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.
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