies in the following elementary lemma.

Lemma 4.3. The induced homomorphism of bigraded algebras

S^{∗}(Coker f) ⊗ Λ^{∗}(Ker f) → H^{∗}(Q(f))

is an isomorphism.

Proof. Choose (non-canonical) splittings

W = Ker f ⊕ U , V = U ⊕ Coker f

in such a way that f identifies the two copies of U. The differential graded algebra Q(f) is isomorphic to the tensor product of a trivial differential graded algebra S^{∗}(Coker f) ⊗ Λ^{∗}(Ker f) and the usual Koszul differential graded algebra corresponding to the vector space U. Our statement follows now from the acyclicity of the usual Koszul algebra.

Remark 4.3.1. In applications, the vector spaces V and W are usually graded and the homomorphism f preserves the grading (or shifts it by a certain integer deg f). In this case Q_{∗}^{∗}(f), and H^{∗}(Q(f)), acquires an additional grading which we call the internal grading.

The product operation in the tri-graded Hopf algebra Ext_{P}^{∗}(Γ^{∗(r)}, S^{∗(j)}) defines a canonical homomorphism of graded vector spaces

V_{j}^{⊗d} → Ext_{P}^{∗}(Γ^{d(r)}, S^{dp^{r−j}(j)}).

Lemma 1.11 (and the fact that V_{j} is concentrated in even degrees) implies that this homomorphism commutes with the action of the symmetric group Σ_{d}, and thus factors to give a map:

S^{d}(V_{j}) → Ext_{P}^{∗}(Γ^{d(r)}, S^{dp^{r−j}(j)}).  (4.3.2)

We shall also consider the analogous map

W_{j}^{⊗d} → Ext_{P}^{∗}(Γ^{d(r)}, Λ^{dp^{r−j}(j)}).

Lemma 1.11 implies that this map commutes with the action of the symmetric group Σ_{d} up to a sign, and thus factors through Λ^{d}(W_{j}), provided that p ≠ 2. In the case p = 2, the fact that this map factors through Λ^{d}(W_{j}) is established below in Lemma 4.4. We shall employ the notation v_{1} · ... · v_{i} (resp. w_{1} ∧ ... ∧ w_{i}) for the image of v_{1} ⊗ ... ⊗ v_{i} ∈ V_{j}^{⊗i} (resp. w_{1} ⊗ ... ⊗ w_{i} ∈ W_{j}^{⊗i}) in Ext_{P}^{∗}(Γ^{i(r)}, S^{ip^{r−j}(j)}) (resp. in Ext_{P}^{∗}(Γ^{i(r)}, Λ^{ip^{r−j}(j)})).

Lemma 4.4. The natural map

W_{j}^{⊗d} → Ext_{P}^{∗}(Γ^{d(r)}, Λ^{dp^{r−j}(j)})

factors through Λ^{d}(W_{j}).

Proof. This result can be obtained as a byproduct of our proof of Theorem 4.5 below. Such a presentation however would make the central part of the argument even harder to follow than it is right now. To avoid this, we give a direct proof of Lemma 4.4 based on the use of the canonical injective resolution of the functor S^{∗(r)} constructed in [F-S, §8].

As was explained above it suffices to consider the case p = 2 and in this case it suffices to show that the square of each homogeneous element w in W_{j} dies in Ext_{P}^{∗}(Γ^{2(r)}, Λ^{2^{r−j+1}(j)}). We start with the special case j = r. First:

(4.4.1) Ext_{P}^{odd}(Γ^{2(r)}, S^{2(r)}) = 0 .

To prove this we use the injective resolution of S^{2(r)} constructed in [F-S, §8] 0 → S^{2(r)} → C^{0} → C^{1} → ... , where

C^{n} = ⨁_{∑ i·m_{i}=n·2^{r−1}^{m_{0}+...+m_{2^{r} −1}=2^{r+1}}} S^{m_{0}} ⊗ ... ⊗ S^{m_{2^{r} −1}}

Theorem 1.7 implies immediately that Hom_{P}(Γ^{2(r)}, C^{n}) = 0 if n is odd.

The cohomology long exact sequence corresponding to the short exact sequence of strict polynomial functors: 0 → Λ^{2(r)} → I^{(r)} ⊗ I^{(r)} → S^{2(r)} → 0 now implies:

(4.4.2) The standard embedding Λ^{2(r)} ↪ I^{(r)} ⊗ I^{(r)} induces injective maps

Ext_{P}^{even}(Γ^{2(r)}, Λ^{2(r)}) ↪ Ext_{P}^{even}(Γ^{2(r)}, I^{(r)} ⊗ I^{(r)}).

To finish the argument in the case j = r it suffices to note now that the composition

(W_{r} ⊗W_{r})^{even} → Ext_{P}^{even}(Γ^{2(r)}, Λ^{2(r)}) ↪ Ext_{P}^{even}(Γ^{2(r)}, I^{(r)}⊗I^{(r)}) = (W_{r} ⊗W_{r})^{even} coincides with 1 + σ (where σ permutes the two factors) and hence is zero on elements of the form w ⊗ w.

Assume now that j < r. Let kz ∈ Ext_{P}^{2^{r−j}−1}(S^{2^{r−j}}, Λ^{2^{r−j}}) be the extension class represented by the Koszul complex and let ϕ_{r−j} : I^{(r−j)} ↪ S^{2^{r−j}} denote the standard embedding. The standard arguments (see [F-S, §4]) show that the left Yoneda multiplication with kz^{(j)} defines an isomorphism (inverse to θ)

Ext_{P}^{∗}(I^{(r)}, S^{2^{r−j}(j)}) ⟶^{kz^{(j)}} Ext_{P}^{∗+2^{r−j}−1}(I^{(r)}, Λ^{2^{r−j}(j)}).

On the other hand ϕ_{r−j}^{(j)}: I^{(r)} ↪ S^{2^{r−j}(j)} induces epimorphisms on Ext_{P}^{∗}(I^{(r)}, −) [F-S, §4]. This shows that each homogeneous element in W_{j} = Ext_{P}^{∗}(I^{(r)}, Λ^{2^{r−j}(j)}) may be written as a Yoneda product w = kz^{(j)} · ϕ_{r−j}^{(j)} · x , for a certain homogeneous element x in Ext_{P}^{∗}(I^{(r)}, I^{(r)}). In these notations we have the following formula:

w ⊗w = ((kz ⊗kz)·(ϕ_{r−j} ⊗ϕ_{r−j}))^{(j)} ·(x⊗x) ∈ Ext_{P}^{∗}(I^{(r)} ⊗I^{(r)}, Λ^{2^{r−j}(j)} ⊗Λ^{2^{r−j}(j)}).

The image of w ⊗ w in Ext_{P}^{∗}(Γ^{2(r)}, Λ^{2^{r−j+1}(j)}) is the Yoneda product m_{Λ}^{(j)} · (w ⊗ w) · c_{Γ}^{(r)}, where m_{Λ} : Λ^{2^{r−j}} ⊗ Λ^{2^{r−j}} → Λ^{2^{r−j+1}} is the multiplication homomorphism and c_{Γ} : Γ^{2} ↪ I ⊗ I is the standard embedding. The long exact sequence of Ext-groups associated to the short exact sequence of strict polyno-

mial functors 0 → Γ^{2(r)} ⟶^{c^{(r)}} I^{(r)} ⊗ I^{(r)} ⟶^{m^{(r)}} Λ^{2(r)} → 0 and the vanishing of m_{Λ}^{(r)} · (x ⊗ x) · c_{Γ}^{(r)} established above show that (x ⊗ x) · c_{Γ}^{(r)} may be written in the form c_{Γ}^{(r)} ·y for some y ∈ Ext_{P}^{∗}(Γ^{2(r)}, Γ^{2(r)}). Now it suffices to note that the element under consideration is a right multiple of (m_{Λ} · (kz ⊗ kz) · (ϕ_{r−j} ⊗ ϕ_{r−j}) · c_{Γ}^{(r−j)})^{(j)} and m_{Λ} · (kz ⊗ kz) · (ϕ_{r−j} ⊗ ϕ_{r−j}) · c_{Γ}^{(r−j)} ∈ Ext_{P}^{∗}(Γ^{2(r−j)}, Λ^{2^{r−j+1}}) = 0 - see [F-S, Proposition 5.4].

Basic computation of Ext-groups

The following theorem, whose proof occupies much of the remainder of this section, formulates the result of our computation.

Theorem 4.5. For all d and all 0 ≤ j ≤ r, the natural homomorphisms

V_{j}^{⊗d} → Ext_{P}^{∗}(Γ^{d(r)}, S^{dp^{r−j}(j)}) and W_{j}^{⊗d} → Ext_{P}^{∗}(Γ^{d(r)}, Λ^{dp^{r−j}(j)})

factor to induce isomorphisms of graded vector spaces

S^{d}(V_{j}) ⟶^{∼} Ext_{P}^{∗}(Γ^{d(r)}, S^{dp^{r−j}(j)}) , Λ^{d}(W_{j}) ⟶^{∼} Ext_{P}^{∗}(Γ^{d(r)}, Λ^{dp^{r−j}(j)}) .

Proof. We proceed by a triple induction. The main induction is on the number j of twists, j ≤ r. The case j = 0 follows from the injectivity of S^{dp^{r}} in P and [F-S, 5.4].

In the sequel we shall assume that 0 < j ≤ r and the result holds for j − 1. The second induction is on d. The cases d = 0, 1 of our theorem are tautological,

so we assume in the sequel that d > 1 and the result holds for all d′ < d. For the induction step, we use the hypercohomology spectral sequences I, II obtained by applying the functor RHom_{P}(Γ^{d(r)}, −) to the De Rham complex Ω_{dp^{r−j+1}}^{•(j−1)}.

Lemma 4.6. All the differentials in the spectral sequence I are trivial and hence I_{1} = I_{∞}.

Proof. The induction hypothesis on j together with Theorem 1.7 gives the following computation of the I_{1}-terms:

I_{1}^{n,m} = \{ \begin{matrix}0 & if n ≢ 0 mod p^{r−j+1} \\ [S^{d−s}(V_{j−1}) ⊗ Λ^{s}(W_{j−1})]^{m} & if n = sp^{r−j+1} \end{matrix}

Here [−]^{m} denotes the m-th homogeneous component with respect to the internal grading. This formula shows that the map of spectral sequences defined by the homomorphism of complexes

(Ω_{p^{r−j+1}}^{•(j−1)})^{⊗d} → Ω_{dp^{r−j+1}}^{•(j−1)}

is surjective on the E_{1}-terms, thus the lemma follows from the first part of Corollary 4.2.

Corollary 4.7.

dim_{k} Ext_{P}^{n}(Γ^{d(r)}, Ω_{p^{r−j+1}}^{•(j−1)}) = ∑_{s=0}^{d} dim_{k} I_{1}^{sp^{r−j+1},n−sp^{r−j+1}} =

= ∑_{s=0}^{d} dim_{k}[S^{d−s}(V_{j−1}) ⊗ Λ^{s}(W_{j−1})]^{n−sp^{r−j+1}} =

= ∑_{s=0}^{d} dim_{k}[S^{d−s}(C_{j}) ⊗ Λ^{s}(K_{j})]^{n−sp^{r−j}}

Here [−]^{n−sp^{r−j+1}} stands for the homogeneous component with respect to the internal grading. The shift in degrees in the last identification comes from natural isomorphisms of graded vector spaces W_{j−1} = K_{j}[p^{r−j+1} − p^{r−j}], V_{j−1} = C_{j}.

Consider now the spectral sequence II

II_{2}^{n,m} = Ext_{P}^{n}(Γ^{d(r)}, S^{dp^{r−j}−m(j)} ⊗ Λ^{m(j)}) ⟹ Ext_{P}^{n+m}(Γ^{d(r)}, Ω_{dp^{r−j+1}}^{•(j−1)}) .

The induction hypothesis on d and Theorem 1.7 allow us to identify all II_{2}^{n,m}-terms except for the two edge rows m = 0, dp^{r−j}:

II_{2}^{n,m} = \{ \begin{matrix}0 & if m ≢ 0 mod p^{r−j} \\ [S^{d−s}(V_{j}) ⊗ Λ^{s}(W_{j})]^{n} & if m = sp^{r−j} , 0 < s < d . \end{matrix}

By dimension considerations all differentials d_{k} with 2 ≤ k ≤ p^{r−j} are trivial. Furthermore, considering the homomorphism of the second hypercohomology spectral sequences induced by the homomorphism of complexes (Ω_{p^{r−j+1}}^{•(j−1)})^{⊗d} → Ω_{dp^{r−j+1}}^{•(j−1)} and using Corollary 4.2 we identify the action of the differential d_{p^{r−j}+1} on decomposable elements as

d_{p^{r−j}+1}(v_{1} · ... · v_{d−s} ⊗ w_{1} ∧ ... ∧ w_{s}) =

= ∑_{i=1}^{s}(−1)^{i−1}v_{1} · ... · v_{d−s} · ∂(w_{i}) ⊗ w_{1} ∧ ... ∧ w^{∧}_{i} ∧ ... ∧ w_{s} ( 4.7.1)

Here v_{1}, ..., v_{d−s} are homogeneous elements in V_{j}, w_{1}, ..., w_{s} are homogeneous elements in W_{j} and ∂ : W_{j} → V_{j} is the differential d_{p^{r−j}+1} in the second hypercohomology spectral sequence corresponding to Ω_{p^{r−j+1}}^{•(j)}, discussed previously. The sign in the formula (4.7.1) is explained by the fact that all v’s are of even total degree, while all w’s are of odd total degree. Thus, d_{p^{r−j}+1} may be viewed as the generalized Koszul differential in Q_{d}^{•}(∂).

Lemma 4.3 together with (4.7.1) implies that we get a canonical homomorphism

S^{d}(C_{j}) = Coker(S^{d−1}(V_{j}) ⊗ W_{j} → S^{d}(V_{j})) →

→ II_{p^{r−j}+2}^{∗,0} = Coker(S^{d−1}(V_{j}) ⊗ W_{j} → Ext_{P}^{∗}(Γ^{d(r)}, S^{dp^{r−j}(j)}) ).

Lemma 4.8. The resulting homomorphism

S^{d}(C_{j}) → II_{p^{r−j}+2}^{∗,0} → II_{∞}^{∗,0} → Ext_{P}^{∗}(Γ^{d(r)}, Ω_{dp^{r−j+1}}^{•(j−1)})

is injective.

Proof. It’s clear from the construction that the homomorphism in question is induced by the composition

S^{d}(V_{j}) → Ext_{P}^{∗}(Γ^{d(r)}, S^{dp^{r−j}(j)}) → Ext_{P}^{∗}(Γ^{d(r)}, Ω_{dp^{r−j+1}}^{•(j−1)})

where the second arrow is defined by the homomorphism of complexes S^{dp^{r−j}(j)} → Ω_{dp^{r−j+1}}^{•(j−1)} (identifying S^{dp^{r−j}(j)} with the zero-dimensional homology group of

Ω_{dp^{r−j+1}}^{•(j−1)}). Consider also the natural homomorphism Ω_{dp^{r−j+1}}^{•(j−1)} → S^{dp^{r−j+1}(j−1)} (projection onto the zero-dimensional component). The composition of these homomorphisms coincides with the standard embedding S^{dp^{r−j}(j)} ↪ S^{dp^{r−j+1}(j−1)}. The induction hypothesis on j shows that Ext_{P}^{∗}(Γ^{d(r)}, S^{dp^{r−j+1}(j−1)}) = S^{d}(V_{j−1}). The composite

S^{d}(C_{j}) → Ext_{P}^{∗}(Γ^{d(r)}, Ω_{dp^{r−j+1}}^{•(j−1)}) → Ext_{P}^{∗}(Γ^{d(r)}, S^{dp^{r−j+1}(j−1)}) = S^{d}(V_{j−1})

is by construction the isomorphism C_{j} ⟶^{∼} V_{j−1} for d = 1 and thus (by its multiplicative nature) is an isomorphism for all d.

Now we can start the final induction – that on the internal index. More precisely we are going to prove, using induction on t, the following statement. As in Corollary 4.7, [−]^{m} denotes the m-th homogeneous component with respect to the internal grading.

Proposition 4.9. With the above notation and assumptions, the following assertions hold for any t ≥ −1.

1_{t}. The natural map [S^{d}(V_{j})]^{t} → Ext_{P}^{t}(Γ^{d(r)}, S^{dp^{r−j}(j)}) is an isomorphism.

2_{t}. The natural map [Λ^{d}(W_{j})]^{t} → Ext_{P}^{t}(Γ^{d(r)}, Λ^{dp^{r−j}(j)}) is an isomorphism.

3_{t}. II_{p^{r−j}+2}^{n,m} = II_{∞}^{n,m} for all n ≤ t − p^{r−j} − 1.

Proof. All the statements are trivial for t = −1, which gives us the base for the inductive argument. Assume now that t ≥ 0 and the statement holds for all t′ < t.

Consider once again the spectral sequence II. Using (4.7.1), we get the following commutative diagram

\begin{matrix}[Q_{d}^{s+1}(∂)]^{n−p^{r−j}−1} & ⟶^{κ} & [Q_{d}^{s}(∂)]^{n} & ⟶^{κ} & [Q_{d}^{s−1}(∂)]^{n+p^{r−j}+1} \\ ↓ & & ↓ & & ↓ \\ II_{p^{r−j}+1}^{n−p^{r−j}−1,(s+1)p^{r−j}} & ⟶^{d_{p^{r−j} +1}} & II_{p^{r−j}+1}^{n,sp^{r−j}} & ⟶^{d_{p^{r−j} +1}} & II_{p^{r−j}+1}^{n+p^{r−j}+1,(s−1)p^{r−j}} \end{matrix} (4.9.1)

Here the top row is a piece of the generalized Koszul complex, corresponding to the homomorphism (shifting degrees by p^{r−j} + 1) of graded vector spaces ∂ : W_{j} → V_{j}. Our induction hypothesis on d immediately implies that the three vertical maps in the above diagram are isomorphisms unless s = 0, 1, d − 1, d. Examining each of these four cases separately and using our induction hypothesis on t, we easily conclude the following:

(4.9.2). The induced map [S^{d−s}(C_{j}) ⊗ Λ^{s}(K_{j})]^{n} → II_{p^{r−j}+2}^{n,sp^{r−j}}, given as the map on cohomology of (4.9.1),

a) is an isomorphism provided that s ≠ 0, 1, d − 1, d or n < t − p^{r−j} − 1

b) is a monomorphism provided that n < t.

Consider the terms of total degree t in II_{p^{r−j}+2}. The differentials that could possibly hit II_{p^{r−j}+2}^{t,0} should come from II^{n,m}-terms with n < t − p^{r−j} − 1. However all such differentials are trivial in view of the inductive assumption 3_{t−1}. Thus II_{p^{r−j}+2}^{t,0} = II_{∞}^{t,0}. The same reasoning shows that II_{p^{r−j}+2}^{t−p^{r−j},p^{r−j}} = II_{∞}^{t−p^{r−j},p^{r−j}}. Moreover for s ≥ 2

II_{∞}^{t−sp^{r−j},sp^{r−j}} = II_{p^{r−j}+2}^{t−sp^{r−j},sp^{r−j}} = [S^{d−s}(C_{j}) ⊗ Λ^{s}(K_{j})]^{t−sp^{r−j}}

according to 3_{t−1} and (4.9.2 a). Thus

dim_{k} Ext_{P}^{t}(Γ^{d(r)}, Ω_{p^{r−j+1}}^{•(j−1)}) = dim_{k} II_{p^{r−j}+2}^{t,0} + dim_{k} II_{p^{r−j}+2}^{t−p^{r−j},p^{r−j}}+

+ ∑_{s=2}^{d} dim_{k}[S^{d−s}(C_{j}) ⊗ Λ^{s}(K_{j})]^{t−sp^{r−j}} (4.9.3)

Finally we have the following inequalities:

dim_{k} II_{p^{r−j}+2}^{t,0} ≥ dim_{k}[S^{d}(C_{j})]^{t} − according to (4.8)
dim_{k} II_{p^{r−j}+2}^{t−p^{r−j},p^{r−j}} ≥ dim_{k}[S^{d−1}(C_{j}) ⊗ K_{j}]^{t−p^{r−j}}

Comparing (4.9.3) with the result given by Corollary 4.7, we conclude that both inequalities above are actually equalities. This shows that the natural homomorphisms [S^{d}(C_{j})]^{t} → II_{p^{r−j}+2}^{t,0} and [S^{d−1}(C_{j}) ⊗ K_{j}]^{t−p^{r−j}} → II_{p^{r−j}+2}^{t−p^{r−j},p^{r−j}} are isomorphisms. Inspecting the diagram (4.9.1) for s = 1, n = t − p^{r−j} − 1 (extended by zeros to the right) and using the Five-Lemma we conclude that [S^{d}(V_{j})]^{t} → Ext_{P}^{t}(Γ^{d(r)}, S^{dp^{r−j}(j)}) is an isomorphism ; in other words, 1_{t} holds. Using this fact we extend easily the conclusion made in (4.9.2a) to degrees n ≤ t − p^{r−j} − 1 :

(4.9.4). [S^{d−s}(C_{j}) ⊗ Λ^{s}(K_{j})]^{n} → II_{p^{r−j}+2}^{n,sp^{r−j}} is an isomorphism for all s provided that n ≤ t − p^{r−j} − 1.

We now prove 3_{t}. Let E denote the second hypercohomology spectral sequence obtained by applying RHom(I^{(r)}, − ) to the De Rham complex Ω_{p^{r−j+1}}^{•(j−1)}. The

natural homomorphism of complexes (Ω_{p^{r−j+1}}^{•(j−1)})^{⊗d} → Ω_{dp^{r−j+1}}^{•(j−1)} defines (in view of Proposition 4.1) a homomorphism of spectral sequences E^{⊗d} → II. We conclude from (4.9.4) that the homomorphism

(E^{⊗d})_{p^{r−j}+2}^{n,m} → II_{p^{r−j}+2}^{n,m}

is surjective for n ≤ t − p^{r−j} − 1. Since the differentials d_{k} with k > p^{r−j} + 1 are trivial in E, they are trivial in E^{⊗d}. We conclude that all the differentials d_{k} in II with k > p^{r−j} + 1 starting at II^{n,m}-terms with n ≤ t − p^{r−j} − 1 are trivial. So II_{p^{r−j}+2}^{n,m} = II_{∞}^{n,m} for these values of n, i.e. 3_{t} holds.

To prove that 2_{t} holds, we consider the spectral sequence E^{∼} (the first spectral

sequence corresponding to Kz_{dp^{r−j}}^{•(j)}). The E^{∼}_{1}-term of this spectral sequence is given (using Theorem 1.7) by the formula

E^{∼}_{1}^{m,n} = Ext_{P}^{n}(Γ^{d(r)},S^{m(j)} ⊗ Λ^{dp^{r−j}−m(j)}) =

= \{ \begin{matrix}0 & if m ≢ 0 mod p^{r−j} \\ [S^{s}(V_{j}) ⊗ Λ^{d−s}(W_{j})]^{n} & if m = sp^{r−j} 0 < s < d \end{matrix}

The differentials d_{1}, ..., d_{p^{r−j}−1} are trivial by dimension considerations. Corollary 4.2 allows to determine the action of d_{p^{r−j}} on decomposable elements, so that as for (4.9.1) we obtain a commutative diagram

\begin{matrix}[Q_{d}^{d−s+1}(θ)]^{n+p^{r−j}−1} & ⟶^{κ} & [Q_{d}^{d−s}(θ)]^{n} & ⟶^{κ} & [Q_{d}^{d−s−1}(θ)]^{n−p^{r−j}+1} \\ ↓ & & ↓ & & ↓ \\ E^{∼}_{p^{r−j}}^{(s−1)p^{r−j},n+p^{r−j}−1} & ⟶^{d_{p^{r−j}}} & E^{∼}_{p^{r−j}}^{sp^{r−j},n} & ⟶^{d_{p^{r−j}}} & E^{∼}_{p^{r−j}}^{(s+1)p^{r−j},n−p^{r−j}+1} \end{matrix} (4.9.5)

Here the top row is a piece of the (acyclic) generalized Koszul complex, corresponding to the isomorphism (shifting degrees by p^{r−j} − 1) of graded vector spaces θ : W_{j} → V_{j} (cf. (4.2.1)). To conclude that the bottom row of (4.9.5) is acyclic it suffices by the acyclicity of the top row to know that the central and the right-hand side vertical arrows are isomorphism. This remark plus our induction hypotheses on d and t implies: (4.9.7) E^{∼}_{p^{r−j}+1}^{sp^{r−j},n} = 0 provided that s ≠ 0, d − 1, d, or n < t.

For k ≥ p^{r−j} + 1, the differential d_{k} : E^{∼}_{k}^{0,t} → E^{∼}_{k}^{k,t−k+1} is trivial by (4.9.7). Thus

E^{∼}_{p^{r−j}+1}^{0,t} = E^{∼}_{∞}^{0,t} = 0. Inspecting finally the diagram (4.9.5) for s = 1, n = t−p^{r−j}+1 (extended by zeros to the left) and using the Five-Lemma we conclude that

[Λ^{d}(W_{j})]^{t} → Ext_{P}^{t}(Γ^{d(r)}, Λ^{dp^{r−j}(j)}) = E^{∼}_{p^{r−j}}^{0,t}

is an isomorphism. This shows that 2_{t} also holds, thus completing the induction step, the proof of Proposition 4.9 and thus of Theorem 4.5.

Corollary 4.10 (Strong Twist Stability). Let A and B be homogeneous strict polynomial functors of degree d. The Frobenius twist map

Ext_{P}^{s}(A^{(m)}, B^{(m)}) → Ext_{P}^{s}(A^{(m+1)}, B^{(m+1)})

is an isomorphism provided that m ≥ log_{p} \frac{s+1}{2}.

Proof. Consider first the special case A = Γ^{d}, B = S^{d}. Theorem 4.3 shows that

Ext_{P}^{∗}(Γ^{d(m)}, S^{d(m)}) = S^{d}(Ext_{P}^{∗}(I^{(m)}, I^{(m)})) .

Since the Frobenius twist map Ext_{P}^{∗}(I^{(m)}, I^{(m)}) → Ext_{P}^{∗}(I^{(m+1)}, I^{(m+1)}) is an isomorphism in degrees ≤ 2p^{m} − 1 [F-S;4.9], we conclude that the induced map S^{d}(Ext_{P}^{∗}(I^{(m)}, I^{(m)})) → S^{d}(Ext_{P}^{∗}(I^{(m+1)}, I^{(m+1)}) is an isomorphism in the same range.

Consider next the special case A = Γ^{d_{1}} ⊗ ⋯ ⊗ Γ^{d_{a}}, B = S^{d′^{1}} ⊗ ⋯ ⊗ S^{d′^{b}}, d_{1} + ⋯ + d_{a} = d′_{1} + ⋯ + d′_{b} = d. In this case, Corollary 1.8 gives us the following formula

Ext_{P}^{∗}(A^{(m)}, B^{(m)}) = _{⨁_{d_{11}+···+d_{1b}=d_{1} d_{11}+···+d_{a1}=d′_{1}_{… …}} ; d_{a1}+···+d_{ab}=d_{a} d_{1b}+···+d_{ab}=d′_{b}} ⨂_{i=1,j=1}^{a b} Ext_{P}^{∗}(Γ^{d_{ij}(m)}, S^{d_{ij}(m)}) .

Thus the result in this case follows from the previous one.

In the general case, we can find resolutions

0 ⟵ A ⟵ P_{•} 0 → B → I^{•}

in which each P_{i} (resp. each I^{j}) is a direct sum of functors of the form Γ^{d_{1}} ⊗ ... ⊗ Γ^{d_{a}} (resp. of the form S^{d′^{1}} ⊗ ... ⊗ S^{d′^{b}}). The result now follows by applying the standard comparison theorem to the map of the hypercohomology spectral sequences converging to Ext_{P}^{∗}(A^{(m)}, B^{(m)}) and Ext_{P}^{∗}(A^{(m+1)}, B^{(m+1)}) respectively.

§5. Ext-groups between the classical functors in the category P.

In this section we continue the computation of Ext-groups in the category P of strict polynomial functors (over an arbitrary fixed field k) between various classical functors. As the reader will see the results obtained in this section are (relatively) easy applications of Theorem 4.5. Again, we begin by fixing a positive integer r.

The general philosophy behind these computations is as follows. There is sort of a hierarchy between the functors Γ^{()}, Λ^{()} and S^{()}: Γ^{()} has a tendency to be projective (it really is if there is no twists), S^{()} has a tendency to be injective and Λ^{()} is in between. We can provide a complete computation for the Ext-groups from a more projective functor to a less projective one but not the other way round. More specifically, we compute all Ext-groups from Γ^{()} (with any number of twists) to Γ^{()}, Λ^{()} and S^{()}, we compute all Ext-groups from Λ^{()} to Λ^{()} and S^{()} and finally from S^{()} to itself; we suspect that there is no comparable computation of the Extgroups from S^{()} to Λ^{()} or Γ^{()} or Ext-groups from Λ^{()} to Γ^{()} (actually it seems that there is just no easy answer for these Ext-groups).

For non-negative integers 0 ≤ j ≤ r, set U_{j} = U_{r,j} = Ext_{P}(Γ^{p^{r−j}(j)}, I^{(r)}). Observe that U_{r} = V_{r} = W_{r} = Ext_{P}^{∗}(I^{(r)}, I^{(r)}); moreover, the duality isomorphism (1.12) defines canonical isomorphisms of graded vector spaces

U_{j} = Ext_{P}^{∗}(Γ^{p^{r−j}(j)}, I^{(r)}) ⟶^{∼} Ext_{P}^{∗}(I^{(r)}, S^{p^{r−j}(j)}) = V_{j}

for any j ≤ r.

For any non-negative integer d the product operations define canonical homomorphisms

U_{j}^{⊗d} = Ext_{P}^{∗}(Γ^{dp^{r−j}(j)}, I^{(r)} ⊗ ... ⊗ I^{(r)}) → Ext_{P}^{∗}(Γ^{dp^{r−j}(j)}, S^{d(r)})
U_{j}^{⊗d} = Ext_{P}^{∗}(Γ^{dp^{r−j}(j)}, I^{(r)} ⊗ ... ⊗ I^{(r)}) → Ext_{P}^{∗}(Γ^{dp^{r−j}(j)}, Λ^{d(r)})

Using these homomorphisms, we formulate our first result, thereby completing the computation of Theorem 4.5 by considering the situation in which the number of twists on the source functor is less (or equal) to the number of twists on the target functor.

Theorem 5.1. For any non-negative integer d the above maps define natural isomorphisms

S^{d}(U_{j}) ⟶^{∼} Ext_{P}^{∗}(Γ^{dp^{r−j}(j)}, S^{d(r)})

Λ^{d}(U_{j}) ⟶^{∼} Ext_{P}^{∗}(Γ^{dp^{r−j}(j)}, Λ^{d(r)}).

Proof. The first isomorphism follows immediately from the Theorem 4.5, in view of the following commutative diagram

S^{d}(U_{j}) ⟶ Ext_{P}^{∗}(Γ^{dp^{r−j}(j)}, S^{d(r)})
=^{∼}↓  =^{∼}↓
S^{d}(V_{j}) ⟶^{∼} Ext_{P}^{∗}(Γ^{d(r)}, S^{dp^{r−j}(j)})

Here the vertical arrows are the duality isomorphisms of (1.12), the bottom horizontal arrow is the isomorphism of the Theorem 4.5 and the top horizontal arrow is the map under consideration.

To prove the second isomorphism, we proceed by induction on d. Consider the first hypercohomology spectral sequence corresponding to the Koszul complex

Kz_{d}^{•(r)} : 0 → Λ^{d(r)} → Λ^{d−1(r)} ⊗ S^{1(r)} → ... → Λ^{1(r)} ⊗ S^{d−1(r)} → S^{d(r)} → 0

E_{1}^{m,n} = Ext_{P}^{n}(Γ^{dp^{r−j}(j)}, Λ^{d−m(r)} ⊗ S^{m(r)}) ⟹ 0

The inductive assumption on d and the already proved part of the theorem give the computation of all the E_{1}-terms except those in the first column. Moreover, we have the following commutative diagram of complexes

0 ⟶ Λ^{d}(U_{j}) ⟶^{κ} Λ^{d−1}(U_{j}) ⊗ U_{j} ⟶^{κ} … ⟶^{κ} S^{d}(U_{j}) ⟶ 0
=^{∼}↓  =^{∼}↓
0 ⟶ E_{1}^{0,∗} ⟶^{d_{1}}  E_{1}^{1,∗}  ⟶^{d_{1}} … ⟶^{d_{1}} E_{1}^{d,∗} ⟶ 0 .

Here the top row is the Koszul complex corresponding to the vector space U_{j}, and the bottom row is the E_{1}-term of our spectral sequence. The acyclicity of the Koszul complex implies immediately that E_{2}^{m,∗} = 0 for m > 1. This implies further vanishing of all higher differentials and hence the identification E_{2} = E_{∞} = 0. Thus the bottom complex in the above diagram is also acyclic, which gives a natural identification Ext_{P}^{∗}(Γ^{dp^{r−j}(j)}, Λ^{d(r)}) = E_{1}^{0,∗} = Λ^{d}(U_{j}).

The duality isomorphism (1.12) together with Theorems 4.5 and 5.1 immediately leads to the following computations.

Corollary 5.2. For all 0 ≤ j ≤ r and all d ≥ 0 we have natural isomorphisms

Λ^{d}(Ext_{P}^{∗}(Λ^{p^{r−j}(j)}, I^{(r)})) ⟶^{∼} Ext_{P}^{∗}(Λ^{dp^{r−j}(j)}, S^{d(r)})
Λ^{d}(Ext_{P}^{∗}(I^{(r)}, S^{p^{r−j}(j)})) ⟶^{∼} Ext_{P}^{∗}(Λ^{d(r)}, S^{dp^{r−j}(j)}) .

The computation of Ext_{P}^{∗}(Γ^{()}, Γ^{()}) in Theorem 5.4 below requires a determination of the coproduct (rather than product) operation in Ext_{P}^{∗}(Γ^{()}, Λ^{()}). Recall that comultiplication in the exterior algebra defines natural homomorphisms (for any 0 ≤ j ≤ r and any d ≥ 0)

Λ^{dp^{r−j}(j)} ↪ Λ^{p^{r−j}(j)} ⊗ ... ⊗ Λ^{p^{r−j}(j)}
Λ^{d(r)} ↪ I^{(r)} ⊗ ... ⊗ I^{(r)}

which induce homomorphisms on Ext-groups

Ext_{P}^{∗}(Γ^{d(r)}, Λ^{dp^{r−j}(j)}) → Ext_{P}^{∗}(Γ^{d(r)}, Λ^{p^{r−j}(j)} ⊗ ... ⊗ Λ^{p^{r−j}(j)}) = W_{j}^{⊗d}
Ext_{P}^{∗}(Γ^{dp^{r−j}(j)}, Λ^{d(r)}) → Ext_{P}^{∗}(Γ^{dp^{r−j}(j)}, I^{(r)} ⊗ ... ⊗ I^{(r)}) = U_{j}^{⊗d}

Corollary 5.3. For all 0 ≤ j ≤ r and all d ≥ 0 the above coproduct operations induce natural isomorphisms of graded vector spaces

Ext_{P}^{∗}(Γ^{d(r)}, Λ^{dp^{r−j}(j)}) ⟶^{∼} Λ^{d}(W_{j})
Ext_{P}^{∗}(Γ^{dp^{r−j}(j)}, Λ^{d(r)}) ⟶^{∼} Λ^{d}(U_{j}) .

Proof. A straightforward computation shows that the composite maps

Λ^{d}(W_{j}) ⟶^{∼}Ext_{P}^{∗}(Γ^{d(r)}, Λ^{dp^{r−j}(j)}) ⟶ W_{j}^{⊗d}
Λ^{d}(U_{j}) ⟶^{∼}Ext_{P}^{∗}(Γ^{dp^{r−j}(j)}, Λ^{d(r)}) ⟶ U_{j}^{⊗d}

coincide with the corresponding natural embeddings.

Now we turn to the computation of Ext_{P}^{∗}(Γ^{()}, Γ^{()}). We introduce one more

basic Ext-space (defined for all 0 ≤ j ≤ r) V^{∼}_{j} = V^{∼}_{r,j} = Ext_{P}^{∗}(I^{(r)}, Γ^{p^{r−j}(j)}). The differential d_{p^{r−j}} in the first hypercohomology spectral sequence, corresponding to the dual Koszul complex

Kz_{p^{r−j}}^{#(j)} : 0 ⟶ Γ^{p^{r−j}(j)} ⟶ Γ^{p^{r−j}−1(j)} ⊗ Λ^{1(j)} ⟶ ⋯ ⟶ Λ^{p^{r−j}(j)} → 0

defines an isomorphism

η : V^{∼}_{j} ⟶^{∼} W_{j}

of graded vector spaces (shifting degrees by p^{r−j} − 1).

In the same way as above, comultiplication in the divided power algebra defines natural homomorphisms

Ext_{P}^{∗}(Γ^{d(r)}, Γ^{dp^{r−j}(j)}) → Ext_{P}^{∗}(Γ^{d(r)}, Γ^{p^{r−j}(j)} ⊗ ... ⊗ Γ^{p^{r−j}(j)}) =

= Ext_{P}^{∗}(I^{(r)}, Γ^{p^{r−j}(j)})^{⊗d} = V^{∼}_{j}^{⊗d}

Ext_{P}^{∗}(Γ^{dp^{r−j}(j)}, Γ^{d(r)}) → Ext_{P}^{∗}(Γ^{dp^{r−j}(j)}, I^{(r)} ⊗ ... ⊗ I^{(r)}) =

= Ext_{P}^{∗}(Γ^{p^{r−j}(j)}, I^{(r)})^{⊗d} = U_{j}^{⊗d}

Lemma 1.11 shows that these homomorphisms are Σ_{d}-invariant and hence their images are contained in the subspaces of Σ_{d}-invariant tensors, i.e. in Γ^{d}(V^{∼}_{j}) and Γ^{d}(U_{j}) respectively.

Theorem 5.4. For all 0 ≤ j ≤ r and all d ≥ 0 the above constructed homomorphisms

Ext_{P}^{∗}(Γ^{d(r)}, Γ^{dp^{r−j}(j)}) ⟶ Γ^{d}(V^{∼}_{j})
Ext_{P}^{∗}(Γ^{dp^{r−j}(j)}, Γ^{d(r)}) ⟶ Γ^{d}(U_{j})

are isomorphisms.

Proof. The proof is essentially the same in both cases. We consider only the first one (which is slightly more complicated). We proceed by induction on d. Consider the first hypercohomology spectral sequence corresponding to the dual Koszul complex

Kz_{dp^{r−j}}^{#(j)} : 0 ⟶ Γ^{dp^{r−j}(j)} ⟶ Γ^{dp^{r−j}−1(j)} ⊗ Λ^{1(j)} ⟶ ... ⟶ Λ^{dp^{r−j}(j)} ⟶ 0
E_{1}^{m,n} = Ext_{P}^{n}(Γ^{d(r)}, Γ^{dp^{r−j}−m(j)} ⊗ Λ^{m(j)}) ⟹ 0

The columns E_{1}^{m,∗} of this spectral sequence with m ≢ 0 mod p^{r−j} are trivial. Furthermore

E_{1}^{sp^{r−j},∗} = Ext_{P}^{∗}(Γ^{d(r)},Γ^{(d−s)p^{r−j}(j)} ⊗ Λ^{sp^{r−j}(j)}) =
= Ext_{P}^{∗}(Γ^{d−s(r)}, Γ^{(d−s)p^{r−j}(j)}) ⊗ Ext_{P}^{∗}(Γ^{s(r)}, Λ^{sp^{r−j}(j)})

Our induction hypothesis on d and Corollary 5.3 show further that comultiplication defines (for s > 0) natural isomorphisms E_{1}^{sp^{r−j},∗} ⟶^{∼} Γ^{d−s}(V^{∼}_{j})⊗Λ^{s}(W_{j}). The first (and the only) non-trivial differential in this spectral sequence is d_{p^{r−j}} : E_{p^{r−j}}^{sp^{r−j},∗} → E_{p^{r−j}}^{(s+1)p^{r−j},∗}. Considering the natural homomorphism of com-

plexes Kz_{dp^{r−j}}^{#(j)} → (Kz_{p^{r−j}}^{#(j)})^{⊗d} (defined by comultiplication in the divided power and exterior algebras) and using Proposition 4.1, we finally get a commutative diagram of complexes

0 ⟶ E_{p^{r−j}}^{0,∗} ⟶^{d_{p^{r−j}}}  E_{p^{r−j}}^{p^{r−j},∗}  ⟶^{d_{p^{r−j}}} ⋯ ⟶^{d_{p^{r−j}}} E_{p^{r−j}}^{dp^{r−j},∗} ⟶ 0
↓  =^{∼}↓  =^{∼}↓
0 ⟶ Γ^{d}(V^{∼}_{j}) ⟶ Γ^{d−1}(V^{∼}_{j}) ⊗ W_{j} ⟶ ⋯ ⟶ Λ^{d}(W_{j}) ⟶ 0.

Here the bottom row is the dual of the generalized Koszul complex for the isomorphism of graded vector spaces η : V^{∼}_{j} → W_{j}. The bottom row being acyclic, we conclude that E_{p^{r−j}+1}^{sp^{r−j},∗} = 0 for s > 1. This implies the vanishing of all higher differentials and hence gives the following relations : 0 = E_{∞}^{0,∗} = E_{p^{r−j}+1}^{0,∗}. Thus the top row in the above diagram is also acyclic and hence the left hand side vertical arrow is also an isomorphism.

The following corollary follows immediately from Theorem 5.4 and the duality isomorphism (1.12).

Corollary 5.5. For any 0 ≤ j ≤ r and any d ≥ 0, we have natural isomorphisms

Ext_{P}^{∗}(S^{dp^{r−j}(j)}, S^{d(r)}) ⟶^{∼} Γ^{d}(Ext_{P}^{∗}(S^{p^{r−j}(j)}, I^{(r)}))
Ext_{P}^{∗}(S^{d(r)}, S^{dp^{r−j}(j)}) ⟶^{∼} Γ^{d}(Ext_{P}^{∗}(I^{(r)}, S^{p^{r−j}(j)}))

To compute Ext^{∗}(Λ^{()}, Λ^{()}) we proceed in the same way as above. For any 0 ≤ j ≤ r and any d ≥ 0, the comultiplication in the exterior algebra defines natural homomorphisms

Ext_{P}^{∗}(Λ^{d(r)}, Λ^{dp^{r−j}(j)}) ⟶ Ext_{P}^{∗}(Λ^{d(r)}, Λ^{p^{r−j}(j)} ⊗ ... ⊗ Λ^{p^{r−j}(j)}) =
= Ext_{P}^{∗}(I^{(r)}, Λ^{p^{r−j}(j)})^{⊗d}
Ext_{P}^{∗}(Λ^{dp^{r−j}(j)}, Λ^{d(r)}) ⟶ Ext_{P}^{∗}(Λ^{dp^{r−j}(j)}, I^{(r)} ⊗ ... ⊗ I^{(r)}) =
= Ext_{P}^{∗}(Λ^{p^{r−j}(j)}, I^{(r)})^{⊗d}

By Lemma 1.11, these homomorphisms are Σ_{d}-invariant and hence their images are contained in Γ^{d}(Ext_{P}^{∗}(I^{(r)}, Λ^{p^{r−j}(j)})) and Γ^{d}(Ext_{P}^{∗}(Λ^{p^{r−j}(j)}, I^{(r)}) respectively.

We may now conclude the following theorem in a manner strictly parallel to that for Theorem 5.4. We leave details to the reader.

Theorem 5.6. For all 0 ≤ j ≤ r and all d ≥ 0 the natural homomorphisms

Ext_{P}^{∗}(Λ^{d(r)}, Λ^{dp^{r−j}(j)}) ⟶ Γ^{d}(Ext_{P}^{∗}(I^{(r)}, Λ^{p^{r−j}(j)}))
Ext_{P}^{∗}(Λ^{dp^{r−j}(j)}, Λ^{d(r)}) ⟶ Γ^{d}(Ext_{P}^{∗}(Λ^{p^{r−j}(j)}, I^{(r)}))

are isomorphisms.

We phrase our next result in terms of chosen vector-space-basis in our Extgroups.

For an integer j, let ϕ_{j} denote the natural embedding ϕ_{j} : I^{(j)} → S^{p^{j}} and let ϕ_{j}^{#} denote the dual homomorphism ϕ_{j}^{#} : Γ^{p^{j}} → I^{(j)}. We recall from [F-S, 4.5.6] that composition with ϕ_{r−j}^{(j)}, j ≤ r, induces an isomorphism

Ext_{P}^{s}(I^{(r)}, I^{(r)}) → Ext_{P}^{s}(I^{(r)}, S^{p^{r−j}(j)})

for s ≡ 0 mod 2p^{r−j}. Let kz_{j} denote the extension class in Ext_{P}^{p^{j}−1}(S^{p^{j}}, Λ^{p^{j}}) represented by the Koszul complex and let kz_{j}^{#} denote the

extension class in Ext_{P}^{p^{j}−1}(Λ^{p^{j}}, Γ^{p^{j}}) represented by the dual Koszul complex. Define e_{r} to be the class in Ext_{P}^{2p^{r−1}}(I^{(r)}, I^{(r)}) defined by the following equation in Ext_{P}^{2p^{r−1}}(I^{(r)}, S^{p^{r−1}(1)}) (cf. [F-S, §4]):

ϕ_{r−1}^{(1)} · e_{r} = d_{p^{r−1}+1}(kz_{r−1}^{(1)} · ϕ_{r−1}^{(1)}) ,

where d_{p^{r−1}+1} is the differential (applied to the E_{p^{r−1}}^{p^{r−1}−1,p^{r−1}}-term) of the second hypercohomology spectral sequence associated to applying RHom(I^{(r)}, −) to the De Rham complex Ω_{p^{r}}^{•}.

Recall that the graded vector space Ext_{P}^{∗}(I^{(r)}, I^{(r)}) is one dimensional in even degrees < 2p^{r} and is zero otherwise. We use the basis of Ext_{P}^{2m}(I^{(r)}, I^{(r)}) given by the Yoneda products

e_{r}(m) = e_{1}^{m_{0}(r−1)} ⋯ e_{r}^{m_{r−1}} m = m_{0} + m_{1}p + ⋯ + m_{r−1}p^{r−1} (0 ≤ m_{i} < p).

Thus, e_{r}(m)^{(1)} = e_{r+1}(m) for all r and 0 ≤ m < p^{r}. It is shown in [F-S, §4] that the elements ϕ_{r−j}^{(j)}e_{r}(mp^{r−j}), 0 ≤ m < p^{j} form a basis of the vector space V_{j} = Ext_{P}^{∗}(I^{(r)}, S^{p^{r−j}(j)}). By duality, dual elements e_{r}(mp^{r−j})ϕ_{r−j}^{#(j)} form a basis of U_{j} = Ext_{P}^{∗}(Γ^{p^{r−j}(j)}, I^{(r)}).

In the following theorem, we summarize the computations of Theorem 4.5 and the preceding results of this section, augmenting these computations by explicitly describing the Hopf algebra structure (cf. Lemma 1.11).

Theorem 5.8. Let j and r be integers, 0 ≤ j ≤ r.

(1) The tri-graded Hopf algebra

Ext_{P}^{∗}(Γ^{∗(j)}, S^{∗(r)})

is a primitively generated polynomial algebra on generators e_{r}(mp^{r−j})ϕ_{r−j}^{#(j)}

in Ext_{P}^{2p^{r−j}m}(Γ^{p^{r−j}(j)}, I^{(r)}), 0 ≤ m < p^{j}, of tri-degree (2p^{r−j}m, p^{r−j}, 1). Similarly, the tri-graded Hopf algebra

Ext_{P}^{∗}(Γ^{∗(r)}, S^{∗(j)})

is a primitively generated polynomial algebra on generators ϕ_{r−j}^{(j)}e_{r}(mp^{r−j})

in Ext_{P}^{2p^{r−j}m}(I^{(r)}, S^{p^{r−j}(j)}), 0 ≤ m < p^{j}, of tri-degree (2p^{r−j}m, 1, p^{r−j}).

(2) The tri-graded Hopf algebra

Ext_{P}^{∗}(Γ^{∗(j)}, Λ^{∗(r)})

is a primitively generated exterior algebra on generators e_{r}(mp^{r−j})ϕ_{r−j}^{#(j)} in

Ext_{P}^{2p^{r−j}m}(Γ^{p^{r−j}(j)}, I^{(r)}), 0 ≤ m < p^{j}, of tri-degree (2p^{r−j}m, p^{r−j}, 1). A similar statement holds by duality for the tri-graded Hopf algebra

Ext_{P}^{∗}(Λ^{∗(r)}, S^{∗(j)}).

(3) The tri-graded Hopf algebra

Ext_{P}^{∗}(Γ^{∗(r)}, Λ^{∗(j)})

is a primitively generated exterior algebra on generators kz_{r−j}^{(j)}ϕ_{r−j}^{(j)}e_{r}(mp^{r−j}) in Ext_{P}^{2mp^{r−j}+p^{r−j}−1}(I^{(r)}, Λ^{p^{r−j}(j)}), 0 ≤ m < p^{j}, of tri-degree (2mp^{r−j} + p^{r−j} − 1, 1, p^{r−j}).

A similar statement holds by duality for the tri-graded Hopf algebra

Ext_{P}^{∗}(Λ^{∗(j)}, S^{∗(r)}).

(4) The tri-graded Hopf algebra

Ext_{P}^{∗}(Γ^{∗(r)}, Γ^{∗(j)})

is a primitively generated divided power algebra on generators (kz_{r−j}^{#}kz_{r−j})^{(j)}ϕ_{r−j}^{(j)}e_{r}(mp^{r−j}) in Ext^{2mp^{r−j}+2p^{r−j}−2}(I^{(r)}, Γ^{p^{r−j}(j)}), 0 ≤ m < p^{j}, of tri-degree (2mp^{r−j} + 2p^{r−j} − 2, 1, p^{r−j}).

A similar statement holds by duality for the tri-graded Hopf algebra

Ext_{P}^{∗}(S^{∗(j)}, S^{∗(r)}).

(5) The tri-graded Hopf algebra

Ext_{P}^{∗}(Γ^{∗(j)}, Γ^{∗(r)})

is a primitively generated divided power algebra on generators e_{r}(mp^{r−j})ϕ_{r−j}^{#(j)} in Ext_{P}^{2mp^{r−j}}(Γ^{p^{r−j}(j)}, I^{(r)}), 0 ≤ m < p^{j}, of tri-degree (2mp^{r−j}, p^{r−j}, 1).

A similar statement holds by duality for the tri-graded Hopf algebra

Ext_{P}^{∗}(S^{∗(r)}, S^{∗(j)}).

(6) The tri-graded Hopf algebra

Ext_{P}^{∗}(Λ^{∗(r)}, Λ^{∗(j)})

is a primitively generated divided power algebra on generators kz_{r−j}^{(j)}ϕ_{r−j}^{(j)}e_{r}(mp^{r−j}) in Ext_{P}^{2mp^{r−j}+p^{r−j}−1}(I^{(r)}, Λ^{p^{r−j}(j)}), 0 ≤ m < p^{j}, of tri-degree (2mp^{r−j} + p^{r−j} − 1, 1, p^{r−j}).

A similar statement holds by duality for the tri-graded Hopf algebra

Ext_{P}^{∗}(Λ^{∗(j)}, Λ^{∗(r)}).

Proof. The construction of the map in Theorem 4.5 implies that the isomorphism

S^{∗}(V_{j}) ⟶^{∼} Ext_{P}(Γ^{∗(r)}, S^{∗p^{r−j}(j)})  (5.8.1)

is multiplicative. This identifies Ext_{P}^{∗}(Γ^{∗(r)}, S^{∗(j)}) as a tri-graded algebra. Finally the elements of V_{j} are primitive by dimension considerations. This proves (1).

The proofs of the remaining assertions follow in a similar, straightforward manner from the computations given earlier in this section.

We derive from Theorem 5.8 the following Corollary which gives a partial answer to a question raised in [F-S, 5.8 ].

Corollary 5.9. The image of e_{r} ∈ Ext_{P}^{2p^{r−1}}(I^{(r)}, I^{(r)}) in Ext_{P}^{2p^{r−1}}(Γ^{p^{r−1}(1)}, S^{p^{r−1}(1)}) equals e_{1}^{p^{r−1}} (up to a non-zero scalar factor). Moreover if one modifies the definition of e_{r} the way it was done in [S-F-B], then the image of e_{r} would be exactly e_{1}^{p^{r−1}}.

Proof. We start by computing the image of e_{r} in Ext_{P}^{2p^{r−1}}(Γ^{p(r−1)}, S^{p(r−1)}). To do so note that the homomorphism Ext_{P}^{2p^{r−1}}(I^{(r)}, I^{(r)}) → Ext_{P}^{2p^{r−1}}(Γ^{p(r−1)}, S^{p(r−1)}) is a homogeneous component of a homomorphism of tri-graded Hopf algebras

ξ : Ext_{P}^{∗}(Γ^{∼}^{∗(r)}, S^{∼}^{∗(r)}) → Ext_{P}^{∗}(Γ^{∗(r−1)}, S^{∗(r−1)})

induced by natural homomorphisms of exponential functors Γ^{∗(r−1)} → Γ^{∼}^{∗(r)}, S^{∼}^{∗(r)} → S^{∗(r−1)}. Here ^{∼} denotes the p-rarefication of an exponential functor, i.e.

A^{∼}^{n} = \{ \begin{matrix}A^{m} & if n = pm \\ 0 & if n ≢ 0 mod p \end{matrix}

Since the element e_{r} ∈ Ext_{P}^{2p^{r−1}}(I^{(r)}, I^{(r)}) is primitive in Ext_{P}^{∗}(Γ^{∼}^{∗(r)}, S^{∼}^{∗(r)}) its image in Ext_{P}^{∗}(Γ^{∗(r−1)}, S^{∗(r−1)}) = S^{∗}(Ext_{P}^{∗}(I^{(r−1)}, I^{(r−1)})) also has to be primitive. The subspace of primitive elements in a (primitively generated) symmetric algebra S^{∗}(V) coincides with a linear span < V, V^{p}, V^{p^{2}}, ... >. From this we derive immediately that the only primitive element of tri-degree (2p^{r−1}, p, p) in S^{∗}(Ext_{P}^{∗}(I^{(r−1)}, I^{(r−1)}) is e_{r−1}^{p}. Thus ξ(e_{r}) is a scalar multiple of e_{r−1}^{p}. Since ξ is multiplicative we conclude further that ξ(e_{r}^{n}) is a scalar multiple of e_{r−1}^{pn} for each n. Repeating this construction r − 1 times we finally conclude that the image of e_{r} in Ext_{P}^{2p^{r−1}}(Γ^{p^{r−1}(1)}, S^{p^{r−1}(1)}) is a scalar multiple of e_{1}^{p^{r−1}}.

We proceed to show now that with the modification made in [S-F-B, 3.4] the image of e_{r} in Ext_{P}^{2p^{r−1}}(Γ^{p(r−1)}, S^{p(r−1)}) is exactly e_{r−1}^{p} (i.e. the corresponding scalar factor is identity). Recall that (with the modification made in [S-F-B]) for any n ≥ 2 the image of e_{r} under the natural homomorphism

Ext_{P}^{2p^{r−1}}(I^{(r)}, I^{(r)}) ⟶ Ext_{GL_{n}}^{2p^{r−1}}(k^{n(r)}, k^{n(r)}) → Ext_{GL_{n(1)}}^{2p^{r−1}}(k^{n(r)}, k^{n(r)}) =
= H^{2pr−1}(GL_{n(1)}, gl_{n}^{(r)}) = Hom_{k}(gl_{n}^{(r)#}, H^{2p^{r−1}}(GL_{n(1)}, k))

is non-trivial [F-S, §6] and coincides with the composition

gl_{n}^{(r)#} ⟶ S^{p^{r−1}}(gl_{n}^{(1)#}) ⟶ H^{2p^{r−1}}(GL_{n(1)}, k)

where the first arrow is the standard embedding and the second one is the edge homomorphism in the May spectral sequence. Consider further the natural homomorphism

Ext_{P}^{2p^{r−1}}(Γ^{p(r−1)}, S^{p(r−1)}) → H^{2p^{r−1}}(GL_{n(1)}, (Γ^{p(r−1)}(k^{n}))^{#} ⊗ S^{p(r−1)}(k^{n})) =

= Hom_{k}(Γ^{p(r−1)}(k^{n}) ⊗ Γ^{p(r−1)}(k^{n#}), H^{2p^{r−1}}(GL_{n(1)}, k))

The image of ξ(e_{r}) under this map coincides with the composite

Γ^{p(r−1)}(k^{n}) ⊗ Γ^{p(r−1)}(k^{n#}) ↠ k^{n(r)} ⊗ k^{n(r)#} =gl_{n}^{(r)#} → S^{p^{r−1}}(gl_{n}^{(1)#}) →
→ H^{2p^{r−1}}(GL_{n(1)}, k)

and, in particular, is non-trivial. Keeping in mind that the tensor product operation

Ext_{GL_{n(1)}}^{∗}(k, k) ⊗ Ext_{GL_{n(1)}}^{∗}(k, k) → Ext_{GL_{n(1)}}^{∗}(k, k)

coincides with the usual product operation in the cohomology ring Ext_{GL_{n(1)}}^{∗}(k, k) = H^{∗}(GL_{n(1)}, k), using the description of the product operation in Ext_{P}^{∗}(Γ^{∗(r−1)}, S^{∗(r−1)}) in terms of the tensor product operation and the multiplicative properties of the May homomorphism S^{∗}(gl_{n}^{(1)}) → H^{2∗}(GL_{n(1)}, k) we easily conclude that the image of e_{r−1}^{p} ∈ Ext_{P}^{2p^{r−1}}(Γ^{p(r−1)}, S^{p(r−1)}) in

Hom_{k}(Γ^{p(r−1)}(k^{n}) ⊗ Γ^{p(r−1)}(k^{n#}), H^{2p^{r−1}}(GL_{n(1)}, k)

is given by the composition

Γ^{p(r−1)}(k^{n}) ⊗ Γ^{p(r−1)}(k^{n#}) ↪ (k^{n(r−1)})^{⊗p} ⊗ (k^{n(r−1)#})^{⊗p} =
= (gl_{n}^{(r−1)})^{⊗p} → (S^{p^{r−2}}(gl_{n}^{(1)}))^{⊗p} ↠ S^{p^{r−1}}(gl_{n}^{(1)}) → H^{2p^{r−1}}(GL_{n(1)}, k)

A straight forward verification shows that the resulting maps Γ^{p(r−1)}(k^{n}) ⊗ Γ^{p(r−1)}(k^{n#}) → H^{2p^{r−1}}(GL_{n(1)}, k) coincide. Since this map is moreover non-trivial we conclude that the constant relating ξ(e_{r}) and e_{r−1}^{p} is equal to identity.

To simplify the notations we adopt the following convention: for homogeneous strict polynomial functors A, B in P we define

Ext_{P→F}^{∗}(A, B) = \underrightarrow{lim}_{m} Ext_{P}^{∗}(A^{(m)}, B^{(m)})

Letting r vary over positive integers and j vary between 0 and r, and setting h = r − j, we have the following consequence of Theorem 5.8.

Corollary 5.10. Let h be a non-negative integer.

(1) The tri-graded Hopf algebra

Ext_{P→F}^{∗}(Γ^{∗}, S^{∗(h)})

is a primitively generated polynomial algebra on generators e(mp^{h})ϕ_{h}^{#} in Ext_{P→F}^{2p^{h}m}(Γ^{p^{h}}, I^{(h)}), 0 ≤ m, of tri-degree (2p^{h}m, p^{h}, 1).

Similarly, the tri-graded Hopf algebra

Ext_{P→F}^{∗}(Γ^{∗(h)}, S^{∗})

is a primitively generated polynomial algebra on generators ϕ_{h}e(mp^{h}) in Ext_{P→F}^{2p^{h}m}(I^{(h)}, S^{p^{h}}), 0 ≤ m, of tri-degree (2p^{h}m, 1, p^{h}).

(2) The tri-graded Hopf algebra

Ext_{P→F}^{∗}(Γ^{∗}, Λ^{∗(h)})

is a primitively generated exterior algebra on generators e(mp^{h})ϕ_{h}^{#} in Ext_{P→F}^{2p^{h}m}(Γ^{p^{h}}, I^{(h)}), 0 ≤ m, of tri-degree (2p^{h}m, p^{h}, 1).

A similar statement holds by duality for the tri-graded Hopf algebra

Ext_{P→F}^{∗}(Λ^{∗(h)}, S^{∗}).

(3) The tri-graded Hopf algebra

Ext_{P→F}^{∗}(Γ^{∗(h)}, Λ^{∗})

is a primitively generated exterior algebra on generators kz_{h}ϕ_{h}e(mp^{h}) in Ext_{P→F}^{2mp^{h}+p^{h}−1}(I^{(h)}, Λ^{p^{h}}), 0 ≤ m, of tri-degree (2mp^{h} + p^{h} − 1, 1, p^{h}).

A similar statement holds by duality for the tri-graded Hopf algebra

Ext_{P→F}^{∗}(Λ^{∗}, S^{∗(h)}).

(4) The tri-graded Hopf algebra

Ext_{P→F}^{∗}(Γ^{∗(h)}, Γ^{∗})

is a primitively generated divided power algebra on generators kz_{h}^{#}kz_{h}ϕ_{h}e_{r}(mp^{h}) in Ext_{P→F}^{2mp^{h}+2p^{h}−2}(I^{(h)}, Γ^{p^{h}}), 0 ≤ m, of tri-degree (2mp^{h} + 2p^{h} − 2, 1, p^{h}).

A similar statement holds by duality for the tri-graded Hopf algebra

Ext_{P→F}^{∗}(S^{∗}, S^{∗(h)}).

(5) The tri-graded Hopf algebra

Ext_{P→F}^{∗}(Γ^{∗}, Γ^{∗(h)})

is a primitively generated divided power algebra on generators e(mp^{h})ϕ_{h}^{#} in Ext_{P→F}^{2mp^{h}}(Γ^{p^{h}}, I^{(h)}), 0 ≤ m, of tri-degree (2mp^{h}, p^{h}, 1).

A similar statement holds by duality for the tri-graded Hopf algebra

Ext_{P→F}^{∗}(S^{∗(h)}, S^{∗}).

(6) The tri-graded Hopf algebra

Ext_{P→F}^{∗}(Λ^{∗(h)}, Λ^{∗})

is a primitively generated divided power algebra on generators kz_{h}ϕ_{h}e(mp^{h}) in Ext_{P→F}^{2mp^{h}+p^{h}−1}(I^{(h)}, Λ^{p^{h}}), 0 ≤ m, of tri-degree (2mp^{h} + p^{h} − 1, 1, p^{h}). A similar statement holds by duality for the tri-graded Hopf algebra

Ext_{P→F}^{∗}(Λ^{∗}, Λ^{∗(h)}).

§6. Ext-groups between the classical functors in the category F.

As we see in this section, the Extension of Scalars formula (Theorem 3.4) and the Comparison Theorem (Theorem 3.10) allow us in many cases to reduce the computation of Ext-groups in the category F to the computation of appropriate Ext-groups in the category P.

Throughout this section k is a finite field with q = p^{N} elements, F = F(k) is the category of functors V_{k}^{f} → V_{k} and P = P(k) is the category of strict polynomial functors. Finally, A^{∗} and B^{∗} are exponential (homogeneous) strict polynomial functors with deg(A^{i}) = deg(B^{i}) = i.

For non negative integers h < N, and j, l, n, let P r_{n}^{h}(j, l) be the set of all sequences j_{0}, j_{1}, …, j_{n−1}, l_{1}, …, l_{n} of non negative integers with

j = j_{0} + ⋯ + j_{n−1} + l_{1}p^{N−h} + ⋯ + l_{n}p^{nN−h}
l = j_{0}p^{h} + ⋯ + j_{n−1}p^{(n−1)N+h} + l_{1} + ⋯ + l_{n}

The obvious embedding P r_{n}^{h}(j, l) ↪ P r_{n+1}^{h}(j, l) (taking j_{n} = l_{n+1} = 0) is an isomorphism, provided that p^{(n+1)N−h} > j and p^{nN+h} > l. We denote by P r^{h}(j, l) the limit set \underrightarrow{lim}_{n} P r_{n}^{h}(j, l).

For any sequence j_{0}, ..., j_{n−1}, l_{1}, ..., l_{n} in P r_{n}^{h}(j, l) consider the following homomorphism

⨂_{s=0}^{n−1} Ext_{P→F}^{∗}(A^{j_{s}(sN+h)}, B^{j_{s}p^{sN+h}}) ⊗ ⨂_{t=1}^{n} Ext_{P→F}^{∗}(A^{l_{t}p^{tN−h}(h)}, B^{l_{t}(tN)}) →

→ ⨂_{s=0}^{n−1} Ext_{F}^{∗}(A^{j_{s}(sN+h)}, B^{j_{s}p^{sN+h}}) ⊗ ⨂_{t=1}^{n} Ext_{F}^{∗}(A^{l_{t}p^{tN−h}(h)}, B^{l_{t}(tN)}) =

= ⨂_{s=0}^{n−1} Ext_{F}^{∗}(A^{j_{s}(h)}, B^{j_{s}p^{sN+h}}) ⊗ ⨂_{t=1}^{n} Ext_{F}^{∗}(A^{l_{t}p^{tN−h}(h)}, B^{l_{t}}) → Ext_{F}^{∗}(A^{j(h)}, B^{l})

Our first result determines Ext_{F}^{∗}(A^{∗}, B^{∗}) in terms of Ext_{P→F}^{∗}(A^{∗}, B^{∗}).

Theorem 6.1. Fix a non-negative integer h less than N. The resulting homomorphism

⨁_{P r^{h}(j,l)} ⨂_{s≥0} Ext_{P→F}^{∗}(A^{j_{s}(sN+h)}, B^{j_{s}p^{sN+h}}) ⊗ ⨂_{t≥1} Ext_{P→F}^{∗}(A^{l_{t}p^{tN−h}(h)}, B^{l_{t}(tN)}) →
→ Ext_{F}^{∗}(A^{j(h)}, B^{l})

is an isomorphism.

Proof. Choose an integer n so that p^{nN−h} > j, p^{nN} > l and consider a tower of finite field extensions L ⊃ K ⊃ k in which each stage is of degree n. Theorem 3.4 gives us a natural isomorphism

K ⊗_{k} Ext_{F(k)}^{∗}(A^{j(h)}, B^{l}) = Ext_{F(K)}^{∗}(A_{K}^{j(h)} ∘ (t ∘ τ), B_{K}^{l}) .

Moreover, according to Lemma 3.6, the functor t∘τ may be identified with I⊕I^{(N)}⊕ ... ⊕ I^{((n−1)N)}. The exponential property of A^{∗} gives now a natural isomorphism

A_{K}^{j(h)} ∘ (t ∘ τ) = ⨁_{j_{0}+...+j_{n−1}=j} A_{K}^{j_{0}(h)} ⊗ ⋯ ⊗ A_{K}^{j_{n−1}((n−1)N+h)} .

Extending scalars further from K to L and applying now the previous procedure to the second variable, we get the following formula:

L ⊗_{k} Ext_{F(k)}^{∗}(A^{j(h)}, B^{l}) = _{⨁_{j_{0}+...+j_{n−1}=j} ; l_{0}+...+l_{n−1}=l}Ext_{F(L)}^{∗}(⨂_{s=0}^{n−1} A_{L}^{j_{s}(sN+h)}, ⨂_{t=0}^{n−1} B_{L}^{l_{t}(tnN)}) .

In view of the exponential properties of A^{∗} and B^{∗}, each summand in the above formula may be further identified (cf. Corollary 1.8) with

⨁_{j_{0,0}+...+j_{0,n−1}=j_{0} l_{0,0}+...+l_{0,n−1}=l_{0}_{… …}} ; j_{n−1,0}+...+j_{n−1,n−1}=j_{n−1} l_{n−1,0}+...+l_{n−1,n−1}=l_{n−1} ⨂_{s,t=0}^{n−1} Ext_{F(L)}^{∗}(A_{L}^{j_{s,t}(sN+h)}, B_{L}^{l_{t,s}(tnN)})

Note further that deg(A_{L}^{j_{s,t}(sN+h)}) = j_{s,t} · p^{sN+h} ≤ j · p^{(n−1)N+h} = \frac{j·card(L)}{p^{(n^{2}−n+1)N−h}} <

card(L) and also deg(B_{L}^{l_{t,s}(tnN)}) = l_{t,s}p^{tnN} ≤ l · p^{(n−1)nN} = \frac{l·card(L)}{p^{nN}} < card(L). Theorem 3.10 and Lemma 3.11 now show that we may replace Ext_{F}^{∗} by Ext_{P→F}^{∗} everywhere in the above formula. This implies, in particular, that the summand corresponding to {j_{t,s}, l_{t,s}}_{t,s} is trivial unless the corresponding degrees coincide, i.e. unless we have the following relations: j_{s,t}p^{sN+h} = l_{t,s}p^{tnN} for all s, t. It’s easy to see from the choice of n that the above relation implies that j_{s,t} = l_{t,s} = 0 unless t = 0 or t = 1. The remaining j_{t,s} and l_{s,t} are related by equations

l_{0,s} = j_{s,0}p^{sN+h} , j_{s,1} = l_{1,s}p^{(n−s)N−h}

Since ∑_{s,t} j_{s,t} = j, ∑_{s,t} l_{s,t} = l, we conclude that (j^{0}, ..., j^{n−1}, l^{1}, ..., l^{n}) = (j_{0,0}, ..., j_{n−1,0}, l_{1,n−1}, ..., l_{1,0}) is an element of P r_{n}^{h}(j, l), i.e. we are left with the direct sum over P r_{n}^{h}(j, l):

L ⊗_{k} Ext_{F(k)}^{∗}(A^{j(h)}, B^{l}) = ⨁_{P r_{n}^{h}(j,l)} L ⊗_{k} ⨂_{s=0}^{n−1} Ext_{P→F}^{∗}(A^{j^{s}(sN+h)}, B^{k^{s}p^{sN+h}})⊗

⊗ ⨂_{t=1}^{n} Ext_{P→F}^{∗}(A^{l^{t}p^{tN−h}(h)}, B^{l^{t}(tN)}).

Finally one checks without difficulty that the composite of the above isomorphism with the extension of scalars in the homomorphism we started with is the identity map.

It is pleasing to reformulate the result of Theorem 6.1 in terms of the corresponding Hopf algebras. Observe that for any s ≥ 0, t ≥ 1 we have natural homomorphisms of tri-graded vector spaces

Ext_{P→F}^{∗}(A^{∗(sN+h)}, B^{∗}) → Ext_{F}^{∗}(A^{∗(sN+h)}, B^{∗}) = Ext_{F}^{∗}(A^{∗(h)}, B^{∗})

Ext_{P→F}^{∗}(A^{∗}, B^{∗(tN−h)}) = Ext_{P→F}^{∗}(A^{∗(h)}, B^{∗(tN)}) → Ext_{F}^{∗}(A^{∗(h)}, B^{∗(tN)}) =

= Ext_{F}^{∗}(A^{∗(h)}, B^{∗})

Furthermore, it is easy to see that the tri-graded space Ext_{P→F}^{∗}(A^{∗(sN+h)}, B^{∗}) is p^{sN+h}-connected with respect to the total degree, whereas Ext_{P→F}^{∗}(A^{∗}, B^{∗(tN−h)}) is p^{tN−h}-connected with respect to the total degree, which allows us to consider the infinite tensor product

⨂_{s≥0} Ext_{P→F}^{∗}(A^{∗(sN+h)}, B^{∗}) ⊗ ⨂_{t≥1} Ext_{P→F}^{∗}(A^{∗}, B^{∗(tN−h)}) .

Suppose now that A^{∗} and B^{∗} are Hopf functors. The product operations in Ext_{F}^{∗}(A^{∗(h)}, B^{∗}) define a natural homomorphism from the above infinite tensor product to Ext_{F}^{∗}(A^{∗(h)}, B^{∗}). Moreover, assuming that A^{∗} (resp. B^{∗}) is ϵ(A)- commutative (resp. ϵ(B)-commutative), the above map is even a homomorphism of Hopf algebras, provided that one endows the tensor product algebra with multiplication and comultiplication taking into account the sign convention given by Lemma 1.11.

Corollary 6.2. Let A^{∗} and B^{∗} be commutative Hopf functors. The natural homomorphism

⨂_{s≥0} Ext_{P→F}^{∗}(A^{∗(sN+h)}, B^{∗}) ⊗ ⨂_{t≥1} Ext_{P→F}^{∗}(A^{∗}, B^{∗(tN−h)}) → Ext_{F}^{∗}(A^{∗(h)}, B^{∗})

is an isomorphism of tri-graded Hopf algebras.

Combining Corollary 6.2 with Corollary 5.9 we get the computation of various important Ext-algebras in the category F.

Theorem 6.3. Let h be a non-negative integer.

(1) The tri-graded Hopf algebra

Ext_{F}^{∗}(Γ^{∗(h)}, S^{∗})

is a primitively generated polynomial algebra on generators ϕ_{h+sN}e(mp^{h+sN}) in Ext_{F}^{2p^{h+sN} m}(I^{(h)}, S^{p^{h+sN}}), 0 ≤ m, 0 ≤ s, of tri-degree

(2p^{h+sN}m, 1, p^{h+sN})  and  generators  e(mp^{tN−h})ϕ_{tN−h}^{#}  in

Ext_{F}^{2p^{tN−h}m}(Γ^{p^{tN−h}(h)}, I), 0 ≤ m, 1 ≤ t, of tri-degree (2p^{tN−h}m, p^{tN−h}, 1).

(2) The tri-graded Hopf algebra

Ext_{F}^{∗}(Γ^{∗(h)}, Λ^{∗})

is a primitively generated exterior algebra on generators kz_{h+sN}ϕ_{h+sN}e(mp^{h+sN}) in Ext_{F}^{2mp^{h+sN} +p^{h+sN} −1}(I^{(h)}, Λ^{p^{h+sr}}), 0 ≤ m, 0 ≤ s, of tri-degree (2mp^{h+sN} + p^{h+sN} − 1, 1, p^{h+sN}) and generators e(mp^{tN−h})ϕ_{tN−h}^{#(h)} in Ext_{F}^{2p^{tN−h}m}(Γ^{p^{tN−h}(h)}, I), 0 ≤ m, 0 ≤ s, of tri-degree (2p^{tN−h}m, p^{tN−h}, 1).

(3) The tri-graded Hopf algebra

Ext_{F}^{∗}(Γ^{∗(h)}, Γ^{∗})

is a primitively generated divided power algebra on generators kz_{h+sN}^{#}kz_{h+sN}ϕ_{h+sN}e(mp^{h+sN}) in Ext_{F}^{2mp^{sN+h}+2p^{sN+h}−2}(I^{(h)}, Γ^{p^{sN+h}}), 0 ≤ m, 0 ≤ s, of tri-degree (2mp^{h+sN} +2p^{h+sN} −2, 1, p^{h+sN}) and generators e(mp^{tN−h})ϕ_{tN−h}^{#(h)} in Ext_{F}^{2mp^{tN−h}}(Γ^{p^{tN−h}(h)}, I), 0 ≤ m, 1 ≤ t, of tri-degree (2mp^{tN−h}, p^{tN−h}, 1).

A similar statement holds by duality for the tri-graded Hopf algebra

Ext_{F}^{∗}(S^{∗(N−h)}, S^{∗}).

(4) The tri-graded Hopf algebra

Ext_{F}^{∗}(Λ^{∗(h)}, Λ^{∗})

is a primitively generated divided power algebra on generators kz_{h+sN}ϕ_{h+sN}e(mp^{sN+h}) in Ext_{F}^{2mp^{sN+h}+p^{sN+h}−1}(I^{(h)}, Λ^{p^{sN+h}}), 0 ≤ m, 0 ≤ s, of tri-degree (2mp^{sN+h} + p^{sN+h} − 1, 1, p^{sN+h}) and generators e(mp^{tN−h})ϕ_{tN−h}^{#(h)}kz_{tN−h}^{#(h)} in Ext_{F}^{2mp^{tN−h}+p^{tN−h}−1}(Λ^{p^{tN−h}(h)}, I), 0 ≤ m, 1 ≤ t, of tri-degree (2mp^{tN−h} + p^{tN−h} − 1, p^{tN−h}, 1)

Appendix: GL-cohomology and Ext_{F}-groups. A. Suslin

The main subject discussed in this appendix is the map on Ext-groups

Ext_{F}^{∗}(P, Q) ⟶ Ext_{k[GL_{n}(k)]}^{∗}(P (k^{n}), Q(k^{n})) = H^{∗}(GL_{n}(k), P (k^{n})^{#} ⊗ Q(k^{n}))

induced by the exact functor F(k) → {k[GL_{n}(k)] − modules} : P ↦ P (k^{n}) (here k[GL_{n}(k)] stands for the group ring of the discrete group GL_{n}(k) over the field k). All GL_{n}(k)-modules considered below are always assumed to be k-vector spaces on which the group GL_{n}(k) acts by k-linear transformations (i.e. k[GL_{n}(k)]- modules), we abbreviate the notation Ext_{k[GL_{n}(k)]}^{∗} to Ext_{GL_{n}(k)}^{∗}. Note that the Ext_{GL_{n}(k)}^{∗}(P (k^{n}), Q(k^{n}))-groups above stabilize with the growth of n, according to a well-known Theorem of W. Dwyer [D], provided that the functors P and Q are finite. We use the notation Ext_{GL(k)}^{∗}(P, Q) for the stable values of the corresponding Ext-groups. The main result proved below is the following Theorem Theorem A.1. Let k be a finite field. Then the natural map

Ext_{F}^{∗}(P, Q) ⟶ Ext_{GL(k)}^{∗}(P, Q)

is an isomorphism for any finite functors P, Q ∈ F = F(k).

Over more general fields one has to compare Ext_{F}^{∗}-groups with the so called stable K-theory (see [B-P 1]). The above theorem shows that the conjecture stated in [B-P 1] (saying that for finite functors stable K-theory equals topological Hochschild homology) holds at least over finite fields. Apparently an appropriate modification of the argument given below should work over any field of positive characteristic. I did not try to work out this more general case, the more so that Ext_{F}-groups don’t seem to be computable unless we are dealing with finite fields.

Theorem A.1 was known to be true (in a more general situation) in case P = Q = I by a work of Dundy and McCarthy [D-M]. For finite fields a purely algebraic proof of this result was given in [F-S]. In a recent preprint [B-P 2] Betley and Pirashvili sketch a different proof of theorem A.1, following the approach developed in [F-S]. The treatment of transfer maps in [B-P 2] does not seem to be correct and before it’s fixed it’s difficult to say whether this approach works or not. Throughout this appendix k = 𝔽_{q} is a finite field of characteristic p (with q elements).

Lemma A.2. For any finite functors A, B ∈ F there is a natural duality isomorphism Ext_{GL(k)}^{∗}(A, B) = Ext_{GL(k)}^{∗}(B^{#}, A^{#}). which makes the following diagram commute

Ext_{F}^{∗}(A, B) ⟶^{∼} Ext_{F}^{∗}(B^{#}, A^{#}))
↓  ↓
Ext_{GL(k)}^{∗}(A, B) ⟶^{∼} Ext_{GL(k)}^{∗}(B^{#}, A^{#})

Proof. For any s ≥ 0 we can find an integer N(s) such that for all n ≥ N(s) we have canonical identifications

Ext_{GL(k)}^{s}(A, B) = Ext_{GL_{n}(k)}^{s}(A(k^{n}),B(k^{n}))
Ext_{GL(k)}^{s}(B^{#}, A^{#}) = Ext_{GL_{n}(k)}^{s}(B^{#}(k^{n}), A^{#}(k^{n})) =
= Ext_{GL_{n}(k)}^{s}(B(k^{n#})^{#}, A(k^{n#})^{#}).

Furthermore for any finite dimensional GL_{n}(k)-modules M, N the exact contravariant functor M ↦ M^{#} defines natural duality isomorphisms (cf. Lemma 1.12)

Ext_{GL_{n}(k)}^{∗}(M, N) ⟶^{∼} Ext_{GL_{n}(k)}^{∗}(N^{#}, M^{#}).

Taking here M = A(k^{n}), N = B(k^{n}) we identify Ext_{GL_{n}(k)}^{s}(A(k^{n}), B(k^{n})) with Ext_{GL_{n}(k)}^{s}(B(k^{n})^{#}, A(k^{n})^{#}) = Ext_{GL_{n}(k)}^{s}(B^{#}(k^{n#}), A^{#}(k^{n#})). Finally for any

GL_{n}(k)-module M consider a new GL_{n}(k)-module M^{∼}, which coincides with M as a k-vector space and the action of GL_{n}(k) on which is obtained from the original one by means of the automorphism

GL_{n}(k) ⟶^{∼} GL_{n}(k) : α ↦ (α^{T})^{−1}.

The functor M ↦ M^{∼} is obviously an equivalence of the category of GL_{n}(k)-modules with itself and hence defines natural isomorphisms

Ext_{GL_{n}(k)}^{∗}(M, N) ⟶^{∼} Ext_{GL_{n}(k)}^{∗}(M^{∼}, N^{∼}).

To finish the proof it suffices to note now that (for any functor A ∈ F and any n)

the GL_{n}(k)-modules A(k^{n})^{∼} and A(k^{n#}) are canonically isomorphic.

Lemma A.3. Let V ∈ V^{f} be a finite dimensional vector space over k. Assume that n ≥ dim V. Then for any GL_{n}(k)-module M we have isomorphisms:

Ext_{GL_{n}(k)}^{∗}(P_{V} (k^{n}), M) = ⨁_{W ⊂V} H^{∗}(( \begin{matrix}1_{n−dimW} & ∗ \\ 0 & GL_{dimW} (k) \end{matrix} ) , M).

Here P_{V} is the projective generator of F, discussed in §1 and W ⊂ V on the right runs through all subspaces of the vector space V.

Proof. P_{V} (k^{n}) = k[Hom(V, k^{n})] is a permutational GL_{n}(k)-module. Two homomorphisms f, g : V → k^{n} are in the same GL_{n}(k)-orbit if and only if Ker f = Ker g. Thus orbits of the action of GL_{n}(k) on Hom(V, k^{n}) are in one to one correspondance with subspaces W ⊂ V. For any W ⊂ V choose a representative f_{W} of the corresponding orbit so that Im f_{W} = k^{n−dim W} - the standard coordinate subspace in k^{n}. It’s clear that the stabilizer of f_{W} coincides with the affine group

( \begin{matrix}1_{n−dimW} & ∗ \\ 0 & GL_{dimW} (k) \end{matrix} )

Thus our statement follows from Shapiro’s Lemma.

Proposition A.4. Let B ∈ F be a finite functor. For any s ≥ 0 there exists N(s) such that for n ≥ N(s)

H^{i}(GL_{n}(k), B(k^{n})) = \{ \begin{matrix}B(0) & for i = 0 \\ 0 & for 0 < i ≤ s \end{matrix}

Proof. This follows immediately from the theorem of Betley [B], stating that over any field k cohomology of GL_{n}(k) with coefficients in B(k^{n}) coincide (for n big enough with respect to the cohomology index) with cohomology of GL_{n}(k) with coefficients in the trivial submodule B(0), and the theorem of Quillen [Q], giving the vanishing of higher dimensional cohomology in case of finite fields.

Let A ∈ F be any functor and let j be a non-negative integer. For any V ∈ V set

H_{A,j}^{i}(V) = H^{i}(Hom(V, k^{j}), A(k^{j} ⊕ V)),

where the action of Hom(V, k^{j}) on A(k^{j} ⊕ V) is defined via the natural embedding

Hom(V, k^{j}) = ( \begin{matrix}1_{j} & Hom(V, k^{j}) \\ 0 & 1_{V} \end{matrix} ) ↪ GL(k^{j} ⊕ V).

Each k-space homomorphism V → V′ defines a homomorphism of (abelian) groups Hom(V′, k^{j}) → Hom(V, k^{j}) and a homomorphism of Hom(V′, k^{j})-modules A(k^{j} ⊕V) → A(k^{j} ⊕V′), thus defining a homomorphism in cohomology H_{A,j}^{i}(V) → H_{A,j}^{i}(V′). One checks immediately that in this way H_{A,j}^{i} becomes a functor from V^{f} to V, i.e an element of F.

Lemma A.5. Assume that the functor A is finite. Then all functors H_{A,j}^{i} are finite as well. Moreover, H_{A,j}^{i}(0) = 0 for i > 0 and H_{A,j}^{0}(0) = A(k^{j}).

Proof. The second part of the statement is obvious. To prove the first one we consider first a special case A = S^{m_{1}} ⊗ ... ⊗ S^{m_{l}}. To simplify notations we use the following multi-index notation: for every multi-index i_{•} = (i_{1}, ..., i_{l}) set

S^{i_{•}} = S^{i_{1}} ⊗ ... ⊗ S^{i_{l}}.

We consider the usual partial ordering on the set of multi-indices, and for each multi-index i_{•} we set |i_{•}| = i_{1} +...+i_{l}. Note that the functor S^{m_{•}} is ”exponential”, more precisely we have the following obvious formula:

S^{m_{•}}(k^{j} ⊕ V) = ⨁_{0≤i_{•}≤m_{•}} S^{m_{•}−i_{•}}(k^{j}) ⊗ S^{i_{•}}(V).

Consider an increasing filtration on S^{m_{•}}(k^{j} ⊕ V) defined by the formula

Φ_{t} = ⨁_{0≤i_{•}≤m_{•}_{|i•|≤t}} S^{m_{•}−i_{•}}(k^{j}) ⊗ S^{i_{•}}(V).

One checks easily that the action of Hom(V, k^{j}) respects this filtration and moreover the action of Hom(V, k^{j}) on subsequent factors of this filtration Φ_{t}/Φ_{t−1} = ⨁_{|i_{•}|=t} S^{m_{•}−i_{•}}(k^{j}) ⊗ S^{i_{•}}(V) is trivial. Consider the spectral sequence defined by the above filtration:

E_{1}^{s,t} = H^{s}(Hom(V, k^{j}), k) ⊗ ⨁_{0≤i_{•}≤m_{•}_{|i•|=t}} S^{m_{•}−i_{•}}(k^{j}) ⊗ S^{i_{•}}(V) ⇒ H_{S^{m•} ,k}^{s+t}(V).

Observe finally that the above filtration and hence also the above spectral sequence depend functorially on V. In particular E_{r}^{s,t} ∈ F for all s, t, r. A.well known computation of cohomology of a finite dimensional vector k-space with trivial coefficients implies that the functors E_{1}^{s,t} are finite. Immediate induction on r shows further that all functors E_{r}^{s,t} 1 ≤ r ≤ ∞ are finite. Finally, the functor H_{S^{m•} ,j}^{i} has a finite filtration all subsequent factors of which are finite and hence is finite as well.

For a general finite A, one can find a resolution

0 → A → A^{0} → A^{1} → ...

in which every functor A^{i} is a finite direct sum of functors of the form S^{m_{•}}. To conclude the proof consider the spectral sequence defined by this resolution

E_{1}^{s,t} = H_{A^{t},k}^{s}(V) ⇒ H_{A,k}^{s+t}(V)

(which depends functorially on V) and apply the same argument as above.

Corollary A.6. Let B ∈ F be a finite functor. Then for every s, j ≥ 0 there exists N(s, j) such that for n ≥ N(s, j)

H^{i}(( \begin{matrix}1_{j} & ∗ \\ 0 & GL_{n−j}(k) \end{matrix} ) , B(k^{n})) = \{ \begin{matrix}B(k^{j}) & if i = 0 \\ 0 & if 0 < i ≤ s \end{matrix}

Proof. Consider the Hochschild-Serre spectral sequence corresponding to the group extension

1 → Hom(k^{n−j}, k^{j}) → ( \begin{matrix}1_{j} & ∗ \\ 0 & GL_{n−j}(k) \end{matrix} ) → GL_{n−j}(k) → 1

E_{2}^{a,b} = H^{a}(GL_{n−j}(k), H_{B,j}^{b}(k^{n−j})) ⇒ H^{a+b}(( \begin{matrix}1_{j} & ∗ \\ 0 & GL_{n−j}(k) \end{matrix} ) , B(k^{n}))

and apply Lemma A.5 and Proposition A.4.

Combining Lemma A.3 with Corollary A.5, we immediately conclude the following

Corollary A.7. Let B be a finite functor. Then for any V ∈ V and any s ≥ 0, there exists N(V, s) such that for n ≥ N(V, s)

Ext_{GL_{n}(k)}^{i}(P_{V} (k^{n}), B(k^{n})) = \{ \begin{matrix}⨁_{W ⊂V} B(V/W) & if i = 0 \\ 0 & if 0 < i ≤ s \end{matrix}

Note that unlike the situation of Lemma A.3 the isomorphism

Hom_{GL_{n}(k)}(P_{V} (k^{n}), B(k^{n})) = ⨁_{W ⊂V} B(V/W)

is now canonical: it associates to a family {b_{W} ∈ B(V/W)}_{W ⊂V} a GL_{n}(k)- equivariant homomorphism

P_{V} (k^{n}) = k[Hom(V, k^{n})] → B(k^{n})

which sends f : V → k^{n} to B(\overline{f})(b_{Ker f}), where \overline{f} : V/Ker f → k^{n} is the induced homomorphism.

If A, B ∈ F are functors for which the sequence Ext_{GL_{n}(k)}^{s}(A(k^{n}), B(k^{n})) stabilizes with the growth of n, then we use the notation Ext_{GL(k)}^{s}(A, B) for the stable value of this Ext-group. Thus Corollary A.7 tells us, in particular, that

Hom_{GL(k)}(P_{V} , B) = ⨁_{W ⊂V} B(V/W)

for a finite B. Note that every k-linear homomorphism V → V′ defines a homomorphism of functors P_{V′} → P_{V} and hence induces also a homomorphism

Hom_{GL(k)}(P_{V} , B) → Hom_{GL(k)}(P_{V′}, B).

This makes Hom_{GL(k)}(P_{V} , B) = ⨁_{W ⊂V} B(V/W) into a functor from V^{f} to itself. One checks easily, that the map ⨁_{W ⊂V} B(V/W) → ⨁_{W′⊂V′} B(V′/W′), corresponding to a k-linear homomorphism f : V → V′ looks as follows: it takes the W -summand on the left to the f(W) summand on the right and the corresponding homomorphism B(V/W) → B(V′/f(W)) coincides with B(\overline{f}), where \overline{f} : V/W → V′/f(W) is the homomorphism induced by f.

More generally, for any (not necessarily finite) functor B ∈ F we can define a new functor aB ∈ F, by setting

(aB)(V) = ⨁_{W ⊂V} B(V/W)

and defining homomorphisms (aB)(f) (for f : V → V′) in the same way as above. Obviously we get in this way an exact functor a : F → F. By the universal property of P_{V} ∈ F we conclude that for any finite functor B we have a natural isomorphism Hom_{GL}(P_{V} , B) = Hom_{F}(P_{V} , aB).

Moreover, for any B ∈ F there is a natural monomorphism B ↪ aB, identifying B(V) with the direct summand in (aB)(V) = ⨁_{W ⊂V} B(V/W) corresponding to the zero subspace. Define a^{∼}B = aB/B. This gives us another exact functor from F to itself. Except for the trivial case B = 0 the functor a^{∼}B is never finite.

Theorem A.8. For any finite functors A, B ∈ F we have a natural identification

Ext_{GL(k)}^{∗}(A, B) = Ext_{F}^{∗}(A, aB).

After this identification, the natural homomorphism Ext_{F}^{∗}(A, B) → Ext_{GL(k)}^{∗}(A, B) corresponds to the homomorphism Ext_{F}^{∗}(A, B) → Ext_{F}^{∗}(A, aB) defined by the natural embedding B ↪ aB.

Proof. Choose a resolution of A:

0 ⟵ A ⟵ P_{0} ⟵ P_{1} ⟵ ...

in which each P_{i} is a finite direct sum of functors of the form P_{V} and consider the spectral sequence

E_{1}^{i,j} = Ext_{GL_{n}(k)}^{i}(P_{j}(k^{n}), B(k^{n})) ⇒ Ext_{GL_{n}(k)}^{i+j}(A(k^{n}), B(k^{n})).

Corollary A.7 shows that for n big, enough with respect to s, all E_{1}^{i,j}-terms with 0 < i ≤ s, 0 ≤ j ≤ s are zero. Thus for n big enough, Ext_{GL_{n}(k)}^{s}(A(k^{n}), B(k^{n})) coincides with the s-th homology group of the complex

0 → Hom_{GL_{n}(k)}(P_{0}(k^{n}), B(k^{n})) = Hom_{F}(P_{0}, aB) →
→ Hom_{GL_{k}(F)}(P_{1}(k^{n}), B(k^{n})) = Hom_{F}(P_{1}, aB) → ...

i.e. coincides with Ext_{F}^{s}(A, aB).

Corollary A.9. For finite functors A, B ∈ F the following conditions are equivalent

(1) The natutal homomorphism Ext_{F}^{∗}(A, B) → Ext_{GL(k)}^{∗}(A, B) is an isomorphism.

(2) The natural homomorphism Ext_{F}^{∗}(A, B) → Ext_{F}^{∗}(A, aB) is an isomorphism.

(3) Ext_{F}^{∗}(A, a^{∼}B) = 0.

We need a natural generalization of the functor a : F → F to the case of bifunctors (and more generally poly-functors). Denote by F_{n} the category whose objects are covariant functors of n variables V^{f} × ... × V^{f} → V (and morphisms are natural transformations). For every functor B ∈ F_{n} define a new functor a_{n}(B) ∈ F_{n} using the formula

(a_{n}B)(V_{1}, ..., V_{n}) = ⨁_{W_{1}⊂V_{1}...W_{n}⊂V_{n}} B(V_{1}/W_{1}, ..., V_{n}/W_{n})

and defining a_{n}B on morphisms in the same way as above. Denote further by

V^{f} ⟶^{D} ; ⟵_{Π}V^{f} × ... × V^{f}

the diagonal and the direct sum functors.

Proposition A.10. For any functors A ∈ F, B ∈ F_{n} we have a natural split monomorphism

Ext_{F}^{∗}(A, a(B ∘ D)) ↪ Ext_{F_{n}}^{∗}(A ∘ Π, a_{n}B).

Proof. The standard adjunction formula (1.7.1) shows that

Ext_{F_{n}}^{∗}(A ∘ Π, a_{n}B) = Ext_{F}^{∗}(A, (a_{n}B) ∘ D)

Now it suffices to note that there are homomorphisms (natural in V)

(a(B ∘ D))(V) = ⨁_{W ⊂V} B(V/W, ..., V/W) ⟶^{i} ; ⟵_{p}((a_{n}B) ∘ D)(V) =

= ⨁_{W_{1}⊂V,...,W_{n}⊂V} B(V/W_{1}, ..., V/W_{n})

Here i takes W -summand on the left to the (W, ..., W)-summand on the right and p takes (W_{1}, ..., W_{n})-summand on the right to the W_{1} +...+W_{n}-summand on the left (and the corresponding maps from one summand to the other are the natural ones). Obviously pi = 1 so that a(B ∘ D) is canonically a direct summand in (a_{n}B) ∘ D.

The following lemma is obvious from the construction

Lemma A.11. For any B_{1}, ..., B_{n} ∈ F we have natural isomorphism of functors

a_{n}(B_{1} ⊠ ... ⊠ B_{n}) = aB_{1} ⊠ ... ⊠ aB_{n}

Corollary A.12. Let A^{•} ∈ F be a (graded) exponential functor. Then for any functors B_{1}, ..., B_{n} ∈ F and any s ≥ 0 we have a natural split monomorphism

Ext_{F}^{∗}(A^{s}, a(B_{1} ⊗ ... ⊗ B_{n})) ↪ ⨁_{s_{1}+...+s_{n}=s} ⨂_{i=1}^{n} Ext^{∗}(A^{s_{i}}, a(B_{i}))

Proof. This follows immediately from Proposition A.10, Lemma A.11 and the Kunneth formula (1.7.2).

Corollary A.13. In conditions and notations of Corollary A.12 assume in addition that the functors B_{i} are without constant term (i.e. B_{i}(0) = 0). Assume further that n > 1 and the natural homomorphisms Ext_{F}^{∗}(A^{s′}, B_{i}) → Ext_{F}^{∗}(A^{s′}, aB_{i}) are isomorphisms for all i and all s′ < s. Then the homomorphism

Ext_{F}^{∗}(A^{s}, B_{1} ⊗ ... ⊗ B_{n}) → Ext_{F}^{∗}(A^{s}, a(B_{1} ⊗ ... ⊗ B_{n}))

is an isomorphism

Proof. This follows immediately from the commutative diagram

Ext_{F}^{∗}(A^{s}, B_{1} ⊗ ... ⊗ B_{n}) ⟶^{=^{∼}} ⨁_{s_{1}+...+s_{n}=s} ⨂_{i=1}^{n} Ext_{F}^{∗}(A^{s_{i}}, B_{i})
↓  ↓=^{∼}
Ext_{F}^{∗}(A^{s}, a(B_{1} ⊗ ... ⊗ B_{n})) ⟶^{↪} ⨁_{s_{1}+...+s_{n}=s} ⨂_{i=1}^{n} Ext^{∗}(A^{s_{i}}, aB_{i})

The right hand side vertical arrow in this diagram is an isomorphism since summands, corresponding to n-tuples (s_{1}, ..., s_{n}) with s_{i} = 0 for some i are trivial (the functors B_{i} and aB_{i} being without constant term).

Now we are prepared to start the proof of the Theorem A.1. We begin with a special case when both functors A and B are additive.

Proposition A.14. The natural homomorphism

Ext_{F}^{∗}(I^{(s)}, I) → Ext_{GL(k)}^{∗}(I^{(s)}, I) = Ext_{F}^{∗}(I^{(s)}, aI)

is an isomorphisms for all s ≥ 0.

Proof. This follows immediately from a Theorem of Dundy and McCarthy [D-M]. Alternatively one can proceede as follows. We may clearly assume that 0 ≤ s < r ( where q = p^{r}). Both sides are zero (by weight considerations) unless s = 0. So assume that s = 0 and consider the following commutative diagram

\underrightarrow{lim}_{m≥0} Ext_{P}^{∗}(I^{(m)}, I^{(m)}) ⟶^{∼} Ext_{GL(k)}^{∗}(I, I)
=^{∼}↓  ^{=}↓
Ext_{F}^{∗}(I, I)  ⟶ Ext_{GL(k)}^{∗}(I, I)

The top horizontal arrow is an isomorphism - this is just a reformulation of theorem 7.6 [F-S]. The left vertical arrow is an isomorphism - this follows from our strong comparison theorem (Theorem 3.10) or directly from computation of both groups presented in [F-L-S] and [F-S] respectively. Thus the bottom horizontal arrow is an isomorphism as well.

Proposition A.15. The natural homomorphisms

Ext_{F}^{∗}(I^{(s)}, S^{n}) → Ext_{F}^{∗}(I^{(s)}, aS^{n})
Ext_{F}^{∗}(I^{(s)}, Λ^{n}) → Ext_{F}^{∗}(I^{(s)}, aΛ^{n})

are isomorphisms for all n and s.

Proof. We proceede by induction on n, using Proposition A.14 as the induction base. Applying the exact functor a^{∼} to the Koszul and De Rham complexes we get the following two complexes:

0 → a^{∼}Λ^{n} → a^{∼}(Λ^{n−1} ⊗ S^{1}) → ... → a^{∼}(Λ^{1} ⊗ S^{n−1}) → a^{∼}S^{n} → 0

0 → a^{∼}S^{n} → a^{∼}(S^{n−1} ⊗ Λ^{1}) → ... → a^{∼}(S^{1} ⊗ Λ^{n−1}) → a^{∼}Λ^{n} → 0

The first of these two complexes is always acyclic, the second one is acyclic if n ≢ 0 mod (p) and has homology isomorphic to a^{∼}((S^{m−i} ⊗ Λ^{i})^{(1)}) = (a^{∼}(S^{m−i} ⊗ Λ^{i}))^{(1)} if n = pm. Induction hypothesis and Corollary A.13 show that Ext_{F}^{∗}(I^{(s)}, (a^{∼}(S^{m−i} ⊗ Λ^{i}))^{(1)}) = 0. Thus the second hypercohomology spectral sequences in both cases consist of zeroes only and hence the limit of the first spectral sequences is zero. On the other hand, Corollary A.13 implies that the E_{1}-term of both first hypercohomology spectral sequences could have only two nonzero columns and hence the differentials

d_{n} : Ext_{F}^{∗}(I^{(s)}, a^{∼}(Λ^{n})) → Ext_{F}^{∗−(n−1)}(I^{(s)}, a^{∼}(S^{n}))
d′_{n} : Ext_{F}^{∗}(I^{(s)}, a^{∼}(S^{n})) → Ext_{F}^{∗−(n−1)}(I^{(s)}, a^{∼}(Λ^{n}))

are isomorphisms. Using an additional induction on the cohomology index we easily conclude the proof.

Applying the duality isomorphisms of Lemma A.2 we derive from Proposition A.15 the following Corollary A.16. The natural homomorphisms

Ext_{F}^{∗}(Γ^{n}, I^{(s)}) → Ext_{F}^{∗}(Γ^{n}, aI^{(s)})
Ext_{F}^{∗}(Λ^{n}, I^{(s)}) → Ext_{F}^{∗}(Λ^{n}, aI^{(s)})

are isomorphisms for all n, s.

Proposition A.17. The natural homomorphisms

Ext_{F}^{∗}(Γ^{m(s)}, S^{n}) → Ext_{F}^{∗}(Γ^{m(s)}, aS^{n})
Ext_{F}^{∗}(Γ^{m(s)}, Λ^{n}) → Ext_{F}^{∗}(Γ^{m(s)}, aΛ^{n})

are isomorphisms for all m, n, s.

Proof. We use induction on n and repeat word by word the argument used in the proof of the Proposition A.15, using now Corollary A.16 as the induction base.

Corollary A.18. The natural homomorphism

Ext_{F}^{∗}(Γ^{m_{1}} ⊗ ... ⊗ Γ^{m_{s}}, S^{n_{1}} ⊗ ... ⊗ S^{n_{t}}) ⟶ Ext_{F}^{∗}(Γ^{m_{1}} ⊗ ... ⊗ Γ^{m_{s}}, a(S^{n_{1}} ⊗ ... ⊗ S^{n_{t}}))

is an isomorphism for all m_{1}, ..., m_{s}, n_{1}, ..., n_{t}.

Proof. Assume first that s = 1. In this case our statement follows from Corollary A.13 and Proposition A.17. Lemma A.2 shows further that the statement holds also (for any s) in case t = 1. The general case may be now obtained in the same way as above, applying Corollary A.13 to the exponential functor \underbrace{Γ^{•} ⊗ ... ⊗ Γ^{•}}_{s}.

End of the Proof of the Theorem A.1.

We can construct resolutions

0 → B → B^{0} → B^{1} → ...
0 ⟵ A ⟵ A_{0} ⟵ A_{1} ⟵ ...

in which all B^{i} are finite direct sums of functors of the form S^{n_{1}} ⊗ ... ⊗ S^{n_{t}} and all A_{i} are finite direct sums of functors of the form Γ^{m_{1}} ⊗ ... ⊗ Γ^{m_{s}}. Comparing the corresponding hypercohomology spectral sequences and using Corollary A.18 we get the desired result.

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