BRIDGELAND–KING–REID–HAIMAN EQUIVALENCE ANDREAS KRUG
Abstract. We give a description of the image of tensor products of tautological bundles on Hilbert schemes of points on surfaces under the Bridgeland–King–Reid–Haiman equivalence. Using this, some new formulas for cohomological invariants of these bundles are obtained. In particular, we give formulas for the Euler characteristic of arbitrary tensor products on the Hilbert scheme of two points and of triple tensor products in general.
1. Introduction
For every smooth quasi-projective surface X over Cthere is a series of associated higher dimensional smooth varieties namely theHilbert schemes of n points on X for n∈N. They are the fine moduli spacesX[n]of zero dimensional subschemes of lengthnof X. Thus, there is a universal family Ξ together with its projections
X ←−−prX Ξ−−−−→prX[n] X[n].
Using this, one can associate to every coherent sheafF on X the so calledtautological sheaf
F[n]on each X[n] given by
F[n]:= prX[n]∗pr∗XF .
It is well known (see [Fog68]) that the Hilbert scheme X[n] of npoints on X is a resolution of the singularities of SnX=Xn/Sn via theHilbert–Chow morphism
µ:X[n]→SnX , [ξ]7→X x∈ξ
`(ξ, x)·x .
For every line bundle L on X the line bundle Ln ∈ Pic(Xn) descends to the line bundle (Ln)Sn on SnX. Thus, there is thenatural line bundle on X[n] given by
DL:=µ∗((Ln)Sn)
for every L ∈ Pic(X). One goal in studying Hilbert schemes of points is to find formulas expressing the invariants of X[n] in terms of the invariants of the surface X. This includes the invariants of the induced sheaves defined above. There are already some results in this area. For example, in [Leh99] there is a formula for the Chern classes ofF[n]in terms of those of F in the case that F is a line bundle. In [Boi05] and [BNW07] the existence of universal formulas, i.e. formulas independent of the surface X, expressing the characteristic classes of any tautological sheaf in terms of the characteristic classes of F is shown and those formulas are computed in some cases. Furthermore, Danila ([Dan01], [Dan07], [Dan00]) and Scala ([Sca09a], [Sca09b]) proved formulas for the cohomology of tautological sheaves, natural line
bundles, and some natural constructions (tensor, wedge, and symmetric products) of these. In particular, there is the formula
H∗(X[n], F[n]⊗ DL)∼= H∗(F ⊗L)⊗Sn−1H∗(L) (1)
for the cohomology of a tautological sheaf twisted by a natural line bundle. We will use and further develop Scala’s approach of [Sca09a] and [Sca09b] which in turn uses theBridgeland– King–Reid–Haiman equivalence (see [BKR01] and [Hai01]). It is an equivalence
Φ := ΦXOInX[n]→Xn: D
b(X[n])−→' Db
Sn(X
n)
between the bounded derived category of X[n] and the bounded derived category of Sn -equivariant sheaves onXn (for details about equivariant derived categories see e.g. [BKR01, Section 4]). The equivalence is given by the Fourier–Mukai transform with kernel the struc-tural sheaf of the isospectral Hilbert scheme InX := (X[n] ×SnX Xn)red. It induces for
E,F ∈Db(X[n]) andi∈Znatural isomorphisms
ExtiX[n](E,F)∼= HomDb(X[n])(E,F[i])∼= HomDb
Sn(X
n)(Φ(E),Φ(F)[i])∼= ExtXi n(Φ(E),Φ(F))Sn.
Furthermore, there is a natural isomorphism Rµ∗F 'Φ(F)Sn (see [Sca09a]) which yields
H∗(X[n],F)∼= H∗(SnX,Φ(F)Sn)∼= H∗(Xn,Φ(F))Sn.
(2)
So instead of computing the cohomology and extension groups of constructions of tautological sheaves on X[n] directly, the approach is to compute them for the image of these sheaves under the Bridgeland–King–Reid–Haiman equivalence. In order to do this we need a good description of Φ(F[n]) ∈DbSn(Xn) for F[n] a tautological sheaf. This was provided by Scala in [Sca09a] and [Sca09b]. He showed that Φ(F[n]) is always concentrated in degree zero. This means that we can replace Φ by its non-derived versionp∗q∗ wherepandq are the projections
from InX to Xn and X[n] respectively, i.e. we have Φ(F[n]) ' p∗q∗(F[n]). Moreover, he
gave for p∗q∗(F[n]) a right resolution CF•. This is a Sn-equivariant complex associated to F concentrated in non-negative degrees whose terms are good sheaves. For us a good Sn -equivariant sheaf on Xn is a sheaf which is constructed out of sheaves on the surfaceX in a not too complicated way. In particular, it should be possible to give a formula for its (Sn -invariant) cohomology in terms of the cohomology of sheaves on X. For example the degree zero term of the complex CF• is CF0 =Ln
i=1pr∗iF. Note that if F is locally free CF0 is, too. Its cohomology is by the K¨unneth formula given by
H∗(Xn, CF0) = H∗(F)⊗H∗(OX)⊗n−1
⊕n
.
The Sn-invariants of the cohomology can be computed as
H∗(Xn, CF0)Sn = H∗(F)⊗Sn−1H∗(O
X)
(For the proof of the Danila–Scala formula (1) in the case L=OX it only remains to show that the invariants of CFp for p ≥ 1 vanish). Let now E1, . . . , Ek be locally free sheaves on X. The associated tautological sheaves Ei[n] on X[n] are again locally free and hence called
tautological bundles. In [Sca09a] it is shown that again
Φ(E1[n]⊗ · · · ⊗Ek[n])'p∗q∗(E1[n]⊗ · · · ⊗Ek[n]).
Furthermore, a description of p∗q∗(⊗iEi[n]) as the E
0,0
∞ term of a certain spectral sequence
of K0 := CE01 ⊗ · · · ⊗C
0
Ek as follows: We construct successively Sn-equivariant morphisms
ϕ`:K`−1 →T` for`= 1, . . . , k, whereK`:= kerϕ` and theT` are good sheaves given by
T`=
M
(M;i,j;a)
S`−1ΩX ⊗
O
t∈M Et
!
ij
⊗ O
t∈[k]\M
pr∗a(t)Et.
The sum is taken over all tuples (M;i, j;a) with M ⊂[k] := {1, . . . , k},|M|=`, i, j ∈ [n], i6=j, anda: [k]\M →[n]. The functor ( )ij is the compositionιij∗p∗ij, whereιij: ∆ij →Xn is the inclusion of the pairwise diagonal and pij: ∆ij →X is the restriction of the projection pri:Xn→X. We show (theorem 4.10) thatKk =p∗q∗(⊗ki=1Ei[n]). If the exact sequences
0→K`→K`−1 →T`
for ` = 1, . . . , k were also exact with a zero on the right, this result would yield directly a description of the cohomology of E1[n]⊗ · · · ⊗Ek[n] via long exact sequences and an explicit formula for its Euler characteristic. Since this is not the case, we have to enlarge the sequences to exact sequences with a zero on the right or at least do the same with the sequences
0→KSn
` →K
Sn
`−1→T
Sn
`
of invariants onSnX. The latter also yields the cohomological invariants since by (2) we have H∗(X[n],⊗
iEi[n])∼= H
∗(SnX, KSn
k ). We are able to get the following results:
• A formula for the cohomology of tensor products of tautological sheaves in the max-imal cohomological degree 2n(proposition 6.1).
• ForE1, . . . , Ek locally free sheaves on a projective surface X and k≤nthe formula
H0(X[n], E1[n]⊗ · · · ⊗Ek[n])∼= H0(E1)⊗ · · · ⊗H0(Ek) (theorem 6.6).
• ForE1, . . . , Ek locally free sheaves on X and arbitrary klong exact sequences
0→K`→K`−1 →T`→T`1 → · · · →T`k−`→0
onX2with good sheavesT`i(proposition 7.7). This yields a description via long exact sequences of the cohomology and an explicit formula for the Euler characteristic of E1[2]⊗· · ·⊗Ek[2](proposition 7.12) and the Euler bicharacteristics between two different tensor products of tautological sheaves onX[2] (proposition 7.13).
• Similar long exact sequences for arbitrary n over the open subset X∗∗n consisting of
points (x1, . . . , xn) where at most twoxi coincide (subsection 7.4).
• For E1, E2, E3 locally free sheaves on X and n ≥ 3 long exact sequences on SnX
whose kernels converge to KSn
3 (subsection 8.2). This yields a description via long
exact sequences of the cohomology and an explicit formula for the Euler characteristic of E1[n]⊗E2[n]⊗E3[n] (corollary 8.11).
We have Φ(F ⊗ DL) ' Φ(F)⊗Ln for every F ∈ Db(X[n]) and L ∈ PicX. Using this, can generalise the results from products of tautological bundles to products of tautological bundles twisted by a natural line bundle by simply tensoring the exact sequences on Xn by Lnor tensoring the exact sequences onSnXby (Ln)Sn. Furthermore, we can generalise the
X[2], we show that tensor products of tautological objects and natural line bundles are never
P2-objects on X[2] forX an abelian surface.
Acknowledgements: Most of the content of this article is also part of the authors PhD thesis. The author wants to thank his adviser Marc Nieper-Wißkirchen for his support. The article was finished during the authors stay at the SFB Transregio 45 in Bonn.
2. Preliminaries
2.1. General notations and conventions.
(i) Let X be a variety. We write ( )∨ = Hom( ,OX) for the operation of taking the dual of a sheaf and ( )v =RHom( ,OX) for the derived dual.
(ii) An empty tensor, wedge or symmetric product of sheaves on X is the sheafOX. (iii) In formulas with enumerations putting the sign ˆ over an element means that this
element is omitted. For example {1, . . . ,ˆ3, . . . ,5} denotes the set {1,2,4,5}. (iv) For a local sectionsof a sheaf F we will often writes∈ F.
(v) For a direct sumV =⊕i∈IVi of vector spaces or sheaves we will write interchangeably ViandV(i) for the summands. For an element or local sections∈V we will writes(i) orsifor its component inV(i). We denote the components of a morphismψ:Z →V by ψ(i) : Z → V(i). Let W = ⊕j∈JWj be an other direct sum and ϕ:V → W a morphism. We will denote the components ofϕ by
ϕ(i→j) =ϕ(i, j) :V(i)→W(j).
(vi) Let ι:Z →X be a closed embedding of schemes and let F ∈QCoh(X) be a quasi-coherent sheaf onX. The symbolF|Zwill sometimes denote the sheafι∗F ∈QCoh(Z) and at other times the sheafι∗ι∗F ∈QCoh(X). The restriction morphism
F →F|Z =ι∗ι∗F
is the unit of the adjunction (ι∗, ι∗). The image of a section s ∈ F under this
morphism is denoted bys|Z.
(vii) Putting the symbol PF over an isomorphism sign means that the isomorphism is given by projection formula.
2.2. Symmetric groups and signs. For any finite set M the symmetric group SM is the group of bijections ofM. Note that we haveS∅ ∼= 1. For two positive integersn < mwe use the notation
[n] ={1,2, . . . , n} , [n, m] ={n, n+ 1, . . . , m}.
Ifn > mwe set [n, m] :=∅. We interpret the symmetric groupSnas the group acting on [n], i.e. Sn=S[n]. For any subsetI ⊂[n] we denote by ¯I = [n]\I its complement in [n]. There
is the group homomorphism sgn :SM → {−1,+1}which is given after choosing a total order <onM by
sgnσ = (−1)#{(i,j)∈M×M|i<j , σ(i)>σ(j)}
forσ∈SM. For two finite totally ordered setsM,Lof the same cardinality we defineuM→L as the unique strictly increasing map. Let now N be totally ordered, m ∈ M ⊂ N and σ∈SN. We define the signs
Lemma 2.1. (i) Let N be a finite set with a total order and M ⊂N. Then
εµ,σ(M)·εσ,M =εµ◦σ,M for allσ, µ∈SN. (ii) Let M ⊂N be as above, m∈M, andσ ∈SN. Then
εσ−1(m),σ−1(M)·εσ,σ−1(M)=εm,M ·εσ,σ−1(M\{m})
2.3. Graded vector spaces and their Euler characteristics. Graded vector spaces are denoted by V∗ := ⊕i∈ZVi[−i]. The symmetric power of a graded vector space is taken in
the graded sense. That means thatSnV∗ are the coinvariants of (V∗)⊗nunder theSn-action given on homogeneous vectors by
σ(u1⊗ · · · ⊗uk) :=εσ,p1,...,pk(uσ−1(1)⊗ · · · ⊗uσ−1(k)).
(3)
Here thepi are the degrees of the ui. The sign εis defined by setting ετ,p1,...,pk = (−1)
pi·pi+1
for the transposition τ = (i , i+ 1) and requiring it to be a homomorphism in σ. The
Euler characteristic of a graded vector space V∗ is given by χ(V∗) := P
i∈Z(−1)idimVi.
More generally, the Euler characteristic of a bounded complexC• of finite dimensional vector spaces is given by
χ(C•) :=X i∈Z
(−1)idimCi =X i∈Z
(−1)idimHi(C•).
Let X be a complete variety. For sheaves E, F ∈Coh(X) or more generally objects E, F ∈ Db(X) we set χ(F) :=χ(H∗(X, F)) andχ(E, F) :=χ(Ext∗(E, F)).
Lemma 2.2. Let V∗ and W∗ be graded vector spaces and m∈N. Then
χ(V∗⊗W∗) :=χ(V∗)·χ(W∗) , χ(SmV∗) =
χ(V∗) +m−1 m
.
3. Image of tautological sheaves under the Bridgeland–King–Reid–Haiman equivalence
3.1. The Bridgeland–King–Reid–Haiman equivalence. From now on letXbe a smooth quasi-projective surface. Forn∈Nwe denote by X[n] theHilbert scheme of n points on the surface X which is the fine moduli space of zero-dimensional subschemes ξ of X of length `(ξ) :=h0(ξ,Oξ) =n. The isospectral Hilbert scheme is given by InX:= (X[n]×SnXXn)red.
Here, the fibre product is defined via the Hilbert-Chow morphism µ: X[n] → SnX and the quotient morphismπ:Xn→SnX =Xn/Sn. Thus, there is the commutative diagram
InX −−−−→p Xn
q
y
yπ
X[n] −−−−→
µ S
nX . (4)
Theorem 3.1 ([BKR01], [Hai01]). The Fourier-Mukai Transform
Φ := ΦXOInX[n]→Xn =Rp∗q
∗
: Db(X[n])→DbSn(Xn)
is an equivalence of triangulated categories.
Corollary 3.2. Let F•,G• ∈Db(X[n]). Then ExtiX[n](F
•,G•)∼= Exti
Xn(Φ(F•),Φ(G•))Sn for
We will abbreviate the functor [ ]Sn◦π
∗: DbSn(Xn)→ Db(SnX) by [ ]Sn. Note that π∗
indeed does not need to be derived sinceπ is finite.
Proposition 3.3 ([Sca09a]). There is a natural isomorphism Rµ∗F• '[Φ(F•)]Sn for every
F• ∈Db(X[n]). This induces isomorphisms
H∗(X[n],F•)∼= H∗(SnX,Φ(F•)Sn)∼= H∗(Xn,Φ(F•))Sn.
3.2. Tautological sheaves.
Definition 3.4. We define the tautological functor for sheaves as
( )[n]:= prX[n]∗(OΞ⊗pr∗X( )) : Coh(X)→Coh(X[n]).
For a sheaf F ∈ Coh(X) we call its image F[n] under this functor the tautological sheaf associated with F. In [Sca09b, Proposition 2.3] it is shown that the functor ( )[n] is exact. Thus, it induces thetautological functor for objects ( )[n]: Db(X)→Db(X[n]). For an object F• ∈Db(X) thetautological object associated to F• is (F•)[n].
Remark 3.5. The tautological functor for objects is isomorphic to the Fourier-Mukai trans-form with kernel the structural sheaf of the universal family, i.e. (F•)[n]'ΦXO→X[n]
Ξ (F •) for
everyF•∈Db(X).
Remark 3.6. If F is locally free of rankkthetautological bundle F[n]is locally free of rank k·n with fibresF[n]([ξ]) = Γ(ξ, F
|ξ) since prX[n]: Ξ→X[n] is flat and finite of degreen.
3.3. The complex C•. We define for I ⊂ [n] with |I| ≥2 the I-th partial diagonal as the reduced closed subvariety given by
∆I={(x1, . . . , xn)∈Xn|xi=xj∀i, j∈I}.
We denote by pri: Xn → X the projection on the i-th factor and by pI: ∆I → X the projection induced by pri for anyi∈I. We denote the inclusion of the partial diagonals into the product byιI: ∆I →Xn. For a coherent sheafF onX we set
FI :=ιI∗p∗IF .
We will sometimes drop the brackets{ } in the notations, e.g we will write
∆12= ∆1,2 = ∆{1,2} , F12=F1,2 =F{1,2}.
To any coherent sheaf F on X we associate aSn-equivariant complex CF• of sheaves on Xn as follows. We set
CF0 = n
M
i=1
p∗iF , CFp = M I⊂[n],|I|=p+1
FI for 0< p < n , CFp = 0 else.
Let s= (sI)|I|=p+1 be a local section of C
p
F. We define the Sn-linearization ofCFp by λσ(s)I :=εσ,I ·σ∗(sσ−1(I)),
whereσ∗ is the flat base change isomorphism from the following diagram withpI◦σ =pσ−1(I)
∆σ−1(I) σ //
ισ−1(I)
∆I pI //
ιI
X
This gives also a Sn-linearization of CF0 using the convention F{i} := p∗iF and ∆{i} := Xn.
Finally, we define the differentials dp:CFp →CFp+1 by the formula
dp(s)J :=
X
i∈J
εi,J·sJ\{i}|∆J .
As one can check using lemma 2.1, CF• is indeed anSn-equivariant complex.
3.4. The image of tautological sheaves under Φ.
Theorem 3.7 ([Sca09b]). (i) For every F ∈Coh(F), the object Φ(F[n]) is cohomologi-cally concentrated in degree zero. Furthermore, the complex CF• is a right resolution of p∗q∗(F[n]). Hence, in DbSn(X
n) there are the isomorphisms
Φ(F[n])'p∗q∗F[n]'CF• .
(ii) For every collection E1, . . . , Ek ∈ Coh(X) of locally free sheaves on X, the object Φ(E1[n] ⊗ · · · ⊗E[kn]) is cohomologically concentrated in degree zero. Furthermore, there is a natural Sn-equivariant surjection
α:p∗q∗E1[n]⊗ · · · ⊗p∗q∗Ek[n]→p∗q∗(E1[n]⊗ · · · ⊗E [n]
k )
whose kernel is the torsion subsheaf. Hence, in DbSn(Xn) there are the isomorphisms
Φ k
O
i=1
Ei[n]
!
'p∗q∗
k
O
i=1
Ei[n]
!
' k
O
i=1
p∗q∗Ei[n]
!
/torsion.
We denote for i∈[k] the augmentation map byγi:p∗q∗(Ei[n])→CE0i.
Proposition 3.8. Let E1, . . . , Ek be locally free sheaves on X. Then there is a natural
isomorphism
p∗q∗(E1[n]⊗ · · · ⊗Ek[n])∼= im(γ1⊗ · · · ⊗γk)⊂CE01 ⊗ · · · ⊗C
0
Ek.
Proof. The sheaf ⊗iCE0
i is locally free and hence torsion-free. Thus, the kernel of⊗iγi must
contain the torsion subsheaf. Since CE1
i is supported on the big diagonal D=∪1≤a<b≤n∆ab
for everyi∈[k], the map⊗iγi is an isomorphism outside ofD. Thus, the kernel is supported onDand hence torsion. In summary, im(⊗iγi) is the quotient of⊗ip∗q∗(Ei[n]) by the torsion
subsheaf, which leads to the identification withp∗q∗(⊗iEi[n]) by theorem 3.7 (ii).
Remark 3.9. The morphism⊗iγi: ⊗ip∗q∗(Ei[n])→ ⊗iCE0i isSn-equivariant since all theγi are equivariant. Thus, the isomorphism of the previous proposition is Sn-equivariant when considering im(⊗iγ1) with the linearization induced by the linearization on ⊗iCE0i. In the
4. Description of p∗q∗(E1[n]⊗ · · · ⊗Ek[n])
In this whole section let E1, . . . , Ek be locally free sheaves on a quasi-projective surfaceX and n ∈ N. We will use proposition 3.8 in order to describe the image of E1[n]⊗ · · · ⊗E[kn] under the Bridgeland–King–Reid–Haiman equivalence as a subsheaf of
K0(E1, . . . , Ek) :=K0 :=
k
O
t=1
CE0t = M a: [k]→[n]
K0(a) , K0(a) =
k
O
t=1
pr∗a(t)Et
We have to keep track of theSn-linearization, since we later want compute theSn-invariant cohomology (see proposition 3.3). In the case that E1 = · · · = Ek = E we also have to keep track of the Sk-action in order to get later also results for the symmetric products of tautological bundles (see remark 3.9).
4.1. Combinatorical notations. We will write multi-indices mostly in the form of maps, i.e. for two positive integersn, k∈N we denote multi-indices withkvalues between 1 and n rather as elements ofI0:= Map([k],[n]) than as elements of [n]k. But sometimes we will switch
between the notations and write a multi-indexa: [k]→ [n] in the form a= (a(1), . . . , a(k)) ora= (a1, . . . , ak). For two mapsa:M →K andb:N →K with disjoint domains we write a]b:M`
N →K for the induced map on the union. IfN ={i}consists of only one element we will also writea]b= (a, i7→b(i)). Forx∈K we writex:M →K for the map which is constantlyx. For a multi-indexa:M → {i < j}with a totally ordered codomain consisting of two elements we introduce the sign
εa:= (−1)#a −1({j})
.
For the preimage sets of one elementiin the codomain ofawe will often write for shorta−1(i) instead of a−1({i}). For 1 ≤` ≤ k we define I` as the set of tuples of the form (M;i, j;a) consisting of a subset M ⊂ [k] with |M| = `, two numbers i, j ∈ [n] with i < j, and a multi-indexa: [k]\M →[n]. Given such a tuple we set ˆM := ˆM(i, j;a) :=M ∪a−1({i, j}) and a| := a|[k]\Mˆ. The data of i, j ∈ [n] with i < j is the same as the subset {i, j} ⊂ [n].
Thus, we will also write (M;{i, j};a) instead of (M;i, j;a). We write Ii,j for the ideal sheaf of ∆i,j inXnand N∆ij = (Ii,j/Ii,j2 )∨ for the normal bundle.
4.2. The case E1 =· · ·=Ek=OX. In the special case thatE1 =· · ·=Ek=OX it is easy to state the result (we will prove it later in the subsections 4.7 and 4.8). Let s ∈ K0 be a
local section with componentss(a)∈K0(a)∼=OXn. For `= 1, . . . , k and (M;i, j;a)∈I` we
set
s(M;i, j;a) := X b:M→{i,j}
εbs(a]b).
Then the subsheaf im(γ1⊗. . . γk)⊂K0 equals
Kk:=
s∈K0 |s(M;i, j;a)∈ Ii,j` for all (M;i, j;a)∈I` and `= 1, . . . , k . (5)
4.3. The case n=k = 2. To illustrate the general construction, we begin with the special case thatn=k= 2 and also assume for simplicity thatE1 =LandE2 =M are line bundles.
Letδ:X→X2 be the embedding of the diagonal ∆ with ideal sheaf I=I12. We have
K0=K0(1,1)⊕K0(1,2)⊕K0(2,1)⊕K0(2,2)
= ((L⊗M)O)⊕(LM)⊕(M L)⊕(O(L⊗M)).
Let s∈K0 be a local section. In case thatM =L=O are trivial the conditions of (5) fors
to be in the subsheafK2 ⊂K0 are
s(1,1) =s(1,2) =s(2,1) =s(2,2) mod I, (6)
s(1,1)−s(1,2)−s(2,1) +s(2,2) = 0 mod I2.
(7)
The first condition is the same as the vanishing of the map
ϕ1:K0 →T1 :=δ∗(L⊗M)⊕4 , s7→
(s(1,1)−s(2,1))|∆
(s(1,2)−s(2,2))|∆
(s(1,1)−s(1,2))|∆
(s(2,1)−s(2,2))|∆
.
This map is defined also for non-trivial L and M. For general n and k the map ϕ1 can be
defined similarly (see subsection 5.4). We setK1:= kerϕ1. The condition (7) (in the case of
trivial line bundles) is the same as the vanishing of the map
ˆ
ϕ2:K0 → OX2/I2 , s7→s(1,1)−s(1,2)−s(2,1) +s(2,2) mod I2.
Since for s ∈ K1 we have s(1,1)−s(1,2)−s(2,1) +s(2,2) ∈ I, the map ˆϕ2 restricts to
ϕ2:K1 → I/I2 withK2= kerϕ2.
Remark 4.1. Letf ∈ O(X). It induces the automorphisma(f) ofK0given by multiplication
by f 1 on K0(1,1) and K0(1,2) and by multiplication by 1f on K0(2,1) and K0(2,2).
It also induces the automorphismb(f) given by given by multiplication byf1 on K0(1,1)
and K0(2,1) and by multiplication by 1f on K0(1,2) and K0(2,2). Both a(f) and b(f)
restrict to automorphisms ofK1. On I/I2 multiplication byf1 and 1f is the same map
which we denote byc(f). We haveϕ2◦a(f) =c(f)◦ϕ2 =ϕ2◦b(f).
Let now L, M again be arbitrary line bundles and U ⊂X an open subset on whichL and M are simultaneously trivial. We define
ϕ2:K1→T2:=δ∗(L⊗M⊗ΩX)∼=δ∗(L⊗M)⊗ I/I2
overU2 as the mapϕ2:K1(O,O)→ I/I of the trivial case under the isomorphisms induced
by the trivialisations of L|U and M|U. Since two different trivialisations differ by elements of O(U)×, the above remark shows that ϕ2|U2 is independent of the chosen trivialisations.
Thus, it glues to a mapϕ2:K1 →T2 on the wholeX2.
4.4. Construction of the T` and ϕ`. For E1, . . . , Ek locally free sheaves on X and ` = 1, . . . , k we define the coherent sheaf
T`(E1, . . . , Ek) :=
M
(M;i,j;a)∈I`
S`−1N∆i,j∨ ⊗ O α∈M
Eα
!
i,j ⊗
O
β∈[k]\M
pr∗a(β)Eβ
under respectively upper index of the objects and morphisms, e.g. we will writen
kT`. We can rewrite the summands as
T`(M;i, j;a) =
S`
−1Ω
X ⊗(
O
α∈Mˆ
Eα)
i,j ⊗
O
β∈[k]\Mˆ
pr∗a(β)Eβ
or as (see [Har77, Chapter II 8])
T`(M;i, j;a) =S`−1(Ii,j/Ii,j2 )⊗
O
α∈Mˆ
Eα
i,j ⊗
O
β∈[k]\Mˆ
pr∗a(β)Eβ
∼
= (Ii,j`−1/Ii,j` )⊗
O
α∈Mˆ Eα
i,j ⊗
O
β∈[k]\Mˆ
pr∗a(β)Eβ
.
As in subsection 3.3 for the terms FI, we get for σ ∈ Sn by flat base change canonical isomorphisms
σ∗:T`(M;σ−1({i, j});σ−1◦a)→σ∗T`(M;i, j;a). Thus, there is aSn-linearization λofT` given on local sectionss∈T` by
λσ(s)(M;i, j;a) =ε`σ,σ−1({i,j})σ∗s(M;σ−1({i, j});σ−1◦a).
Remark 4.2. Forσ= (i j) the mapσ∗:N∆ij →N∆ij is given by multiplication with−1 (see
[Kru11, section 4]). Thus, σ∗:T`(M;i, j;a) → T`(M;i, j;σ−1◦a) is given by multiplication with (−1)`−1. Together with the sign ε`
σ,σ−1({i,j})this makes σ act by−1 onT`(M;i, j;a) for every tuple (M;i, j;a) such that a−1({i, j}) =∅, i.e. if ˆM =M.
If E1 =· · ·=Ek we define aSk-action onT` by setting
(µ·s)(M;i, j;a) :=µ·s(µ−1(M);i, j;a◦µ)
forµ ∈Sk. The action of µ on the right-hand side is given by permuting the factors Et of the tensor product. Since the two linearizations commute, they give aSn×Sk-linearization of T`. We will now successively define Sn- respectively Sn×Sk-equivariant morphisms ϕ`:K`−1 →T`, where
K0(E1, . . . , Ek) :=K0 :=
k
O
t=1
CE0t = M a: [k]→[n]
K0(a) , K0(a) =
k
O
t=1
pr∗a(t)Et
and K` := ker(ϕ`) for ` = 1, . . . , k. We consider K0 with the Sn-linearization λ given by λσ(s)(a) :=σ∗s(σ−1◦a) and, if all theEtare equal, with theSk-action (µ·s)(a) :=µ·s(a◦µ) (see also remark 3.9). For (M;i, j;a)∈I` we set
I0 ⊃I(M;i, j;a) :=
c: [k]→[n]|c(M)⊂ {i, j}, c|[k]\M =a =
a]b|b:M → {i, j}
and defineK`−1(M;i, j;a) as the image ofK`−1 under the projection
K`−1 ,→K0= M
c∈I0
K0(c)→
M
c∈I(M;i,j;a)
K0(c)
For s ∈ K`−1 ⊂ K0 the above means that s(c) for c /∈ I(M;i, j;a) does not contribute to
ϕ`(s)(M;i, j;a). We assume first that for all t∈Mˆ =M ∪a−1({i, j}) the bundles Et equal the trivial line bundle, i.e. Et=OX. Then for allb:M → {i, j} we have
K0(a]b) =H , T`(M;i, j;a) = (Ii,j`−1/Ii,j` )⊗H , H:=
O
t∈[k]\Mˆ
pr∗a(t)Et.
Thus, for a local sections∈K`−1 the componentss(a]b)∈K`−1(M;i, j;a) are all sections
of the same locally free sheafH and we can define
ϕ`(M;i, j;a)(s) :=
X
b:M→{i,j}
εbs(a]b) modIij` ·H
where εb = (−1)#{t|b(t)=j}. Inductively, the map ϕ`(M;i, j;a) is well defined, which means thatϕ`(M;i, j;a)(s)∈ Ii,j`−1·H, since if we take any m∈M we have
ϕ`(M;i, j;a)(s) =
X
b:M\{m}→{i,j}
εbs(a]b, m7→i) −
X
b:M\{m}→{i,j}
εbs(a]b, m7→j).
Both sums occurring are elements ofIi,j`−1·H since because ofs∈kerϕ`−1 we have
ϕ`−1(s)(M\ {m};i, j;a, m7→i) =ϕ`−1(s)(M\ {m};i, j;a, m7→j) = 0 mod Ii,j`−1·H .
Let now Et for t ∈ Mˆ be the trivial vector bundle of rank rt. Then for b:M → {i, j} we have K0(a]b) = (⊕αH) and T` = (Iij`−1/Iij`)⊗(⊕αH). Here the index α goes through all
multi-indices (αt|t∈Mˆ) with 1≤αt≤rt. Now we can defineϕ`(M;i, j;a) component-wise: The components ϕ`(M;i, j;a)(α, α0) are zero if α 6= α0 and coincide with the ϕ`(M;i, j;a) from the trivial line bundle case ifα=α0.
Remark 4.3. Let f be an automorphism of Et = OX⊕rt for t ∈ M. It induces the auto-ˆ morphism pr∗c(t)f of K0(c) for c ∈ I0. On T`(M;i, j;a) the automorphisms pr∗i f and pr∗jf coincide. The morphism ϕ`(M;i, j;a) commutes with the automorphisms induced by f on its domain and codomain.
This observation allows us to define ϕ`(M;i, j;a) in the case of general locally free sheaves as follows. We choose an open covering {Um}m of X such that on every open set Um all the Et are simultaneously trivial, say with trivialisations µm,t: Et|Um
∼
=
−→ Ort
Um. Let
prij := pri×prj: Xn → X2. Then the trivialisations µm,t for t ∈Mˆ induce over pr−ij1(Um2) isomorphismsK0(a]b)∼= (⊕αH) andT`(M;i, j;a)∼= (Iij`−1/Iij`)⊗(⊕αH). We define the re-striction ofϕ`(M;i, j;a) to pr−ij1(Um2) under these isomorphisms as the morphismϕ`(M;i, j;a) from the case of trivial vector bundles. It is independent of the chosen trivialisations by the above remark. Thus, theϕ`(M;i, j;a) defined over the pr−ij1(Um2) for varyingmglue together. Since the pr−ij1(Um2) cover the partial diagonal ∆ij, which is the support ofT`(M;i, j;a), this defines ϕ`(M;i, j;a) globally. Using lemma 2.1 one can check that the morphisms ϕ` are indeed equivariant.
4.5. Restriction along open immersions. In this subsection letM be a normal variety. For any open subvariety ι:U ,→ M with codim(M \U, M) ≥ 2 and F ∈ Coh(X) a locally free sheafuF:F
∼
=
−→ι∗ι∗F is an isomorphism. HereuF is the unit of the adjunction (ι∗, ι∗),
Lemma 4.4. Let N be an other normal variety and f:N →M be a proper morphism such that alsocodim(N \f−1(U), N)≥2 holds. Then there is a natural isomorphism
ι∗ι∗f∗E ∼=f∗E
for every locally free sheaf E onN. Proof. Due to the flat base change
f−1(U) ι 0 −−−−→ N
f0
y
yf
U −−−−→
ι M
we get indeedι∗ι∗f∗E∼=ι∗f∗0ι0∗E∼=f∗ι0∗ι0∗E∼=f∗E.
Lemma 4.5. Let ι:U →X be any open immersion and let
0→F0 →F →F00
be an exact sequence in Coh(X) such that uF and uF00 are isomorphisms. Then uF0 is also
an isomorphism.
Proof. Since open immersions are flat, the functor ι∗ι∗ is left-exact. Therefore, there is the
commutative following diagram with exact horizontal sequences:
0 −−−−→ F0 −−−−→ F −−−−→ F00
y
∼
= y
∼
= y
0 −−−−→ ι∗ι∗F0 −−−−→ ι∗ι∗F −−−−→ ι∗ι∗F00.
Since the last two vertical maps are isomorphisms, the first one is, too.
4.6. The open subsetX∗∗n. As done in [Dan00] and [Sca09a], we consider the following open
subvarieties ofXn,SnX, andX[n]. LetW ⊂SnXbe the closed subvariety of unordered tuples
Pn
i=1xi with the property that |{x1, . . . , xn}| ≤n−2, i.e
W =π
[
|J|=|K|=2,J6=K
∆J∩∆K
.
We setSnX∗∗:=SnX\W,X∗∗n :=π−1(SnX∗∗),X∗∗[n]:=µ−1(SnX∗∗) and
InX∗∗=q−1(X∗∗[n]) =p−1(X∗∗n) = (X
[n]
∗∗ ×SnX
∗∗ X n
∗∗)red.
In summary, there is the following open immersion of commutative diagrams
InX∗∗
p∗∗
−−−−→ X∗∗n
q∗∗
y
yπ∗∗
X∗∗[n] −−−−→
µ∗∗
SnX∗∗
ι ,→
InX −−−−→p Xn
q
y
yπ
X[n] −−−−→
µ S
where we denote all four open immersions ( )∗∗,→( ) byι. For a sheaf or complex of sheaves
F on Xn,SnX,X[n] orInX we write F∗∗ for its restriction to the appropriate open subset.
The codimensions of the complements are at least two. More precisely, we have
codim(Xn\X∗∗n, Xn) = codim(SnX\SnX∗∗, SnX) = 4,
codim(X[n]\X∗∗[n], X[n]) = codim(InX\InX∗∗, InX) = 2.
Lemma 4.6. Let E1, . . . , Ek be locally free sheaves on X. Then on Xn there is a natural
isomorphism
ι∗ι∗p∗q∗(E1[n]⊗ · · · ⊗Ek[n])∼=p∗q∗(E1[n]⊗ · · · ⊗Ek[n]).
Proof. We apply lemma 4.4 with f =p and U =X∗∗n.
4.7. Description of p∗q∗(E1[n]⊗ · · · ⊗Ek[n])∗∗.
Proposition 4.7. Let E1, . . . , Ek be locally free sheaves on X. Then on the open subset X∗∗n ⊂Xn there is the following equality of subsheaves of K0∗∗:
Kk∗∗=p∗q∗(E1[n]⊗ · · · ⊗Ek[n])∗∗.
Proof. We will often drop the indices ( )∗∗ in this proof. Using proposition 3.8 it suffices to
show thatKk= im(γE1⊗ · · · ⊗γEk). For fixed 1≤i≤j≤nand`∈[k] we denote byϕ`(i, j)
the direct sum of all ϕ`(M;i, j;a) with (M;i, j;a)∈ I`. On the open subset X∗∗n ⊂Xn the
pairwise diagonals ∆i,j do not intersect. We denote the big diagonal by D =∪1≤i<j≤n∆i,j. Then X∗∗n is covered by the open subsets Vi,j := (X∗∗n \D)∪∆i,j. Hence, we can test the equality on the Vi,j whereT`(M;u, v;a) = 0 for all {u, v} 6={i, j} and thus
K`=K`(i, j) :=∩`k=1kerϕk(i, j)
holds. We will assume without loss of generality the case that i= 1 andj= 2. We consider as in the construction of the ϕ` an open covering {Um}m of X on which all the Et are simultaneously trivial. Since bothKkand im(γE1⊗· · ·⊗γEk) equalK0onV12\∆12=X
n
∗∗\D,
it is sufficient to show the equality on every member of the covering of ∆12 given by
Um×Um×Um3× · · · ×Umn.
Since on these open sets the maps ϕ`(1,2) are defined as the mapsϕ`(1,2)(O⊕Xr1, . . . ,O
⊕rk
X ) under the trivializations, we may assume that all theEtare trivial vector bundles of rankrt, i.e. Et=O⊕Xrt. Since in this case theϕ`as well asγE1⊗ · · · ⊗γEk are defined component-wise,
we may assume thatE1 =· · ·=Et=OX. By theorem 3.7 (i) a sectionx∈p∗q∗OX[n]⊂CO0X
over V12 is of the form x = (x(1), x(2), . . . , x(n)) with x(α) ∈ pr∗αOX ∼= OXn for α ∈ [n]
and x(1)|∆12 =x(2)|∆12. For a sections∈ K0 and a multi-index a: [k]→ [n] we denote the
component ofsin
K0(a) = pra(1)OX ⊗ · · · ⊗pra(k)OX ∼=OXn
by s(a). The image of a pure tensor x1⊗ · · · ⊗xk ∈ (p∗q∗OX[n])⊗k under the k-th power of
γ=γOX is given by
γ⊗k(x1⊗ · · · ⊗xk)(a) =x1(a(1))· · ·xk(a(k))∈ OXn.
For a tuple (M, a) with∅ 6=M ⊂[k],a: [k]\M →[n], ands∈K0 we set
s(M, a) = X
b:M→[2]
Then for a sections∈K0 being a section ofKk=Kk(1,2) is equivalent to the condition that s(M, a)∈ I|M|for each tuple (M, a) as above, whereI :=I
12 (see subsection 4.2).
Example 4.8. Before continuing the general proof we consider again the case n = k = 2 (see also subsection 4.3). Let x = x1 ⊗x2 ∈ (p∗q∗OX[2])⊗2 and s = γ⊗2(x) ∈ K0. Because
of x2(1)−x2(2) ∈ I, we have s(1,1)−s(1,2) = x1(1)x2(1)−x1(1)x2(2) ∈ I. The other
equalities moduloI of (6) are verified the same way. Furthermore,
s(1,1)−s(1,2)−s(2,1) +s(2,2) =x1(1)x2(1)−x1(1)x2(2)−x1(2)x2(1) +x1(2)x2(2)
= (x1(1)−x1(2)) (x2(1)−x2(2))∈ I2.
This verifies (7) and thus showsK2 ⊃imγ⊗2. For the other direction let s∈K2 be arbitrary.
We set x := ss(1(1,,1)1)
⊗ 11
∈ (p∗q∗OX[2])⊗2 and t := s−γ⊗2x. Then t(1,1) = 0 which gives
t(1,2), t(2,1)∈ Iand thusy:= t(20,1)
⊗ 11
+ 11
⊗ t(10,2)
∈(p∗q∗O[2]X)⊗2. Settingu:=t−γ⊗2y
we get u(1,1) = u(1,2) = u(2,1) = 0. Hence, u(2,2) ∈ I2 by (7) and we may write
u(2,2) =P
ifi·gi with fi, gi∈ I. Then z:=Pi
0
fi
⊗ g0
i
∈(p∗q∗O[2]X)⊗2 withγ⊗2(z) =u.
In order to show the inclusion im(γ⊗k)⊂K
kin general, letx=x1⊗· · ·⊗xk∈(p∗q∗OX[n])⊗k
and s =γ⊗k(x). We show by induction over |M| that s(M, a) ∈ I|M| for each pair (M, a).
ForM ={t} we have
s({t}, a) =s(t7→1, a)−s(t7→2, a) = (xt(1)−xt(2))·
Y
i∈[k]\{t}
xi(a(i))
which is indeed a section ofI sincext(1)|∆12 =xt(2)|∆12. For an arbitraryM ⊂[k] we choose
anm∈M and set ˜
x=x1⊗ · · · ⊗xm−1⊗x˜m⊗xm+1⊗ · · · ⊗xk
with ˜xm(j) = 1 for everyj∈[n]. We also set ˜s=γ⊗k(˜x). With this notation
s(M, a) = (xm(1)−xm(2))·˜s(M\ {k}, a, m7→1).
By induction we have ˜s(M \ {k}, a, m 7→ 1) ∈ I|M|−1 and thus s(M, a) ∈ I|M|. For the
inclusion Kk ⊂ imγ⊗k we need the following lemma, where we are still working over V12.
Remember that fora: [k]→[n] we write ˆM(a) :=a−1({1,2}) anda|:=a|[k]\Mˆ(a).
Lemma 4.9. Let s ∈ Kk be a local section and a: [k] → [n] such that s(b) = 0 for all b: [k] → [n] with ( ˆM(b), b|) = ( ˆM(a), a|) and |b−1({2})| < |a−1({2})|. Then there exists a local section x ∈ (p∗q∗OX[n])⊗k such that γ⊗k(x)(a) = s(a) and γ⊗k(x)(c) = 0 for all multi-indices c: [k]→ [n] with the property that ( ˆM(c), c|) 6= ( ˆM(a), a|) or with the property that there exists ani∈Mˆ(a) = ˆM(c) with c(i) = 1 and a(i) = 2.
Proof. We assume for simplicity that ˆM(a) = [u] anda−1({2}) = [v] with 1≤v≤u≤k, i.e. ais of the form
a= (2, . . . ,2,1, . . . ,1, a(u+ 1), . . . , a(k)).
By the assumptions
Iv 3s([v], a|[v+1,k]) =s([v],1]a|) =
X
b: [v]→[2]
Now we can writes(a) =P
α∈Ayα,1· · ·yα,v as a finite sum with all yα,β ∈ I. We denote by ej the section of CO0X withej(h) =δjh. Then the section
x= X
α∈A
yα,1e2⊗ · · · ⊗yα,ve2⊗(e1+e2)⊗ · · · ⊗(e1+e2)⊗ea(u+1)⊗ · · · ⊗ea(k)
is indeed in (p∗q∗O[Xn])⊗k and has the desired properties.
Let ≺be any total order on the set of tuples (M, a) withM ⊂[k] and a: [k]\M →[3, n] and let CM,v be any total order on the set of subsets of M of cardinality v. We define a total order < on the set I0 = Map([k],[n]) by setting b < a if ( ˆM(b), b|) ≺ ( ˆM(a), a|) or
if ( ˆM(b), b|) = ( ˆM(a), a|) and |b−1(2)| < |a−1(2)| or if ( ˆM(b), b|) = ( ˆM(a), a|) =: M and
|b−1(2)|=|a−1(2)|=:v andb−1(2)C
M,va−1(2). For s∈Kk letabe the minimal multi-index withs(a)6= 0, i.e. s(b) = 0 for allb < a. Then lemma 4.9 yields ax∈(p∗q∗O[Xn])⊗k such that
ˆ
s=s−γ⊗k(x) fulfils ˆs(b) = 0 for allb≤a. Thus, by induction over the setI
0 with the order
<, indeed, s∈im(γ⊗k) which completes the proof of proposition 4.7.
4.8. Description of p∗q∗(E1[n]⊗ · · · ⊗E [n]
k ). The result of the last subsection carries over directly to the wholeXn.
Theorem 4.10. LetE1, . . . , Ekbe locally free sheaves onX. Then onXnthere is the equality Kk=p∗q∗(E1[n]⊗ · · · ⊗Ek[n])
of subsheaves of K0.
Proof. Since K0 is locally free and codim(Xn\X∗∗n, Xn) = 4 we have ι∗K0∗∗ = K0.
Fur-thermore the direct summands of T` are push forwards of locally free sheaves on the partial diagonals ∆i,j. Since
codim(∆ij\(∆ij∩X∗∗n),∆i,j) = 2
we get by lemma 4.4 that ι∗T`∗∗ = T` for all `∈ [k]. Using lemma 4.5 we get by induction thatι∗K`∗∗=K` for `∈[k]. In particular
Kk =ι∗Kk∗∗4=.7ι∗
p∗q∗(E1[n]⊗ · · · ⊗Ek[n])∗∗
4.6
= p∗q∗(E1[n]⊗ · · · ⊗Ek[n]).
Corollary 4.11. There are natural isomorphisms µ∗(E1[n]⊗ · · · ⊗Ek[n])∼=KkSn and
H∗(X[n], E1[n]⊗ · · · ⊗Ek[n])∼= H∗(Xn, Kk)Sn.
Proof. This follows by the previous theorem together with proposition 3.3 and theorem 3.7.
5. Invariants of K0 and the T`
5.1. Danila’s lemma. The following lemma was used by Danila in [Dan01] in order to simplify the computation of invariants. LetG be a finite group acting transitively on a set I. Let G also act on a variety X. Let M be a G-sheaf on X admitting a decomposition M = ⊕i∈IMi such that for any i ∈ I and g ∈ G the linearization λ restricted to Mi is an isomorphism λg: Mi
∼
=
Lemma 5.1. Let π:X →Y be a G-invariant morphism of schemes. Then for all i∈I the projectionM →Mi induces an isomorphism (π∗M)G
∼
=
→(π∗Mi)StabG(i).
Proof. The inverse is given on local sectionsmi∈(π∗Mi)StabG(i)bymi 7→ ⊕[g]∈G/StabG(i)g·mi
with g·mi ∈Mg(i).
Remark 5.2. LetG,I, andM be as above andN =⊕j∈JNj a second equivariant sheaf such thatG acts transitively on J and such that g:Nj
∼
=
−→g∗Ng(j) for allj ∈J. Let ϕ: M → N
be an equivariant morphism with components ϕ(i, j) : Mi → Nj. Then for fixed i ∈ I and j∈J the mapϕG under the isomorphisms MG∼=MiStab(i) and NG ∼=NjStab(j) of lemma 5.1 is given by (see also [Sca09a, Appendix B])
ϕG:MiStab(i)→NjStab(j) , m7→ X
[g]∈Stab(i)\G
ϕ(g(i), j)(g·m).
Remark 5.3. Danila’s lemma can also be used to simplify the computation of invariants if Gdoes not act transitively onI. In that case letI1, . . . , Ik be theG-orbits inI. ThenGacts transitively onI` for every 1≤`≤kand the lemma can be applied to every MI` =⊕i∈I`Mi
instead of M. Choosing representatives i`∈I` yieldsMG∼=Lk`=1M
StabG(i`)
i` .
5.2. Orbits and their isotropy groups on the sets of indices. For`= 1, . . . , kwe have the decompositionsK0=La∈I0K0(a) and T`=
L
I`T`(M;i, j;a) with
I0= Map([k],[n]) , I` =
(M;i, j;a)|M ⊂[k],#M =` ,1≤i < j ≤n , a: [k]\M →[n] .
TheSn–linearizations of K0 andT` induce actions onI0 and I` given forσ ∈Sn by
σ·a=σ◦a , σ·(M;{i, j};a) = (M;σ({i, j});σ◦a)µ−1).
Remark 5.4. We choose a total order≺on the set of subsets of [k] such that∅is the maximal element. EverySn-orbit of I0 has a unique representativea such that
a−1(1)≺a−1(2)≺ · · · ≺a−1(n).
We denote the set of these representatives by J0. For a∈ I0 the isotropy group is given by
StabSn(a) =S[n]\im(a). Fora∈J0(1) we have [n]\im(a) = [maxa+ 1, n]. EverySn-orbit of
I` has a unique representative of the form (M; 1,2;a) such thata−1(1)≺a−1(2) and
a−1(3)≺a−1(4)≺ · · · ≺a−1(n).
We denote the set of these representatives byJ`. Furthermore, we set
ˆ I`:=
(M;i, j;a)∈I`|a−1({i, j})6=∅ , Jˆ` :=J`∩Iˆ`.
We will often use the identification (M;a)∼= (M; 1,2;a)∈J` in the notations. The isotropy group of a tuple (M;i, j;a)∈I` withQ:={i, j} ∪im(a) and ¯Q= [n]\Q is given by
StabSn(M;i, j;a) = (
SQ¯ if (M;i, j;a)∈Iˆ`, S{i,j}×SQ¯ if (M;i, j;a)∈/ Iˆ`.
5.3. The sheaves of invariants and their cohomology. Lemma 5.5. There is a natural isomorphism
KSn 0 ∼=
M
a∈J0
K0(a)S[maxa+1,n].
Proof. This follows from lemma 5.1 and remark 5.4 (i). Lemma 5.6. For `= 1, . . . , k there are natural isomorphisms
TSn
` ∼=
M
(M;a)∈Jˆ`(1)
T`(M;a)S[max(a,2)+1,n].
Proof. By lemma 5.1 and remark 5.4 (ii) we have TSn
` =
M
(M;a)∈J`
T`(M;a)Stab(M;a).
Let (M;a)∈J`\Jˆ`. Then τ = (1 2)∈Stab(M;a) acts on T`(M;a) by (−1)`+`−1 =−1 (see
remark 4.2) which makes the invariants vanish.
Corollary 5.7. The sheaf TSn
k is zero and thus
µ∗(E1[n]⊗ · · · ⊗Ek[n])∼=p∗q∗(E1[n]⊗ · · · ⊗Ek[n])Sn ∼=KkS−n1.
Proof. The set ˆJk is empty. The isomorphisms follow by corollary 4.11.
Remark 5.8. For a subset Q⊂[n] with |Q|=q and ¯Q= [n]\Qwe denote by XQ×SQ¯X the quotient of Xn by the SQ¯-action. It is isomorphic to Xq ×Sn−qX. We denote by
πQ: XQ×SQ¯X → SnX the morphism induced by the quotient morphism π:Xn → SnX. Under the identificationXQ×SQ¯X =∼Xq×Sn−q it is given by
(x1, . . . , xq,Σ)7→x1+· · ·+xq+ Σ.
Let a ∈ I0 ((M;i, j;a) ∈ Iˆ`) and Q = im(a) (Q = {i, j} ∪im(a)), and ¯Q = [n]\Q. The sheavesK0(a)SQ¯ (T`(M;i, j;a)SQ¯) in the two lemmas above are considered as sheaves on the Sn-quotientSnX, i.e. they are abbreviations
K0(a)SQ¯ := (π∗K0(a))SQ¯ , T`(M;i, j;a)SQ¯ := (π∗T`(M;i, j;a))SQ¯
But we can also take theSQ¯-invariants already on theSQ¯-quotientXQ×SQ¯X and consider
K0(a)SQ¯ and T`(M;i, j;a)SQ¯ as sheaves on this variety. With this notation we have (π∗K0(a))SQ¯ =πQ∗(K0(a)SQ¯) , (π∗T`(M;i, j;a))SQ¯ =πQ∗(T`(M;i, j;a)SQ¯). We denote for m ∈ Q by pm:XQ ×S
¯
QX → X the projection induced by the projection prm:Xn→X. For I ⊂Q we have the closed embedding ∆I×SQ¯X ⊂XQ×SQ¯X which is theSQ¯-quotient of the closed embedding ∆J ⊂Xn. Then the sheaves of invariants considered as sheaves onXQ×SQ¯X are given by
K0(a)SQ¯ = O
m∈Q t∈a−1(m)
p∗mEt , T`(M;i, j;a)SQ¯ =
O
t∈M∪a−1({i,j})
Et
ij
⊗ O
m∈Q\{i,j}
t∈a−1(m)
p∗mEt.
In particular,K0(a)SQ¯ is still locally free. The sheafT(M;i, j;a)SQ¯ can also be considered
as a sheaf on its support ∆ij ×S
¯
For the following, remember that we interpret an empty tensor product of sheaves on the surfaceX as the structural sheafOX.
Lemma 5.9. (i) For every a∈I0 there is a natural isomorphism
H∗(Xn, K0(a))∼= H∗
O
t∈a−1(1)
Et
⊗ · · · ⊗H
∗
O
t∈a−1(n)
Et
.
(ii) For every a ∈ J0 the invariant cohomology H∗(Xn, K0(a))S[maxa+1,n] is naturally
isomorphic to
H∗
O
t∈a−1(1)
Et
⊗ · · · ⊗H
∗
O
t∈a−1(maxa)
Et
⊗Sn
−maxaH∗(O
X).
(iii) For every(M;i, j;a)∈I` the cohomologyH∗(Xn, T`(M;i, j;a))is naturally
isomor-phic to
H∗
S`−1ΩX ⊗ O
t∈Mˆ(a)
Et
⊗ O
m∈[n]\{i,j}
H∗
O
t∈a−1(m)
Et
.
(iv) For(M;a)∈Jˆ` the invariant cohomology H∗(Xn, T`(M;a))S[max(a,2)+1,n] is naturally
isomorphic to
H∗
S`
−1Ω
X ⊗
O
t∈Mˆ(a)
Et ⊗ maxa O m=3 H∗ O
t∈a−1(m)
Et
⊗Sn
−max(a,2)H∗(O
X).
Proof. The natural isomorphisms in (i) and (iii) are the K¨unneth isomorphisms. The asser-tions (ii) and (iv) follow from the fact that the natural Sn-linearization ofOXn induces the
action on H∗(Xn,OXn)∼= H∗(OX)⊗ngiven by permuting the tensor factors together with the
cohomoligcal signεσ,p1,...,pn (see section 2.3).
Lemma 5.10. Let X be projective.
(i) For every a∈J0 the Euler characteristic of the invariants of K0(a) is given by
χ
K0(a)S[maxa+1,n] = maxa Y m=1 χ O
t∈a−1(m)
Et
·
χ(OX) +n−maxa−1 n−maxa
.
(ii) For every (M;a)∈Jˆ` the Euler characteristic χ(T`(M;a)S[max(a,2)+1,n]) is given by
χ
S`−1ΩX ⊗ O
t∈Mˆ(a)
Et · maxa Y m=3 χ O
t∈a−1(m)
Et
· χ(O
X) +n−max(a,2)−1 n−max(a,2)
.
5.4. The map ϕ1 on cohomology and the cup product. We consider the morphism
ϕ1:K0 →T1 defined in subsection 4.4 and for a∈I0 and ({t};i, j;b)∈I1 its components
ϕ1(a→({t};i, j;b)) : K0(a)→T1({t};i, j;b).
The morphism ϕ1(a→({t};i, j;b)) is non-zero only if a|[k]\{t} =b and a(t)∈ {i, j}. In this
case it is given byεa(t),{i,j} times the morphisms given by restricting sections to the pairwise
diagonal ∆ij. For two sheavesF, G∈Coh(X) (or more generally two objects in Db(X)) the composition
H∗(X, F)⊗H∗(X, G)∼= H∗(X2, F G)→H∗(X, F⊗G)
of the K¨unneth isomorphism and the map induced by the restriction to the diagonal equals the cup product. Thus the mapH∗(ϕ1(a,({t};i, j;a|[k]\{t}))) is given in terms of the natural
isomorphisms of lemma 5.9 by sending
v1⊗ · · · ⊗vn∈H∗
O
t∈a−1(1)
Et
⊗ · · · ⊗H∗
O
t∈a−1(n)
Et
to the class
(vi∪vj)⊗v1⊗· · ·⊗ˆvi⊗· · ·⊗vˆj⊗· · ·⊗vn∈H∗
O
t∈a−1({i,j})
Et
⊗ O
m∈[n]\{i,j}
H∗
O
t∈a−1(m)
Et
.
Remember (theorem 3.7 (i)) that there are the augmentation morphismsγi:p∗q∗Ei[n]→CE0i
and thatK0=⊗ki=1CE0i. We consider the composition
k
O
i=1
H∗(Xn, p∗q∗Ei[n]) ∪
−→H∗(Xn, k
O
i=1
p∗q∗Ei[n])
H∗(⊗iγi)
−−−−−→H∗(Xn, K0).
Taking (factor-wise) theSn-invariants we get the map (see formula (1) of the introduction)
Ψ : k
O
i=1
H∗(X[n], Ei[n])∼= k
O
i=1
H∗(Ei)⊗Sn−1H∗(OX)
→H∗(Xn, K0)Sn.
This map coincides with theSn-invariant cup product ⊗iH∗(CE0
i)
Sn → H∗(⊗
iCE0i)
Sn. The
inclusion p∗q∗(⊗iE
[n]
i )Sn ⊂ K
Sn
0 induces a map H
∗(X[n],⊗
iE
[n]
i ) → H
∗(Xn, K
0)Sn. Since
im(⊗iγi) =⊗ip∗q∗Ei[n](proposition 3.8), the image of Ψ is a subset of the image of this map.
In degree zero the map H0(X[n],⊗iEi[n]) → H0(Xn, K0)Sn is an inclusion. Thus, we have
im(Ψ)⊂H0(X[n],⊗iEi[n]). Let X be projective. In this case H0(OX) = h1i ∼=C where 1 is the function with constant value one. Thus, we have fora∈I0 the formula (see lemma 5.9)
H0(Xn, K0(a))∼= O
m∈im(a)
H0
O
r∈a−1(m)
Er
and the action of StabSn(a) = Sima on this vector spaces is the trivial one, which means H0(Xn, K
0(a))Sima = H0(Xn, K0(a)). Now, for
xi ∈H0(Ei[n])∼= H
0(E
anda∈J0(1) we have
Ψ(x1⊗ · · · ⊗xk)(a) =
O
m∈im(a)
(∪r∈a−1(m)xr).
(8)
6. Cohomology in the highest and lowest degree
6.1. Cohomology in degree 2n. Let for `∈[k] be B` := im(ϕ`)⊂ T`, i.e. we have exact sequences
0→K`→K`−1→B`→0. (1)
SinceT` is the push-forward of a sheaf on D, the subsheafB` is, too. Since dimD= 2(n−1) we have Hi(Xn, B`) = 0 for i = 2n−1,2n. By the long exact sheaf cohomology sequence associated to (1)
· · · →H2n−1(B`)→H2n(K`)→H2n(K`−1)→H2n(B`)→0
we see that H2n(K`) = H2n(K`−1). By induction we get H2n(K`) = H2n(K0). Using corollary
4.11 and lemma 5.9 this yields the following formula for the cohomology of tensor products of tautological bundles in the maximal degree.
Proposition 6.1.
H2n(X[n], E1[n]⊗ · · · ⊗Ek[n])∼= H2n(Xn, K0)Sn ∼= M
a∈J0
maxa
O
r=1
H2( O t∈a−1(r)
Et)⊗Sn−maxaH2(OX).
Remark 6.2. In the case thatX is projective, the above proposition is the same as [Kru11, Remark 6.22] by Serre duality.
6.2. Global sections for n≥k and X projective. In this subsection we assume that X is projective. We will generalise the formula given in [Dan07] for the global sections of tensor powers of a tautological sheaf associated to a line bundles to a formula for tensor products of arbitrary tautological bundles.
Lemma 6.3. Let a: [k]→[n], t∈[k], i=a(t), and j∈[n]\im(a). Then under the natural isomorphisms of lemma 5.9 the mapH0(ϕ1)(a→({t};{i, j};a|[k]\{t}))corresponds toεa(t),{i,j} times the identity on ⊗m∈im(a)H0(⊗r∈a−1(m)Er).
Proof. This follows by the formula for H∗(ϕ1)(a→({t};{i, j};a|[k]\{t})) of subsection 5.4 and
the fact thatv∪1 =v for everyv∈H0(⊗r∈a−1(i)Er).
Lemma 6.4. Let m= min(n, k). Then everys∈ker H0(ϕ1)is determined by its components
s(a)∈H0(K0(a)) for a∈I0 with|im(a)|=m.
Proof. We use induction overw:=m−im(b) with the hypothesis that s(b) is determined by the values of the s(a) with |im(a)|=m. Clearly, the hypothesis is true for w = 0. So now let b: [n] → [k] with |im(b)| < min(n, k). Such a map b is neither injective nor surjective. Thus, we can choosej∈[n]\im(a) and a pairt, t0 ∈[k] witht6=t0 and i:=b(t) =b(t0). We define ˜b: [k]→[n] by ˜b|[k]\{t} =b|[k]\{t} and ˜b(t) =j. Then im(˜b) = im(b)∪ {j} which gives
|im(˜b)|=|im(b)|+ 1. We have
0 = H0(ϕ1)(s)({t};{i, j};b|[k]\{t})
Since H0(ϕ1)(b → ({t};{i, j};b|[k]\{t})) is an isomorphism by the previous lemma, s(b) is
determined bys(˜b) which in turn is determined by the values ofs(a) with |im(a)|= m by
the induction hypothesis.
Since the functor of taking global sections is left-exact, the inclusions
p∗q∗(E1[n]⊗ · · · ⊗Ek[n]) =Kk⊂K1 ⊂K0
induce inclusions
H0(X[n], E1[n]⊗ · · · ⊗Ek[n])⊂H0(Xn, K1)Sn ⊂H0(Xn, K0)Sn.
Furthermore, H0(Xn, K1) = ker(H0(ϕ1)) holds.
Lemma 6.5. Let n≥k. Then the projection
H0(Xn, K0)Sn ∼= M
a∈J0
H0(Xn, K0(a))→H0(Xn, K0(1,2, . . . , k))
induces an isomorphism
H0(Xn, K1)Sn
∼
=
−→H0(Xn, K0(1,2, . . . , k))
as well as an isomorphism
H0(X[n], E1⊗ · · · ⊗Ek)
∼
=
−→H0(K0(1,2, . . . , k)).
Proof. By the previous lemma, the map H0(K1)Sn → H0(K0(1,2, . . . , k)) is injective. Thus
it is left to show that for each
x=x1⊗ · · · ⊗xk∈H0(Xn, K0(1,2, . . . , k)) = H0(E1)⊗ · · · ⊗H0(Ek)
there exists an s ∈ H0(X[n], E1[n] ⊗ · · · ⊗ Ek[n]) ⊂ H0(Xn, K1)Sn ⊂ H0(Xn, K0)Sn with
s(1,2, . . . , k) =x. We can consider eachxt as a section of H0(X[n], Et[n])∼= H0(Et). Then by formula (8) of the previous sections= Ψ(x1⊗ · · · ⊗xn) has the desired property.
Theorem 6.6. For n≥k there is a natural isomorphism
H0(X[n], E1[n]⊗ · · · ⊗Ek[n])∼= H0(E1)⊗ · · · ⊗H0(Ek).
Proof. This follows from the lemmas 6.5 and 5.9.
7. Tensor products of tautological bundles on X[2] and X∗∗[n]
We want to enlarge the exact sequences
0→K` →K`−1
ϕ`
−→T`
to long exact sequences with a 0 on the right. We will do this first onX2. Since the pairwise diagonals are disjoint on X∗∗n, long exact sequences on X∗∗n can be obtained later from this
7.1. The complexes R`•. For a set M = {t1 < · · · < ts} ⊂ [k] of cardinality s we will consider the standard representation%M ∼=%s of SM ∼=Ss as the subspace of %k ⊂Ck with basis
ζM1 :=et1−et2, ζ
2
M :=et2−et3, . . . , ζ
s−1
M :=ets−1−ets.
For M ⊂N we denote the inclusion by ιM→N:%M → %N but will also often omit it in the following. For`= 1, . . . , k and i= 0, . . . , k−`we set
I`i:=(M;a)|M ⊂[k],#M =`+i , a: [k]\M →[2] , Ri`:= M
(M;a)∈Ii `
∧`−1%M(a)
where%M(a) =%M for everya. We define differentialsdi`:Ri` →R i+1
` fors∈Ri` by di`(s)(M;a) := X
i∈M
εi,MιM\{i}→M(s(M\ {i};a, i7→1)−s(M\ {i};a, i7→2)).
We have indeed defined complexesR•` for`= 1. . . , k since
(d◦d)(s)(M;a) =X i∈M
εi,MιM\{i}→M d(s)(M \ {i};a, i7→1)−d(s)(M \ {i};a, i7→2)
=X
i∈M
X
j∈M\{i}
εi,Mεj,M\{i}ιM\{i,j}→M
X
b:{i,j}→[2]
εbs(M \ {i, j};a]b)
vanishes by the fact thatεi,Mεj,M\{i}=−εj,Mεi,M\{j} for alli, j∈M. We define aSk-action on everyRi` by setting
(σ·s)(M;a) :=εσ,σ−1(M)σ·s(σ−1(M);a◦σ).
TheSk-action on the right-hand side is the exterior power of the action on%k. It maps indeed ∧`−1%
σ−1(M) to∧`−1%M. This makes eachR•` into aSk-equivariant complex, since fori∈M
the term
s(σ−1(M\ {i});a◦σ, σ−1(i)7→1)−s(σ−1(M\ {i});a◦σ, σ−1(i)7→2)
occurs in (σ·d(s))(M;a) with the sign εσ−1(i),σ−1(M)·εσ,σ−1(M) and in d(σ·s)(M;a) with
the signεi,M ·εσ,σ−1(M\{i}). Both signs are equal by 2.1. Note that for (M;a) ∈I`0 we have
∧`−1%
M(a)∼=C. We will denote the canonical base vectorζM1 ∧ · · · ∧ζM`−1 bye(M;a). We also
define
R−`1:= M a: [k]→[2]
C(a) , C(a) =C
together with theSk-equivariant map ˜ϕ` =d−`1:R
−1
` →R0` given by
˜
ϕ`(s)(M;a) =
X
b:M→[2]
εbs(a]b)
·e(M;a).
TheSk-action on R−`1 is given by (σ·s)(a) =s(σ−1◦a). We set
˜
R`•:= 0→R−`1→R•`
= 0→R−`1→R`0→ · · · →Rk`−` →0
.
We make this complex also S2-equivariant by defining the action of τ = (1 2) in degree−1
by (τ·s)(a) :=a(τ−1◦a) =a(τ ◦a) and in degreei≥0 by
We will sometimes write a k as a left lower index of the occurring objects and morphisms, e.g. kR`i, if we want to emphasise a chosen value of k.
Proposition 7.1. For every `= 1, . . . , k the complex R˜•` is cohomologically concentrated in degree−1, i.e. the sequence R−`1 →R0` →R1` → · · · →Rn`−`→0 is exact.
Proof. We will divide the proof into several lemmas. We will often omit certain indices in the notation, when we think that it will not lead to confusion. For `= 1 the complex ˜R•1 is isomorphic to ( ˜C•)⊗k[1], where ˜C• is the complex concentrated in degree 0 and 1 given by
0→C⊕C→C→0 ,
a b
7→a−b .
Since the complex ˜C• has only cohomology in degree zero and the tensor product is taken over the field C, it follows that ˜R•1 is indeed cohomologically concentrated in degree −1. We go on by induction over` assuming that the proposition is true for all values smaller than `.
Lemma 7.2. Let t∈R0`. Then d`0(t) = 0if and only if for every (N;a)∈I`1 and every pair
i, j∈N the following holds:
t(N\ {i};a, i7→1)−t(N \ {i};a, i7→2) =t(N\ {j};a, j 7→1)−t(N \ {j};a, j7→2).
Here for (M;b)∈I`0 we use the notationt(M;b) =t(M;b)·e(M;b), i.e. we denote byt(M;b)
also its preimage under the canonical isomorphism C∼=∧`−1%
M.
Proof. Let N = {n1 < · · · < n`+1} ⊂ [k]. The above formula holds for every pair i, j ∈ N
if and only if it holds for every pair of neighbors. Thus we may assume that i = nh and j=nh+1 withh∈[`]. The wedge product∧`−1%N\{nh} is spanned by the vector
eN\{nh} =ζ 1
N\{nh}∧ · · · ∧ζ
`−1
N\{nh} = (
ζN2 ∧ · · · ∧ζN` forh= 1, ζN1 ∧ · · · ∧(ζNh−1+ζNh)∧ · · · ∧ζN` else.
Thus, fort∈R0` the coefficient ofζN1 ∧ · · · ∧ζch
N∧ · · · ∧ζN` of d(t)(N, a)∈ ∧`
−1%
N equals
εnh,N t(N\ {nh};a, nh 7→1)−t(N\ {nh};a, nh 7→2)
+εnh+1,N t(N \ {nh+1};a, nh+17→1)−t(N\ {nh+1};a, nh+1 7→2)
which proves the lemma.
The inclusion im( ˜ϕ) ⊂ ker(d0`) follows since for s ∈ R−`1 and t = ˜ϕ(s) both sides of the equation in the above lemma equal
X
b:N→[2]
εb·s(a]b).
We will actually show a bit more than im( ˜ϕ) ⊃ker(d0`) in the next lemma. We decompose R−`1 into U`⊕S` with
U`={s∈R−`1 |s(a) = 0∀a: |a−1({2})| ≤`−1}=hea|a−1({2})≥`i
S`={s∈R−`1 |s(a) = 0∀a: |a
−1({2})| ≥`}=he
a|a−1({2})≤`−1i.
Lemma 7.3. The mapϕ˜`|U`:U`→ker(d 0
Proof. We first show the injectivity. Lets∈U with ˜ϕ(s) = 0. We show that s(a) is zero for everyaby induction overα=|a−1({2})|. Forα=`we set M =a−1({2}) and get
0 = ˜ϕ(s)(M; 1) =s(a).
We may now assume that s(b) = 0 for every b: [k] → [2] with b−1({2}) < α. Then for M ⊂a−1({2}) with|M|=`we obtain
0 = ˜ϕ(s)(M;a|[k]\M) =s(a).
For the surjectivity we precede by induction onk. Fork=`the map ˜ϕsends the basis vector e2 of the one-dimensional vector spaceU with a factor of (−1)` to the basis vectore[`] of the
one-dimensional vector spaceT`0. Let now be k > `. We set
V0={a: [k]→[2]|a(k) = 2,#a−1({2}) =`},
V1={a: [k]→[2]|a(k) = 1,#a−1({2})≥`},
V2={a: [k]→[2]|a(k) = 2,#a−1({2})≥`+ 1},
W0={(M, a)∈kI`0 |k∈M, a= 1}, W1 ={(M, a)∈kI`0|k /∈M, a(k) = 1},
W2={(M, a)∈kI`0 |k /∈M, a(k) = 2}, W3={(M, a)∈kI`0 |k∈M, a6= 1},
We define subspaces ofU respectivelyR`0 by
hVii=hea|a∈Vii , hWii=he(M;a)|(M;a)∈Wii
which yields U =hV0i ⊕ hV1i ⊕ hV2i and T0
` =hW0i ⊕ hW1i ⊕ hW2i ⊕ hW3i. We denote by
˜
ϕ(i, j) the component of ˜ϕ` given by the composition
hVii →U
˜
ϕ
−→R0` → hWji.
Let now t∈ker(kd0`). We define s0 ∈ hV0i by s0(a) = t(a−1({2}); 1) fora∈ V0. This yields
˜
ϕ(0,0)(s0) = t|hW0i. We set ˜t := t−ϕ(s˜ 0). The components ˜ϕ(1,0), ˜ϕ(2,0), ˜ϕ(1,2), and
˜
ϕ(2,1) are all zero. There are canonical bijections
V1
∼
=
−→ {a: [k−1]→[2]|#a−1({2})≥`}←∼=−V2,
W1
∼
=
−→ {(M, a)|M ⊂[k−1],#M =`, a: [k−1]\M →[2]}←∼=−W2
given by dropping a(k). They induce linear isomorphisms hV1i ∼= k−1U` ∼= hV2i as well as
hW1i ∼= k−1T`0 ∼= hW2i under which the linear maps ˜ϕ(1,1) and ˜ϕ(2,2) both correspond to
k−1ϕ. Thus, by induction there are˜ s1∈ hV1iand s2 ∈ hV2i such that ˜ϕ(1,1)(s1) = ˜t|hW1i and
˜
ϕ(2,2)(s2) = ˜t|hW2i. Definings∈kU by s|hVii=si fori= 0,1,2 we get
˜
ϕ(s)|hW0i⊕hW1i⊕hW2i=t|hW0i⊕hW1i⊕hW2i.
It is left to show that the equation also holds on hW3i. For this we show that for every
x ∈ kerd0` with x|hW0i⊕hW1i⊕hW2i = 0 also x(M;a) = 0 for every (M, a) ∈ W3 holds. We
use induction overα =|a−1({2})|. For α = 0 the tuple (M;a) is an element ofW
0. Hence,
x(M;a) = 0 holds. We now assume that x(M;b) = 0 for every tuple (M;b) ∈ W3 with
|b−1({2})|< α=|a−1({2})| holds and choose ani∈[k]\M witha(i) = 2. Applying lemma
7.2 toN =M∪ {i} we get
The first term on the right hand side is zero by induction hypothesis and the second and third are zero since they are coefficients ofsinW1 respectively W2.
Lemma 7.4. For all k, `∈N withk≥` we have
k−`
X
i=0
(−1)i2k−`−i
k `+i
`+i−1 `−1
= k
X
j=`
k j
.
Proof. Fork=`both sides of the equation equal 1. Insertingk+ 1 forkwe get by induction k−`+1
X
i=0
(−1)i2k−`−i+1
k+ 1 `+i
`+i−1 `−1
= k−`+1
X
i=0
(−1)i2k−`−i+1
k `+i
+
k `+i−1
`+i−1 `−1
=2 k−`
X
i=0
(−1)i2k−`−i
k `+i
`+i−1 `−1
+ k−`+1
X
i=0
(−1)i2k−`−i+1
k `+i−1
`+i−1 `−1
=2 k
X
j=`
k j
+ 2k−`+1
k `−1
k−`+1 X
i=0
(−2)−i
k−`+ 1 i
=2 k
X
j=`
k j
+ 2k−`+1
k `−1
(1−1 2)
k−`+1
=2 k
X
j=`
k j + k `−1
= k+1 X
j=`
k j + k j−1
= k+1 X
j=`
k+ 1 j
.
Remark 7.5. Counting the cardinality of the base we get dim(kU`) =
k
X
j=`
k j
.
On the other hand we have
dim(kRi`) = 2k
−`−i
k `+i
`+i−1 `−1
