Koszul duality and extensions of exponential functors Marcin Cha_lupnik Instytut Matematyki, University of Warsaw ul. Banacha 2, 02-097 Warsaw, Poland e-mail: mchal@mimuw.edu.pl Abstract We study Koszul duality in the category of strict polynomial func- tors. We compute Koszul duals for various functors and apply these results to the problem of calculating Ext-groups between exponential functors. The main application is a full description of the Ext-groups between twisted exterior and divided powers and between twisted sym- metric and divided powers. 1 Introduction Duality between symmetric and exterior powers (called the Koszul duality) appears in many contexts in algebra. In the category of functors (or repre- sentations of general linear groups) it has suggestive combinatorial interpre- tation. Namely, the symmetric power functor is a Schur functor associated to a Young diagram consisting of a single column while the exterior power is that associated to a one rowed diagram. Thus it is quite natural to ask for a natural operation which would relate a Schur functor associated to a Young diagram ~ to the one associated to the conjugate diagram e~(ie. the diagram whose rows are columns of ~). In the present article we construct such an operation. Like in other situations where Koszul duality appears, our construction has best properties when considered on the level of derived 1 category. More specifically, we define a self-equivalence of the derived category of the category of strict polynomial functors which sends a Schur functor to a Weyl functor associated to the conjugate diagram. I should say that such equivalence was constructed by Donkin [Do] in the cat- egory of modules over the Schur algebra which is isomorphic to the category of strict polynomial functors. Our approach is independent of that of Donkin and, I think, it deserves some attention, since it is much more elementary. But the main purpose of this paper is different. Thanks to good homological properties of the category of strict polynomial functors (which are completely invisible from the level of the Schur algebra) we are able to compute for some interesting functors. Moreover, the Koszul duality turns out to be a valuable tool in homological calculations in the functor category. Using it we compute Ext-groups between various exponential functors completing calcu- lations started in [FFSS]. Apart from these concrete applications, the Koszul duality, I believe, offers some conceptual explanation of effectiveness of the method of ü ntwisting functorsü sed so successfully in [FS], [FFSS], and implicitly in the pioneering [FLS]. The article is organized as follows. In Section 2 after giving some moti- vation we introduce our duality and establish its basic properties. These facts (probably with the exception of Fact 2.6 concerning twists) may be ob- tained by translating results of Donkin into our context. In section 3 we deal with one nontrivial example. We compute (Theorem 3.2) cohomology of the Koszul dual of the twisted divided power functor. In Section 4 we compute the Ext-groups between twisted exterior and divided powers (Corollaries 4.3, 4.4) and between twisted symmetric and divided powers (Corollary 4.5). All these calculations are surprisingly easy consequences of Theorem 3.2. 2 Duality Our working category will be the category P dof homogeneous strict polyno- mial functors of degree d over a fixed field k (see [FS, Sect. 2]). We preserve all conventions and terminology from [C1] and [C2], but for the convenience of the reader we shall recall some strict polynomial functors which will be used frequently in this article. First of all we have a functor jI 2 P 1, the j-th direct sum: (V ; V j). We will also consider the following homogeneous strict polynomial functors 2 of degree d: the d-th tensor power Id (V ; V d); the d-th symmetric power Sd (V ; (V d) d); the d-th divided power Dd (V ; (V d) d); the d-th exterior power d (V ; ((V d)alt) d ' ((V d)alt) d).PMore generally, for a Young diagram ~ = (~1, . .,.~k) of weight d (ie. j~j = d), we put S~ := S~1 . . .S~k and we define similarly D~ and ~. An important fact is that the family {S~} forms a set of injective cogenerators of P d (analo- gously {D~} forms a set of projective ones) [FS, p. 18]. Having these functors we are able to define a family of functors which are of great importance in representation theory. Namely, for a Young diagram ~ we define the Schur functor S~ as the image of a composition ~ - ! Id - ! Se~(cf. [ABW], Sect. II.1). There is a useful contravariant duality # in P d, called the Kuhn duality. We put F #(V ) to be (F (V *))*. It is easy to see that (Sd)# = Dd, whereas d is self-dual. The dual of S~ is called the Weyl functor W~ (it may be explicitly defined as the image of a composition De~-! Id -! ~). A formal definition of dualising functor , which I am going to give, may look rather unmotivated (at least for somebody not familiar with tilting modules). Thus I would like to present some elementary considerations which have led me to it. Looking at the definitions of Schur and Weyl functors one can see certain symmetry in them. Namely, the so called Straightening Rule [ABW, Sect. II.2, II. 3] provides an explicit presentation of S~ as a kernel of certain map from Se~to a sum of products of symmetric powers. But it is easy to see that the Weyl functor We~is the kernel of the äs me map" from e~to a sum of products of exterior powers (I will make this statement precise in a moment). For this reason my initial approach to the Koszul duality was the following. Since any functor F may be presented as the kernel of a map between sums of S~, we can put naive(F ) to be the kernel of the äs me map" between sums of ~. There are many equivalent ways of explaining what the äs me map"means. In fact there is an isomorphism 0 ~ ~0 Hom P d(S~, S~ ) ' Hom P d( , ). These groups are easy to compute (see eg. [FFSS, Cor. 1.8] or [AB, Sect. 6]) and compare. For example it is easy to 0 see that every element of Hom P d(S~, S~ ) is a composition of transformations of three simple types (perhaps tensored with identities): the multiplication Sa Sb -! Sa+b, the comultiplication Sa+b -! Sa Sb and the transposition Sa Sb -! Sb Sa. An analogous fact holds for exterior powers (which also form a Hopf algebra) and the practical way of establishing an isomorphism 3 between spaces of transformations is to say that we send the multiplication in the symmetric power to the multiplication in the exterior power etc. A more intrinsic description of this isomorphism (from which immediately fol- lows its functoriality) may be obtained using language of d-functors and transformations developed in [C1, Sect. 3]. Namely, [C1, Lemma 3.4] says 0 that for any OE 2 Hom Pd(S~, S~ ) there exists a unique d-transformation eOE2 Hom F ~ ~0 e d d(s , s ) such that OE = OE(V ). Then the äs me map" between exterior powers is just eOE((V d)alt). This än ive Koszul duality" works well when k has characteristic 0 but in positive characteristic it may kill objects. For example, the Frobenius twist I(1)(the functor which associates to a space the same space but with the action of scalars induced by the Frobenius auto- morphism), is the kernel of the comultiplication Sp -! Sp-1 S1. But the corresponding comultiplication in the exterior power p -! p-1 1 is a monomorphism. More careful analysis of this example reveals that when we extend our duality in an obvious way to complexes of symmetric powers then we will find our lost object: it was not killed but shifted (see Fact 2.6). This suggests that if we want our duality to be an isomorphism we should extend it to the derived category. The definition just given (we apply degreewise the naive duality) is perfectly correct, but it is not convenient in practice since it is usually impossible to describe explicitly resolution of a functor by symmetric powers (for example it is an open problem even for Schur functors). For this reason we will give a more sophisticated definition. It re- lies on the observation ([C1, Th. 5.1] or rather its additive counterpart) that Ext*( dOjI, S~) = Hom ( dOjI, S~) = d(A0j), where A0j= Hom (jI, I) = kj. Of course this construction is functorial in S~ and it follows from [C1, Th. 5.* *1] 0 e d alt that it takes OE 2 Hom Pd(S~, S~ ) to OE((V ) ). Thus we can say that the value of on kj is just RHom ( dOjI, F ). Since it is easy to generalize the * *for- mula from [C1, Th. 5.1] to the form Ext*( dO PV , S~) = Hom ( dO PV , S~) = d(V ), where PV 2 P 1is given by the formula PV (W ) := Hom k(V, W ), we can give elegant purely algebro-homological definition of . Definition 2.1 We define a functor : DP d -! DP d by the formula (F )(V ) := RHom ( d O PV , F ), where DP d denotes the category of finite cohomological complexes of objects Pd modulo quasiisomorphisms (we recall that P d has finite homological di- 4 mension and that every object has a finite injective resolution by (sums of products of) symmetric powers (cf. [To]). Of course, this invariant definition is much more convenient than that re- ferring to resolutions, but the former one is sometimes useful in concrete computations. Fact 2.2 For every Young diagram ~ of weight d, (S~) = We~. Proof: Since H0 = naiveand, as we have explained earlier, naive(S~) = We~, it remains to show that Hj (S~) = 0 for j > 0. But since d O PV is a direct sum of functors ~, it suffices to show that Extj( ~, S~) = 0 for every ~, ~. The last problem, by the Decomposition Spectral Sequence [C1, Cor. 0 2.5], may be reduced to showing that Ext j( d , S~0) = 0. But this follows from the general fact that Extj(W~0, S~0) = 0 for every ~0, ~0and j > 0 [CPS, proof of Th. 3.11]. || In order to show that is an isomorphism we describe explicitly its inverse e, which may be thought of as projective version of duality. Definition 2.3 A functor e : DP d -! DP d is defined by the formula e(F )(V ) := (RHom (F, d O PV ))# . It is easy to see that e takes divided powers to exterior ones, hence it satisf* *ies the dual version of Fact 2.2: e(W~) = Se~. Corollary 2.4 e O = Id= O e. Proof: Since e O (S~) = e ( ~) = S~, this composition is an identity on any complex of symmetric powers. But every complex is quasiisomorphic to a complex of symmetric powers. For the second composition we do the same with divided powers. || Thus we have shown that is a self-equivalence of DP d. Of course for k of characteristic 0 P is semisimple and = naiveis just a self-equivalence of P d. Therefore from now on we will assume that our ground field k is of positive characteristic p. 5 Corollary 2.5 For every pair of diagrams ~, ~0 Ext*Pd(S~, S~0) = Ext*Pd(We~, We~0) = Ext*Pd(Se~0, Se~). Proof: The first equality follows from the fact that is an isomorphism. The second equality is just the Kuhn duality. I have used it in order to come back to Schur functors. || This fact corresponds to Corollary 3.9 in [Do], and as Donkin pointed out was known even earlier [AB, Th. 7.7]. The last fact in this section allows us to extend Corollary 2.5 to twisted Schur functors, which seems to be a new result. Fact 2.6 For any F 2 P d: (F (i)) = ( (F ))(i)[d(pi- 1)], (F G) = (F ) (G), (we take convention (C[l])k := Ck+l, hence Hom *(C[l], C0[l0]) = Hom *(C, C0)[l* *0- l]. i d(i) i Proof: As we know from [FFSS, Th. 5.8], Ext *( dp , S ) = k[d(p - 1)]. i ~(i) Thus, arguing like in untwisted case, we obtain Ext *( dp O PV , S ) = i ~(i) ~(i)(V )[d(pi - 1)]. Hence RHom ( dp O PV , S ) is formal and we get i ~(i) * dpi ~(i) ~(i) i (Sd(i)) := RHom ( dp O P, S ) ' Ext ( O P, S ) = [d(p - 1)] = ( (S~))(i)[d(pi - 1)]. In order to get our assertion for general F we take 0 a resolution of F by symmetric powers 0 - ! F - ! S~ -! . . .After i- times twisting and applying RHom ( d O PV , -) we get an exact sequence of complexes 0(i) 0 -! RHom ( d O PV , F (i)) -! RHom ( d O PV , S~ ) -! . . . Since starting from the second spot we have formal complexes with cohomol- ogy concentrated in degree d(pi- 1), we get a quasiisomorphism 0(i) i RHom ( d O PV , F (i)) ' ~ (V )[d(p - 1)] -! . . . functorial in V . But the right-hand side is just ( (F ))(i)[d(pi- 1)]. The proof for tensor product is straightforward. || 6 3 Calculation of H (Dd(i)) There is the fundamental asymmetry in the duality coming from the fact that we define it using (at least implicitly) injective resolutions. Thus one should not expect as easy description of values of on projective objects as we have got for injective or Schur objects. In the present section we deal with the problem of computing (Dd). This complex seems to be far from being formal, but its cohomology still has a reasonable description. Since O PV has decomposition (not functorial in V ) into a sum of ~, then, by the Decomposition Spectral Sequence, the main ingredient in computation of H (Dd) will be the calculation of Ext*( d, Dd). Already this computation is nontrivial in contrast to the situation considered in [FFSS] where the starting point ie. determination of Ext-groups between untwisted functors was tautological. Since we are going to proceed by induction using Koszul and De-Rham complexes (like in [FLS], [FS], [FFSS]), we are forced to consider i l(i) d-l(i) * *(i) the four-graded object Extk( dp , D ). Luckily, since and D*(i)are Hopf algebras so are Ext-groups between them (cf. ([Ku, Sect. 5], [FFSS, Lemma 1.10]), which greatly helps to organize computations. Since * *(*) *-*(*) structural arrows in the Hopf algebra Ext *( *p , D ) preserve i *(i) index i we may fix it and describe the trigraded algebras Ext *( *p , D*-*(i)) for each i separately. Our terminology slightly differs here from that of [FFSS, Sect. 1]. We just call a Hopf algebra n-graded when our object has n indices which can vary independently. The advantage of convention taken in [FFSS] is that the commutativity relations [FFSS, Lemma 1.11] take more elegant form with it, but since we do not use them explicitly, I decided not to increase the number of indices artificially. i *(i) *-*(i) Fact 3.1 The trigraded Hopf algebra Ext*( *p , D ) is isomorphic to D*(x(i)) D*(ff(i)s) *(fi(i)s), for primitive generators: i-1 pi 1(i) o x(i)2 Extp ( , ), s+i-2 ps+i ps(i) o ff(i)s2 Extp ( , D ), for s = 1, 2 . . . s+i-1 ps+i ps(i) o fi(i)s2 Extp ( , D ), for s = 0, 1 . . . 7 Proof: Let Kd and Rd be the Kuhn duals of respectively Koszul and De- Rham complexes (we recall that (Kd)l = (Rd)l = d-l Dl but the Koszul complex is equipped with a homological differential whereas De-Rham com- plex with a cohomological one (cf. [FS, Sect. 4])). We will consider spec- i d(i) * dpi d(i) tral sequences converging to hExt *( dp , K ) and hExt ( , R ) call- ing them (twisted) Koszul and De-Rham spectral sequences (the first one, of course, converges to 0). Observe that since the structural maps in the O O Hopf algebra * D* ( indicates that we take graded tensor product of Hopf algebras (cf. [ABW, Sect. V.1]) commute with the Koszul and De- Rham differentials [C2, Sect. 3], the Koszul and De-Rham spectral sequences are sequences of Hopf algebras ie. the structural maps in the Hopf algebra i *(i) *-*(i) Ext*( *p , D ) commute with differentials in spectral sequences. i d-*(i) *(i) We will compute Ext*( dp , D ) by induction on d. For needs of in- duction we should also understand Koszul and De-Rham spectral sequences. Since our sequences are sequences of Hopf algebras, they are determined by the action of differentials on primitive generators ff(i)s, fi(i)s, x(i). In th* *e course of induction we will show that o All differentials in the second De-Rham spectral sequence converging i d(i) to hExt *( dp , R ) are trivial. o In the first De-Rham spectral sequence, we have ffi(ff(i)s) = ffi(fi(i)s) = 0, ffi(x(i)) = fi(i)0. s+i ps(i) o In the first Koszul spectral sequence converging to hExt *( p , K ), s-ps-1(i) (i) (i) ps(i) we have @(ff(i)s) = xp fis-1, @(fis ) = x , @(x) = 0. i 1(i) * pi 1(i) We start with d = 1. Then, by [FFSS, Th. 5.8], Ext*( p , ) = Ext ( , D * *) = k[pi-1] and we choose our generators x(i), fi(i)0in such a way that @(fi(i)0) =* * x(i) (of course, here ffi = @-1). Now we turn to the induction step. Let p > 2. Let us first state explic- itly, which conditions we should check (in addition to counting dimensions of course) to control the structure of Hopf algebra and differentials in the spectral sequences. In fact, there is anything to do only for d = ps. Here we should show that two new indecomposable primitive elements appear (these are our candidates for ff(i)sand fi(i)s) and compute their differentials. Fact * *that fi(i)sgenerates an exterior algebra follows from the parity of its multigrading (see [FFSS, Lemma 1.11]) but the structure of a summand generated by ff(i)s 8 depends on components of higher degrees. Thus we need not to care about ff(i)sat the moment but we should check some facts concerning the previous s-s0 generators ff(i)s0. Namely, we should show that for s0 < s, (ff(i)s0)p = 0 * *(in a divided power algebra we should carefully differ between the kth power of generator ff(i)s0which we denote by (ff(i)s0)k and the canonical nonzero element * * s-s0(i) ffk(i)s0(in fact (ff(i)s0)k = k! . ffk(i)s0)), and that we can choose an elemen* *t ffps0 s-s0(i) (i) s-s0 in such a way that (ffps0 ) = (ffs0) p where is a component in co- s0+i ps-s0 ps+i multiplication on Ext-groups induced by ( p ) -! . Let us start with the case d 6= 0, 1(mod p). Here we need only to compute i d(i) dimensions. The only nontrivial thing to show is that Ext *( dp , D ) = 0. i d(i) Look at the first Koszul spectral sequence converging to hExt *( dp , K ). It follows from ffi1(x(i)) = fi(i)0and the induction assumption that ffi1 yield* *s an isomorphism between the (d0p)th and (d0p + 1)th column in the first term of the sequence. This shows that the last column must be trivial. Let now d be divisible by p. The same analysis of the first De-Rham spec- tral sequence reveals that only the last column survives. An obvious di- i d(i) mension counting shows then, that this group (ie. Ext *( dp , D )) has the desired (graded) dimension if and only if all differentials in the second De- Rham spectral sequence are trivial. But the second term of this sequence is known by the induction assumption (for i + 1). Thus the triviality of differentials follows from the induction assumption unless d = ps where we a priori do not know this for generators ff(i+1)s-1, fi(i+1)s-1. But these ele* *ments lie in the last column and the triviality of differentials on them follows im- mediately from dimension argument. It remains to check facts concerning Hopf structure andsdifferentials+which,iagain, are nontrivial only for d = ps. s+i ps(i) ps+i-1 ps+i ps(i) Observe that Ext p -2( p , D ), and Ext ( , D ) are one di- s+i-1 ps+i ps(i) mensional and that the Koszul differential sends Ext p ( , D ) to s+i-ps ps+i ps(i) ps(i) (i) Extp ( , ) where x belongs to. Hence we put fis to the s(i) (i) preimage of xp under this differential. In order to choose ffs in a similar s+i-2 ps+i ps(i) manner we should show that the Koszul differential sends Extp ( , D ) s+i-ps+ps-1-1 ps+i ps-ps-1(i) ps-1(i) (i) ps-ps-1(i) to Extp ( , D ) where fis-1 x lives. s-ps-1(i) To this end, observe that fi(i)s-1 xp is a cycle with respect to the Kos* *zul s+i-2 ps+i p* *s(i) differential, and that the element which kills it must lie in Extp ( , D * * ) by dimension argument. In this way we have defined ff(i)sand fi(i)swith the expected action of the Koszul differential. Fact that they are primitive in- 9 decomposable, and that both De-Rham differentials act on them trivially follows from dimension argument. Then we turn to the analysis of genera- s-s0 tors ff(i)s0for s0 < s. Let us start with the observation that (ff(i)s0)p =* * 0 0 since the Koszul differential on a ps-s th power must be zero and our ele- ment lies in the last column, so there is nothing to kill it. Hence, there ex- s-s0(ps0+i-2)ps+i ps(i) ists a nonzero indecomposable element in Extp ( , D ) which s-s0(i) will be our candidate for ffps0 . But to finish the proof, we should also s-s0(i) (i) s-s0 show that (ffps0 ) = (ffs`) p To this end, it suffices to show that s-s0(i) (i) s s0 @(ffps0 ) = ffs0 xp -p (i)(up to scalar by which we can always modify s-s0(i) our definition of ffps0 ). But this follows from the fact that the latter ele- ment is a Koszul cycle and cannot be killed by any decomposable element by the induction assumption. This finishes the proof for d divisible by p. The remaining case d = 1(mod p) is easy. We need only to compute dimensions. The right answer follows from the fact that the De-Rham differential gives an isomorphism between the (d - 1)-th and dth column. For p = 2 only minor modifications of the proof are needed. The only prob- lem is that our Hopf algebra is now just commutative and hence, we cannot a priori rule out the possibility that ff(i)sgenerates exterior algebra or fi(i)s* *gen- erates divided powers or eg. (truncated) polynomials. But in our situation we still know that the pth (ie. second) powers of our candidates for genera- tors vanish and our method for deciding whether given element is primitive also works. || Now we turn to computing H (Dd(i)). In fact, thanks to Fact 2.6, it would suffice to compute H (Dd), but since we were already forced to deal with twists in Fact 3.1, computation of H (Dd(i)) will take no additional work. i d(i) dpi Thus our task is to compute Ext*( dp O PV , D ). Since O PV is a direct sumNof ~ and, by the Exponential Formula [C1, Sect. 2], Ext*( ~, Dd(i)) = * ~k ~k=pi(i) kExt ( , D ), Fact 3.1 provides us all needed computational input. The only problem is to organize results in a functorial way (we recall that the i ~ decomposition of dp O PV into a sum of is not functorial in V ). Again it will be easier to describe the entire exponential functor H (D*(i)). Let us rewrite the computation achieved in Fact 3.1 in a suggestive form: O Ext*( *, D*) = *(fi(i)0) *(fi(i)s) D*(ff(i)s), s 1 10 (observe that all products are finite in any multidegree). Roughly speaking, in order to get the functor H (D*(i)) we should replace in the above formula generators ff(i)sand fi(i)sby a space V (s+i). Theorem 3.2 There is an isomorphism of exponential functors O : Fi,0 Fi,s Gi,s-! H (D*(i)) s 1 i+s-1 ps(i) where Fi,s(V ) := *(i+s)(V ) for V (i+s)placed in Hp (D ), and Gi,s(V * *) := i+s-2 ps(i) D*(i+s)(V ) for V (i+s)placed in Hp (D ). i d(i) Proof: We shall describe quite explicitly. Let v 2 V and fl 2 Ext( dp , D ). i d(i) We define the element fl(v) 2 Ext( dp O PV , D ) in the following way. The element v determines the transformation OEv : PV - ! I in an obvious way i (we send f : V -! W to f(v)). Hence dp (OEv) is a transformation from i dpi dpi * dp O PV to and we put fl(v) to be ( (OEv)) (fl). This construction is clearly functorial in v and it is easy to see that thesassignment+vi ; s+i ps(i) ff(i)s(v) produces a transformation from I(s+i)to Ext p -2( p O P, D ) and analogously v ; fi(i)s(v) determines a transformation from I(s+i)to s+i-1 ps+i ps(i) Extp ( OP, D ). Then we define on the factor Gi,sby the formula vd11.. ...vdkk7! ffd1(i)s(v1).. ...ffdk(i)s(vk) and analogously on Fi,sby the f* *ormula v1 ^ . .^.vd 7! fi(i)s(v1) ^ . .^.fi(i)s(vd). Thanks to Fact 3.1 the transforma- tion is well defined. To show that it is an isomorphism we observe that is an exponential transformation (ie. it is compatible with decompositions of functors applied to direct sums). But an exponential transformation is essentially determined by its action on a one dimensional space. Lemma 3.3 Let : A* -! B* be an exponential transformation. If (k) is an isomorphism, so is the entire transformation . Proof: is an obvious double induction on dimension and degree. || Thus it suffices to show that (k) is an isomorphism. Let us take 1d 2 Dd(k) = Gi,s(k). Then (1d) = ffd(i)s(1) = ff(i)s. Analogously (even easier) we show an isomorphism on factors Fi,s. This completes the proof of Theorem 3.2. || Although computing H (Dd(i)) is sufficient for applications, it would be in- teresting to describe (Dd(i)) (of course, in a way from which it would be 11 clear how to obtain cohomology, which is not the case for the resolution of Dd constructed in [To]). Let us consider the simplest nontrivial example Dp. Then Kp provides a p-acyclic resolution of Dp and it is easy to derive from it that (Dp) is Rp with removed the degree 0 component. This example suggests that (Dd) is built out of d and some De-Rham complexes. Since the same example shows that (Dd) is not formal we can only hope that Conjecture 3.4 H (Dd) can be realized as a complex which is isomorphic up to filtration to the complex O * R*(s). s 0 In fact this conjecture suggested me the way of organizing results in Theorem 3.2. It seems that (Dd) consists of än ive part" which is d (warning: this is not exactly naive(Dd) when p = 2) and some additional öh mological part" in which De-Rham complexes are involved. 4 Applications to exponential functors As we know from [C1, Th. 4.3], the groups Ext*(Dd(j), F (j)) are computable for any F . The Koszul duality will allow us to extend this class of calculatio* *ns significantly. For example: Ext*( d(j), F (j)) = Hom *( ( d(j)), (F (j))) = hExt*(Dd(j), (F )(j)). By this method we will compute the Ext-groups be- tween twisted exterior and divided powers and between twisted symmetric and divided powers. In order to make our considerations precise, we start with stating some im- mediate consequences of [C1, Th. 4.3] which were not formulated explicitly there. Let P (j)ddenote the full subcategory of P dpjconsisting of j-times twisted functors. Since the embedding P d- ! P dpjis faithfully full, P (j)dis an abelian category which may be identified with the image of this embed- ding. Fact 4.1 For any F 2 P dthere is an isomorphism Ext *(Dd(j), F (j)) ' F (Aj), where Aj = Ext *(I(j), I(j)) (it is a one dimensional space in degrees 2k for k = 0, . .,.pj - 1 and trivial elsewhere), which is functorial in F ie. for any 12 transformation OE : F - ! G the induced map (OE(j))* : Ext *(Dd(j), F (j)) -! Ext*(Dd(j), G(j)) corresponds to OE(Aj). Moreover, for any bounded above complex C of objects of P (j)d, we have hExt *(Dd(j), C) = HC(Aj), (all Ext-groups are taken in P dpj). Proof: Description of Ext-groups is just [C1, Th. 4.3] for ~ = (d) (I discuss this example also at the end of [C1, Sect. 5]). In [C1] I examined functo- riality of this description with respect to the first variable, which is rather delicate question. Functoriality in F follows immediately from the machin- ery developed in [C1, Sect. 4]. There is only one subtlety, which was also discussed more thoroughly in [C1]. Strictly speaking we not just apply F to Aj but its injective symmetrization fin to Ajd. The difference is that fin carries information about grading. In other words, choosing fin allows to extend F to a functor on graded vector spaces which a priori may be done in many ways. In practice, in order to find fin one should present F as a kernel of a map between symmetric powers which we regard as functors on graded spaces in an obvious way. All this has a practical consequence which otherwise could be easily overlooked. Namely, it is easy to check that an injective symmetrization of I(i)is not just the identity but the pi-functor is s i(i)which multiplies degrees of components by pi(ie. (i(i)(V ))p := V and is trivial elsewhere). Thus I(i)(Aj) has nontrivial components in degrees 2kpi for k = 0, . .,.pj - 1. For the second part of Fact 4.1 we observe that the functor Ext *(Dd(j), -) is exact on sequences of objects of P (j)d. Thus, since all cycles, boundaries and cohomology spaces in C belong to P (j)d(because it is a full subcategory), all connecting homomorphisms which could produce nontrivial higher differ- entials in the hyperExt spectral sequences converging to hExt (Dd(j), C) are trivial, hence these spectral sequences collapse. || This fact shows that in order to compute Ext *( d(j), F (j)) we only need to know cohomology of (F ). Corollary 4.2 For any F 2 P d Ext *( d(j), F (j)) = H (F )(Aj). 13 Proof: As we have already observed, Ext*( d(j), F (j)) = hExt *(Dd(j), (F )(j)* *), and then we apply Fact 4.1 (in fact it is crucial that preserves the family of complexes of objects of P (j)dwhich is a consequence of Fact 2.6). || In the remainder of this section we will apply Corollary 4.2 to various expo- 0pi(j) d0(i+j) nential functors. We start with computing Ext *( d , D ). For this 0(i) we take Corollary 4.2 for d = d0pi and F = Dd . Gathering up results for various d0 we get Corollary 4.3 There is an isomorphism of bigraded Hopf algebras i(j) *(i+j) * O * * 0 Ext*( *p , D ) ' (Bi,j,0) (Bi,j,s) D (Bi,j,s). s 1 i+s(j) ps(i+j) The space Bi,j,s Ext *( p , D ) is one dimensional in Ext-degrees pi+s- 1 + 2kpi+s for k = 0, . .,.pj - 1 and trivial elsewhere. i+s(j) ps(i+j) The space B0i,j,s Ext *( p , D ) is one dimensional in Ext-degrees pi+s- 2 + 2kpi+s for k = 0, . .,.pj - 1 and trivial elsewhere. Thus we have computed Ext-groups between larger exterior and smaller di- i(j) vided powers. Trying to compute Ext *( *(i+j), D*p ) we face the problem i(j) that (D*p ) need not to be (i + j)-times twisted, hence we cannot apply i(j) Corollary 4.2 and it may be not that easy to compute Ext*(D*(i+j), (D*p )). We bypass this difficulty with the aid of projective version of the Koszul dual- i(j) ity. Applying it to the groups under consideration we get Ext*( d(i+j), Ddp )* * = i(j) * d(i+j) dpi(j) i * dpi(j) d* *(i+j) i Hom *( e( d(i+j)), e(Ddp )) = Ext (S , )[d(p -1)] = Ext ( , D * * )[d(p - 1)] reducing the problem to that we have already solved. Therefore, the result is Corollary 4.4 There is an isomorphism of bigraded Hopf algebras i(j) * O * * 0 Ext *( *(i+j), D*p ) ' (Ci,j,0) (Ci,j,s) D (Ci,j,s). s 1 s(i+j) pi+s(j) The space C0i,j,s Ext *( p , D ) is one dimensional in Ext-degrees pi+s+ ps(pi- 1) - 1 + 2kpi+s for k = 0, . .,.pj - 1 and trivial elsewhere. s(i+j) pi+s(j) The space C0i,j,s Ext *( p , D ) is one dimensional in Ext-degrees pi+s+ ps(pi- 1) - 2 + 2kpi+s for k = 0, . .,.pj - 1 and trivial elsewhere. 14 i(j) d(i+j) Our last task will be computation of Ext*(Sdp , D ). Applying twice i(j) 2 d(i+j) we replace these groups by Ext *(Ddp , (D )). Thus we should com- pute H 2(D*). This would be easy if we knew that H 2(D*) = H (H (D*)). For this we should show that all differentials in the second spectral se- i d(i) *(i) quence converging to hExt *( dp , (D )) are trivial. Since (D ) is an exponential functor (in the category of complexes) it suffices to show the triviality of differentials on primitive generators of the Hopf algebra i *(i) Ext*( *p , H (D )). But this is obvious by dimension argument. Therefore we get H 2(D*(i)) = H (H (D*(i))) which is by Theorem 3.2 equal to O O Gi,0,0 Gi,s,0 Fi,s,t Gi,s,t. s 1 s 1,t 1 In the above formula Gi,s,t(V ) := D*(i+s+t)(V ) for V (i+s+t)placed in i+s+t-1+pt(pi+s-2)2 ps+t(i) 2(pi+s-1) 2 ps(i) Hp (D ) for t 1 and in H ( (D ) for t = 0; i+s+t-2+pt(pi+s-2)2 ps* *+t(i) and Fi,s,t(V ) := *(i+s+t)(V ) for V (i+s+t)placed in Hp (D * * ). This leads to the following description of the Ext-groups. Corollary 4.5 There is an isomorphism of bigraded Hopf algebras i(j) *(i+j) * 0 O * 0 O * * 0 Ext*(S*p , D ) ' D (Ci,j,0,0) D (Ci,j,s,0) (Ci,j,s,t) D (Ci,j,* *s,t). s 1 s 1,t 1 i+s+t(j) ps+t(i+j) The space C0i,j,s,t Ext *(Sp , D ) is one dimensional in Ext- degrees pi+s+t- 1 + pt(pi+s- 2) + 2kpi+s+tfor k = 0, . .,.pj - 1 and trivial elsewhere, for t 1; and is one dimensional in Ext-degrees 2(pi+s-1)+2kpi+sfor k = 0, . .,.pj-1 and trivial elsewhere, for t = 0. i+s+t(j) ps+t(i+j) The space C0i,j,s,t Ext *(Sp , D ) is one dimensional in Ext- degrees pi+s+t- 2 + pt(pi+s- 2) + 2kpi+s+tfor k = 0, . .,.pj - 1 and trivial elsewhere. The Ext-groups between smaller symmetric and larger divided powers are exactly the same, since by the Kuhn duality i(j) *(i+j) * *(i+j) *pi(j) Ext*(S*p , D ) = Ext (S , D ). Computation of Ext-groups between exponential functors in the category P immediately leads to a paralel calculation in the category F of functors in 15 a naive sense over a finite field k, by methods of [FFSS, Sect. 6]. We recall that by [FFSS, Cor. 6.2] it suffices to compute the groups Ext*P!F (F, G) := colimjExt*P(F (j), G(j)) for certain functors F, G. It is easy to see that in o* *ur context we obtain the groups Ext *P!F just putting in formulae in Cor. 4.3, 4.4, 4.5 instead of spaces Bi,j,setc. the limit spaces (with respect to j) Bi,setc. in which we drop the condition k pj - 1. Hence, by [FFSS, Cor. 6.2], we get complete calculations in F (to make formulae simpler we restrict ourselves to the case of prime field k, leaving the general case to the interested reader). Corollary 4.6 Let |k| = p. Then we have isomorphisms of trigraded Hopf algebras: Ext *F( *, D*) ' O O (( *(Bi,0) *(Bi,s) D*(B0i,s))) i 0 s 1 O O ( *(Ci,0) *(Ci,s) D*(C0i,s)), i>0 s 1 and Ext *F(S*, D*) ' O O O (D*(C0i,0,0) D*(C0i,s,0) *(Ci,s,t) D*(C0i,s,t)) i 0 s 1 s 1,t 1 O O O (D*(C0i,0,0) D*(C0i,s,0) *(Ci,s,t) D*(C0i,s,t)), i>0 s 1 s 1,t 1 where multidegrees of generators are as in Cor. 4.3, 4.4, 4.5. Acknowledgments: I wish to thank Stanis_law Betley for careful reading a draft of this article. References [AB] K. Akin, D. Buchsbaum, Characteristic-free representation theory of the general linear group II. Homological Considerations, Advances in Math. 72 (1988), 171-210. 16 [ABW] K. Akin, D. Buchsbaum, J. Weyman, Schur Functors and Schur Complexes, Advances in Math. 44 (1982), 207-278. [C1] M. Cha_lupnik, Extensions of strict polynomial functors, preprint. [C2] M. Cha_lupnik, Schur-De-Rham complex and its cohomology, preprint. [CPS] E. Cline, B. Parshall, L. 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