SCHEMES ANDREAS KRUG
Abstract. We give formulas for the extension groups between tautological sheaves and more generally between tautological objects twisted by natural line bundles on the Hilbert scheme of points on a smooth quasi-projective surface. As a consequence, we observe that a tautological object can never be a spherical orPn-object. We also provide a description of
the Yoneda products.
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 fine moduli spacesX[n] of zero dimensional subschemes of lengthn ofX. Thus, there is a universal family Ξ⊂X×X[n] 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 .
More generally, for any object of the bounded derived categoryF ∈Db(X) using the Fourier-Mukai transform with kernel the structural sheaf of Ξ yields thetautological object
F[n]:= ΦOXΞ→X[n](F) ∈D
b(X[n]).
It is well known (see [Fog68]) that the Hilbert schemeX[n] of npoints on X is a resolution of the singularities ofSnX=Xn/Sn via theHilbert-Chow morphism
µ:X[n]→SnX , [ξ]7→X
x∈ξ
`(ξ, x)·x , `(ξ, x) := dimCOξ,x.
For everyL ∈PicX the line bundle Ln ∈Pic(Xn) descends to the line bundle π∗(Ln)Sn on SnX where π:Xn → SnX is the quotient morphism. Thus, we can define on X[n] the natural line bundle DL associated to the line bundleL onX as
DL:=µ∗(π∗(Ln)Sn).
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 surfaceX. 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 thatF is a line bundle. In [BNW07] the existence of universal formulas, i.e. formulas independent of the surfaceX, expressing characteristic classes of any tautological sheaf in terms of characteristic classes ofF is shown and those formulas are computed in some cases. Furthermore, Danila
([Dan00], [Dan01], [Dan07]) and Scala ([Sca09a], [Sca09b]) proved formulas for the cohomol-ogy of tautological sheaves, natural line bundles, and some natural constructions (tensor, wedge, and symmetric products) of these. In particular,
H∗(X[n], F[n]⊗ DL)∼= H∗(F ⊗L)⊗Sn−1H∗(L) (1.1)
for the cohomology of a tautological sheaf twisted by a natural line bundle. In this article we compute extension groups between tautological sheaves and more generallytwisted tautological objects, i.e. tautological objects tensorised with natural line bundles. Our main theorem is the existence of natural isomorphisms of graded vector spaces
Ext∗(E[n]⊗ DL, F[n]⊗ DM)∼= Ext
∗(E⊗L, F ⊗M)⊗Sn−1Ext∗(L, M)⊕
Ext∗(E⊗L, M)⊗Ext∗(L, F ⊗M)⊗Sn−2Ext∗(L, M) for objectsE, F ∈Db(X) and line bundles L, M ∈Pic(X). We also give a similar formula for Ext∗(E[n]⊗ DL,DM). SinceDOX =OX[n], by settingL=M =OX the extension groups and the cohomology of the dual of non-twisted tautological bundles occur as special cases. As an application we show that, for X a projective surface with trivial canonical bundle, twisted tautological objects are never spherical orPn-objects in Db(X[n]).
We will use Scala’s approach of [Sca09a], which in turn uses the Bridgeland–King–Reid equivalence. Let G be a finite group acting on a smooth quasi-projective variety M. A G-cluster on M is a zero-dimensional closed G-invariant subscheme Z of M where Γ(Z,OZ) equipped with the induced G-action is isomorphic to the regular representation CG. The Nakamura G-Hilbert scheme HilbG(M) is defined as the irreducible component of the fine moduli space ofG-clusters onM that contains the points corresponding to free orbits. Bridge-land, King and Reid proved in [BKR01] that under some requirements HilbG(M) is a crepant resolution of the quotient M/G. Furthermore, in this case the G-equivariant Fourier-Mukai transform with kernel the structural sheaf of the universal familyZ of G-clusters
Φ := ΦOZ: D
b(HilbG(M))→Db G(M)
between the bounded derived category of HilbG(M) and the equivariant bounded derived category DbG(M) = Db(CohG(M)) ofM is an equivalence of triangulated categories. Hence, Φ is called the Bridgeland–King–Reid equivalence. Haiman proved in [Hai01] that X[n] is isomorphic as a resolution ofSnX to HilbSn(Xn) and that the isospectral Hilbert scheme
InX:= X[n]×SnX Xn
red
can be taken as the universal family of Sn-clusters. Furthermore, the conditions of the Bridgeland–King–Reid theorem are satisfied in this situation. Thus, there is the equivalence
Φ := ΦXOInX[n]→Xn: D
b(X[n]) ∼=
−→DbSn(Xn).
Using this, we can compute extension groups on X[n] as Sn-invariant extension groups on
Xn after applying the Bridgeland-King-Reid equivalence, i.e. for E,F ∈Db(X[n]) we have ExtiX[n](E,F)∼= Ext
i
Xn(Φ(E),Φ(F))Sn. The right-hand side can be rewritten further as
So instead of computing the extension groups of tautological sheaves and objects on X[n]
directly, the approach is to compute them for the image of these objects under the Bridgeland– King–Reid equivalence. In order to do this we need a good description of Φ(F[n])∈Db
Sn(X n)
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 version p∗q∗ where p and q 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 forp∗q∗(F[n]) a right resolutionCF•. This is aSn-equivariant complex associated toF concentrated in non-negative degrees whose terms are of the form
C(F) :=CF0 = n
M
i=1
pr∗i F , CFp = M I⊂[n],|I|=p+1
FI for 0< p < n , CFp = 0 else.
Here, [n] :={1, . . . , n}andFIdenotes the sheafιI∗p∗IF, whereιI: ∆I →Xnis the inclusion of the partial diagonal andpI: ∆I →Xis the morphism induced by the projection pri:Xn→X for any i∈I.
The main step in the proof of the formula for the extension groups is the computation of [π∗RHomXn(Φ(E[n]),Φ(F[n])]Sn in the case of tautological bundles, i.e. forE, F locally free sheaves on X. We show that
[π∗RHom(Φ(E[n]),Φ(F[n]))]Sn ∼= [π∗Hom(C0
E, CF0)]Sn. (1.3)
In particular, the Sn-invariants of the higher sheaf-Ext vanish. The isomorphism in degree zero is shown using the fact that the supports of the terms CEp forp > 0 have codimension at least two and the normality of the variety Xn. For the vanishing of the higher derived sheaf-Homs we use the spectral sequences associated to the bifunctor
Hom( , ) := [π∗HomXn( , )]Sn.
We can generalise (1.3) to arbitrary objectsE•, F•∈Db(X) by taking locally free resolutions onXand using some formal arguments for derived functors. Since Φ(F ⊗ DL)∼= Φ(F)⊗Ln
holds for every objectF ∈Db(X[n]), we can generalise further to twisted tautological objects and get
RHom Φ((E•)[n]⊗ DL),Φ((F•)[n]⊗ DM)Sn ∼=RHom C(E•)⊗Ln, C(F•)⊗MnSn.
Then we can compute the desired extension groups by (1.2) as the cohomology of the object on the right and get the main theorem.
There is also a similar formula for the extension groups between two natural line bundles. Furthermore, we can apply our arguments in order to generalise (1.1) to get a formula for the cohomology of twisted tautological objects. So now there are formulas for Ext∗X[n](E,F)
whenever both of E,F ∈ Db(X[n]) are either twisted tautological objects or natural line bundles. In the last section we describe how to compute all the possible Yoneda products in terms of these formulas.
2. Preliminaries
General notation and conventions. Throughout the text, X will be a smooth quasi-projective surface overC,n≥2 an integer andX[n]the Hilbert scheme ofnpoints onX. We write ( )∨=Hom( ,OM) for the dual of a sheaf and ( )v=RHom( ,OM) for the derived dual. We will often write the symbol “PF” above an isomorphism symbol to indicate that the isomorphism is given by the projection formula. We also use “lf” to indicate that the given isomorphism is because of some sheaf being locally free and thus a tensor product or sheaf-Hom functor needs not be derived or commutes with taking cohomology. 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 that SnV∗ are the coinvariants of (V∗)⊗n under the
Sn-action given on homogeneous vectors by
σ(s1⊗ · · · ⊗sn) :=εσ,p1,...,pk(sσ−1(1)⊗ · · · ⊗sσ−1(n)).
(2.1)
Here thepi are the degrees of thesi. The signε is defined by setting ετ,p1,...,pk = (−1)
pi·pi+1
for the transpositionτ = (i , i+ 1) and requiring it to be a homomorphism inσ. Since the graded vector spaces we will consider are defined over C, the coinvariants under this action coincide with the invariants under the isomorphism
s1· · ·sn7→ 1
n!
X
σ∈Sn
εσ,p1,...,pk(sσ−1(1)⊗ · · · ⊗sσ−1(n)).
(2.2)
Derived bifunctors. LetA,BandCbe abelian categories andF:A×B → Cbe an additive bifunctor left exact in both arguments. We assume that there is a full additive subcategory
J of Bsuch that for every B ∈ J the functorF( , B) :A → C is exact and for every A∈ A
the subcategory J is a F(A, )-adapted class. Under these assumptions the right derived bifunctorRF: D+(A)×D+(B)→D+(C) exists (see [KS06, Section 13.4]. An example were the above assumptions are fulfilled is for any variety M the functor
Hom: QCoh(M)◦×QCoh(M)→QCoh(M)
where we can chooseJ as the class of all injective sheaves (see [Har66, Lemma II 3.1]). We also consider the bifunctor
F•: Kom(A)×Kom(B)→Kom(C) , F•(A•, B•) := totF(A•, B•).
Proposition 2.1. Under the assumptions from above let A• ∈ D+(A) and B• ∈ D+(B) be complexes such that RqF(Ai, Bj) = 0 for all q 6= 0 and all pairs i, j ∈ Z. Then we have
RF(A•, B•)∼=F•(A•, B•).
Equivariant sheaves and derived categories. LetGbe a finite group acting on a variety
M. We denote by CohG(M) the category of equivariant coherent sheaves and by DbG(M) its bounded derived category. If the action of G on M is trivial, we denote the the functor of invariants by ( ) : CohG(M)→Coh(M). Since our varieties are defined overC, this functor is exact and thus gives a functor ( ) : DbG(M)→Db(M) without deriving. For details about equivariant sheaves, derived categories, and functors we refer to [BKR01, Section 4].
Let Gact transitively on a set I and let (F, λ)∈CohG(M) decomposing into F =⊕i∈IFi such that for every i ∈I and g ∈G the linearisation λg restricted to Fi is an isomorphism
λg:Fi
∼
=
Lemma 2.2 ([Dan01]). Let π:M →Y be a G-invariant morphism of schemes. Then for all
i∈I the projection F →Fi induces an isomorphism (π∗F)G
∼
=
−→(π∗Fi)StabG(i).
Proof. The inverse is given on local sections si ∈ (π∗Fi)StabG(i) by si 7→ ⊕[g]∈G/StabG(i)g·si
withg·si∈Fg(i).
Remark 2.3. The lemma can also be used to simplify the computation of invariants if G
does not act transitively on I. In that case let I1, . . . , Ik be the G-orbits in I. Then G acts transitively onI` for every 1 ≤`≤k and the lemma can be applied to every FI` =⊕i∈I`Fi instead ofF. Choosing representatives i`∈I` yields (π∗F)G ∼=⊕k`=1(π∗Fi`)StabG(i`).
There is an equivariant version of Grothendieck duality (see [LH09]). We need the case of a regular closedG-equivariant embeddingj:Z →M. We consider the normal bundle sequence 0 → TZ → TM|Z → NZ/M → 0. As TZ as well as TM|Z are canonically G-linearised, the cokernelNZ/M inherits a G-linearizationλ. In the following,NZ/M is always considered as a
G-sheaf with the linearisation λ.
Proposition 2.4. Let codimM(Z) =c, i.e. NZ/M is locally free of rank c. (i) There is in DbG(M) the isomorphism (j∗OZ)v ∼=j∗(∧cN
Z/M)[−c].
(ii) Forq ∈Z there areG-equivariant isomorphisms Extq(j∗OZ, j∗OZ)∼=j∗(∧qN Z/M). (iii) Let j! := Lj∗( )⊗ ∧cN
Z/M[−c] : DbG(M) → DbG(Z). Then there are in DbG(M) for
E ∈DbG(Z) and F ∈DbG(M) natural isomorphisms
Rj∗RHomZ(E, j!F)∼=RHomM(Rj∗E,F).
Proof. The proposition is proved in chapter 28 of [LH09] in the more general framework of diagrams of schemes. How to obtain schemes with a group action as a special case is explained
in the introduction and chapter 29.
Lemma 2.5. For everyF ∈CohG(Z)and every q∈Zthere areG-equivariant isomorphisms
L−qj∗j∗F :=H−q(Lj∗j∗F)∼=F ⊗ ∧qNZ/M∨ .
Proof. We first proof the special case F =OZ. By proposition 2.4 (iii)
j∗(Lj∗j∗OZ⊗ ∧cNZ/M[−c])∼=j∗j!j∗OZ ∼=j∗RHomZ(OZ, j!j∗OZ)∼=RHomM(j∗OZ, j∗OZ).
Now taking the (c−q)-th cohomology on both sides by 2.4 (ii) yields
j∗(L−qj∗j∗OZ⊗ ∧cNZ/M)∼=j∗(∧c−qNZ/M).
Using the fact that j∗: CohG(Z)→CohG(M) is fully faithful we can cancel it from the iso-morphism. Tensoring with∧cN∨
Z/M gives the result. For generalF, we can use the projection formula twice to get
j∗Lj∗j∗F ∼=j∗(Lj∗j∗F ⊗ OZ)∼=j∗F ⊗Lj∗OZ ∼=j∗(F⊗LLj∗j∗OZ).
Partial diagonals and the standard representation. Letn≥2 be an integer. We write [n] := {1, . . . , n}. For I ⊂ [n] with |I| ≥ 2 let SI be the group of bijections of I and let ¯
I = [n]\I. We also writeXI for the varietyX|I| equipped with the canonicalS
I action. We setS(I) :=SI×SI¯. TheI-th partial diagonal is defined as the reduced subvariety ofXngiven
by ∆I := {(x1, . . . , xn) ∈ Xn |xi =xj for alli, j ∈I}. It is an S(I)-equivariant subvariety isomorphic to X ×XI¯. We denote its inclusion by ιI: ∆I → Xn. It is a regular closed embedding of codimension 2(|I| −1). We denote the normal bundle by NI. Furthermore, let pI: ∆I → X be the restriction of the projection pri:Xn → X for any i ∈I. For every coherent sheafF ∈Coh(X) we set
FI :=ιI∗p∗IF ∼=ιI∗(FOXI¯)∈CohS(I)(Xn).
The S(I)-linearisation of FI is given by the natural SI¯-linearisation of OXI¯ and the trivial
SI-action. The natural representation CI of SI is the vector space C|I| with SI acting by permuting the vectors of the standard base. The subspace of invariants is one-dimensional and given by the vectors of the form (v, . . . , v). Thestandard representation is defined as the quotient by this subspace. So we have the short exact sequence 0 → C → CI → %
I → 0 where the first map is the diagonal embedding.
Lemma 2.6. Under the identification ∆I ∼= X ×XI¯ we have NI ∼= (TX ⊗%I)OXI¯ as
S(I)-sheaves.
Proposition 2.7 ([Sca09a, Appendix B]). Let V be a two-dimensional vector space with a basis consisting of vectors u and v. Then the space of invariants [∧i(V ⊗%n)]Sn is one-dimensional if0≤i≤2(n−1)is even and zero ifi is odd. In the even casei= 2`the space of invariants[∧i(V ⊗%n)]Sn is spanned by the image of the vectorα`∈ ∧i(V ⊗Cn) under the projection induced by the projection Cn→%n where α=Pni=1uei∧vei ∈ ∧2(V ⊗Cn).
Corollary 2.8. For 0 ≤ ` ≤ |I| −1 there are isomorphisms [∧2`NI]SI ∼= ω−`
X OXI¯. In particular, the SI-action on the highest wedge power∧2(|I|−1)NI is trivial. Furthermore, the
SI-invariants of ∧kN
I vanish if kis odd.
Remark 2.9. In the hypothesis of corollary 2.8, after pushing forward byιI we get
ιI∗NI ∼= (TX)I⊗%I , ιI∗(∧2`NI)SI ∼= (ωX−`)I. Also note that we get together with proposition 2.4 (iii) for|I|=p+ 1 that
(ιI∗ι!IOXn)SI =∼ιI∗ι!IOXn ∼= (ω−p
X )I[−2p].
The complex C•. For every F ∈Coh(X) we define the complexCF• as follows. Let
C(F) :=CF0 := n
M
i=1
p∗iF , CFp := M I⊂[n],|I|=p+1
FI⊗aI for 0< p < n , CFp := 0 else.
HereaI denotes the alternating representation ofSI, i.e. the one-dimensional representation on whichσ ∈SI acts by multiplication by sgn(σ). There is a canonicalSn-linearisationλp of everyCFp such that λpσ maps FI isomorphically toσ∗Fσ(I)for everyσ ∈Snand that restricts to theS(I)-linearisation ofFI⊗aI for everyI ⊂[n]. The differentials are given by
dp(s)J :=
X
i∈J
(−1)#{j∈J|j<i}·sJ\{i}|∆J
Tautological sheaves. We define thetautological 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 to 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 isF[n].
The Bridgeland–King–Reid–Haiman equivalence. Letµ:X[n]→SnX :=Xn/S n the Hilbert-Chow morphism andπ:Xn→SnXthe quotient morphism. We consider the isospec-tral Hilbert scheme InX := (X[n]×
SnX Xn)red ⊂X[n]×Xn together with the commutative diagram
InX −−−−→p Xn
q
y
yπ
X[n] −−−−→
µ S nX
where p andq are the projections from X[n]×Xn to the factors. We define the Bridgeland– King–Reid–Haiman equivalence by
Φ :=Rp∗◦q∗: Db(X[n])→DbSn(Xn).
Theorem 2.10. The functorΦ is an equivalence of triangulated categories.
Proof. The follows from the results of [BKR01] and [Hai01]. See also [Sca09a, Section 1] for
a summary.
Corollary 2.11. For E,F ∈ Db(X[n]) we have ExtiX[n](E,F) ∼= ExtiXn(Φ(E),Φ(F))Sn for every i∈Z.
Theorem 2.12 ([Sca09b, Theorem 16]). For every F ∈Coh(X) the object Φ(F[n]) is coho-mologically concentrated in degree zero. Furthermore, the complexCF• is a right resolution of
p∗q∗(F[n]). Hence, inDbSn(Xn) there are natural isomorphisms Φ(F[n])∼=p∗q∗F[n]∼=CF•.
Theorem 2.13 ([Sca09b, Theorem 24]). Let E1, . . . , Ek∈Coh(X) be locally free sheaves on
X. Then Φ(⊗k i=1E
[n]
i ) is cohomologically concentrated in degree zero and hence isomorphic to p∗q∗(⊗k
i=1E [n]
i ) in DbSn(X
n). Furthermore, p∗q∗(⊗k i=1E
[n]
i ) ∼= E
0,0
∞ where E∞0,0 is on the
limiting sheet of the spectral sequence
E1p,q= M i1+···+ik=p
Tor−q(CEi11, . . . , C
ik
Ek) =⇒E
n=Hn(C•
E1⊗
L· · · ⊗LC•
Ek).
The differentials of the spectral sequence are induced by the differentials of the complexesCEi• .
Remark 2.14. Let E1, . . . , Ek be locally free and F an arbitrary coherent sheaf on X. By theorem 2.12 we can make the identification p∗q∗F[n] = ker(dF0 : CF0 → CF1). Thus, for U ⊂ Xn open we can describe the space of sections of p∗q∗F[n] over U as the sections
s= (s1, . . . , sn) ofCF0 withsi|∆ij =sj|∆ij for every distinct i, j∈[n] or alternatively
The differential on the 1-level of the spectral sequence of the previous theorem
d01,0:E10,0 = k
O
i=1
CEi0 →E11,0= k
M
`=1
CE`1 ⊗ O
i∈[k]\{`} CEi0
is given by the direct sum of the maps d0E` ⊗id⊗
i∈[k]\{`}CEi0 . For every a: [k] → [n] we set
E(a) := pr∗a(1)E1⊗ · · · ⊗pra∗(k)Ek. ThenE10,0 =
L
a: [k]→[n]E(a). Forsa local section ofE 0,0 1
we denote its component inE(a) bys(a). Thensis inE20,0 = kerd01,0 if and only if for every 1≤i < j ≤n the condition
s(a)|∆ij =s(˜a)|∆ij (2.3)
holds for every pair a,˜a: [k]→ [n] such that a|[k]\{`} = ˜a|[k]\{`} and {a(`),˜a(`)} = {i.j} for
some`∈[k]. Iterated application of condition (2.3) gives
s(a)|∆ij =s((i j)◦a)|∆ij for everya: [k]→[n] which says thats|∆ij is (i j)-invariant. This shows that every local section of
p∗q∗(E1[n]⊗ · · · ⊗Ek[n]) =E∞0,0⊂E20,0
has the property that its restriction to ∆ij is (i j)-invariant for every 1≤i < j≤n.
3. Extension groups of twisted tautological objects
We will use the Bridgeland-King-Reid equivalence to compute the extension groups of certain objectsE,F ∈Db(X[n]) by the formula (see corollary 2.11)
ExtiX[n](E,F)∼= Ext
i
Xn(Φ(E),Φ(F))Sn. Setting ( )Sn := ( )Sn◦π∗ we can rewrite the right-hand side as
ExtiXn(Φ(E),Φ(F)))Sn ∼= Hi SnX, RHom
Xn(Φ(E),Φ(F))Sn.
The functors π∗ and ( )Sn are indeed exact and hence need not be derived. We will first compute the inner termRHomXn(Φ(E),Φ(F))Sn. We abbreviate the occurring bifunctor by
Hom( , ) :=HomXn( , )Sn and set Exti =RiHom. Because of the exactness of ( )Sn we have
RHom( , )∼= [RHomXn( , )]Sn.
Computation of the Homs.
Lemma 3.1. Let X be a normal variety together with U ⊂M an open subvariety such that codim(X\U, X) ≥2. Given two locally free sheaves F and G on X and a subsheaf E ⊂F
with E|U = F|U the maps a: Hom(F, G) → Hom(E, G) and ˆa: Hom(F, G) → Hom(E, G), given by restricting the domain of the morphisms, are isomorphism.
Proof. It suffices to show that a is an isomorphism. Since every open V ⊂ X is again a normal variety with V \(U ∩V) of codimension at least 2, it will follow that ˆa is also an isomorphism. We construct the inverseb: Hom(E, G) → Hom(F, G) of a. For a morphism
ϕ:E →Gthe morphism b(ϕ) sends s∈F(V) to the unique section inG(V) which restricts
We denote by D the big diagonal inXn, i.e. the reduced subvariety
D= [
1≤i<j≤n
∆ij ={(x1, . . . , xn)|#{x1, . . . , xn}< n} ⊂Xn,
and byU =Xn\Dits open complement in Xn.
Proposition 3.2. Letk∈Nand E1, . . . , Ek, F ∈Coh(X) be locally free sheaves. Then there are natural isomorphisms
(i) Hom(Φ(E1[n]⊗ · · · ⊗Ek[n]),OXn)∼=Hom(CE0
1 ⊗ · · · ⊗C 0
Ek,OXn), (ii) Hom(Φ(E1[n]⊗ · · · ⊗Ek[n]),Φ(F[n]))∼=Hom(CE0
1 ⊗ · · · ⊗C 0
Ek, CF0).
Proof. By theorem 2.13 we have Φ(E1[n]⊗ · · · ⊗Ek[n]) =∼ p∗q∗(E1[n]⊗ · · · ⊗Ek[n]). Moreover,
p∗q∗(⊗k i=1E
[n]
i ) can be identified with the subsheafE
0,0
∞ ofE10,0 =CE0
1⊗ · · · ⊗C 0
Ek. Since the termsE1p,q of the spectral sequence are for p≥1 all supported onD, we have
p∗q∗(E[1n]⊗ · · · ⊗Ek[n])|U = (CE01 ⊗ · · · ⊗C 0
Ek)|U.
Since Xn is normal and D of codimension 2, lemma 3.1 yields the first isomorphism of the proposition (even before taking invariants). For the second one we consider an open subset
W ⊂SnX and aSn-equivariant morphism
ϕ:p∗q∗(E1[n]⊗ · · · ⊗Ek[n])|π−1W →CF0|π−1W.
By remark 2.14 the morphismϕ factorises overp∗q∗F|[πn−]1W. This yields
Hom(p∗q∗(E1[n]⊗ · · · ⊗Ek[n]), p∗q∗F[n])∼=Hom(p∗q∗(E1[n]⊗ · · · ⊗Ek[n]), CF0)
∼
=Hom(CE01 ⊗ · · · ⊗CEk0 , CF0)
where we have used Lemma 3.1 again for the second isomorphism.
Vanishing of the higher Exti(Φ(F[n]),OXn).
Lemma 3.3. Let F be a locally free sheaf onX. Then [(CFp)v]Sn ∼= 0 for p >0. Proof. LetI ⊂[n] with|I|=p+ 1. By equivariant Grothendieck duality
FIv∼=RHomXn(ιI∗p∗IF,OXn)
2.4 ∼
= ιI∗RHom∆I(p
∗
IF, ι!IOXn)
2.9 ∼
= (F∨⊗ωX−p)I[−2p].
Note that by lemma 2.8 the SI-action on FIv is indeed the trivial one. Now,
[(CFp)v]Sn ∼= [ M
|I|=p+1
(FI⊗aI)v]Sn
2.2 ∼
= [F[vp+1]⊗a[p+1]]S([p+1]).
Every transposition in S[p+1] is acting on F[vp+1]⊗a[p+1] by −1 which makes the invariants
vanish.
Proposition 3.4. For every locally free sheaf F ∈ Coh(X) there is a natural isomorphism [(Φ(F[n]))v]Sn ∼= [(C0
F)∨]Sn in Db(SnX). In particular, Exti(Φ(F[n]),OXn) = 0 for i >0. Proof. Indeed, [Φ(F[n])v]Sn 2∼= [(.12 C•
F)v]Sn
3.3 ∼
= [(CF0)v]Sn ∼= [(lf C0
Vanishing of the higher Exti(Φ(E[n]),Φ(F[n])). Throughout this subsection let E and F
be locally free sheaves on X. In order to compute the invariants of the higher extension sheaves we will use the spectral sequenceAassociated to the functorHom( , p∗q∗F[n]) given
by (see e.g. [Huy06, Remark 2.67])
Ap,q1 =Extq(CE−p, p∗q∗F[n]) =⇒Am=Extm(CE•, p∗q∗F[n])∼=Extm(Φ(E[n]),Φ(F[n])).
The terms in the −k-th column of A1 in turn are computed by the spectral sequence B(k)
associated to Hom(CEk, ) and given by
B(k)p,q1 =Extq(CEk, CFp) =⇒B(k)m =Extm(CEk, CF•)∼=Extm(CEk, p∗q∗F[n]).
Here as direct summands terms of the formExtq(EI⊗aI, FJ⊗aJ) for∅ 6=I, J ⊂[n] occur. In the case that I ={i}orJ ={j}consist of just one element we set EI= pr∗i E orFJ = pr∗jF, respectively. Furthermore, we set ∆{i} := Xn =: ∆∅ and N{i} := 0 =: N∅. We also set
∧00 :=O
Xn. The latter is needed for the statement of lemma 3.5 also to be true in the cases
|I| ≤1 and|I∩J| ≤1. Let for p, q ∈ {0, . . . , n−1} be M(k, p) the set of pairs (I, J) with
I, J ⊂[n],|I|=k+ 1, and|J|=p+ 1. Then Sn acts onM(k, p) byσ·(I, J) = (σ(I), σ(J)). The stabiliser of (I, J) is given by
S(I, J) :=S(I)∩S(J) =SI\J×SJ\I×SI∩J ×SI∪J.
The orbits are the set of pairs such that the value of |I ∩J| is constant. We have the commutative diagram of closed regular embeddings
∆I ιI ## G G G G G G G G G j
∆I∩∆J b 99 s s s s s s s s s s a %% K K K K K K K K K
K ∆I∩J ιI∩J //
Xn.
∆J i OO ιJ ;; w w w w w w w w w (3.1)
The subvarieties ∆I and ∆J intersect transversely inside ∆I∩J. We denote the inclusion of ∆I∩∆J into Xn by t.
Lemma 3.5. For (I, J)∈M(k, p) we have an S(I, J)-equivariant isomorphism
Extq(EI, FJ) =t∗
((E∨)I⊗FJ)|∆I∩∆J ⊗(∧
2kN
I)|∆I∩∆J ⊗(∧
2k−qN∨
I∩J)|∆I∩∆J
.
(3.2)
Proof. We will prove (3.2) with the roles of q and 2k−q interchanged. In DbS(I,J)(Xn) we have the isomorphisms
RHomXn(EI, FJ)∼=EIv⊗LFJ
2.4 ∼
= ιI∗(p∗IE∨⊗ ∧2kNI)⊗LιJ∗p∗JF[−2k]
PF ∼
= ιI∗
p∗IE∨⊗ ∧2kNI⊗Lι∗IιJ∗p∗JF
[−2k].
Taking cohomology in degree 2k−q we get
Ext2k−q(EI, FJ)∼=ιI∗
p∗IE∨⊗ ∧2kNI⊗L−qι∗IιJ∗p∗JF
.
By diagram (3.1),Lι∗IιJ∗p∗JF ∼=Lj∗Lι∗I∩JιI∩J∗i∗p∗JF and by lemma 2.5
L−qι∗I∩JιI∩J∗i∗p∗JF ∼=i∗p
∗
JF⊗ ∧qN
∨
I∩J
PF ∼
= i∗ p∗JF ⊗i∗∧qNI∨∩J
.
By the base change formula [Kuz06, Corollary 2.27] we have Lj∗i∗∼=b∗La∗. This yields
Lj∗L−qι∗I∩JιI∩J∗i∗p∗JF ∼=b∗La
∗
p∗JF⊗i∗∧qNI∨∩J
∼lf
=b∗a∗ p∗JF⊗i∗∧qNI∨∩J
.
In particular,Lj∗L−qι∗I∩JιI∩J∗i∗p∗JF is cohomologically concentrated in degree zero for every
q ∈Z. Using the spectral sequence associated to the functorj∗ (see e.g. [Huy06, p. 81]) this gives
L−qι∗IιJ∗p∗JF ∼=j
∗
L−qι∗I∩JιI∩J∗i∗p∗JF ∼=b∗ (p
∗
JF)|∆I∩∆J ⊗(∧ qN∨
I∩J)|∆I∩∆J
.
Plugging this into (3.3) and using again the projection formula yields the result.
Lemma 3.6. Let |I\J| ≥2 or |J\I| ≥2. Then Extq(EI⊗aI, FJ⊗aJ)S(I,J)= 0for every
q ∈Z.
Proof. By the previous lemma we see that the action ofSI\J×SJ\I onExtq(EI, FJ) is trivial (see also corollary 2.8). Since
Extq(EI⊗aI, FJ ⊗aJ)∼=Extq(EI, FJ)⊗aI⊗aJ ∼=Extq(EI, FJ)⊗aI\J ⊗aJ\I
asS(I, J)-sheaves, there is a transposition acting by−1 as soon as|I\J| ≥2 or|J\I| ≥2.
This makes the invariants vanish.
Lemma 3.7. Let k∈ {0, . . . , n−1}. All the termsB(k)p,q1 vanish unlessp∈ {k−1, k, k+ 1}
and q ∈ {0,2, . . . ,2k} is even.
Proof. LetO(k, p) be a system of representatives of theSn-orbits of M(k, p). By lemma 2.2
(3.4)
B(k)p,q1 =Extq(CEk, CFp)∼= M M(k,p)
Extq(EI⊗aI, FJ⊗aJ)
Sn
∼
= M
O(k,p)
Extq(EI, FJ)⊗aI\J⊗aJ\I
S(I,J) .
Now if p /∈ {k−1, k, k+ 1} we have|I\J| ≥2 or|J\I| ≥2 for all (I, J) ∈M(k, p) which gives the vanishing by lemma 3.6. The only factor of (3.2) on which SI∩J acts non-trivially is ∧2k−qN∨
I∩J. Thus, the vanishing forq odd follows by corollary 2.8.
Lemma 3.8. For k= 0, . . . , n−1 and q= 1, . . . , k all the sequences 0→B(k)1k−1,2q→B(k)k,12q→B(k)k1+1,2q→0
are exact. Furthermore B(k)k1−1,0 vanishes and B(k)k,10 →B(k)k1+1,0 is surjective.
Proof. In order to compute the terms of B(k) we use formula (3.4) and the fact that by lemma 3.6 only the summands such that neither|I\J| ≥2 nor|J\I| ≥2 are non-vanishing. Note in the following that for the extremal cases k = 0, n−1 some of the occurring terms vanish but nevertheless the proof remains valid in these cases. We see that the only Sn-orbit in M(k, k−1) contributing to (3.4) is represented by ([k+ 1],[k]). Using lemma 3.5 and corollary 2.8 we get
B(k)k1−1,2q∼=Ext2q(E[k+1], F[k]))
S[k]×S[k+1] ∼=
Hom(E, F)⊗ω−Xq S
forq = 1, . . . , kandB(k)k1−1,0 = 0. InM(k, k) there are two orbits such that|I\J|,|J\I| ≤1; namely those represented by ([k+ 1],[k+ 1]) and ([k+ 1],[k]∪ {k+ 2}). InM(k, k+ 1) there is one such orbit represented by ([k+ 1],[k+ 2]). This gives
B(k)k,12q ∼=
Hom(E, F)S[k+1]
[k+1] forq= 0,
Hom(E, F)⊗ω−XqS[k+1]
[k+1] ⊕ Hom(E, F)⊗ω
−q X
S[k+2]
[k+2] forq= 1, . . . , k,
B(k)k1+1,2q ∼= Hom(E, F)⊗ωX−qS[k+2]
[k+2] forq= 0, . . . , k,
We first consider the case thatq = 1, . . . , k. We show that under the above isomorphisms the component
ϕ: Hom(E, F)⊗ωX−qS[k[+1]k+1] → Hom(E, F)⊗ωX−qS[k[+1]k+1]
ofdk1−1,2q is an isomorphism (already before takingS[k+1]-invariants). For this we can argue locally and thus may assume thatE=F =OX. We have the following commutative diagram where the vertical maps are isomorphisms and the horizontal maps are induced by ϕ
[Lk+1
i=1 Ext2q(O∆[k+1],O∆[k+1]\{i})]
S[k+1] −−−−→ϕ1 Ext2q(O
∆[k+1],O∆[k+1])
S[k+1]
∼
=
y
y ∼
=
[Lk+1
i=1(ω
−k
X )[k+1]⊗ ∧2(k−q)N∆[∨k+1]\{i}|∆[k+1])]
S[k+1] −−−−→ϕ2 [(ω−k
X )[k+1]⊗ ∧2(k−q)N∆∨[k+1]]
S[k+1]
Since the differentialdk1−1,2qis induced by the differentialdk−1:COk−1
X →C k
OX, the components ofϕ1are induced by the restriction mapsO∆[k+1]\{i} → O∆[k+1]. By a local computation using
Koszul complexes (see also [Sca09a, Corollary B.4]) it follows that the components ofϕ2 are
given by the inclusionsN∆[∨
k+1]\{i}|∆[k+1] →N
∨
[k+1]. Under the isomorphism of lemma 2.6 these,
in turn, are given by the inclusions%[k+1]\{i} →%[k+1]. It follows by [Sca09a, Lemma B.6(4)]
thatϕis indeed an isomorphism. By the same arguments, the component
Hom(E, F)⊗ωX−qS[k+2]
[k+2] → Hom(E, F)⊗ω
−q X
S[k+2] [k+2]
of dk,12q is an isomorphism for q = 1, . . . , k. These two facts together show that the short exact sequences in the rows 2,4, . . . ,2kare indeed exact. Consider now the caseq= 0. Then, one is left with the component
(Hom(E, F))S[k[+1]k+1] →(Hom(E, F))S[k[+2]k+2]
of dk,10 which is given by the restriction map. We have to show that it is still surjective after taking the invariants. Since SnX can be covered by open subsets SnU with U ⊂ X open and affine, it is sufficient that the map is surjective on the level of sections overSnU which
is shown in [Sca09a, Lemma 5.1.2].
Corollary 3.9. Let E, F be locally free sheaves on X and k= 0. . . , n−1. Then the object
RHom(CEk, p∗q∗F[n])is cohomologically concentrated in degreek, i.e. Extm(CEk, p∗q∗F[n]) = 0 for m6=k.
Proposition 3.10. LetE andF be locally free sheaves onX. ThenRHom(Φ(E[n]),Φ(F[n]))
is cohomologically concentrated in degree zero and there is a natural isomorphism
RHom(Φ(E[n]),Φ(F[n]))∼=Hom(CE0, CF0) in Db(SnX). Proof. As mentioned above we use the spectral sequence
Ap,q1 =Extq(CE−p, p∗q∗F[n]) =⇒Am=Extm(CE•, p∗q∗F[n])∼=Extm(Φ(E[n]),Φ(F[n])).
By the above corollary the 1-sheet of A is concentrated on the diagonal p+q = 0. Thus
Am= 0 for m6= 0. This yields
[RHom(Φ(E),Φ(F))=∼Hom(p∗q∗(E[n]), p∗q∗(F[n]))3∼=.2Hom(CE0, CF0).
From tautological bundles to tautological objects. Remark 3.11. The functor C = C0 = ⊕n
i=1p∗i: Coh(X) → CohSn(X
n) is exact, since the
projectionspi are flat. Thus, it descents to the functorC =⊕ni=1pi∗: Db(X)→DbSn(X n) on
the level of the derived categories.
Proposition 3.12. Let E•, F•∈Db(X). Then there are natural isomorphisms
RHom Φ((E•)[n]),Φ((F•)[n])∼=RHom(C(E•), C(F•)), RHom(Φ((E•)[n]),OXn)∼=RHom(C(E•),OXn). Proof. Taking locally free resolutions A• ∼=E• andB•∼=F• gives
Φ((E•)[n])∼= Φ((A•)[n])2∼=.12p∗q∗((A•)[n]) , Φ((F•)[n])∼= Φ((B•)[n])2∼=.12p∗q∗((B•)[n]) Now, for every pair i, j∈Zby proposition 3.10 we have
Extq p∗q∗((Ai)[n]), p∗q∗((Bi)[n])= 0
for q6= 0. Thus, we can apply proposition 2.1 to the situation of the bifunctor
Hom( , ) = [ ]Sn◦π∗◦ Hom( , ) : QCoh
Sn(X n)◦×
QCohSn(Xn)→QCoh(SnX) and obtain using proposition 3.10 that
RHom Φ((E•)[n]),Φ((F•)[n]) ∼
=RHom p∗q∗((A•)[n]), p∗q∗((B•)[n])
2.1 ∼
= Hom• p∗q∗((A•)[n]), p∗q∗((B•)[n])
3.10 ∼
= Hom•(C(A•), C(B•))
lf ∼
=RHom(C(A•), C(B•))
3.11 ∼
= RHom(C(E•), C(F•)).
Natural line bundles. At the level of Picard groups, there is a homomorphism which associates to any line bundle on X thenatural line bundle onX[n] given by
D: PicX→PicX[n] , L7→ DL:=µ∗((Ln)Sn).
Here the Sn-linearisation of Ln is given by the canonical isomorphisms p∗σ−1(i)L∼= σ∗p∗iL,
i.e. it is given by permutation of the factors.
Remark 3.13. By [DN89, Theorem 2.3] the sheaf of invariants ofLn is the descent ofLn, i.e. Ln∼=π∗((Ln)Sn).
Remark 3.14. The homomorphismDmaps the trivial line bundle to the trivial line bundle and the canonical line bundle to the canonical line bundle, i.e. DOX ∼=OX[n] andDωX ∼=ωX[n]
(see [NW04, Proposition 1.6]).
Lemma 3.15. Let L be a line bundle onX.
(i) For everyF ∈Db(X[n])there is a natural isomorphismΦ(F ⊗ DL)∼= Φ(F)⊗Ln in DSn(X
n). In particular, Φ(D
L)∼=Ln.
(ii) For every G ∈DbSn(Xn) there is the isomorphism (G ⊗Ln)Sn ∼= (G)Sn⊗(Ln)Sn. Proof. By remark 3.13 we have q∗DL ∼= q∗µ∗(Ln)Sn =∼ p∗π∗(Ln)Sn ∼=p∗Ln. Using this, we get indeed natural isomorphisms
Rp∗q∗(F ⊗ DL)∼=Rp∗(q∗F ⊗q∗DL)∼=Rp∗ q∗F ⊗p∗LnPF∼= Rp∗q∗(F)⊗Ln.
Since Φ(OX[n]) ∼= OXn (see [Sca09a, Proposition 1.3.3]), we get Φ(DL) ∼= Ln as a special case. For (ii) we remember that the functor ( )Sn: Db
Sn(X
n)→Db(SnX) is an abbreviation of the composition ( )Sn◦π∗. Then
π∗(G ⊗Ln)Sn ∼ =
h
π∗(G ⊗π∗(Ln)Sn)iSn PF∼= hπ∗(G)⊗(Ln)SniSn ∼= (π∗G)Sn⊗(Ln)Sn.
We call an object of the form E[n]⊗ DL ∈Db(X[n]) with E ∈Db(X) and L a line bundle onX atwisted tautological object.
Proposition 3.16. Let E[n]⊗ DL and F[n]⊗ DM be twisted tautological objects. Then there are natural isomorphisms
RHom Φ(E[n]⊗ DL),Φ(F[n]⊗ DM)
∼
=RHom C(E)⊗Ln, C(F)⊗Mn, RHom Φ(E[n]⊗ DL),Φ(DM)∼
=RHom C(E)⊗Ln, Mn .
Proof. We will only show the first isomorphism since the proof of the second one is very similar. We have indeed
h
RHom Φ(E[n]⊗ DL),Φ(F[n]⊗ DM)iSn
3.15 ∼
=
h
RHom Φ(E[n])⊗Ln,Φ(F[n])⊗MniSn
∼
=
h
3.15 ∼
= RHom(Φ(E[n]),Φ(F[n]))Sn⊗((Ln)∨)Sn⊗(Mn)Sn
3.12 ∼
= RHom(C(E), C(F))Sn⊗((Ln)∨)Sn⊗(Mn)Sn
3.15 ∼
= RHom(C(E), C(F))⊗(Ln)∨⊗MnSn
∼
= [RHom C(E)⊗Ln, C(F)⊗Mn]Sn.
Global Ext-groups.
Theorem 3.17. Let X be a smooth quasi-projective complex surface, n≥2, E, F ∈Db(X), and L, M ∈ PicX. Then the extension groups of the associated twisted tautological objects are given by the following natural isomorphisms of graded vector spaces:
Ext∗(E[n]⊗ DL, F[n]⊗ DM)∼= Ext
∗
(E⊗L, F⊗M)⊗Sn−1Ext∗(L, M)⊕
Ext∗(E⊗L, M)⊗Ext∗(L, F⊗M)⊗Sn−2Ext∗(L, M),
Ext∗(E[n]⊗ DL,DM)∼= Ext∗(E⊗L, M)⊗Sn−1Ext∗(L, M).
Proof. Using the previous proposition and the considerations at the beginning of this section, the extension groups are given by
Ext∗ E[n]⊗ DL, F[n]⊗ DM
∼
= Ext∗ Φ(E[n]⊗ DL),Φ(F[n]⊗ DM)
Sn
∼
= H∗(SnX, h
RHom(Φ(E[n]⊗ DL),Φ(F[n]⊗ DM))
iSn )
∼
= H∗(SnX,
RHom(C(E)⊗Ln, C(F)⊗Mn)Sn )
∼
=H∗ Xn, RHom(C(E)⊗Ln, C(F)⊗Mn)Sn .
Applying lemma 2.2 we get
RHom(C(E)⊗Ln, C(F)⊗Mn)Sn
∼
=
M
i,j∈[n]
RHom(p∗iE⊗Ln, p∗jF⊗Mn)
Sn
∼
=RHom(p∗1E⊗Ln, p∗1F⊗Mn)S[1]
⊕
RHom(p∗1E⊗Ln, p∗2F ⊗Mn)S[2] .
(∗)
Using the compatibility of the derived sheaf-Hom with pullbacks gives
RHom(p∗1E⊗Ln, p∗1F⊗Mn)∼=RHom(E⊗L, F ⊗M)Hom(L, M)n−1 and also
RHom(p∗1E⊗Ln, p∗2F⊗Mn)
∼
=RHom(E⊗L, M)Hom(L, F⊗M)Hom(L, M)n−2.
Now by the K¨unneth formula
∼
=H∗(RHom(E⊗L, F⊗M))⊗H∗(Hom(L, M))⊗n−1S[1] ∼
=Ext∗(E⊗L, F ⊗M)⊗Ext∗(L, M)⊗n−1S[1]
∼
= Ext∗(E⊗L, F ⊗M)⊗Sn−1Ext∗(L, M).
Doing the same for the other direct summand in (∗) yields the result. The proof of the second
formula is again similar and therefore omitted.
Remark 3.18. The above formulas are functorial in E,F, L, and M, as well as automor-phisms of X and pull-backs along open immersionsU ⊂X.
Remark 3.19. By setting L = M = OX we get the following formulas for non-twisted tautological objects
Ext∗(E[n], F[n])= Ext∼ ∗(E, F)⊗Sn−1H∗(OX)⊕H∗(Ev)⊗H∗(F)⊗Sn−2H∗(OX) Ext∗(E[n],OX)∼= H∗(X[n],(E[n])v)∼= H∗(Ev)⊗Sn−1H∗(OX).
Remark 3.20. LetF ∈Db(X). Computations similar to the ones performed in the previous subsections show that Φ(F[n]⊗ DL)Sn ∼= (C(F)⊗Ln)Sn (see [Sca09b, Theorem 31] for the case that F ∈ Coh(X)). As a consequence, for every twisted tautological object F[n]⊗ DL there is a natural isomorphism of graded vector spaces
H∗(X[n], F[n]⊗ DL)∼= H∗(F ⊗L)⊗Sn−1H∗(L)
(see [Dan01] for the case of tautological bundles and [Sca09b] for the case of tautological sheaves). Moreover, the fact that D∨L ⊗ DM ∼= DHom(L,M) for L, M ∈ Pic(X) yields the
formula
Ext∗(DL, F[n]⊗ DM)∼= Ext∗(L, F ⊗M)⊗Sn−1Ext∗(L, M).
In the case thatX is projective one can also directly deduce the formula of theorem 3.17 for Ext∗(E[n]⊗ D
L,DM) by Serre duality from this formula using that DωX =ωX[n].
Remark 3.21. Because of Φ(DL)∼=Ln we get for L, M ∈PicX natural isomorphisms Ext∗(DL,DM)∼= Ext∗(Ln, Mn)Sn ∼=SnExt∗(L, M).
Remark 3.22. Let forE1, . . . , Ek, F be locally free sheaves on X. Using proposition 3.2 we also get formulas for Ext0(E1[n]⊗ · · · ⊗Ek[n],OX[n]) as well as for Ext0(E
[n]
1 ⊗ · · · ⊗E [n]
k , F[n]). For`∈NandM a set letP(M, `) denote the set of partitionsI ={I1, . . . , I`}ofM of length
`. Then
Hom(E1[n]⊗ · · · ⊗Ek[n],OX[n])∼=
M
I∈P([k],`),`≤n `
O
s=1
H0(⊗i∈IsEi∨)⊗Sn
−`H0(O
X)
!
and Hom(E1[n]⊗ · · · ⊗Ek[n], F[n]) is naturally isomorphic to
M
M⊂[k]
I∈P([k]\M,`),`≤n−1
Hom(⊗i∈MEi, F)⊗ `
O
s=1
H∗(⊗i∈IsEi∨)⊗Sn−`−1H0(OX)
! .
Again, there are similar formulas for the sheaves twisted by natural line bundles. However, these formulas cannot be generalised directly to formulas for Ext∗ since the corresponding
4. Consequences
Spherical and Pn-objects. Let M be a smooth projective variety with canonical bundle
ωM. We call an objectE ∈Db(M) anωM-invariant object ifE ⊗ωM ∼=E. A spherical object in Db(M) is anωM-invariant objectE with the property
Exti(E,E) =
(
C ifi= 0,dimM, 0 else,
i.e. Ext∗(E,E) ∼= H∗(SdimM,C), where Sn denotes the real n-sphere. A Pn-object in the derived category Db(M) is a ωM-invariant object F such that there is an isomorphism of graded algebras Ext∗(F,F)=∼H∗(Pn,C), where the multiplication on the left is given by the Yoneda product and on the right by the cup product. In particular, we have
Exti(E,E) =
(
C if 0≤i≤2nis even 0 ifiis odd
By Serre duality the dimension of M must be 2n as soon as Db(M) contains a Pn-object. Spherical andPn-objects are of interest because they induce automorphisms of Db(M) (see [ST01] and [HT06]). For a smooth projective surfaceX withωX =OX the canonical bundle on X[n] is also trivial (see remark 3.14). Hence, the property of being ωX[n]-invariant is
automatically satisfied for every object in Db(X[n]). Thus, one could hope that there are tautological objects induced by some special objects in Db(X) that are spherical or Pn -objects. But this is not the case by the following proposition.
Proposition 4.1. Let X be a smooth projective surface with trivial canonical line bundle and
n≥2. Then twisted tautological objects on X[n] are never spherical or Pn-objects.
Proof. Let E[n]⊗ DL be a non-zero twisted tautological object onX[n]. Then using the fact that Hom(L, L)∼=OX we have
Ext∗(E[n]⊗ DL, E[n]⊗ DL)∼= Ext∗(E[n], E[n]). Thus, we may assume L=OX. By remark 3.19
Ext∗(E[n], E[n])= Ext∼ ∗(E, E)⊗Sn−1H∗(OX)⊕H∗(Ev)⊗H∗(E)⊗Sn−2H∗(OX). Because of ωX = OX, we have by Serre duality h2(OX) = h0(OX) = 1. The vector space Ext0(E, E) is non-zero since it contains the identity. Thus, Ext0(E, E)⊗Sn−1H∗(OX) con-tributes non-trivially to the degrees 0,2, . . . ,2n−2 of Ext∗(E[n], E[n]). But by Serre duality also Ext2(E, E) is non-vanishing. Thus Ext2(E, E) ⊗Sn−1H∗(OX) also contributes non-trivially to the degrees 2,4, . . . ,2n. This yields exti(E[n], E[n]) ≥2 for i= 2,4,2n−2 which shows thatE[n]⊗ DL is indeed neither a spherical nor aPn-object.
Yoneda products. Theorem 3.17 and the remarks 3.20 and 3.21 together give formulas for Ext∗X[n](E,F) in terms of extension groups on the surface X whenever E,F ∈ Db(X[n]) are
tautological objects or natural line bundles, we get the commutative diagram
Ext∗(F,G)×Ext∗(E,F) −−−−→Yon Ext∗(E,G)
∼
=
y
y ∼
=
Ext∗(Φ(F),Φ(G))Sn×Ext∗(Φ(E),Φ(F))Sn −−−−→Yon Ext∗(Φ(E),Φ(G))Sn
∼
=
y
y ∼
=
Ext∗(Ψ(F),Ψ(G))Sn×Ext∗(Ψ(E),Ψ(F))Sn −−−−→Yon Ext∗(Ψ(E),Ψ(G))Sn.
The formulas in terms of the extension groups on the surfaceX are obtained by computing Ext∗(Ψ(E),Ψ(F)) using the K¨unneth isomorphism and the isomorphism of Danila’s lemma 2.2. Following these isomorphisms, one gets explicit formulas for the Yoneda products. There will occur signs resulting from the graded commutativity of the cup product: letE1, . . . , Ek∈ Db(X). Then forσ ∈Sk the map
σ: H∗(Xk, E1· · ·Ek)→H∗(Xk, Eσ−1(1)· · ·Eσ−1(k))
induced by the action ofσ on Xk is given under the K¨unneth isomorphism by
σ(s1⊗ · · · ⊗sk) =εσ,p1,...,pk(sσ−1(1)⊗ · · · ⊗sσ−1(k)).
(4.1)
(compare (2.1)). Here, pi is the cohomological degree of si, i.e. si ∈ Hpi(X, Ei). Let
E, E0, E00∈Db(X) andF, F0, F00∈Db(Y). Then the Yoneda pairing
◦ : Ext∗(E0F0, E00F00)×Ext∗(EF, E0F0)→Ext∗(EF, E00F00) is given under the K¨unneth isomorphism for s1 ∈ Extp1(E0, E00), s2 ∈ Extp2(F0, F00), t3 ∈
Extp3(E, E0), andt
4 ∈Extp4(F, F0) by
(s1⊗s2)◦(t3⊗t4) = (−1)p2p3(s1◦t3)⊗(s2◦t4).
(4.2)
One way to see the need for the sign (−1)p2p3 is to define the Yoneda product via the cup
product (see e.g. [HL10, Section 10.1.1]). In order to capture these signs we use the follow-ing notation: Let m ∈ N and s1, . . . , sm be homogeneous elements of graded vector spaces
V1∗, . . . , Vm∗ of degrees p1, . . . , pm, i.e. si ∈ Vipi. Let T be a mathematical expression (of some vector in some vector space) involving thesi such that after erasing all symbols in the expression besides the si we are left with sσ−1(1)sσ−1(2)· · ·sσ−1(m) for some σ ∈ Sm. Then
we set [T] :=εσ,p1,...,pm·T. With this notation, (4.1) reads
σ(s1⊗ · · · ⊗sk) = [sσ−1(1)⊗ · · · ⊗sσ−1(k)].
We use the notation also in the case that some of the si are instead denoted by ti. For example, (4.2) can be written as
(s1⊗s2)◦(t3⊗t4) = [(s1◦t3)⊗(s2◦t4)].
More generally, let X1, . . . , Xn be varieties and Ei, Ei0, Ei00 ∈ Db(Xi) for i = 1, . . . , n. Then the Yoneda pairing
Ext∗(ni=1E
0
i,ni=1E
00
i)×Ext
∗
(ni=1Ei,ni=1E
0
i)→Ext
∗
(ni=1Ei,ni=1E
00
i)
is given under the K¨unneth isomorphism by
Furthermore, letP
i∈ITi be some finite sum such that the Ti have the same property as T from above. Then we setP•i∈ITi:=Pi∈I[Ti]. For example, (2.2) reads
s1· · ·sn7→ 1
n!
• X
σ∈Sn
sσ−1(1)⊗ · · · ⊗sσ−1(n).
We will explicitly state and prove the formula for the Yoneda product only in the case that all three objects involved are twisted tautological objects
E =E[n]⊗ DL , F =F[n]⊗ DM , G=G[n]⊗ DN.
The other seven cases can be done very similarly. For E, F ∈Db(X) and L, M ∈Pic(X) we set
P(E, L, F, M) := Ext
∗
(E⊗L, F ⊗M)⊗Sn−1Ext∗(L, M)⊕
Ext∗(E⊗L, M)⊗Ext∗(L, F ⊗M)⊗Sn−2Ext∗(L, M).
We consider the elements
ϕ⊗s2· · ·sn
η⊗x⊗t3· · ·tn
∈P(F, M, G, N)∼= Ext∗(F[n]⊗ DM, G[n]⊗ DN),
ϕ0⊗s02· · ·s0n η0⊗x0⊗t0
3· · ·t0n
∈P(E, L, F, M)∼= Ext∗(E[n]⊗ DL, F[n]⊗ DM).
In order to use the sign convention from above we set
s1 :=ϕ , t1 :=η , t2:=x , s01 :=ϕ
0
, t01 :=η0, t02 :=x0 , sn+i :=s0i, tn+i:=t0i
and assume that all thesi andti are homogeneous. Now we can compute the Yoneda product ϕ⊗s2···sn
η⊗x⊗t3···tn
◦ ϕ0⊗s02···s
0 n η0⊗x0⊗t0
3···t0n
in Ext∗(E[n]⊗ DL, G[n] ⊗ DN) and express it as an element in
P(E, L, G, M).
Proposition 4.2. The Yoneda product is given by 1
(n−1)!
• X
σ∈S[2,n]
(ϕϕ0)⊗(s2s0σ−1(2))· · ·(sns0σ−1(n))
+ 1
(n−2)!
• X
τ∈S[3,n]
(xη0)⊗(ηx0)·(t3t0τ−1(3))· · ·(tnt0τ−1(n))
⊕ 1
(n−1)!
• X
β∈S[2,n]
(ηϕ0)⊗(xs0β−1(2))⊗(t3s0β−1(3))· · ·(tns0β−1(n))
+ 1
(n−1)!
• X
γ∈S[2,n]
(sγ−1(2)η0)⊗(ϕx0)⊗(sγ−1(3)t03)· · ·(sγ−1(n)t0n)
+ 1
(n−2)!
• X
i=3,...,n α∈S[3,...,n]
(tiη0)⊗(xtα0−1(i))⊗(ηx
0
)·(t3t0α−1(3))· · ·(ti\t0α−1(i))· · ·(tnt
0
Proof. The element ϕ⊗s2···sn
η⊗x⊗t3···tn
∈P(F, M, G, N) corresponds to the element
1 (n−1)!
• X
σ∈Sn
sσ−1(1)⊗ · · · ⊗sσ−1(n)⊕
1 (n−2)!
• X
τ∈Sn
tτ−1(1)⊗ · · · ⊗tτ−1(n)
in Ext∗(CF0 ⊗Mn, CG0 ⊗Nn). The coefficients are coming from the canonical isomorphism
SkV → S
kV (see (2.2), note that the isomorphism of lemma 2.2 does not involve such co-efficients). The analogue holds for ϕ0⊗s02···s0n
η0⊗x0⊗t0
3···t0n
. The summand sσ−1(1) ⊗ · · · ⊗sσ−1(n) is
an element of Ext∗(pr∗σ(1)F ⊗Mn,pr∗σ(1)G⊗Nn) and tτ−1(1) ⊗ · · · ⊗tτ−1(2) an element
of Ext∗(pr∗τ(1)F ⊗Mn,pr∗
τ(2)G⊗Nn). There are five types of composable pairs of the
components of the classes in Ext∗(CF0 ⊗Mn, CG0 ⊗Nn)×Ext∗(CE0 ⊗Ln, CF0 ⊗Mn),
namely
pr∗i E⊗Ln→pr∗i F ⊗Mn→pr∗i G⊗Nn,pr∗i E⊗Ln→pr∗jF⊗Mn→pr∗i G⊗Nn,
pr∗i E⊗Ln→pr∗i F ⊗Mn→pr∗jG⊗Nn,pr∗i E⊗Ln→pr∗jF⊗Mn→pr∗jG⊗Nn,
pr∗i E⊗Ln→pr∗jF ⊗Mn→pr∗kG⊗Nn
withi, j, k ∈[n] pairwise distinct. Thus, the Yoneda product in Ext∗(CE0 ⊗Ln, CG0 ⊗Nn) looks like this :
1 (n−1)!2
• X
σ,σ0∈S n σ(1)=σ0(1)
⊗n
i=1(sσ−1(i)s0σ0−1(i)) +
1 (n−2)!2
• X
τ,τ0∈S n τ(2)=τ0(1),τ(1)=τ0(2)
⊗n
i=1(tτ−1(i)t0τ0−1(i))
+ 1
(n−1)!(n−2)!
• X
τ,σ0∈Sn τ(1)=σ0(1)
⊗ni=1(tτ−1(i)s0σ0−1(i)) +
1
(n−1)!(n−2)!
• X
σ,τ0∈Sn σ(1)=τ0(2)
⊗ni=1(sσ−1(i)t0τ0−1(i))
+ 1
(n−2)!2
• X
τ,τ0∈Sn τ(1)=τ0(2),τ(2)6=τ0(1)
⊗ni=1(tτ−1(i)t0τ0−1(i)) .
(6)
The first term of (6) is aSn-invariant element of⊕n i=1Ext
∗(p∗
iE⊗Ln, p
∗
iG⊗Nn). Danila’s isomorphism (lemma 2.2) is simply the projection to one summand. Thus, it maps the first term of (6) to
1 (n−1)!2
• X
σ,σ0∈Sn σ(1)=σ0(1)=1
⊗ni=1(sσ−1(i)s0σ0−1(i))∈Ext
∗
(p∗1E⊗Ln, p∗1G⊗Nn)
Under the isomorphismSn−1Ext∗(L, N)
∼
=
−→Sn−1Ext∗(L, N) this element is mapped to 1
(n−1)!
• X
σ∈S[2,n]
(ϕϕ0)⊗(s2s0σ−1(2))· · ·(sns0σ−1(n))∈Ext∗(E⊗L, G⊗N)⊗Sn−1Ext∗(L, N)
Remark 4.3. LetE, F ∈Db(X) and M ∈PicX. For every morphism
ϕ∈HomDb(X)(E, F[i])∼= Exti(E, F)
the induced morphism ϕ[n]⊗idDM ∈Ext
i(E[n]⊗ D
M, F[n]⊗ DM) corresponds to
ϕ⊗idM· · ·idM 0
∈P(E, M, F, M).
Setting M = OX shows that the tautological functor ( )[n]: Db(X) → Db(X[n]) is faithful but not full.
References
[BKR01] Tom Bridgeland, Alastair King, and Miles Reid. The McKay correspondence as an equivalence of derived categories.J. Amer. Math. Soc., 14(3):535–554 (electronic), 2001.
[BNW07] Samuel Boissi`ere and Marc A. Nieper-Wißkirchen. Universal formulas for characteristic classes on the Hilbert schemes of points on surfaces.J. Algebra, 315(2):924–953, 2007.
[Dan00] Gentiana Danila. Sections du fibr´e d´eterminant sur l’espace de modules des faisceaux semi-stables de rang 2 sur le plan projectif.Ann. Inst. Fourier (Grenoble), 50(5):1323–1374, 2000.
[Dan01] Gentiana Danila. Sur la cohomologie d’un fibr´e tautologique sur le sch´ema de Hilbert d’une surface.
J. Algebraic Geom., 10(2):247–280, 2001.
[Dan07] Gentiana Danila. Sections de la puissance tensorielle du fibr´e tautologique sur le sch´ema de Hilbert des points d’une surface.Bull. Lond. Math. Soc., 39(2):311–316, 2007.
[DN89] J.-M. Drezet and M. S. Narasimhan. Groupe de Picard des vari´et´es de modules de fibr´es semi-stables sur les courbes alg´ebriques.Invent. Math., 97(1):53–94, 1989.
[Fog68] John Fogarty. Algebraic families on an algebraic surface.Amer. J. Math, 90:511–521, 1968. [Hai01] Mark Haiman. Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J. Amer.
Math. Soc., 14(4):941–1006 (electronic), 2001.
[Har66] Robin Hartshorne.Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20. Springer-Verlag, Berlin, 1966.
[HL10] Daniel Huybrechts and Manfred Lehn.The geometry of moduli spaces of sheaves. Cambridge Math-ematical Library. Cambridge University Press, Cambridge, second edition, 2010.
[HT06] Daniel Huybrechts and Richard Thomas.P-objects and autoequivalences of derived categories.Math. Res. Lett., 13(1):87–98, 2006.
[Huy06] Daniel Huybrechts.Fourier-Mukai transforms in algebraic geometry. Oxford Mathematical Mono-graphs. The Clarendon Press Oxford University Press, Oxford, 2006.
[KS06] Masaki Kashiwara and Pierre Schapira. Categories and sheaves, volume 332 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2006.
[Kuz06] A. G. Kuznetsov. Hyperplane sections and derived categories. Izv. Ross. Akad. Nauk Ser. Mat., 70(3):23–128, 2006.
[Leh99] Manfred Lehn. Chern classes of tautological sheaves on Hilbert schemes of points on surfaces.Invent. Math., 136(1):157–207, 1999.
[LH09] Joseph Lipman and Mitsuyasu Hashimoto. Foundations of Grothendieck duality for diagrams of schemes, volume 1960 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 2009.
[NW04] Marc Nieper-Wißkirchen.Chern numbers and Rozansky-Witten invariants of compact hyper-K¨ahler manifolds. World Scientific Publishing Co. Inc., River Edge, NJ, 2004.
[Sca09a] Luca Scala. Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles.Duke Math. J., 150(2):211–267, 2009.
[Sca09b] Luca Scala. Some remarks on tautological sheaves on Hilbert schemes of points on a surface.Geom. Dedicata, 139:313–329, 2009.
[ST01] Paul Seidel and Richard Thomas. Braid group actions on derived categories of coherent sheaves.
Universit¨at Bonn, Institut f¨ur Mathematik
