General Linear and Functor Cohomology Over Finite Fields

Annals of Mathematics, 150 (1999), 663–728

General linear and functor cohomology over nite elds

By 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 elds of positive characteristic p. The reader familiar with the representation theory of algebraic objects will recognize the importance of an understanding of Ext-groups. For example, the existence of nonzero Ext-groups of positive degree is equivalent to the existence of objects which are not “direct sums” of simple objects. Indeed, a knowledge of Ext-groups provides considerable knowledge of compound objects. In the study of modular representation theory of nite Chevalley groups such as GLn(Fq), Ext-groups play an even more central role: it has been shown in [CPS] that a knowledge of certain Ext1-groups is sucient to prove Lusztig’s Conjecture concerning the dimension and characters of irreducible representations. We consider two dierent categories of functors, the category F(Fq)ofall functors from nite dimensional Fq-vector spaces to Fq-vector spaces, where Fq is the nite eld of cardinality q, and the category P(Fq) of strict polynomial functors of nite degree as dened in [F-S]. The category P(Fq) presents several advantages over the category F(Fq) 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 3.10 of Ext-groups in the category P(Fq) with Ext-groups in the category F(Fq). Weaker forms of this theorem have been known to us since 1995 and to S. Betley independently

Partially supported by the C.N.R.S., 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.

664 Franjou, Friedlander, Scorichenko, and Suslin

(see its use in [B2, §3]). This early work apparently inspired the paper of N. Kuhn [K2] as well as the present paper. The calculation (for an arbitrary eld k of positive characteristic) of the ExtP(k)-groups from a Frobenius-twisted divided power functor to a Frobenius- twisted symmetric power functor is presented in Theorem 4.5. This calculation is extended in Section 5 to various other calculations of ExtP(k)-groups between divided powers, exterior powers, and symmetric powers. This leads to similar ExtF(Fq)-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 (HomF(Fq)-case) from a divided power functor 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(Fp). The nal result, proved by the last-named author in the appendix, is the proof of equality of the ExtF(Fq)-groups and ExtGL∞(Fq)-groups associated to nite functors. This has a history of its own that will be briey 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 dened and split over the prime eld and to general (nite 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 suciently many times to the rational modules, and provided that the nite eld is suciently large. One consequence of our work is a strong stability result for the eect of iterating Frobenius twists (Corollary 4.10); it applies to the Ext-groups of rational modules arising from strict polynomial functors of nite degree. A second consequence is an equally precise lower bound for the order of the nite eld Fq required to compare these “stably twisted” rational Ext-groups with the Ext-groups computed for the innite general linear group GL∞(Fq). For explicit calculations of Ext- groups for GLn(Fq) for various fundamental GLn(Fq)-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 nite degree on - nite dimensional k-vector spaces and further recalls the relationship of P to the category of rational representations of the general linear group. The in- vestigation of the forgetful functor P(k) →F(k) is begun by observing that HomP(k)(P, Q) = HomF(k)(P, Q) for strict polynomial functors of degree d provided that cardinality of the eld k is at least d. Theorem 1.7 presents the key

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 665 property for Ext-groups involving functors of exponential type; this property enables our Ext-group calculations and appears to have no evident analogue for GLn (with n nite). In Section 2, we employ earlier work of N. Kuhn and the rst named author to provide a rst comparison of Extn (P, Q) and Extn (P, Q). P(Fq) F(Fq) 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 eect of an extension L/k of nite elds on ExtF (P, Q)-groups when P and Q are strict polynomial functors. d(r) dprj (j) For every integer d, we compute in Theorem 4.5, ExtP(k)( ,S ) d(r) dprj (j) d d d and ExtP(k)( , ), where , S , denote the d-fold divided power, symmetric power, and exterior power functor respectively. These computations are fundamental, for d (respectively its dual Sd) is projective (resp. injective) in P(k) and tensor products of i (resp. Si) of total degree d consti- tute a family of projective generators (resp. injective cogenerators) for Pd(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 nite 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 dierentials in hypercohomology spectral sequences as Koszul dierentials. 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 ExtP(k)-groups with the strong comparison theorem of

Section 3 and our understanding of base change for ExtF(Fq)-groups. The result is various complete calculations of Ext (, ), as tri-graded Hopf algebras. F(Fq) The appendix, written by the last-named author, demonstrates a natural ∞ ∞ isomorphism Ext (A, B) → Ext (A(F ),B(F )) for nite functors A, F GL(Fq) q q 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 denitions and basic properties of the category P of strict polynomial functors of nite degree (on k-vector spaces) introduced in [F-S] and to be used in subsequent sections. We

666 Franjou, Friedlander, Scorichenko, and Suslin also contrast the category P with the category F of all functors from nite dimensional k-vector spaces to k-vector spaces. Strict polynomial functors were introduced in order to study rational cohomology of the general linear groups GLn over k (i.e., cohomology of comodules for the Hopf algebra k[GLn], the coordinate algebra of the algebraic group GLn). 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 dened on vector spaces over a eld k of characteristic p>0. Indeed, in subsequent sections we specialize further to the case in which k is a nite eld. We let V = Vk denote the category of k-vector spaces and k-linear homomorphisms and we denote by Vf the full subcategory of nite dimensional k-vector spaces. A polynomial map T : V → W between nite dimensional vector spaces is dened to be a morphism of the corresponding ane schemes over k: Spec(S(V #)) → Spec(S(W #)) (where Spec(S(V #)) is the ane scheme associated to the symmetric algebra over k of the k-linear dual of V ). Equiva- lently, 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 ∈ Sd(V #) W . A polynomial map between nite dimensional vector spaces is uniquely determined by its associated set-theoretic function from V to W provided that k is innite; 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 eld is innite.

Strict polynomial functors

We recall [F-S, 2.1] that a strict polynomial functor P : Vf →Vf is the following collection of data: for any V ∈Vf a vector space P (V ) ∈Vf ; for any f V,W in V , a polynomial map PV,W : Homk(V,W) → Homk(P (V ),P(W )). These polynomial maps should satisfy appropriate compatibility conditions similar to the ones used in the usual denition of a functor. A strict polynomial functor is said to be homogeneous of degree d if PV,W : Homk(V,W) → f Homk(P (V ),P(W )) is homogeneous of degree d for each pair V,W in V .A strict polynomial functor is said to be of nite degree provided that the degrees f of the polynomial maps PV,W are bounded independent of V,W ∈V . Replacing the polynomial maps PV,W :Homk(V,W)→Homk(P (V ),P(W )) by the associated set-theoretic functions, we associate to each strict polynomial functor P a functor in the usual sense Vf →Vf (which we usually denote by the same letter P ). The above remarks about polynomial maps imply that over innite elds strict polynomial functors may be viewed as functors (in the usual sense) which satisfy an appropriate additional property. However, over nite elds (and a fortiori over more general base rings), this is no longer the

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 667 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 eld k) the abelian category of strict polynomial functors of nite degree. One easily veries that P splits as a direct sum dPd, where Pd 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 d, d d the d-fold symmetric power functor S (dened as the d-coinvariants of ), d d and the d-fold divided power functor (dened as the d-invariants of ). d d The functor S is injective, and is projective in Pd. Let : k → k denote the pth-power map. The Frobenius twist functor I(1) sends a vector space V to the base change of V via the map (which we denote by V (1)). So dened, I(1) is a strict polynomial functor homogeneous (1) th of degree p; IV,W is the p -power map (1) (1) # # (1) p # Homk(V ,W ) = (Homk(V,W) ) → S (Homk(V,W) ) p # (1) (1) viewed as an element of S (Homk(V,W) ) Homk(V ,W ). For any functor G : Vf →V, we dene G(1) as G I(1). (The reader should consult [F-S] for details.) 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 eld extension and V is a k-vector space, let VK denote K k V with the evident K vector space structure. If P is a strict polynomial functor over k, then the base change PK is dened by setting PK (VK )=P (V )K and setting the polynomial map → (PK )VK ,WK : HomK (VK ,WK ) HomK (P (VK ),P(WK )) to be the base change of PV,W as a morphism of ane 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 nite degree over a eld k. For any eld extension k → K, there is a natural isomorphism of K-vector spaces ExtP(K)(PK ,QK ) = ExtP(k)(P, Q) k K.

If P is a strict polynomial functor, then for any n>0 the vector space n P (k ) inherits a natural structure of a rational GLn-module. (In fact, functoriality of P implies that P (kn) inherits a natural structure of a rational Mn-module whose restriction provides the rational GLn-structure). We recall the following relationship between Ext-groups in the category P and in the category of rational GLn-modules.

668 Franjou, Friedlander, Scorichenko, and Suslin

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

→' n n ExtP (P, Q) ExtGLn (P (k ),Q(k )) induced by the exact functor sending a strict polynomial functor to its value on kn.

The Ext-groups of the previous theorem can also be computed as the Ext-groups of the the classical Schur algebra S(n, d) (with n d as above). An explicit determination of the cohomological dimension of S(n, d) is given in [T]. By a theorem of H. Andersen (cf. [J, II, 10.14]), the Frobenius twist induces an injection on rational Ext-groups: for any two nite dimensional rational GLn-modules M and N, the natural map induced by the Frobenius twist (which we view as an exact functor on the category of rational GLn-modules) → (1) (1) ExtGLn (M,N) ExtGLn (M ,N ) 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:

(1) (1) ExtP (P, Q) → ExtP (P ,Q ).

Consider now the abelian category F of functors Vf →V. If we need to indicate the base eld k explicitly, we shall denote this category by F(k). The forgetful functor P→Fis clearly exact, thereby inducing a natural map on Ext-groups ExtP (P, Q) → ExtF (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 Pd, the natural inclusion HomP (P, Q) ,→ HomF (P, Q) is an isomorphism.

Proof. Let f ∈ HomF (P, Q) be a homomorphism of functors. To check that f is a homomorphism of strict polynomial functors, we have to verify that

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 669 for any V,W in Vf the following diagram of polynomial maps commutes:

→QV,W Homk(V,W) Homk(Q(V ),Q(W ))   PV,W y yfV

fW Homk(P (V ),P(W )) → Homk(P (V ),Q(W )) . To do so, we observe that both compositions are homogeneous polynomial maps of degree d from Homk(V,W)toHomk(P (V ),Q(W )) whose values at all rational points coincide. Finally we observe that if a eld k 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, dene its dierence functor (F )by

(F )(V ) = Ker{F (V k) → F (V )}.

The functor F is said to be polynomial if the rth dierence functor r(F ) vanishes for r suciently large. The Eilenberg-MacLane degree of a polynomial functor F is the least integer d such that 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 d, etc. More generally one checks immediately that for any strict polynomial functor P in Pd its image under the forgetful functor P→Fis a polynomial of Eilenberg-MacLane degree less than or equal to d. Following N. Kuhn [K1] we say that a functor F in F is nite if it is of nite Eilenberg-MacLane degree and takes values in Vf . The previous remarks imply that the image in F of any strict polynomial functor P ∈P is nite. f For any vector space V ∈V dene a functor PV ∈Fby the formula PV (W )=k[Homk(V,W)]. The Yoneda Lemma shows that for any Q in F we have a natural isomorphism HomF (PV ,Q)=Q(V ). This implies immediately that the functor PV is projective in F. A functor P ∈Fis said to be of nite type if it admits an epimorphism from a nite direct sum of functors of the form PV . To say that a functor Q ∈Fadmits a projective resolution of nite type is clearly equivalent to saying that Q admits a resolution each term of which is isomorphic to a nite direct sum of functors of the form PV .On several occasions we shall need the following useful fact.

670 Franjou, Friedlander, Scorichenko, and Suslin

Proposition 1.5 ([S], [F-L-S, 10.1]). Assume that the eld k is nite. Then every nite functor Q in F admits a projective resolution of nite type. A clear dierence between the two categories of functors F and P appears when we consider the Frobenius twist. If k is perfect, then the Frobenius map : k → k is an isomorphism so that the Frobenius twist ()(1) becomes (1) invertible when viewed in F. Indeed, if k is the prime eld Fp, then I = I (1) in F; note that even for k = Fp,() is not invertible in P. Observe that if k is perfect, then for any P , Q in F we have a natural (1) (1) isomorphism ExtF (P, Q) → ExtF (P ,Q ). Hence for any strict polynomial functors P , Q in P we get a natural map (r) (r) → (r) (r) lim→ ExtP (P ,Q ) lim→ ExtF (P ,Q ) = ExtF (P, Q) . r r Theorem 3.10 gives conditions for this map to be an isomorphism.

Exponential functors

An exponential functor is a graded functor A =(A0,A1,... ,An,...) from Vf to Vf together with natural isomorphisms Mn A0(V ) = k, An(V W ) = Am(V ) An m(W ) ,n>0 . m=0 Lemma 1.6. Let A be an exponential functor.

(1) The functors A1, A2, ... are without constant term, i.e. Ai(0)=0for i>0. (2) The natural maps Mn An(V )=An(V ) A0(W ) ,→ Am(V ) An m(W )=An(V W ) , m=0 Mn An(V W )= Am(V ) An m(W ) →→ An(V ) A0(W )=An(V ) m=0 coincide (up to an automorphism of the functor An) with the map induced n by the inclusion of the rst factor A (i1) and the map induced by the n projection onto the rst factor A (p1) respectively. (3) The Eilenberg-MacLane degree of the functor An is at most n and is equal to n provided that A1 =0.6 In particular the functors An are nite.

Proof. (1) The exponential condition shows that for n>0wehavean isomorphism An(0) = An(00) = A0(0)An(0)An(0)A0(0) = An(0)An(0) . Thus dim An(0) 2 dim An(0) and hence dim An(0)=0.

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 671

n n (2) Denote the homomorphisms in question by iV,W and pV,W respectively. The functoriality of all the maps involved implies the commutativity of the following diagram:

in n →V,0 n n A (V ) A (V 0) = A (V )   n =y yA (i1)

in An(V ) →V,W An(V W ) . n Now it suces to note that (according to (1)) i,0 is an automorphism of the n n functor A . The same reasoning applies to pV,W . (3) The exponential condition and (2) show that the functor (An)is isomorphic to the direct sum A1(k)An1 An(k). Immediate induction on n now concludes the proof.

Typical examples of exponential functors are given by the symmetric algebra S =(S0,S1,...Sn,...), the exterior algebra =(0, 1,... ,n,...) and the divided power algebra =(0, 1,... ,n,...). Another exponential functor L =(L0,L1,... ,Ln,...) is obtained as the quotient of the symmetric power algebra by the ideal of pth powers; it coincides with the exterior algebra when p =2. Note also that if we dene the tensor product of graded functors via the usual formula Mn (A B )n = Am Bn m, m=0 then the tensor product of two exponential functors is again exponential. Clearly, the Frobenius twist of an exponential functor is again exponential. The previous denition 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 provides an important tool enabling computations of functor cohomology. From now on we assume (if not specied otherwise) that the base eld k is nite.

Theorem 1.7. Let A be an exponential functor. For any B, C in F, there exist natural isomorphisms Mn n m nm ExtF (A ,B C)= ExtF (A ,B) ExtF (A ,C). m=0

672 Franjou, Friedlander, Scorichenko, and Suslin

Furthermore, if B and C take values in the category Vf then there also exist natural isomorphisms Mn n m nm ExtF (B C, A )= ExtF (B,A ) ExtF (C, A ). m=0 Similarly, if A is an exponential strict polynomial functor (of nite degree), then for any B, C in P there are natural isomorphisms Mn n m nm ExtP (A ,B C)= ExtP (A ,B) ExtP (A ,C) m=0 Mn n m nm ExtP (B C, A )= ExtP (B,A ) ExtP (C, A ). m=0

Proof. In case of the category F the rst part is proved in [F, 1.4.2] (we recall the proof below). The same argument gives the second part. Alterna- tively, the second part follows from the rst by the duality isomorphism 1.12. The proof for the category P is identical, using the theory of strict polynomial bifunctors of nite 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 nite) elds. Let biF denote the abelian category of bifunctors Vf Vf →V. Consider a pair of adjoint (on both sides) functors →D Vf Vf Vf . Here is the direct sum functor (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 biF. P7→P → F biF. QDQ 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 biF we have natural isomorphisms ExtF (P, Q D) = ExtbiF (P ,Q) , ExtF (Q D, P) = ExtbiF (Q, P ) . For any functors B, C in F we dene their external tensor product B C ∈ biF 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: PV PW = P(V,W), give us the following Kunneth-type formula:

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 673

(1.7.2) Assume that A1, A2 are functors in F which admit projective resolutions of nite type; then for any B, C in F there are natural isomorphisms ExtbiF (A1 A2,B C) = ExtF (A1,B) ExtF (A2,C). Using (1.7.1) and the exponential property of A we get isomorphisms: n n n ExtF (A ,B C) = ExtF (A , (B C) D) = ExtbiF (A ,B C) Mn Mn m nm m nm = ExtbiF ( A A ,B C)= ExtbiF (A A ,B C). m=0 m=0 Finally we observe that all functors Ai are nite according to Lemma 1.6 and hence admit projective resolutions of nite type (see Proposition 1.5), so that we may use (1.7.2) to conclude the proof. Observe that for homogeneous strict polynomial functors An, B and C, the sum in Theorem 1.7 cannot have more than one nonzero term. Thus, in the case of the category P, Theorem 1.7 generalizes [F-S, Prop. 5.2]. This, and the injectivity of the symmetric powers in P, make computing Ext-groups in the category P easier than in the category F. Applying Theorem 1.7 to the exponential functors A| {z A} and n |B {z B} we obtain the following corollary. m Corollary 1.8. Let A, B be exponential functors. For any nonnegative integers k1,... ,kn; l1,... ,lm there are natural isomorphisms k k l l ExtF (A 1 A n ,B 1 B m ) M nmO k l = ExtF (A s,t ,B t,s ) .

k1,1++k1,m=k1 kn,1++kn,m=kn s=1,t=1 l1,1++l1,n=l1 lm,1++lm,n=lm Similarly, if A and B are exponential strict polynomial functors of nite degree, then there are corresponding natural isomorphisms obtained by replacing ExtF by ExtP . All our examples of exponential functors are bi-algebras. Indeed, for any exponential functor A, the natural maps

Ai+j () Ai(V ) Aj(V ) ,→ Ai+j(V V ) → Ai+j(V ) , Ai+j () Ai+j(V ) → Ai+j(V V ) →→ Ai(V ) Aj(V ) dene natural product and coproduct operations Ai Aj → Ai+j, Ai+j → Ai Aj. Here, is the sum map (x, y) 7→ x + y and is the diagonal map x 7→ (x, x).

674 Franjou, Friedlander, Scorichenko, and Suslin

Denition 1.9. A is a Hopf exponential functor provided that for any V in Vf the above product operation makes A(V ) into a (graded) associative k-algebra with unit 1 ∈ A0(V )=k. The name Hopf functor is justied by the following lemma. Lemma 1.10. Assume that A is an Hopf exponential functor. Then for any V in Vf the above operations make A(V ) into a (graded) Hopf algebra with co-unit ε : A(V ) → A0(V )=k. Proof. Note rst that the (right) multiplication by 1 ∈ A0(V ) coincides with the natural isomorphism n n n 0 → n n iV,0 : A (V )=A (V ) A (0) A (V 0) = A (V ) (cf. the proof of Lemma 1.6 (2)). In the same way the right comultiplication by ε coincides with the natural isomorphism n n n → n 0 n pV,0 : A (V )=A (V 0) A (V ) A (0) = A (V ). n n 1 ∈ 0 Since pV,0 =(iV,0) we conclude that 1 A (V ) is a unit if and only if ε is a co-unit of A(V ). Observe next that the natural homomorphism Am(V ) Anm(W ) → An(V W ) in the denition of the exponential functor may be expressed in terms of the product operation as the composition

m nm A (i1)A (i2) Am(V ) An m(W ) → Am(V W ) An m(V W ) →mult An(V W ).

This remark implies immediately that the associativity of A(V ) (for all V )is equivalent to the fact that the two possible identications of graded tri-functors A(U) A(V ) A(W ) and A(U V W ) coincide. Now the verication 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

comult→ A (V ) A (V )  A (V )   comulty comult1y A(V ) A(V ) 1comult→ A(V ) A(V ) A(V ).

However one checks easily that both compositions coincide with the homo- A() morphism A(V ) → A(V V V )=A(V ) A(V ) A(V ), where :V → V V V is the diagonal map.

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 675

We say that the exponential functor A is commutative (respectively, skew-commutative) if for all nonnegative integers i, j and every V in Vf the map : V V → V V ,(x, y) 7→ (y, x) gives rise to a commutative diagram (resp. a diagram commutative up to a sign (1)ij):

i j → i+j A (V )  A (V ) A (V V )   T y A()y Aj(V ) Ai(V ) → Ai+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 ,, S, L and their Frobenius twists are Hopf exponential functors and that ε()=ε(S)=ε(L) = +1, ε()=1. Assume that A and B are exponential functors (respectively exponential strict polynomial functors of nite degree). In this case the tri-graded vector space Ext (A ,B ) (a notation to denote ExtF (A ,B ) as well as ExtP (A ,B )) acquires a natural structure of a (tri-graded) bi-algebra. The product operation is dened as the composition

Ext (A ,B ) Ext (A ,B ) → Ext (A A ,B B ) = Ext (A ,B ) → Ext (A D, B D) → Ext (A ,B ).

Q7→QD Here the second arrow is induced by the exact functor biF →F and the last arrow is dened by the adjunction homomorphisms : I → D and : D → I. Equivalently the product of the classes e ∈ Exti(An,Bm), 0 0 0 e0 ∈ Exti (An ,Bm ) may be described as the image of the tensor product 0 0 0 e e0 ∈ Exti+i (An An ,Bm Bm ) under the homomorphism Ext(An 0 0 0 0 An ,BmBm ) → Ext(An+n ,Bm+m ) induced by the product operation Bm 0 0 0 0 Bm → Bm+m and the coproduct operation An+n → An An . Finally the product operation may be also described (using the isomorphisms of Theorem 1.7) as eigher 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 dened as the composition

Ext (A ,B ) → Ext (A ,B ) = Ext (A A ,B B ) = Ext (A ,B ) Ext (A ,B ).

676 Franjou, Friedlander, Scorichenko, and Suslin

The coproduct operation may be also described (using the isomorphisms of Theorem 1.7) as either 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 nite 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 st+ ε(A) 1 ik+ ε(B) 1 jl the following diagrams commute up to a sign (1) 2 2 :

s i j t k l →mult s+t i+k j+l Ext (A ,B )  Ext (A ,B ) Ext (A ,B )   =y =y

Extt(Ak,Bl) Exts(Ai,Bj) →mult Exts+t(Ai+k,Bj+l)

s+t i+k j+l comult→ s i j t k l Ext (A ,B ) Ext (A ,B )  Ext (A ,B )   =y =y

Exts+t(Ai+k,Bj+l) comult→ Extt(Ak,Bl) Exts(Ai,Bj) .

Dual functors

For a functor P : Vf →V, dene its dual P # : Vf →Vby P #(V )= P (V #)#. Two examples of this duality which will be important in what follows are (Sd)# =d , (d)# =d. The contravariant functor # : F→Fis clearly exact and hence for any P , Q in # # # F we get a natural duality homomorphism : ExtF (P, Q) → ExtF (Q ,P ). The same construction obviously applies to the category P as well.

Lemma 1.12. Assume that P, Q ∈Ftake values in the category Vf . Then the duality homomorphism

# # # : ExtF (P, Q) → ExtF (Q ,P ) is an isomorphism. Similarly for any P, Q ∈P the duality homomorphism

# # # : ExtP (P, Q) → ExtP (Q ,P ) is an isomorphism.

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 677

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

# # # # ## ## ExtF (P, Q) → ExtF (Q ,P ) → ExtF (P ,Q ) = ExtF (P, Q) n is the identity map. Let e ∈ ExtF (P, Q) be represented by an extension

0 → Q → Pn →→P1 → P → 0. Then e## is represented by the extension → ## → ## →→ ## → ## → 0 Q Pn P1 P 0 and our statement follows from the commutativity of the diagram → → → → → → 0 Q Pn ... P1 P 0     =y y y =y → ## → ## → → ## → ## → 0 Q Pn ... P1 P 0 . The case of the category P is trivial since in this case # : P→Pis an anti-equivalence.

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

Lemma 1.13. Let A,B be Hopf exponential functors (resp. Hopf exponential strict polynomial functors of nite 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. The 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 the next, we shall restrict our attention to the case in which the base eld k is a nite N eld k = Fq of characteristic p, with q = p . We show for strict polynomial functors A, B of degree d that the natural map i (m) (m) i (m) (m) i ExtP (A ,B ) → ExtF (A ,B ) = ExtF (A, B) is an isomorphism provided that m and q are suciently large compared to d and i. In the following section, we show that the condition on q can be weakened to the simple condition that q d.

678 Franjou, Friedlander, Scorichenko, and Suslin

We begin with the following vanishing theorem which complements the d fact that symmetric power functors S are injective in Pd. The proof of this theorem uses the niteness of the base eld k in an essential way.

Theorem 2.1 ([F, §4]). Let A : Vf →Vf be a nite functor of Eilenberg- MacLane degree d. For any i>0,

i ph ExtF (A, S )=0 i+2 d1 provided that h logp 2 +[p1 ].

h h+1 Proof. By [F, 4.1.2], the natural embedding Sp (l+1) ,→ Sp (l) induces an isomorphism

i ph(l+1) i ph+1(l) ExtF (A, S ) → ExtF (A, S ) i+2 d1 provided that h logp 2 +[p1 ]. Applying this remark N times we see that (for the same values of h) the natural embedding

h h h+1 h+N Sp = Sp (N) ,→ Sp (N 1) ,→,→ Sp induces an isomorphism

i ph i ph+N ExtF (A, S ) → ExtF (A, S ). On the other hand, the vanishing theorem of N. Kuhn [K3, Th. 1.1] shows that

i ph+kN lim→ ExtF (A, S )=0 k0 for any functor A admitting a projective resolution of nite type. Since each nite functor A admits such a resolution (see Proposition 1.5), this completes the proof.

The following classical lemma was 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

k n = n0 + n1p + + nkp (0 ni

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

Sn comult→ Sm Sn m →mult Sn .

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 679

n comult m nm n Explicitly, S → S S sends an n-fold product x1 xn ∈ S (V ) to the sum indexed by n/m nm of tensors of the form xi1 xim n xim+1 xin . Composition of these homomorphisms is multiplication by m , n m nm n so S is a direct summand in S S provided that m is prime to p. One n easily veries that 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 s +2 d 1 h log +[ ] , and n + + n 0modph. p 2 p 1 1 k Then for all 0

m Proof. Using Lemma 2.2, we are easily reduced to the case where ni = p i isapowerofp for all i. By Theorem 1.7, M Ok m d pm1 pmk d p j ExtF (A ,S S )= ExtF (A j ,S ).

d1++dk=d j=1 Since the functor A0 is constant, all its Ext-groups to functors without constant term are trivial; thus the summand corresponding to the k-tuple (d1,... ,dk)is trivial provided that dj = 0 for some j. Consider now a summand corresponding to a k-tuple (d1,... ,dk) with all dj > 0. In this case we have inequalities dj d (k 1). On the other hand, an easy induction on k establishes that m1 m h k1 if p + + p k 0modp , then mj h [ ] for all j. Consequently, p 1 k 1 s +2 d 1 k 1 s +2 d 1 m h log + log + j . j p 1 p 2 p 1 p 1 p 2 p 1 i d pmj Theorem 2.1 shows that ExtF (A j ,S ) = 0 for 0

Proof. This follows immediately from Proposition 2.3 since (d1 dk )(m) is a direct summand in Ad, where A is the exponential functor (m) (| {z }) . k

680 Franjou, Friedlander, Scorichenko, and Suslin

Corollary 2.4 easily implies our rst, and weakest, comparison of the Ext- groups ExtP and ExtF . A rst form of this proposition was proved by E. Friedlander (1995, unpublished) and by S. Betley independently [B2], and discussion of T. Pirashvili with the rst author.

Proposition 2.5. Let A and B be strict polynomial functors homogeneous of degree d, over a eld with q elements. Assume that s +2 d 1 m log +[ ] ,q dpm. p 2 p 1 Then the canonical homomorphisms

i (m) (m) i (m) (m) i ExtP (A ,B ) → ExtF (A ,B ) = ExtF (A, B) are isomorphisms for all 0 i s.

d1 d Proof. Consider rst the special case A = k , d1 ++dk = d. (m) The functor B admits an injective resolution in Pdpm 0 → B(m) → I in which every Ij is a direct sum of functors of the form Sn1 Snl m with n1 + + nl = dp [F-S, §2]. We employ the hypercohomology spectral sequence ij i (m) j ⇒ i+j (m) (m) E1 = ExtF (A ,I )= ExtF (A ,B ). ij Corollary 2.4 shows that E1 = 0 for all 0

0 A P

d d in which each Pi is a direct sum of functors of the form 1 k , d1 + + dk = d. This gives us two hypercohomology spectral sequences ij i (m) (m) ⇒ i+j (m) (m) PE1 = ExtP (Pj ,B )= ExtP (A ,B ) , ij i (m) (m) ⇒ i+j (m) (m) FE1 = ExtF (Pj ,B )= ExtF (A ,B ) and a homomorphism of spectral sequences PE → FE. According to the special ij → ij ij case considered above, the homomorphisms PE1 FE1 on E1 -terms are isomorphisms for all j and all 0 i s. The standard comparison theorem for i (m) (m) i (m) (m) spectral sequences thus implies that ExtP (A ,B ) → ExtF (A ,B )= i ExtF (A, B) is an isomorphism for 0 i s.

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 681

Combining Proposition 2.5 with the injectivity of Frobenius twists established in Corollary 1.3, we obtain the following theorem asserting that ExtP - groups stabilize with respect to repeated Frobenius twists. The lower bound of s the number of twists required to stabilize ExtP (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 s (m) (m) s (m+1) (m+1) ExtP (A ,B ) → ExtP (A ,B ) s+2 d1 is injective for all m and is an isomorphism for m logp 2 +[p1 ]. Proof. Injectivity is given by Corollary 1.3. To prove the asserted surjec- tivity, we pick a nite extension E/k with more than dpm+1 elements. Consider the following commutative diagram

Exts (A(m),B(m)) → Exts (A(m+1),B(m+1)) P(E) E E P(E) E E   =y =y s (m) (m) →= s (m+1) (m+1) ExtF(E)(AE ,BE ) ExtF(E)(AE ,BE ). 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 s (m) (m) → s (m+1) (m+1) ExtP(k)(A ,B ) ExtP(k)(A ,B ). Using Theorem 2.6, we signicantly strengthen Proposition 2.5 by hav- ing the bound on q (the cardinality of our base eld) 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 (the weak comparison theorem). Let A and B be strict polynomial functors homogeneous of degree d, over a eld with q elements. s+2 d 1 m0 Let m0 be the least integer logp 2 +[p1 ]. Assume that q dp . Then for any m m0 the canonical homomorphisms i (m) (m) i (m) (m) i ExtP (A ,B ) → ExtF (A ,B ) = ExtF (A, B) are isomorphisms for all 0 i s.

The following is an immediate corollary of Theorem 2.7.

682 Franjou, Friedlander, Scorichenko, and Suslin

Corollary 2.8. In conditions and notation of Theorem 2.7, the canonical homomorphism i (m) (m) → i lim→ ExtP (A ,B ) ExtF (A, B) m is an isomorphism for 0 i s, provided that q dpm0 .

3. Extension of scalars and Ext-groups

The aim of this section is to understand the eect on ExtF -groups of base change from P(k)toP(K), where k → K is an extension of nite elds. In Theorem 3.9, we see that this eect is simply one of base change provided that the cardinality of the base eld 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 (nite) base eld is weakened to the condition that the cardinality be greater or equal to the degree of the strict polynomial functors involved. For any nite extension k → K of elds, we have pairs of functors

Vf t Vf V t V K → k K → k where is the forgetful (or restriction) functor and t is the extension of scalars (or induction) functor. Clearly, both t and are exact and t is left adjoint to . The following proposition is less evident and presumably less well known. One can informally describe this result as saying that induction equals coin- duction. We include a sketch of proof for the sake of completeness.

Proposition 3.1. Let k → K be a nite extension of elds. A choice of a nonzero k-linear homomorphism T : K → k determines adjunction transformations → → : t IVk ,: IVK t which establish that t is right adjoint to (as functors on either V or Vf ).

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

K K → k (a, b) 7→ T (ab) .

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 683

We dene Xn # ∈ x = ei ei K k K, i=1

# # where e1,... , en is a basis for K over k and e1 ,... , en is the dual basis with respect to the above bilinear form. For any V in Vk, we dene

V = T k 1V : t(V )=K k V → V ; for any W in VK , we dene

W = x ():W → t (W )=K k W

(i.e., multiplication by x ∈ K k K). One easily checks that x is independent of the choice of the basis e1,... , en. This, in turn, easily implies that

(3.1.1) x (a 1 1 a)=0∈ K k K ∀a ∈ K, so that W is K-linear. Similarly, one checks that

(3.1.2) (T 1K )(x)=(1K T )(x)=1∈ K which easily implies that

HomK (W, K k V ) → Homk(W, V ) Homk(W, V ) → HomK (W, K k V )

7→ V 7→ (1K k )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 and t determine adjoint (on both sides) pairs

t F Vf V →F (k) = Funct( k , k) (K/k) Vf V →F Vf V = Funct( k , K ) (K) = Funct( K , K ) . t Here t and are given by the composition with t and on the left, whereas t and are given by composition with t and 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.

684 Franjou, Friedlander, Scorichenko, and Suslin

Proposition 3.2. Let k → K be a nite extension of elds. For any functors A in F(K), B in F(K/k), C in F(k), there are natural isomorphisms of graded K vector-spaces: ExtF(K/k)(B,t A) = ExtF(K)( B,A), ExtF(K/k)(B,tC) = ExtF(k)(B,C), ExtF(K/k)(t A, B) = ExtF(K)(A, B), ExtF(K/k)(tC, B) = ExtF(k)(C, B).

Corollary 3.3. Let k → K be a nite extension of elds. Let A ∈ F(K), A0 ∈F(k) be functors such that t A = tA0 ∈F(K/k). For any C in F(k), we have natural isomorphisms of graded K-vector spaces K k ExtF(k)(A0,C) = ExtF(K)(A, tC), K k ExtF(k)(C, A0) = ExtF(K)( tC, A).

Proof. According to Proposition 3.2 we have natural isomorphisms ExtF(K)(A, tC) = ExtF(K/k)(t A, tC) = ExtF(K/k)(tA0,tC) = ExtF(k)(A0,tC). It now suces to observe that the functor tC is given by the formula V 7→ K k C(V ). If we forget about the action of K, then this is just a direct sum of nitely many copies of C, so that the natural homomorphism → K k ExtF(k)(A0,C) ExtF(k)(A0,tC) is an isomorphism of graded K-vector spaces. Using Corollary 3.3, we obtain the following understanding of the eect of base change on ExtF -groups, an understanding which we later compare to the corresponding property of ExtP -groups, as given by 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 ExtF(k)(P, Q) = ExtF(K)(PK , t QK ) = ExtF(K)( t PK ,QK ).

Proof. This follows immediately from Corollary 3.3, when we take into account the obvious formula t PK = tP ∈F(K/k).

Remark 3.4.1. It is easy to see from the discussion above that the isomorphisms K k ExtF(k)(P, Q) = ExtF(K)(PK , t QK ) = ExtF(K)(PK ,QK (t )) , K k ExtF(k)(P, Q) = ExtF(K)( t PK ,QK ) = ExtF(K)(PK (t ),QK )

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 685 are obtained as K-linear extensions of the canonical k-linear maps → ExtF(k)(P, Q) ExtF(K)(PK ,QK (t )) , → ExtF(k)(P, Q) ExtF(K)(PK (t ),QK ) , which may be described as follows. The exact functor t : F(k) →F(K) de- → nes a canonical homomorphism t :ExtF(k)(P, Q) ExtF(K)( tP, tQ) = ExtF(K)( t PK , t QK ) = ExtF(K)(PK (t ),QK (t )). Furthermore, left adjointness of t to denes a functorial homomorphism : t → I, whereas right adjointness of t to denes a functorial homomorphism : I → t . The homomorphisms in question are the compositions of the homomorphism induced by t with homomorphisms 7→ XX PK (→) ExtF(K)(PK (t ),QK (t )) ExtF(K)(PK ,QK (t )) , 7→ XQK () X→ ExtF(K)(PK (t ),QK (t )) ExtF(K)(PK (t ),QK ) dened by the functor homomorphisms PK ():PK → PK (t ) and QK () : QK (t ) → QK respectively. The following theorem relates the eects of the base change on Ext-groups in the categories P and F. Theorem 3.5. Let k → K be a nite extension of elds and let P , Q be in P(k). The following diagram commutes (3.51) K Ext (P, Q) → Ext (P ,Q ) k P(k) P(K) K K   y y → K k ExtF(k)(P, Q) ExtF(K)(PK ,QK (t )). 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 ExtP(K)(PK ,QK ) ExtF(K)(PK ,QK ) and the homomorphism on Ext-groups induced by the functor homomorphism QK () : QK 7→ QK (t ). Similarly, there is a commutative diagram (3.5.2) K Ext (P, Q) → Ext (P ,Q ) k P(k) P(K) K K   y y → K k ExtF(k)(P, Q) ExtF(K)(PK (t ),QK ).

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

x :0→ Q → Xm →→X1 → P → 0.

686 Franjou, Friedlander, Scorichenko, and Suslin

The extension t(x) coincides with t (xK ). Note further that we have a canonical homomorphism of extensions

t(x ):0→ Q (t ) → → P (t ) → 0 K K x K x   QK () PK ()

xK :0→ QK → → PK → 0.

The existence of such a homomorphism gives us the desired equality

t(x) PK ()= t (xK ) PK ()=QK () xK .

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 nite elds the Ext-groups ExtF(K)(PK ,QK (t )) occurring in the lower right corner of the commutative square (3.5.1).

Lemma 3.6. Let k be a nite eld with q = pN elements and let k → K be a nite eld extension of degree n. The corresponding functor t : VK →VK coincides with I I(N) I((n1)N). Under this identication, the adjunction homomorphism : I → t (respectively, : t → 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 → t → I is the identity endomorphism.

Proof. The functor in question is given by the formula W 7→ K k W = (K k K)K W . The K K bimodule K k K is canonically isomorphic to the direct sum ∈Gal(K/k) K, where K is K with the standard right K-module structure and the left K-module structure is dened via . In case is the th lN power of the Frobenius automorphism, the K-vector space K K W coincides with W (lN). These remarks allow us to identify t with I I(N) ((n1)N) I . The homomorphism W : K k W → W is given by the formula aw 7→ 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 =IdK . Finally the homomorphism 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 -components of x with =Id6 K are trivial. Finally the property (3.1.2) implies easily that the IdK -component of x is equal to 1.

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 687

As we now show, injectivity of ExtP → ExtF is an easy consequence of Theorems 2.7 and 3.5 provided that we work over a nite base eld. (The hypothesis of niteness of the base eld is required in the proof of Theorem 2.7.)

Corollary 3.7. Let k be a nite eld and consider strict polynomial functors P and Q of nite degree. The canonical homomorphism s → s ExtP(k)(P, Q) ExtF(k)(P, Q) is injective for all s.

Proof. According to Theorem 2.7, we may nd an extension K/k and s (m) (m) → an integer m 0 such that the homomorphism ExtP(K)(PK ,QK ) s ExtF(K)(PK ,QK ) 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 compo- s (m) (m) → s (m) (m) sition of an isomorphism ExtP(K)(PK ,QK ) ExtF(K)(PK ,QK ) and a X7→XQ(m)() s (m) (m) K → s (m) (m) split monomorphism ExtF(K)(PK ,QK ) ExtF(K)(PK ,QK s (m) (m) → s (m) (t )) , we conclude that the map ExtP(k)(P ,Q ) ExtF(k)(P , (m) s Q ) = ExtF(k)(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 ExtF -groups applied to strict polynomial functors.

Proposition 3.8. Let k → K be a nite extension of elds. For P and Q in P(k), there is a natural “extension-of -scalars” k-homomorphism → ExtF(k)(P, Q) ExtF(K)(PK ,QK ) which ts in a commutative square of K-linear maps K Ext (P, Q) → Ext (P ,Q ) k P(k) P(K) K K   (3.8.1) y y → K k ExtF(k)(P, Q) ExtF(K)(PK ,QK ) whose upper horizontal arrow is the isomorphism of Proposition 1.1 and whose vertical maps are the standard homomorphisms.

Proof. Let : t → I be the adjunction homomorphism establishing the left adjointness of t to . The extension-of-scalars homomorphism → ExtF(k)(P, Q) ExtF(K)(PK ,QK ) is dened to be the composition of the ho- → momorphism ExtF(k)(P, Q) ExtF(K)(PK ,QK (t )) of Theorem 3.4 with

688 Franjou, Friedlander, Scorichenko, and Suslin the map 7→ XQK () X→ ExtF(K)(PK ,QK (t )) ExtF(K)(PK ,QK ).

Theorem 3.5 and the fact that QK () QK ()=QK ( ) = Id imply 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 ExtF(k)(P, Q) ExtF(K)(PK ,QK ) provided that P and Q are homogeneous strict polynomial functors of degree d and q = |k|d.

Theorem 3.9. Let k be a nite eld with q = pN elements and let k → K be a degree n extension of nite elds. Consider strict polynomial functors P , Q homogeneous of degree d. If q d, then the extension-of -scalars homomorphism → K k ExtF(k)(P, Q) ExtF(K)(PK ,QK ) is an isomorphism.

Proof. Theorem 3.4 shows that the left-hand side coincides with ExtF(K)(PK ,QK (t )). After this identication our homomorphism coincides QK () with the map induced by the functor homomorphism QK (t ) → QK . This functor homomorphism is split by QK ():QK → QK (t ), which 0 gives us a direct sum decomposition QK (t )=QK QK . To prove the 0 theorem, we must show that ExtF(K)(PK ,QK )=0. Assume rst that there exists an exponential strict polynomial functor Q such that each Qi is homogeneous of degree i and Q = Qd. By Lemma 3.6 and the exponential property of Q, we have a natural direct sum decomposition M ((n 1)N) d0 dn 1 ((n 1)N) QK (t )=QK (I I )= QK (QK ) . d0++dn1=d 0 The summand corresponding to the n-tuple (d, 0,... ,0) is QK and hence QK coincides with the direct sum over n-tuples (d0,... ,dn1) =(6 d, 0,... ,0). The summand in the above direct sum decomposition indexed by (d0,... ,dn1) n1 is homogeneous of degree d0 + d1q + + dn1q = d + d1(q 1) + + n1 dn1(q 1). On the other hand PK is homogeneous of degree d. Since all Ext-groups in the category FK between two homogeneous functors are zero unless their degrees are congruent modulo qn 1 (see e.g. [K1, 3.3]), it suces to note now that for any n-tuple =(6 d,...,0) we have inequalities n1 n1 n1 n 0

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 689

In the general case, we have an injective resolution 0 → Q → Q in l l which each functor Qi is a direct sum of functors of the form S 1 S d , 0 l1 +...+ld = d. Since the functor P(k) →F(K), Q 7→ QK is obviously exact, we get a resolution → 0 → 0 0 Q K (Q)K . 0 The above special case shows that ExtF(K)(PK , (Qi)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 nite eld with q elements and further let P and Q be homogeneous strict polynomial functors of degree d. If q d, then the canonical homomorphism (m) (m) → lim→ ExtP(k)(P ,Q ) ExtF(k)(P, Q) m is an isomorphism (in all degrees ).

Proof. For any s 0 we can nd, according to Corollary 2.8, a nite exten- (m) (m) sion K/k such that the standard homomorphism lim ExtP (P ,Q ) → →m (K) K K ExtF(K)(PK ,QK ) is an isomorphism for all 0 s. According to Proposi- tion 3.8 and Theorem 3.9 we have a commutative diagram (m) (m) (m) (m) K k lim ExtP (P ,Q ) → lim ExtP (P ,Q ) →m (k) →m (K) K K   y =y → K k ExtF(k)(P, Q) ExtF(K)(PK ,QK ) 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 nite eld with q elements. Assume that P, Q ∈ P(k) are homogeneous strict polynomial functors with 0 deg P =6 deg Q

Proof. Vanishing of the left-hand side is obvious and vanishing of the right hand side follows from the fact that deg P 6 deg Q mod (q 1); cf. [K1, 3.3].

690 Franjou, Friedlander, Scorichenko, and Suslin

d(r) dprj (j) d(r) dprj (j) 4. Computation of ExtP ( ,S ) and ExtP ( , )

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 nite degree on the category of nite dimensional vector spaces over an arbitrary (but xed) eld k. The starting point is the determination of the Ext-groups (r) prj (j) (r) prj (j) Vj = Vr,j = ExtP (I ,S ) and Wj = Wr,j = ExtP (I , ). This was achieved in [F-S, 4.5, 4.5.1] by use of 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:

(j1) → dprj+1(j1) → dprj11(j1) dprj+1 :0 S S rj+1 1(j 1) →dp (j 1) → 0 (j) → dprj (j) → dprj 1(j) 1(j) → dprj (j) → Kzdprj :0 S S 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 rj rj Sdp (j) and dp (j) simultaneously, and the Koszul complex serves to link tightly Ext-groups 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) nontrivial dierential in our hypercohomology spectral sequence as tting into a generalized Koszul complex. The map involved is nothing but the nontrivial dierential for the case d = 1, which relates the graded vector spaces Vj and Wj. The same feature reappears when one com- putes the rst hypercohomology spectral sequence for the Koszul complex; this has only one nonzero dierential, which we identify as the dierential for the generalized Koszul complex associated once again to the graded map from Wj to Vj arising in the case d =1.

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 691

Hypercohomology spectral sequences In a familiar manner, there are two spectral sequences associated to applying RHomP (d(r), ) to a complex of functors F in P. The rst has n,m m d(r) n n,m an E1 -term ExtP ( ,F ), whereas the second has an E2 -term n d(r) m ExtP ( , H (F )). Since the Koszul complex is exact, the second spectral (j) sequence associated to Kz dprj is trivial (and hence the abutment of the rst one is zero). We shall use the notation E for the rst spectral sequence asso- (j) ciated to the Koszul complex Kz dprj : n,m m d(r) n(j) dprj n(j) ⇒ E1 = ExtP ( ,S )= 0. Furthermore, we shall use the notation I (resp. II) for the rst (resp. second) (j1) spectral sequence associated to the de Rham complex dprj+1 : n,m m d(r) dprj+1n(j1) n(j1) ⇒ n+m d(r) (j1) I1 = ExtP ( ,S )= ExtP ( , dprj+1 ) , n,m n d(r) dprj m(j) m(j) ⇒ n+m d(r) (j1) II2 = ExtP ( ,S )= ExtP ( , dprj+1 ). To compute the dierentials in the spectral sequences E, I, II, we shall use the following proposition. Proposition 4.1. Let T1 ,T2 be bounded below complexes of homoge- r r neous strict polynomial functors of degrees d1p and d2p respectively. If d = d1 +d2, then the rst (respectively, the second) spectral sequence associated d(r) to applying the functor RHomP ( , ) to the complex T1 T2 is naturally isomorphic to the tensor product of the rst (resp., second) spectral sequences di(r) associated to applying the functors RHom( , ) to Ti . → → Proof. Let T1 I1 ,T2 I2 be the Cartan-Eilenberg resolutions of → T1 and T2 respectively. An immediate verication shows that Tot(T1 T2 ) I1 I2 is the Cartan-Eilenberg resolution of Tot(T1 LT2 ) (here I1 I2 should be considered as a bicomplex with (I I )n,m = 1 2 n1+n2=n,m1+m2=m n1,m1 n2,m2 I1 I2 ). The two spectral sequences in question are the two spectral sequences of the bicomplex d(r) HomP ( ,I1 I2 ) which can be identied by use of Theorem 1.7 with d1(r) d2(r) HomP ( ,I1 ) HomP ( ,I2 ). The result now follows immediately [M, 11, §3, Ex. 6]. The following corollary makes explicit the determination of the dierentials in the tensor product of spectral sequences as considered in Proposi- tion 4.1.

692 Franjou, Friedlander, Scorichenko, and Suslin

Corollary 4.2. All dierentials of the rst hypercohomology spectral sequence associated to applying the functor RHomP (d(r), ) to the complex (j1) d (prj+1 ) are trivial. In the second hypercohomology spectral sequence associated to applying (j1) P d(r) d the functor RHom ( , ) to (prj+1 ) (respectively, the rst hypercohomology spectral sequence associated to applying the functor RHomP (d(r), ) (j) d to (Kz prj ) ), all dierentials except for dpr j +1 (resp. except for dpr j ) are trivial. The only nontrivial dierential sends a decomposable, homogeneous element v1 vd to:

Xd (i) (1) v1 ∂(vi) vd i=1 where ∂ is the only nonzero dierential in the second hypercohomology spec- (j1) P (r) tral sequence associated to applying the functor RHom (I , ) to prj+1 (resp. the rst hypercohomology spectral sequence associated to applying (j) P (r) RHom (I , ) to Kz prj ) and (i) denotes the number of terms of odd total degree among v1,... ,vi1.

Proof. This follows immediately from Proposition 4.1 and the known in- formation about dierentials in the spectral sequences associated to applying (j1) (j) P (r) § the functor RHom (I , )toprj+1 and Kz prj ; see [F-S, 4].

Generalized Koszul complexes

The graded vector space Vj is concentrated in even degrees. It is one dimensional in degrees s 0mod2prj,0 s<2pr, and zero otherwise [F-S, 4.5]. The graded vector space Wj is related to Vj by two homomorphisms. The rst one, which we denote by , is the (only nonzero) dierential

(r) prj (j) → (pr j 1) (r) prj (j) (4.2.1) := dprj : ExtP (I , ) ExtP (I ,S ) in the rst hypercohomology spectral sequence associated to applying the func- (r) (j) tor RHom(I , ) to the Koszul complex Kz prj . This homomorphism is an isomorphism, shifting degrees by prj 1, and gives an inverse to the Yoneda multiplication by the Koszul complex. This implies, in particular, that Wj is concentrated in even degrees for p =6 2 and in odd degrees for p =2,j

(r) prj (j) → +pr j +1 (r) prj (j) (4.2.2) ∂ := dprj +1 : ExtP (I , ) ExtP (I ,S )

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 693 in the second hypercohomology spectral sequence associated to applying the (r) (j1) functor RHom(I , ) to the de Rham complex prj+1 . This homomorphism ts into an exact sequence ∂ 0 → Wj1 → Wj → Vj → Vj1 → 0 which can be interpreted in terms of edge homomorphisms of the corresponding spectral sequence (see [F-S, §4]). Here the homomorphism Vj → Vj1 is the +1 natural map of degree zero induced by the embedding Spr j (j) ,→ Spr j (j1), whereas the homomorphism is the Yoneda multiplication by a class in pr j+1pr j prj+1(j1) prj (j) ExtP ( , ) obtained by truncating the de Rham (j1) rj complex pjj+1 at its last non-vanishing cohomology degree (namely, p ) and using Cartier’s Theorem. In particular, increases degrees by prj+1 prj. We shall use the notation Kj (resp. Cj) for the kernel (resp. cokernel) of ∂, so that the previous exact sequence gives natural isomorphisms of graded vector spaces rj rj+1 Wj1[p p ] → Kj = Ker ∂, Coker ∂ = Cj → Vj1. In order to analyze the spectral sequences E˜ and II, we shall observe that iterations of and ∂ form “generalized Koszul complexes” in the following sense. Let f : W → V be a homomorphism of nite dimensional k-vector spaces. Set M i di i i Qd(f)=S (V ) (W ) ,Q(f)= Qd(f). 0id 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 dierential (of cohomological degree 1), dened in terms of f and product and coproduct operations as follows:

1 comult i di i Sd i(V ) → di i1 : Qd(f)=S (V ) (W ) S (V ) W (W ) 1f1 → Sd i(V ) V i 1(W ) mult1 i 1(W→) di+1 i1 i1 S (V ) (W )=Qd (f). We refer to the resulting complex Qd(f) as the generalized Koszul complex for f. One checks easily that 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 identications 1 0 H1(Q(f)) = Ker f H1(Q(f)) = Coker f.

694 Franjou, Friedlander, Scorichenko, and Suslin

The usefulness of this formalism lies in the following elementary lemma. Lemma 4.3. The induced homomorphism of bigraded algebras S (Coker f) (Ker f) → H (Q(f)) is an isomorphism. Proof. Choose (noncanonical) splittings W = Ker f U, V= U Coker f in such a way that f identies the two copies of U. The dierential graded algebra Q(f) is isomorphic to the tensor product of a trivial dierential graded algebra S(Coker f) (Ker f) and the usual Koszul dierential 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)), acquire an additional grading which we call the internal grading.

(r) (j) The product operation in the tri-graded Hopf algebra ExtP ( ,S ) denes a canonical homomorphism of graded vector spaces d → d(r) dprj (j) Vj ExtP ( ,S ).

Lemma 1.11 (and the fact that Vj is concentrated in even degrees) implies that this homomorphism commutes with the action of the symmetric group d, and thus factors to give a map: d d(r) dprj (j) (4.3.2) S (Vj) → ExtP ( ,S ). We shall also consider the analogous map d → d(r) dprj (j) Wj ExtP ( , ). Lemma 1.11 implies that this map commutes with the action of the symmetric d group d up to a sign, and thus factors through (Wj), provided that p =2.6 d In the case p = 2, the fact that this map factors through (Wj) is established below in Lemma 4.4. We shall employ the notation v1 vi (resp. w1 ∧ ∧ ∈ i ∈ i wi) for the image of v1 vi Vj (resp. w1 wi Wj )in i(r) iprj (j) i(r) iprj (j) ExtP ( ,S ) (resp. in ExtP ( , )). Lemma 4.4. The natural map d → d(r) dprj (j) Wj ExtP ( , ) d factors through (Wj).

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 695

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 suces to consider the case p = 2 and in this case it suces to show that the square of each homogeneous element w in Wj 2(r) 2rj+1(j) dies in ExtP ( , ). We start with the special case j = r. First:

odd 2(r) 2(r) (4.4.1) ExtP ( ,S )=0. To prove this we use the injective resolution of S2(r) constructed in [F-S, §8] 0 → S2(r) → C0 → C1 → ... , where M Cn = Sm0 Sm2r1 . r+1 m0P+ +m2r1=2 r1 imi=n2

Theorem 1.7 implies immediately that HomP (2(r),Cn)=0ifn is odd. The cohomology long exact sequence corresponding to the short exact sequence of strict polynomial functors: 0 → 2(r) → I(r) I(r) → S2(r) → 0 now implies:

(4.4.2) The standard embedding 2(r) ,→ I(r) I(r) induces injective maps

even 2(r) 2(r) even 2(r) (r) (r) ExtP ( , ) ,→ ExtP ( ,I I ).

To nish the argument in the case j = r it suces to note now that the composition

even even 2(r) 2(r) (Wr Wr) → ExtP ( , ) even 2(r) (r) (r) even ,→ ExtP ( ,I I )=(Wr Wr) coincides with 1 + (where permutes the two factors) and hence is zero on elements of the form w w. 2r j 1 2rj 2rj Assume now that j

(r) 2rj (j) kz (j) +2rj 1 (r) 2rj (j) ExtP (I ,S ) → ExtP (I , ).

(j) (r) → 2rj (j) On the other hand rj: I , S induces epimorphisms on (r) ExtP (I , ) [F-S, §4]. This shows that each homogeneous element in Wj = (r) 2rj (j) (j) (j) ExtP (I , ) may be written as a Yoneda product w = kz rj x,

696 Franjou, Friedlander, Scorichenko, and Suslin

(r) (r) for a certain homogeneous element x in ExtP (I ,I ). With this notation we have the following formula: (j) (r) (r) 2rj (j) 2rj (j) ww =((kz kz )(rj rj)) (xx) ∈ ExtP (I I , ). 2(r) 2rj+1(j) The image of w w in ExtP ( , ) is the Yoneda product (j) (r) 2rj 2rj → 2rj+1 m (w w) c , where m : is the multiplica- 2 → tion homomorphism and c : , I I is the standard embedding. The long exact sequence of Ext-groups associated to the short exact sequence of (r) (r) strict polynomial functors 0 → 2(r) c→ I(r) I(r) m→ 2(r) → 0 and (r) (r) (r) the vanishing of m (x x) c established above show that (x x) c (r) ∈ 2(r) 2(r) may be written in the form c y for some y ExtP ( , ). Now it suces to note that the element under consideration is a right multiple of (rj) (j) (rj) ∈ (m (kz kz ) (rj rj) c ) and m (kz kz ) (rj rj) c 2(rj) 2rj+1 ExtP ( , ) = 0; see [F-S, Prop. 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 d → d(r) dprj (j) d → d(r) dprj (j) Vj ExtP ( ,S ) and Wj ExtP ( , ) factor to induce isomorphisms of graded vector spaces d d(r) dprj (j) d d(r) dprj (j) S (Vj) → ExtP ( ,S ) , (Wj) → ExtP ( , ).

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 [F-S, 2.10, 5.4] and duality. In the sequel we shall assume that 0 1 and the result holds for all d0

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 697

Here []m denotes the mth homogeneous component with respect to the internal grading. This formula shows that the map of spectral sequences dened by the homomorphism of complexes (given by the products in symmetric and exterior algebras) (j1) d → (j1) (prj+1 ) dprj+1 is surjective on the E1-terms; thus the lemma follows from the rst part of Corollary 4.2.

Corollary 4.7. d X n d(r) (j1) spr j+1,nspr j+1 dimk ExtP ( , prj+1 )= dimk I1 s=0 Xd ds s nsprj+1 = dimk[S (Vj1) (Wj1)] s=0 Xd ds s nsprj = dimk[S (Cj) (Kj)] . s=0 Here []nspr j+1 stands for the homogeneous component with respect to the internal grading. The shift in degrees in the last identication comes from natural rj+1 rj isomorphisms of graded vector spaces Wj1 = Kj[p p ],Vj1 = Cj. Consider now the spectral sequence II: n,m n d(r) dprj m(j) m(j) ⇒ n+m d(r) (j1) II2 = ExtP ( ,S )= ExtP ( , dprj+1 ). The induction hypothesis on d and Theorem 1.7 allow us to identify all n,m rj II2 -terms except for the two edge rows m =0,dp : 6 rj n,m 0ifm 0modp II2 = ds s n rj [S (Vj) (Wj)] if m = sp , 0

Here v1,... ,vds are homogeneous elements in Vj, w1,... ,ws are homoge- → neous elements in Wj and ∂ : Wj Vj is the dierential dprj +1 in the second

698 Franjou, Friedlander, Scorichenko, and Suslin

(j) hypercohomology spectral sequence corresponding to prj+1 , discussed previ- ously. 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, dprj +1 may be viewed as the generalized Koszul dierential in Qd(∂). Lemma 4.3 together with (4.7.1) implies that we get a canonical homomorphism

d d1 d S (Cj) = Coker(S (Vj) Wj → S (Vj)) → ,0 d1 → d(r) dprj (j) IIprj +2 = Coker(S (Vj) Wj ExtP ( ,S )) . Lemma 4.8. The resulting homomorphism d → ,0 → ,0 → d(r) (j1) S (Cj) IIprj +2 II∞ ExtP ( , dprj+1 ) is injective.

Proof. It is clear from the construction that the homomorphism in question is induced by the composition d → d(r) dprj (j) → d(r) (j1) S (Vj) ExtP ( ,S ) ExtP ( , dprj+1 )

rj where the second arrow is dened by the homomorphism of complexes Sdp (j) → (j1) dprj (j) dprj+1 (identifying S with the zero-dimensional homology group of (j1) (j1) → dprj+1(j1) dprj+1 ). Consider also the natural homomorphism dprj+1 S (projection onto the zero-dimensional component). The composition of these homomorphisms coincides with the standard embedding Sdpr j (j) ,→ dprj+1(j1) d(r) S . The induction hypothesis on j shows that ExtP ( , dprj+1(j1) d S )=S (Vj1). The composite

d d(r) (j1) d(r) dprj+1(j1) d → → S (Cj) ExtP ( , dprj+1 ) ExtP ( ,S )=S (Vj 1) is by construction the isomorphism Cj → Vj1 for d = 1 and thus (by its multiplicative nature) is an isomorphism for all d.

Now we can start the nal 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 mth 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. d t t d(r) dprj (j) 1t. The natural map [S (Vj)] → ExtP ( ,S ) is an isomorphism. d t t d(r) dprj (j) 2t. The natural map [ (Wj)] → ExtP ( , ) is an isomorphism. n,m n,m rj 3t.IIprj +2 =II∞ for all n t p 1.

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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 t0

d d npr j 1,(s+1)pr j pr j +1→ n,spr j pr j +1→ n+pr j +1,(s1)pr j IIprj +1 IIprj +1 IIprj +1 . Here the top row is a piece of the generalized Koszul complex, corresponding to the homomorphism (shifting degrees by prj + 1) of graded vector spaces ∂ : Wj → Vj. Our induction hypothesis on d immediately implies that the three vertical maps in the above diagram are isomorphisms unless s =0, 1, d1,d. Examining each of these four cases separately and using our induction hypothesis on t, we easily conclude the following:

ds s n → n,spr j (4.9.2) The induced map [S (Cj) (Kj)] IIprj +2 , given as the map on cohomology of (4.9.1), a) is an isomorphism provided that s =06 , 1,d 1,dor n

Consider the terms of total degree t in IIprj +2. The dierentials that t,0 n,m rj could possibly hit IIprj +2 should come from II -terms with n

tsprj ,sprj tspr j ,spr j ds s tsprj II∞ =IIprj +2 =[S (Cj) (Kj)] according to 3t1 and (4.9.2 a). Thus (4.9.3) t d(r) (j1) t,0 tpr j ,pr j dimk ExtP ( , prj+1 ) = dimk IIprj +2 + dimk IIprj +2 Xd ds s tsprj + dimk[S (Cj) (Kj)] . s=2 Finally we have the following inequalities: t,0 d t dimk IIprj +2 dimk[S (Cj)] , according to (4.8), tpr j ,pr j d1 tprj dimk IIprj +2 dimk[S (Cj) Kj] .

700 Franjou, Friedlander, Scorichenko, and Suslin

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 homo- d t → t,0 d1 tprj → tpr j ,pr j morphisms [S (Cj)] IIprj +2 and [S (Cj) Kj] IIprj +2 are isomorphisms. Inspecting the diagram (4.9.1) for s =1,n = t prj 1 (extended by zeros to the right) and using the Five-Lemma we conclude that d t t d(r) dprj (j) [S (Vj)] → ExtP ( ,S ) is an isomorphism ; in other words, 1t holds. Using this fact we extend easily the conclusion made in (4.9.2a) to degrees n t prj 1:

ds s n → n,spr j (4.9.4) [S (Cj) (Kj)] IIprj +2 is an isomorphism for all s provided that n t prj 1.

We now prove 3t. Let E denote the second hypercohomology spectral se- (r) (j1) quence obtained by applying RHom(I , ) to the de Rham complex prj+1 . (j1) d → (j1) The natural homomorphism of complexes (prj+1 ) dprj+1 denes (in view of Proposition 4.1) a homomorphism of spectral sequences Ed → II. We conclude from (4.9.4) that the homomorphism d n,m → n,m (E )prj +2 IIprj +2 rj rj is surjective for n t p 1. Since the dierentials dk with k>p +1 are trivial in E, they are trivial in Ed. We conclude that all the dierentials rj n,m rj dk in II with k>p + 1 starting at II -terms with n t p 1 are n,m n,m trivial. So IIprj +2 =II∞ for these values of n, i.e. 3t holds. To prove that 2t holds, we consider the spectral sequence E (the rst (j) spectral sequence corresponding to Kz dprj ). The E1-term of this spectral sequence is given (using Theorem 1.7) by the formula m,n n d(r) m(j) dprj m(j) E1 = ExtP ( ,S ) 0ifm 6 0modprj = s ds n rj [S (Vj) (Wj)] if m = sp 0

The dierentials d1,... ,dprj 1 are trivial by dimension considerations. Corol- lary 4.2 allows us to determine the action of dprj on decomposable elements, so that as for (4.9.1) we obtain a commutative diagram (4.9.5) ds+1 n+prj 1 → ds n → ds1 nprj +1 [Qd ()] [Qd ()] [Qd ()]    y y y

d d (s1)pr j ,n+pr j 1 →pr j spr j ,n →pr j (s+1)pr j ,npr j +1 Eprj Eprj Eprj . Here the top row is a piece of the (acyclic) generalized Koszul complex, corresponding to the isomorphism (shifting degrees by prj 1) of graded vector

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 701 spaces : Wj → Vj (cf. (4.2.1)). To conclude that the bottom row of (4.9.5) is acyclic it suces by the acyclicity of the top row to know that the central and the right-hand side vertical arrows are isomorphisms. This remark plus our induction hypotheses on d and t imply:

spr j ,n 6 (4.9.6) Eprj +1 =0provided that s =0,d 1, d,orn

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

s (m) (m) s (m+1) (m+1) ExtP (A ,B ) → ExtP (A ,B ) s+1 is an isomorphism provided that m logp 2 . Proof. Consider rst the special case A =d, B = Sd. Theorem 4.5 shows that d(m) d(m) d (m) (m) ExtP ( ,S )=S (ExtP (I ,I )).

(m) (m) (m+1) (m+1) Since the Frobenius twist map ExtP (I ,I ) → ExtP (I ,I )isan isomorphism in degrees 2pm 1 [F-S;4.9], we conclude that the induced d (m) (m) d (m+1) (m+1) map S (ExtP (I ,I )) → S (ExtP (I ,I ) is an isomorphism in the same range. 0 0 Consider next the special case A =d1 da , B = Sd 1 Sd b , 0 0 d1 + + da = d 1 + + d b = d. In this case, Corollary 1.8 gives us the following formula M abO (m) (m) d (m) d (m) ExtP (A ,B )= ExtP ( ij ,S ij ). 0 d11++d1b=d1 d11++da1=d 1 i=1,j=1 ...... 0 da1++dab=da d1b++dab=d b Thus the result in this case follows from the previous one. In the general case, we can nd resolutions 0 A P 0 → B → I

702 Franjou, Friedlander, Scorichenko, and Suslin

j d in which each Pi (resp. each I ) is a direct sum of functors of the form 1 0 0 da (resp. of the form Sd 1 Sd b ). The result now follows by applying the standard comparison theorem to the map of the hypercohomology (m) (m) (m+1) (m+1) spectral sequences converging to ExtP (A ,B ) and ExtP (A ,B ) 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 xed eld 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 xing a positive integer r. The general philosophy behind these computations is as follows. There is a sort of hierarchy between the functors (),() and S():() has a tendency to be projective (it really is if there are no twists), S() has a tendency to be injective and () 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 specically, we compute all Ext-groups from () (with any number of twists) to (),() and S(), we compute all Ext-groups from () to () and S() and nally from S() to itself; we suspect that there is no comparable computation of the Ext-groups from S() to () or () or Ext- groups from () to () (actually it seems that there is just no easy answer for these Ext-groups). prj (j) (r) For nonnegative integers 0 j r, set Uj = Ur,j = ExtP ( ,I ). (r) (r) Observe that Ur = Vr = Wr = ExtP (I ,I ); moreover, the duality isomorphism (1.12) denes canonical isomorphisms of graded vector spaces

prj (j) (r) (r) prj (j) Uj = ExtP ( ,I ) → ExtP (I ,S )=Vj for any j r. For any nonnegative integer d the product operations dene canonical homomorphisms d dprj (j) (r) (r) → dprj (j) d(r) Uj = ExtP ( ,I I ) ExtP ( ,S ) , d dprj (j) (r) (r) → dprj (j) d(r) Uj = ExtP ( ,I I ) ExtP ( , ). Using these homomorphisms, we formulate our rst 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.

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 703

Theorem 5.1. For any nonnegative integer d the above maps dene natural isomorphisms

d dprj (j) d(r) S (Uj) → ExtP ( ,S ) , d dprj (j) d(r) (Uj) → ExtP ( , ).

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

d → dprj (j) d(r) S (Uj) ExtP (  ,S )   =y =y d d(r) dpr j (j) S (Vj) → ExtP ( ,S ). Here the vertical arrows are the duality isomorphisms of (1.12), the bottom horizontal arrow is the isomorphism of Theorem 4.5 and the top horizontal arrow is the map under consideration. To prove the second isomorphism, we proceed by induction on d. Con- sider the rst hypercohomology spectral sequence corresponding to the Koszul complex

(r) → d(r) → d1(r) 1(r) →→ 1(r) d1(r) → d(r) → Kz d :0 S S S 0 ,

m,n n dprj (j) dm(r) m(r) ⇒ E1 = ExtP ( , S )= 0. The inductive assumption on d and the already-proved part of the theorem give the computation of all the E1-terms except those in the rst column. Moreover, we have the following commutative diagram of complexes

→ d → d1 → → d → 0 (Uj) (Uj) Uj ... S (Uj) 0   =y =y

→ 0, →d1 1, →d1 →d1 d, → 0 E1 E1 ... E1 0.

Here the top row is the Koszul complex corresponding to the vector space Uj, and the bottom row is the E1-term of our spectral sequence. The acyclicity m, of the Koszul complex implies immediately that E2 = 0 for m>1. This implies further vanishing of all higher dierentials and hence the identication E2 = E∞ = 0. Thus the bottom complex in the above diagram is also acyclic, dprj (j) d(r) 0, d which gives a natural identication ExtP ( , )=E1 = (Uj). The duality isomorphism (1.12) together with Theorems 4.5 and 5.1 immediately leads to the following statements.

704 Franjou, Friedlander, Scorichenko, and Suslin

Corollary 5.2. For all 0 j r and all d 0 there are natural isomorphisms

d prj (j) (r) dprj (j) d(r) (ExtP ( ,I )) → ExtP ( ,S ) , d (r) prj (j) d(r) dprj (j) (ExtP (I ,S )) → ExtP ( ,S ).

() () The computation of ExtP ( , ) in Theorem 5.4 below requires a deter- () () mination of the coproduct (rather than product) operation in ExtP ( , ). Recall that comultiplication in the exterior algebra denes natural homomorphisms (for any 0 j r and any d 0)

rj rj rj dp (j) ,→ p (j) p (j) , d(r) ,→ I(r) I(r) which induce homomorphisms on Ext-groups d(r) dprj (j) → d(r) prj (j) prj (j) d ExtP ( , ) ExtP ( , )=Wj , dprj (j) d(r) → dprj (j) (r) (r) d ExtP ( , ) ExtP ( ,I I )=Uj . Corollary 5.3. For all 0 j r and all d 0 the above coproduct operations induce natural isomorphisms of graded vector spaces:

d(r) dprj (j) d ExtP ( , ) → (Wj), dprj (j) d(r) d ExtP ( , ) → (Uj).

Proof. A straightforward computation shows that the composite maps d → d(r) dprj (j) → d (Wj) ExtP ( , ) Wj , d → dprj (j) d(r) → d (Uj) ExtP ( , ) Uj coincide with the corresponding natural embeddings.

() () Now we turn to the computation of ExtP ( , ). We introduce one more (r) prj (j) basic Ext-space (dened for all 0 j r) V j = V r,j = ExtP (I , ). The dierential dprj in the rst hypercohomology spectral sequence, corresponding to the dual Koszul complex #(j) → prj (j) → prj 1(j) 1(j) →→ prj (j) → Kz prj :0 0 denes an isomorphism : V j → Wj of graded vector spaces (shifting degrees by prj 1).

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 705

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

d(r) dprj (j) d(r) prj (j) prj (j) ExtP ( , ) → ExtP ( , ) (r) prj (j) d d = ExtP (I , ) = V j , dprj (j) d(r) dprj (j) (r) (r) ExtP ( , ) → ExtP ( ,I I ) prj (j) (r) d d = ExtP ( ,I ) = Uj . Lemma 1.11 shows that these homomorphisms are -invariant and hence their d d images are contained in the subspaces of d-invariant tensors, i.e. in (V j) d and (Uj) respectively. Theorem 5.4. For all 0 j r and all d 0 the above constructed homomorphisms d(r) dprj (j) d ExtP ( , ) → (V j) , dprj (j) d(r) d ExtP ( , ) → (Uj) are isomorphisms.

Proof. The proof is essentially the same in both cases. We consider only the rst one (which is slightly more complicated). We proceed by induction on d. Consider the rst hypercohomology spectral sequence corresponding to the dual Koszul complex #(j) → dprj (j) → dprj 1(j) 1(j) →→ dprj (j) → Kz dprj :0 0 ,

m,n n d(r) dprj m(j) m(j) ⇒ E1 = ExtP ( , )= 0. m, 6 rj The columns E1 of this spectral sequence with m 0modp are trivial. Furthermore

spr j , d(r) (ds)prj (j) sprj (j) E1 = ExtP ( , ) ds(r) (ds)prj (j) s(r) sprj (j) = ExtP ( , ) ExtP ( , ).

Our induction hypothesis on d and Corollary 5.3 show further that comultipli- rj sp , → ds s cation denes (for s>0) natural isomorphisms E1 (V j) (Wj). The rst (and the only) nontrivial dierential in this spectral sequence is sprj , (s+1)pr j , → dpr j : Eprj Eprj . Considering the natural homomorphism of #(j) → #(j) d complexes Kz dprj (Kz prj ) (dened by comultiplication in the divided power and exterior algebras) and using Proposition 4.1, we nally get a commutative diagram of complexes

706 Franjou, Friedlander, Scorichenko, and Suslin

d d d → 0, →pr j pr j , →pr j →pr j dpr j , → 0 Eprj Eprj Eprj 0       y =y =y d d1 d 0 → (V j) → (V j) Wj → → (Wj) → 0.

Here the bottom row is the dual of the generalized Koszul complex for the isomorphism of graded vector spaces : V j → Wj. Since the bottom row is sprj , acyclic, we conclude that Eprj +1 = 0 for s>1. This implies the vanishing 0, of all higher dierentials and hence gives the following relations : 0 = E∞ = 0, Eprj +1. 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, there are natural isomorphisms

dprj (j) d(r) d prj (j) (r) ExtP (S ,S ) → (ExtP (S ,I )) ,

d(r) dprj (j) d (r) prj (j) ExtP (S ,S ) → (ExtP (I ,S )).

To compute Ext((), ()) we proceed in the same way as above. For any 0 j r and any d 0, the comultiplication in the exterior algebra denes natural homomorphisms

d(r) dprj (j) d(r) prj (j) prj (j) ExtP ( , ) → ExtP ( , )

(r) prj (j) d = ExtP (I , ) ,

dprj (j) d(r) dprj (j) (r) (r) ExtP ( , ) → ExtP ( ,I I )

prj (j) (r) d = ExtP ( ,I ) .

By Lemma 1.11, these homomorphisms are d-invariant and hence their im- d (r) prj (j) d prj (j) (r) ages are contained in (ExtP (I , )) and (ExtP ( ,I ) 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.

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 707

Theorem 5.6. For all 0 j r and all d 0 the natural homomorphisms d(r) dprj (j) d (r) prj (j) ExtP ( , ) → (ExtP (I , )) ,

dprj (j) d(r) d prj (j) (r) ExtP ( , ) → (ExtP ( ,I )) are isomorphisms. We phrase our next result in terms of a chosen vector-space basis in our Ext-groups. (j) pj For an integer j, let j denote the natural embedding j : I → S and # # pj → (j) let j denote the dual homomorphism j : I . We recall from [F-S, (j) 4.5.6] that composition with rj, j r, induces an isomorphism s (r) (r) s (r) prj (j) ExtP (I ,I ) → ExtP (I ,S )

rj pj 1 pj pj for s 0mod2p . Let kz j denote the extension class in ExtP (S , ) # represented by the Koszul complex and let kz j denote the pj 1 pj pj extension class in ExtP ( , ) represented by the dual Koszul complex. 2pr 1 (r) (r) Dene er to be the class in ExtP (I ,I ) dened by the following equation 2pr 1 (r) pr1(1) in ExtP (I ,S ) (cf. [F-S, §4]):

(1) (1) (1) (5.7) r1 er = dpr1+1(kz r1 r1) ,

pr11,pr1 where dpr 1+1 is the dierential (applied to the Epr1 -term) of the second hypercohomology spectral sequence associated to applying RHom(I(r), )to the de Rham complex pr . (r) (r) Recall that the graded vector space ExtP (I ,I ) is one dimensional in r 2m (r) (r) even degrees < 2p and is zero otherwise. We use the basis of ExtP (I ,I ) given by the Yoneda products m0(r 1) mr1 er(m)=e1 er , r1 m = m0 + m1p + + mr1p (0 mi

(1) r Thus, er(m) = er+1(m) for all r and 0 m

708 Franjou, Friedlander, Scorichenko, and Suslin

Theorem 5.8. Let j and r be integers,0 j r. (1) The tri-graded Hopf algebra (j) (r) ExtP ( ,S )

rj #(j) is a primitively generated polynomial algebra on generators er(mp )rj 2pr j m prj (j) (r) j rj rj in ExtP ( ,I ), 0 m

(j) rj is a primitively generated polynomial algebra on generators rjer(mp ) 2pr j m (r) prj (j) j rj rj in ExtP (I ,S ), 0 m

rj #(j) is a primitively generated exterior algebra on generators er(mp )rj 2pr j m prj (j) (r) j rj rj in ExtP ( ,I ), 0 m

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 709

is a primitively generated divided power algebra on generators rj #(j) 2mpr j prj (j) (r) j er(mp )rj in ExtP ( ,I ), 0 m

(r) (j) ExtP (S ,S ).

(6) The tri-graded Hopf algebra

(r) (j) ExtP ( , )

is a primitively generated divided power algebra on generators (j) (j) rj 2mpr j +pr j 1 (r) prj (j) j kz rjrjer(mp ) in ExtP (I , ), 0 m

(j) (r) ExtP ( , ).

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

(r) prj (j) (5.8.1) S (Vj) → ExtP ( ,S )

(r) (j) is multiplicative. This identies ExtP ( ,S ) as a tri-graded algebra. Fi- nally the elements of Vj 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].

2pr 1 (r) (r) Corollary 5.9. The image of er ∈ ExtP (I ,I ) in 2pr 1 pr1(1) pr1(1) pr 1 ExtP ( ,S ) equals e1 (up to a nonzero scalar factor). More- over if one modies the denition of er the way it was done in [S-F-B], then pr1 the image of er is exactly e1 .

2pr 1 p(r1) Proof. We start by computing the image of er in ExtP ( , p(r1) 2pr 1 (r) (r) S ). To do so, note that the homomorphism ExtP (I ,I ) → 2pr 1 p(r1) p(r1) ExtP ( ,S ) is a homogeneous component of a homomorphism of tri-graded Hopf algebras (r) (r) (r1) (r1) : ExtP ( , S ) → ExtP ( ,S )

710 Franjou, Friedlander, Scorichenko, and Suslin induced by natural homomorphisms of exponential functors (r1) → (r), S(r) → S(r1). Here denotes the p-rarefaction of an exponential functor, i.e. Am if n = pm An = 0ifn 6 0modp.

2pr 1 (r) (r) (r) (r) Since the element er ∈ ExtP (I ,I ) is primitive in ExtP ( , S ) (r1) (r1) (r1) (r1) its image in ExtP ( ,S )=S (ExtP (I ,I )) also has to be primitive. The subspace of primitive elements in a (primitively generated) 2 symmetric algebra S(V ) coincides with a linear span . From this we derive immediately that the only primitive element of tri-degree r1 (r1) (r1) p (2p ,p,p)inS (ExtP (I ,I )iser1.Thus(er) is a scalar multiple p n of er1. Since is multiplicative we conclude further that (er ) is a scalar pn multiple of er1 for each n. Repeating this construction r 1 times we nally 2pr 1 pr1(1) pr1(1) conclude that the image of er in ExtP ( ,S ) is a scalar multiple pr1 of e1 . We proceed to show now that with the modication made in [S-F-B, 3.4] 2pr 1 p(r1) p(r1) p the image of er in ExtP ( ,S ) is exactly er1 (i.e. the corresponding scalar factor is the identity). Recall that (with the modication made in [S-F-B]) for any n 2 the image of er under the natural homomorphism 2pr 1 (r) (r) 2pr 1 n(r) n(r) 2pr 1 n(r) n(r) ExtP (I ,I ) → Ext (k ,k ) → Ext (k ,k ) GLn GLn(1) 2pr1 (r) (r)# 2pr1 = H (GLn(1),gln ) = Homk(gln ,H (GLn(1),k)). is nontrivial [F-S, §6] and coincides with the composition (r)# → pr1 (1)# → 2pr1 gln S (gln ) H (GLn(1),k) where the rst arrow is the standard embedding and the second one is the edge homomorphism in the May spectral sequence. Consider further the natural homomorphism 2pr 1 p(r1) p(r1) → 2pr1 p(r1) n # p(r1) n ExtP ( ,S ) H (GLn(1), ( (k )) S (k )) p(r1) n p(r1) n# 2pr1 = Homk( (k ) (k ),H (GLn(1),k)) .

The image of (er) under this map coincides with the composite p(r 1)(kn) p(r 1)(kn#) →→ kn(r) kn(r)# (r)# → pr1 (1)# → 2pr1 = gln S (gln ) H (GLn(1),k) and, in particular, is nontrivial. Keeping in mind that the tensor product operation Ext (k, k) Ext (k, k) → Ext (k, k) GLn(1) GLn(1) GLn(1)

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 711 coincides with the usual product operation in the cohomology ring Ext (k, k)=H(GL ,k), using the description of the product operation GLn(1) n(1) (r1) (r1) in ExtP ( ,S ) in terms of the tensor product operation and the mul- (1) → 2 tiplicative properties of the May homomorphism S (gln ) H (GLn(1),k), p ∈ 2pr 1 p(r1) p(r1) we easily conclude that the image of er1 ExtP ( ,S )in

p(r1) n p(r1) n# 2pr1 Homk( (k ) (k ),H (GLn(1),k) is given by the composition

p(r 1)(kn) p(r 1)(kn#) ,→ (kn(r 1)) p (kn(r 1)#) p (r1) p → pr2 (1) p →→ pr1 (1) → 2pr1 =(gln ) (S (gln )) S (gln ) H (GLn(1),k).

A straightforward verication shows that the resulting maps p(r1)(kn) p(r1) n# → 2pr1 (k ) H (GLn(1),k) coincide. Since this map is moreover non- p trivial we conclude that the constant relating (er) and er1 is equal to the identity.

To simplify the notation, we adopt the following convention: for homogeneous strict polynomial functors A, B in P we dene (m) (m) ExtP→F(A, B) = lim→ ExtP (A ,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 nonnegative integer.

(1) The tri-graded Hopf algebra (h) ExtP→F( ,S )

h # is a primitively generated polynomial algebra on generators e(mp )h in 2phm ph (h) h h ExtP→F( ,I ), 0 m, of tri-degree (2p m, p , 1). Similarly, the tri-graded Hopf algebra (h) ExtP→F( ,S )

h is a primitively generated polynomial algebra on generators he(mp ) in 2phm (h) ph h h ExtP→F(I ,S ), 0 m, of tri-degree (2p m, 1,p ). (2) The tri-graded Hopf algebra (h) ExtP→F( , )

712 Franjou, Friedlander, Scorichenko, and Suslin

h # is a primitively generated exterior algebra on generators e(mp )h in 2phm ph (h) h h ExtP→F( ,I ), 0 m, of tri-degree (2p m, p , 1). A similar statement holds by duality for the tri-graded Hopf algebra

(h) ExtP→F( ,S ).

(3) The tri-graded Hopf algebra

(h) ExtP→F( , )

h is a primitively generated exterior algebra on generators kz hhe(mp ) in 2mph+ph1 (h) ph h h h ExtP→F (I , ), 0 m, of tri-degree (2mp + p 1, 1,p ). A similar statement holds by duality for the tri-graded Hopf algebra

(h) ExtP→F( ,S ).

(4) The tri-graded Hopf algebra

(h) ExtP→F( , )

is a primitively generated divided power algebra on generators # h 2mph+2ph2 (h) ph kz h kz hher(mp ) in ExtP→F (I , ), 0 m, of tri-degree (2mph +2ph 2, 1,ph). A similar statement holds by duality for the tri-graded Hopf algebra

(h) ExtP→F(S ,S ).

(5) The tri-graded Hopf algebra

(h) ExtP→F( , )

h # is a primitively generated divided power algebra on generators e(mp )h 2mph ph (h) h h in ExtP→F( ,I ), 0 m, of tri-degree (2mp ,p , 1). A similar statement holds by duality for the tri-graded Hopf algebra

(h) ExtP→F(S ,S ).

(6) The tri-graded Hopf algebra

(h) ExtP→F( , )

h is a primitively generated divided power algebra on generators kz hhe(mp ) 2mph+ph1 (h) ph h h h in ExtP→F (I , ), 0 m, of tri-degree (2mp + p 1, 1,p ). A similar statement holds by duality for the tri-graded Hopf algebra

(h) ExtP→F( , ).

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 713

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 re- duce 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 nite eld with q = pN elements, F = F(k) Vf →V P P is the category of functors k k and = (k) is the category of strict polynomial functors. Finally, A and B are exponential (homogeneous) strict polynomial functors with deg(Ai) = deg(Bi)=i. h For nonnegative integers h

Nh nNh j = j0 + + jn1 + l1p + + lnp , h (n1)N+h l = j0p + + jn1p + l1 + + ln. h → h The obvious embedding Prn(j, l) , Prn+1(j, l) (taking jn = ln+1 =0)isan isomorphism, provided that p(n+1)Nh >jand pnN+h >l. We denote by h h Pr (j, l) the limit set lim Prn(j, l). →n h For any sequence j0,... ,jn1,l1,... ,ln in Prn(j, l) consider the following homomorphism

On1 On j (sN+h) j psN+h l ptNh(h) l (tN) ExtP→F(A s ,B s ) ExtP→F(A t ,B t ) s=0 t=1 On1 On j (sN+h) j psN+h l ptNh(h) l (tN) → ExtF (A s ,B s ) ExtF (A t ,B t ) s=0 t=1 On1 On j (h) j psN+h l ptNh(h) l = ExtF (A s ,B s ) ExtF (A t ,B t ) s=0 t=1 j(h) l → ExtF (A ,B ).

Our next result determines ExtF (A ,B ) in terms of ExtP→F(A ,B ).

Theorem 6.1. Fix a nonnegative integer h less than N. The resulting homomorphism M O O j (sN+h) j psN+h l ptNh(h) l (tN) ExtP→F(A s ,B s ) ExtP→F(A t ,B t ) Prh(j,l) s0 t1 j(h) l → ExtF (A ,B ) is an isomorphism.

714 Franjou, Friedlander, Scorichenko, and Suslin

Proof. Choose an integer n so that pnNh >j, pnN >land consider a tower of nite eld extensions L K k in which each stage is of degree n. Theorem 3.4 gives a natural isomorphism j(h) l j(h) l K k ExtF(k)(A ,B ) = ExtF(K)(AK (t ),BK ). Moreover, according to Lemma 3.6, the functor t may be identied with I I(N) I((n1)N). The exponential property of A gives now a natural isomorphism M j(h) j0(h) jn1((n 1)N+h) AK (t )= AK AK . j0++jn1=j Extending scalars further from K to L and applying now the previous proce- dure to the second variable, we get the following formula:

M On1 On1 j(h) l js(sN+h) lt(tnN) L k ExtF(k)(A ,B )= ExtF(L)( AL , BL ). j0++jn1=j s=0 t=0 l0++ln1=l

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

M On1 js,t(sN+h) lt,s(tnN) ExtF(L)(AL ,BL ). j0,0++j0,n1=j0 l0,0++l0,n1=l0 s,t=0 ...... jn1,0++jn1,n1=jn1 ln1,0++ln1,n1=ln1

Note further that js,t(sN+h) sN+h (n1)N+h j card(L) deg(A )=js,t p j p = < card(L) L p(n2n+1)Nh and also

l card(L) deg(Blt,s(tnN))=l ptnN l p(n 1)nN = < card(L). L t,s pnN Theorem 3.10 and Lemma 3.11 now show that we may replace ExtF by ExtP→F everywhere in the above formula. This implies, in particular, that the summand corresponding to {jt,s,lt,s}t,s is trivial unless the corresponding degrees sN+h tnN coincide, i.e. unless we have the following relations: js,tp = lt,sp for all s, t. It is easy to see from the choice of n that the above relation implies that js,t = lt,s = 0 unless t =0ort = 1. The remaining jt,s and ls,t are related by equations sN+h (ns)Nh l0,s = js,0p ,js,1 = l1,sp .

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 715 P P 0 n1 1 n Since s,t js,t = j, s,t ls,t = l, we conclude that (j ,... ,j ,l ,... ,l )= h (j0,0,... ,jn1,0,l1,n1,... ,l1,0) is an element of Prn(j, l), i.e. we are left with h the direct sum over Prn(j, l): M On1 j(h) l js(sN+h) kspsN+h L k ExtF(k)(A ,B )= L k ExtP→F(A ,B ) h Prn(j,l) s=0 On ltptNh(h) lt(tN) ExtP→F(A ,B ). t=1 Finally one checks without diculty 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

(sN+h) (sN+h) (h) ExtP→F(A ,B ) → ExtF (A ,B ) = ExtF (A ,B ) (tNh) (h) (tN) (h) (tN) ExtP→F(A ,B ) = ExtP→F(A ,B ) → ExtF (A ,B ) (h) = ExtF (A ,B ).

(sN+h) Furthermore, it is easy to see that the tri-graded space ExtP→F(A ,B ) is psN+h-connected with respect to the total degree, whereas (tNh) tNh ExtP→F(A ,B )isp -connected with respect to the total degree, which allows us to consider the innite tensor product O O (sN+h) (tNh) ExtP→F(A ,B ) ExtP→F(A ,B ). s0 t1 Suppose now that A and B are Hopf functors. The product operations (h) in ExtF (A ,B ) dene a natural homomorphism from the above innite (h) tensor product to ExtF (A ,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 O O (sN+h) (tNh) (h) ExtP→F(A ,B ) ExtP→F(A ,B ) → ExtF (A ,B ) s0 t1 is an isomorphism of tri-graded Hopf algebras.

716 Franjou, Friedlander, Scorichenko, and Suslin

Combining Corollary 6.2 with Corollary 5.9 we are able to compute various important Ext-algebras in the category F.

Theorem 6.3. Let h be a nonnegative integer.

(1) The tri-graded Hopf algebra

(h) ExtF ( ,S )

is a primitively generated polynomial algebra on generators h+sN 2ph+sN m (h) ph+sN h+sN e(mp ) in ExtF (I ,S ), 0 m, 0 s, of h+sN h+sN tNh # tri-degree (2p m, 1,p ) and generators e(mp )tNh in 2ptN hm ptNh(h) tNh tNh ExtF ( ,I), 0 m, 1 t, of tri-degree (2p m, p , 1). (2) The tri-graded Hopf algebra

(h) ExtF ( , )

is a primitively generated exterior algebra on generators h+sN 2mph+sN +ph+sN 1 (h) ph+sr kz h+sN h+sN e(mp ) in ExtF (I , ), 0 m, 0 s, of tri-degree (2mph+sN + ph+sN 1, 1,ph+sN ) and generators tNh #(h) 2ptN hm ptNh(h) e(mp )tNh in ExtF ( ,I), 0 m, 0 s, of tri-degree (2ptNhm, ptNh, 1). (3) The tri-graded Hopf algebra

(h) ExtF ( , )

is a primitively generated divided power algebra on generators # h+sN 2mpsN+h+2psN+h2 (h) psN+h kz h+sN kz h+sN h+sN e(mp ) in ExtF (I , ), 0 m, 0 s, of tri-degree (2mph+sN +2ph+sN 2, 1,ph+sN ) and gen- tNh #(h) 2mptN h ptNh(h) erators e(mp )tNh in ExtF ( ,I), 0 m,1 t, of tri-degree (2mptNh,ptNh, 1). A similar statement holds by duality for the tri-graded Hopf algebra

(Nh) ExtF (S ,S ).

(4) The tri-graded Hopf algebra

(h) ExtF ( , )

is a primitively generated divided power algebra on generators sN+h 2mpsN+h+psN+h1 (h) psN+h kz h+sN h+sN e(mp ) in ExtF (I , ), 0 m, 0 s, of tri-degree (2mpsN+h + psN+h 1, 1,psN+h) and generators tNh #(h) #(h) 2mptN h+ptN h1 ptNh(h) e(mp )tNhkz tNh in ExtF ( ,I), 0 m, 1 t, of tri-degree (2mptNh + ptNh 1,ptNh, 1).

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 717

Appendix: GL-cohomology and ExtF -groups By A. Suslin

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

Ext (P, Q) → Ext (P (kn),Q(kn)) = H (GL (k),P(kn)# Q(kn)) F k[GLn(k)] n

n induced by the exact functor F(k) →{k[GLn(k)] modules} : P 7→ P (k ) (here k[GLn(k)] stands for the group ring of the discrete group GLn(k) over the eld k). All GLn(k)-modules considered below are always assumed to be k-vector spaces on which the group GLn(k) acts by k-linear transformations (i.e. k[GL (k)]-modules); we abbreviate the notation Ext to Ext . n k[GLn(k)] GLn(k) Note that the Ext (P (kn),Q(kn))-groups above stabilize with the growth GLn(k) of n, according to a well-known theorem of W. Dwyer [D], provided that the functors P and Q are nite. We use the notation ExtGL(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 nite eld. Then the natural map

→ ExtF (P, Q) ExtGL(k)(P, Q) is an isomorphism for any nite functors P, Q ∈F= F(k).

Over more general elds one has to compare ExtF -groups with the so- called stable K-theory (see [B-P]). The above theorem shows that the conjecture stated in [B-P] (saying that for nite functors stable K-theory equals topological Hochschild homology) holds at least over nite elds. Apparently an appropriate modication of the argument given below should work over any eld of positive characteristic. I did not try to work out this more general case, the more so that ExtF -groups do not seem to be computable unless we are dealing with nite elds. Theorem A.1 was known to be true (in a more general situation) in case P = Q = I by a work of Dundas and McCarthy [D-M]. For nite elds a purely algebraic proof of this result was given in [F-S]. In a recent paper [B2], S. Betley gives a dierent proof of Theorem A.1, following the approach developed in [F-S]. Throughout this appendix k = Fq is a nite eld of characteristic p (with q elements).

Lemma A.2. For any nite functors A, B ∈F there is a natural duality # # isomorphism ExtGL(k)(A, B) = ExtGL(k)(B ,A ) which makes the following

718 Franjou, Friedlander, Scorichenko, and Suslin diagram commute → # # ExtF(A, B) ExtF (B ,A ))   y y → # # ExtGL(k)(A, B) ExtGL(k)(B ,A ).

Proof. For any s 0 we can nd an integer N(s) such that for all n N(s) we have canonical identications Exts (A, B) = Exts (A(kn),B(kn)) GL(k) GLn(k) Exts (B#,A#) = Exts (B#(kn),A#(kn)) GL(k) GLn(k) = Exts (B(kn#)#,A(kn#)#). GLn(k)

Furthermore for any nite dimensional GLn(k)-modules M,N the exact contravariant functor M 7→ M # denes natural duality isomorphisms (cf. Lemma 1.12) Ext (M,N) → Ext (N #,M#). GLn(k) GLn(k) Taking here M = A(kn),N = B(kn), we identify Exts (A(kn),B(kn)) with GLn(k) s n # n # s # n# # n# ExtGL (k)(B(k ) ,A(k ) ) = ExtGL (k)(B (k ),A (k )). Finally for any n n GLn(k)-module M consider a new GLn(k)-module M, which coincides with M as a k-vector space. The action of GLn(k)onM˜ is obtained from the original one by means of the automorphism T 1 GLn(k) → GLn(k): 7→ ( ) . The functor M 7→ M is obviously an equivalence of the category of GLn(k)- modules with itself and hence denes natural isomorphisms Ext (M,N) → Ext (M,N). GLn(k) GLn(k) To nish the proof it suces to note now that (for any functor A ∈Fand n n# any n) the GLn(k)-modules A(k ) and A(k ) are canonically isomorphic. Lemma A.3. Let V ∈Vf be a nite dimensional vector space over k. Assume that n dim V . Then for any GLn(k)-module M we have isomorphisms: M n 1ndim W ExtGL (k)(PV (k ),M)= H ( ,M). n 0GLdim W (k) W V

Here PV is the projective generator of F, discussed in Section 1 and W V on the right runs through all subspaces of the vector space V .

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 719

n n Proof. PV (k )=k[Hom(V,k )] is a permutation GLn(k)-module. Two n homomorphisms f,g : V → k are in the same GLn(k)-orbit if and only if n Ker f = Ker g. Thus orbits of the action of GLn(k) on Hom(V,k ) are in one-to-one correspondence with subspaces W V . For any W V choose ndim W a representative fW of the corresponding orbit so that Im fW = k , n the standard coordinate subspace in k . It is clear that the stabilizer of fW coincides with the ane group 1 n dim W . 0GLdim W (k) Thus our statement follows from Shapiro’s Lemma.

Proposition A.4. Let B ∈F be a nite functor. For any s 0 there exists N(s) such that for n N(s) i n B(0) for i =0 H (GLn(k),B(k )) = 0 for 0

Proof. This follows immediately from the theorem of Betley [B1], stating n that over any eld k cohomology of GLn(k) with coecients in B(k ) coincides (for n big enough with respect to the cohomology index) with cohomology of GLn(k) with coecients in the trivial submodule B(0), and also from the theorem of Quillen [Q], giving the vanishing of higher dimensional cohomology in case of nite elds.

Let A ∈Fbe any functor and let j be a nonnegative integer. For any V ∈V set i i j j HA,j(V )=H (Hom(V,k ),A(k V )), where the action of Hom(V,kj)onA(kj V ) is dened via the natural embedding 1 Hom(V,kj) Hom(V,kj)= j ,→ GL(kj V ). 01V Each k-space homomorphism V → V 0 denes a homomorphism of (abelian) groups Hom(V 0,kj) → Hom(V,kj) and a homomorphism of Hom(V 0,kj)-modules A(kj V ) → A(kj V 0), thus dening a homomorphism in cohomology i → i 0 i HA,j(V ) HA,j(V ). One checks immediately that in this way HA,j becomes a functor from Vf to V, i.e. an element of F. Lemma i A.5. Assume that the functor A is nite. Then all functors HA,j i 0 j are nite as well. Moreover, HA,j(0)=0for i>0 and HA,j(0) = A(k ).

720 Franjou, Friedlander, Scorichenko, and Suslin

Proof. The second part of the statement is obvious. To prove the rst one we consider rst a special case A = Sm1 Sml . To simplify notation we use the following multi-index notation: for every multi-index i =(i1,... ,il) set Si = Si1 Sil . We consider the usual partial ordering on the set of multi-indices, and for m each multi-index i we set |i| = i1 + + il. Note that the functor S is “exponential”; more precisely we have the following obvious formula: M Sm (kj V )= Sm i (kj) Si (V ). 0im Consider an increasing ltration on Sm (kj V ) dened by the formula M mi j i t = S (k ) S (V ). 0im |i|t One checks easily that the action of Hom(V,kj) respects this ltration and j moreover theL action of Hom(V,k ) on subsequent factors of this ltration m i j i t/t 1 = |i|=t S (k ) S (V ) is trivial. Consider the spectral sequence dened by the above ltration: M s,t s j mi j i ⇒ s+t E1 = H (Hom(V,k ),k) S (k ) S (V ) HSm ,k(V ). 0im |i|=t Observe nally that the above ltration and hence also the above spectral s,t sequence depend functorially on V . In particular Er ∈Ffor all s, t, r.A well-known computation of cohomology of a nite dimensional vector k-space s,t with trivial coecients [C] implies that the functors E1 are nite. Immediate s,t induction on r shows further that all functors Er ,1 r ∞, are nite. i Finally, the functor HSm ,j has a nite ltration all subsequent factors of which are nite and hence is nite as well. For a general nite A, one can nd a resolution [K1, Th. 5.1] 0 → A → A0 → A1 → ... in which every functor Ai is a nite direct sum of functors of the form Sm . To conclude the proof consider the spectral sequence dened by this resolution s,t s ⇒ s+t E1 = HAt,k(V ) HA,k (V ) (which depends functorially on V ) and apply the same argument as above. Corollary A.6. Let B ∈F be a nite functor. Then for every s, j 0 there exists N(s, j) such that for n N(s, j) 1 B(kj)ifi =0 Hi( j ,B(kn)) = 0GLnj(k) 0if0

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 721

Proof. Consider the Hochschild-Serre spectral sequence corresponding to the group extension nj j 1j 1 → Hom(k ,k ) → → GLnj(k) → 1 , 0GLnj(k) a,b a b nj ⇒ a+b 1j n E2 = H (GLnj(k),HB,j(k )) H ( ,B(k )) , 0GLnj(k) 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 nite functor. Then for any V ∈Vand any s 0, there exists N(V,s) such that for n N(V,s) L i n n W V B(V/W)ifi =0 Ext (PV (k ),B(k )) = GLn(k) 0if0

Note that unlike the situation of Lemma A.3, the isomorphism M n n HomGLn(k)(PV (k ),B(k )) = B(V/W) W V is now canonical: it associates to a family {bW ∈ B(V/W)}W V aGLn(k)- equivariant homomorphism n n n PV (k )=k[Hom(V,k )] → B(k ) n n which sends f : V → k to B(f)(bKer f ), where f : V/Ker f → k is the induced homomorphism. If A, B ∈F are functors for which the sequence Exts (A(kn),B(kn)) GLn(k) s stabilizes with the growth of n, then we use the notation ExtGL(k)(A, B) for the stable value of this Ext-group. Thus Corollary A.7 tells us, in particular, that M Y HomGL(k)(PV ,B)= B(V/W)= B(V/W) W V W V for a nite B. Note that every k-linear homomorphism V → V 0 denes a homomorphism of functors PV 0 → PV and hence induces also a homomorphism

→ 0 HomGL(k)(PV ,B) HomGL(k)(PV ,B). L Vf This makes HomGL(k)(PV ,B)= W LV B(V/W) into a functorL from to → 0 0 itself. One checks easily that the map W V B(V/W) W 0V 0 B(V /W ) corresponding to a k-linear homomorphism f : V → V 0 looks as follows: the W 0- component of the image is obtained from the f 1(W 0)-component of the

722 Franjou, Friedlander, Scorichenko, and Suslin pre-image by applying the homomorphism B(f):B(V/f1(W 0)) → B(V 0/W 0) where f : V/f1(W 0) → V 0/W 0 is the homomorphism induced by f. More generally, for any (not necessarily nite) functor B ∈Fwe can dene a new functor aB ∈F, by setting M (aB)(V )= B(V/W) W V and dening homomorphisms (aB)(f) (for f : V → V 0) in the same way as above. Obviously we get in this way an exact functor a : F→F. By the universal property of PV ∈F, we conclude that for any nite functor B we have a natural isomorphism HomGL(PV ,B) = HomF (PV ,aB). Moreover, for any B ∈Fthere is a natural monomorphism B,→ aB, map- L ping B(V ) diagonally to (aB)(V )= W V B(V/W). Dene aB = aB/B. This gives us another exact functor from F to itself. Except for the trivial case B = 0, the functor aB is never nite.

Theorem A.8. For any nite functors A, B ∈F there is a natural identication ExtGL(k)(A, B) = ExtF (A, aB). After this identication, the natural homomorphism ExtF (A, B) → → ExtGL(k)(A, B) corresponds to the homomorphism ExtF (A, B) ExtF (A, aB) dened by the natural embedding B,→ aB.

Proof. Choose a resolution of A:

0 A P0 P1 in which each Pi is a nite direct sum of functors of the form PV and consider the spectral sequence Ei,j = Exti (P (kn),B(kn)) ⇒ Exti+j (A(kn),B(kn)). 1 GLn(k) j GLn(k) i,j Corollary A.7 shows that for n big enough with respect to s, all E1 -terms with 0

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

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 723

→ (1) The natural homomorphism ExtF (A, B) ExtGL(k)(A, B) is an isomorphism. (2) The natural homomorphism ExtF (A, B) → ExtF (A, aB) is an isomorphism. (3) ExtF (A, aB)=0.

We need a natural generalization of the functor a : F→Fto the case of bi-functors (and more generally poly-functors). Denote by Fn the category whose objects are covariant functors of n variables Vf Vf →V(and whose morphisms are natural transformations). For every functor B ∈Fn dene a new functor an(B) ∈Fn using the formula M (anB)(V1,... ,Vn)= B(V1/W1,... ,Vn/Wn)

W1V1WnVn and dening anB on morphisms in the same way as above. Denote further by →D Vf Vf Vf the diagonal and the direct sum functors.

Proposition A.10. For any functors A ∈F,B ∈Fn there is a natural split monomorphism → ExtF (A, a(B D)) , ExtFn (A ,anB).

Proof. The standard adjunction formula (1.7.1) shows that ExtFn (A ,anB) = ExtF (A, (anB) D). Now it suces to note that there are homomorphisms (natural in V ) M →i (a(B D))(V )= B(V/W,...,V/W) ((anB) D)(V ) p W V M = B(V/W1,... ,V/Wn).

W1V,... ,WnV

Here the composition of the (W1,... ,Wn)-indexed projection on the right with i equals the W1 ∩∩Wn-indexed projection on the left followed by the natural map

B(V/W1 ∩∩Wn, ...,V/W1 ∩∩Wn) → B(V/W1,... ,V/Wn); and the composition of the W -indexed projection on the left with p equals the (W,...,W)-indexed projection on the right. Obviously pi = 1 so that a(BD) is canonically a direct summand in (anB) D.

724 Franjou, Friedlander, Scorichenko, and Suslin

The following lemma is obvious from the construction

Lemma A.11. For any B1,... ,Bn ∈F there is a natural isomorphism of functors an(B1 Bn)=aB1 aBn.

Corollary A.12. Let A ∈F be a (graded) exponential functor. Then for any functors B1,... ,Bn ∈Fand any s 0 we have a natural split monomorphism M On s si ExtF (A ,a(B1 Bn)) ,→ Ext (A ,a(Bi)). s1++sn=s i=1

Proof. This follows immediately from Proposition A.10, Lemma A.11 and the Kunneth formula (1.7.2). Corollary A.13. With conditions and notation of Corollary A.12 assume in addition that the functors Bi are without constant term (i.e. Bi(0) s0 = 0). Assume further that n>1 and the natural homomorphisms ExtF (A ,Bi) s0 0 → ExtF (A ,aBi) are isomorphisms for all i and all s

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 725

0 s

Proposition A.15. The natural homomorphisms

(s) n (s) n ExtF (I ,S ) → ExtF (I ,aS )

(s) n (s) n ExtF (I , ) → ExtF (I ,a ) are isomorphisms for all n and s.

Proof. We proceed 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 → an → a(n 1 S1) →→ a(1 Sn 1) → aSn → 0 ,

0 → aSn → a(Sn 1 1) →→ a(S1 n 1) → an → 0.

The rst of these two complexes is always acyclic, the second one is acyclic if n 6 0mod(p) and has homology isomorphic to a((Smi i)(1))=(a(Smi i))(1) if n = pm. The induction hypothesis and Corollary A.13 show that (s) mi i (1) ExtF (I , (a(S )) ) = 0. Thus the second hypercohomology spectral sequences in both cases consist of zeroes only and hence the limit of the rst spectral sequences is zero. On the other hand, Corollary A.13 implies that the E1-term of both rst hypercohomology spectral sequences could have only two nonzero columns and hence the dierentials (s) n (n1) (s) n dn : ExtF (I , a( )) → ExtF (I , a(S )), 0 (s) n → (n1) (s) n dn : ExtF (I , a(S )) ExtF (I , a( )) are isomorphisms. Using an additional induction on the cohomology index we easily conclude the proof.

726 Franjou, Friedlander, Scorichenko, and Suslin

Applying the duality isomorphisms of Lemma A.2 we derive from Propo- sition A.15 the following:

Corollary A.16. The natural homomorphisms

n (s) n (s) ExtF ( ,I ) → ExtF ( ,aI ), n (s) n (s) ExtF ( ,I ) → ExtF ( ,aI ) are isomorphisms for all n, s.

Proposition A.17. The natural homomorphisms

m(s) n m(s) n ExtF ( ,S ) → ExtF ( ,aS ), m(s) n m(s) n ExtF ( , ) → ExtF ( ,a ) 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

m m n n m m n n ExtF ( 1 s ,S 1 S t ) → ExtF ( 1 s ,a(S 1 S t )) is an isomorphism for all m1,... ,ms,n1,... ,nt.

Proof. Assume rst 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. We may now obtain the general case in the same way as above, applying Corollary A.13 to the exponential functor| {z }. s End of the Proof of Theorem A.1. We can construct resolutions

0 → B → B0 → B1 → ... ,

0 A A0 A1 in which all Bi are nite direct sums of functors of the form Sn1 Snt m ms and all Ai are nite direct sums of functors of the form 1 . Comparing the corresponding hypercohomology spectral sequences and using Corollary A.18 we get the desired result.

The following is an immediate corollary of Theorem A.1 and the explicit stability bound of van der Kallen [vdK].

GENERAL LINEAR AND FUNCTOR COHOMOLOGY 727

Corollary A.19. Let k be a nite eld and let P, Q ∈F= F(k) be nite functors each of degree d. Then the natural map

Extm(F, Q) → Extm (P (kn),Q(kn)) F GLn(k) is an isomorphism whenever n 2m +2d.

University of Nantes, Nantes, France E-mail address: [email protected] Northwestern University, Evanston, IL E-mail addresses: [email protected] [email protected] [email protected]

References

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(Received November 13, 1997)