EXTENSION GROUPS OF TAUTOLOGICAL SHEAVES ON HILBERT 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 or n -object. We also provide a description of the Yoneda products.

P

1. Introduction

C

For every smooth quasi-projective surface X over there is a series of associated higher dimensional smooth varieties namely the Hilbert schemes of n points on X for n ∈ . They are fine moduli spaces X [n] of zero dimensional subschemes of length n of X. Thus, there is a universal family Ξ ⊂ X × X [n] together with its projections pr

pr

N

[n]

X X ←−X− Ξ −−− −→ X [n] .

Using this, one can associate to every coherent sheaf F on X the so called tautological sheaf F [n] on each X [n] given by F [n] := prX [n] ∗ pr∗X F . More generally, for any object of the bounded derived category F ∈ Db (X) using the FourierMukai transform with kernel the structural sheaf of Ξ yields the tautological object [n]

F [n] := ΦX→X (F ) ∈ Db (X [n] ) . OΞ It is well known (see [Fog68]) that the Hilbert scheme X [n] of n points on X is a resolution of the singularities of S n X = X n /Sn via the Hilbert-Chow morphism X µ : X [n] → S n X , [ξ] 7→ `(ξ, x) · x , `(ξ, x) := dimC Oξ,x . x∈ξ

For every L ∈ Pic X the line bundle ∈ Pic(X n ) descends to the line bundle π∗ (Ln )Sn on S n X where π : X n → S n X is the quotient morphism. Thus, we can define on X [n] the natural line bundle DL associated to the line bundle L on X as Ln

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 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 of F [n] in terms of those of F in the case that F is a line bundle. In [BNW07] the existence of universal formulas, i.e. formulas independent of the surface X, expressing characteristic classes of any tautological sheaf in terms of characteristic classes of F is shown and those formulas are computed in some cases. Furthermore, Danila 1

2

ANDREAS KRUG

([Dan00], [Dan01], [Dan07]) 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, (1.1)

H∗ (X [n] , F [n] ⊗ DL ) ∼ = H∗ (F ⊗ L) ⊗ S n−1 H∗ (L)

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 generally twisted 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 ) ⊗ S n−1 Ext∗ (L, M )⊕ Ext∗ (E ⊗ L, M ) ⊗ Ext∗ (L, F ⊗ M ) ⊗ S n−2 Ext∗ (L, M )

for objects E, F ∈ Db (X) and line bundles L, M ∈ Pic(X). We also give a similar formula for Ext∗ (E [n] ⊗ DL , DM ). Since DOX = OX [n] , by setting L = 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 or n -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 Gcluster 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 G . The Nakamura G-Hilbert scheme HilbG (M ) is defined as the irreducible component of the fine moduli space of G-clusters on M that contains the points corresponding to free orbits. Bridgeland, 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 family Z of G-clusters

P

C

Φ := ΦOZ : Db (HilbG (M )) → DbG (M ) between the bounded derived category of HilbG (M ) and the equivariant bounded derived category DbG (M ) = Db (CohG (M )) of M 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 of S n X to HilbSn (X n ) and that the isospectral Hilbert scheme
I n X := X [n] ×S n X X n 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 [n]

n

∼ =

→X Φ := ΦX : Db (X [n] ) − → DbSn (X n ) . OI n X

Using this, we can compute extension groups on X [n] as Sn -invariant extension groups on X n after applying the Bridgeland-King-Reid equivalence, i.e. for E, F ∈ Db (X [n] ) we have ExtiX [n] (E, F) ∼ = ExtiX n (Φ(E), Φ(F))Sn . The right-hand side can be rewritten further as (1.2)

ExtiX n (Φ(E), Φ(F)))Sn ∼ = Hi (S n X, [π∗ R HomX n (Φ(E), Φ(F)]Sn ) .

EXTENSION GROUPS OF TAUTOLOGICAL SHEAVES ON HILBERT SCHEMES

3

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] ) ∈ DbSn (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 I n X to X n 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 of the form C(F ) := CF0 =

n M i=1

pr∗i F

,

CFp =

M

FI

for 0 < p < n

,

CFp = 0

else.

I⊂[n], |I|=p+1

Here, [n] := {1, . . . , n} and FI denotes the sheaf ιI∗ p∗I F , where ιI : ∆I → X n is the inclusion of the partial diagonal and pI : ∆I → X is the morphism induced by the projection pri : X n → X for any i ∈ I. The main step in the proof of the formula for the extension groups is the computation of [π∗ R HomX n (Φ(E [n] ), Φ(F [n] )]Sn in the case of tautological bundles, i.e. for E, F locally free sheaves on X. We show that (1.3)

[π∗ R Hom(Φ(E [n] ), Φ(F [n] ))]Sn ∼ = [π∗ Hom(CE0 , CF0 )]Sn .

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 for p > 0 have codimension at least two and the normality of the variety X n . For the vanishing of the higher derived sheaf-Homs we use the spectral sequences associated to the bifunctor Hom( , ) := [π∗ HomX n ( , )]Sn . We can generalise (1.3) to arbitrary objects E • , F • ∈ Db (X) by taking locally free resolutions on X and using some formal arguments for derived functors. Since Φ(F ⊗ DL ) ∼ = Φ(F) ⊗ Ln b [n] holds for every object F ∈ D (X ), we can generalise further to twisted tautological objects and get
S
Sn R Hom Φ((E • )[n] ⊗ DL ), Φ((F • )[n] ⊗ DM ) n ∼ . = R Hom C(E • ) ⊗ Ln , C(F • ) ⊗ M n 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. The author wants to thank his advisor Marc Nieper-Wißkirchen for his support. He thanks Malte Wandel for interesting and helpful discussions about the topic of this paper and Ciaran Meachan for carefully reading a version of this article and giving useful comments. He also thanks the referee for his numerous corrections and suggestions which helped to significantly improve this text.

4

ANDREAS KRUG

2. Preliminaries General notation and conventions. Throughout the text, X will be a smooth quasiprojective surface over , n ≥ 2 an integer and X [n] the Hilbert scheme of n points on X. We write ( )∨ = Hom( , OM ) for the dual of a sheaf and ( )v = R Hom( , 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∈Z V i [−i]. The symmetric power of a graded vector space is taken in the graded sense. That means that S n V ∗ are the coinvariants of (V ∗ )⊗n under the Sn -action given on homogeneous vectors by

C

σ(s1 ⊗ · · · ⊗ sn ) := εσ,p1 ,...,pk (sσ−1 (1) ⊗ · · · ⊗ sσ−1 (n) ) .

(2.1)

Here the pi are the degrees of the si . The sign ε is 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. Since the graded vector spaces we will consider are defined over , the coinvariants under this action coincide with the invariants under the isomorphism 1 X (2.2) εσ,p1 ,...,pk (sσ−1 (1) ⊗ · · · ⊗ sσ−1 (n) ) . s1 · · · sn 7→ n!

C

σ∈Sn

Derived bifunctors. Let A, B and C be abelian categories and F : A×B → C be an additive bifunctor left exact in both arguments. We assume that there is a full additive subcategory J of B such that for every B ∈ J the functor F ( , 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 bifunctor RF : 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 choose J 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 • ) := tot F (A• , B • ) .

Proposition 2.1. Under the assumptions from above let A• ∈ D+ (A) and B • ∈ D+ (B) be complexes such that Rq F (Ai , B j ) = 0 for all q = 6 0 and all pairs i, j ∈ . Then we have RF (A• , B • ) ∼ = F • (A• , B • ).

Z

Equivariant sheaves and derived categories. Let G be 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 over , 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 G act transitively on a set I and let (F, λ) ∈ CohG (M ) decomposing into F = ⊕i∈I Fi such that for every i ∈ I and g ∈ G the linearisation λg restricted to Fi is an isomorphism

C

∼ =

λg : Fi − → g ∗ Fg(i) . Then the G-linearisation of F restricts to a StabG (i)-linearisation of Fi .

EXTENSION GROUPS OF TAUTOLOGICAL SHEAVES ON HILBERT SCHEMES

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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 with g · 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 on I` for every 1 ≤ ` ≤ k and the lemma can be applied to every FI` = ⊕i∈I` Fi instead of F . 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 closed G-equivariant embedding j : 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 cokernel NZ/M inherits a G-linearization λ. 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∗ (∧c NZ/M )[−c]. (ii) For q ∈ there are G-equivariant isomorphisms Extq (j∗ OZ , j∗ OZ ) ∼ = j∗ (∧q NZ/M ). b b ! ∗ c (iii) Let j := Lj ( ) ⊗ ∧ NZ/M [−c] : DG (M ) → DG (Z). Then there are in DbG (M ) for E ∈ DbG (Z) and F ∈ DbG (M ) natural isomorphisms

Z

Rj∗ R HomZ (E, j ! F) ∼ = R HomM (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 every F ∈ CohG (Z) and every q ∈

Z there are G-equivariant isomorphisms

∨ L−q j ∗ j∗ F := H−q (Lj ∗ j∗ F ) ∼ . = F ⊗ ∧q NZ/M

Proof. We first proof the special case F = OZ . By proposition 2.4 (iii) j∗ (Lj ∗ j∗ OZ ⊗ ∧c NZ/M [−c]) ∼ = j∗ j ! j∗ OZ ∼ = j∗ R HomZ (OZ , j ! j∗ OZ ) ∼ = R HomM (j∗ OZ , j∗ OZ ) . Now taking the (c − q)-th cohomology on both sides by 2.4 (ii) yields j∗ (L−q j ∗ j∗ OZ ⊗ ∧c NZ/M ) ∼ = j∗ (∧c−q NZ/M ) . Using the fact that j∗ : CohG (Z) → CohG (M ) is fully faithful we can cancel it from the iso∨ morphism. Tensoring with ∧c NZ/M gives the result. For general F , we can use the projection formula twice to get j∗ Lj ∗ j∗ F ∼ = j∗ (Lj ∗ j∗ F ⊗ OZ ) ∼ = j∗ F ⊗L j∗ OZ ∼ = j∗ (F ⊗L Lj ∗ j∗ OZ ) . We have already proven that the Lq j ∗ j∗ OZ are locally free and hence F ⊗ ( )-acyclic. Thus, after taking the −q-th cohomology on both sides and cancelling j∗ we get the result. 

6

ANDREAS KRUG

Partial diagonals and the standard representation. Let n ≥ 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 write X I for the variety X |I| equipped with the canonical SI action. We set S(I) := SI ×SI¯. The I-th partial diagonal is defined as the reduced subvariety of X n given by ∆I := {(x1 , . . . , xn ) ∈ X n | xi = xj for all i, j ∈ I}. It is an S(I)-equivariant subvariety ¯ isomorphic to X × X I . We denote its inclusion by ιI : ∆I → X n . 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 : X n → X for any i ∈ I. For every coherent sheaf F ∈ Coh(X) we set ∼ ιI∗ (F  O I¯) ∈ CohS(I) (X n ) . FI := ιI∗ p∗ F = I

X

The S(I)-linearisation of FI is given by the natural SI¯-linearisation of OX I¯ and the trivial SI -action. The natural representation I of SI is the vector space |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). The standard representation is defined as the quotient by this subspace. So we have the short exact sequence 0 → → I → %I → 0 where the first map is the diagonal embedding.

C

C

C

C

¯ Lemma 2.6. Under the identification ∆I ∼ = X × X I we have NI ∼ = (TX ⊗ %I )  OX I¯ 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 onedimensional if 0 ≤ i ≤ 2(n − 1) is even and zero if i is odd. In the even case i = 2` the space of invariants [∧i (V ⊗ %n )]Sn is spanned by the image of the α` ∈ ∧i (V ⊗ n ) under the Pvector n n projection induced by the projection → %n where α = i=1 uei ∧ vei ∈ ∧2 (V ⊗ n ).

C

C

C

−`  OX I¯. In Corollary 2.8. For 0 ≤ ` ≤ |I| − 1 there are isomorphisms [∧2` NI ]SI ∼ = ωX 2(|I|−1) particular, the SI -action on the highest wedge power ∧ NI is trivial. Furthermore, the SI -invariants of ∧k NI vanish if k is odd.

Remark 2.9. In the hypothesis of corollary 2.8, after pushing forward by ιI we get −` )I . ιI∗ NI ∼ = (TX )I ⊗ %I , ιI∗ (∧2` NI )SI ∼ = (ωX Also note that we get together with proposition 2.4 (iii) for |I| = p + 1 that −p (ιI∗ ι! OX n )SI ∼ = ιI∗ ι! OX n ∼ = (ω )I [−2p] . I

The complex

C •.

C(F ) := CF0 :=

n M i=1

I

X

For every F ∈ Coh(X) we define the complex CF• as follows. Let p∗i F

,

CFp :=

M

FI ⊗ aI

for 0 < p < n

,

CFp := 0

else.

I⊂[n] , |I|=p+1

Here aI denotes the alternating representation of SI , i.e. the one-dimensional representation on which σ ∈ SI acts by multiplication by sgn(σ). There is a canonical Sn -linearisation λp of every CFp such that λpσ maps FI isomorphically to σ ∗ Fσ(I) for every σ ∈ Sn and that restricts to the S(I)-linearisation of FI ⊗ aI for every I ⊂ [n]. The differentials are given by X dp (s)J := (−1)#{j∈J|j
i∈J

which makes CF• into a Sn -equivariant complex (see [Sca09a, Remark 2.2.2] for details).

EXTENSION GROUPS OF TAUTOLOGICAL SHEAVES ON HILBERT SCHEMES

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Tautological sheaves. 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 to F . In [Sca09b, Proposition 2.3] it is shown that the functor ( )[n] is exact. Thus, it induces the tautological functor for objects ( )[n] : Db (X) → Db (X [n] ). For an object F ∈ Db (X) the tautological object associated to F is F [n] . The Bridgeland–King–Reid–Haiman equivalence. Let µ : X [n] → S n X := X n /Sn the Hilbert-Chow morphism and π : X n → S n X the quotient morphism. We consider the isospectral Hilbert scheme I n X := (X [n] ×S n X X n )red ⊂ X [n] × X n together with the commutative diagram p I n X −−−−→ X n    π qy y X [n] −−−−→ S n X µ

where p and q are the projections from King–Reid–Haiman equivalence by

X [n]

× X n to the factors. We define the Bridgeland–

Φ := Rp∗ ◦ q ∗ : Db (X [n] ) → DbSn (X n ) . Theorem 2.10. The functor Φ 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) ∼ = ExtiX n (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 cohomologically concentrated in degree zero. Furthermore, the complex CF• is a right resolution of p∗ q ∗ (F [n] ). Hence, in DbSn (X n ) 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 [n] X. Then Φ(⊗ki=1 Ei ) is cohomologically concentrated in degree zero and hence isomorphic [n] [n] 0,0 0,0 to p∗ q ∗ (⊗ki=1 Ei ) in DbSn (X n ). Furthermore, p∗ q ∗ (⊗ki=1 Ei ) ∼ = E∞ where E∞ is on the limiting sheet of the spectral sequence M E1p,q = Tor−q (CEi11 , . . . , CEikk ) =⇒ E n = Hn (CE• 1 ⊗L · · · ⊗L CE• k ) . i1 +···+ik =p

The differentials of the spectral sequence are induced by the differentials of the complexes CE• i . 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(d0F : CF0 → CF1 ). Thus, for U ⊂ X n open we can describe the space of sections of p∗ q ∗ F [n] over U as the sections s = (s1 , . . . , sn ) of CF0 with si|∆ij = sj|∆ij for every distinct i, j ∈ [n] or alternatively p∗ q ∗ F [n] (U ) = {s ∈ CF0 (U ) | s|∆ij is (i j)-invariant ∀ 1 ≤ i < j ≤ n} .

8

ANDREAS KRUG

The differential on the 1-level of the spectral sequence of the previous theorem d0,0 1 :

E10,0

=

k O

CE0 i

→

E11,0

=

i=1

k  M

O

CE1 ` ⊗

`=1

CE0 i



i∈[k]\{`}

is given by the direct sum of the maps d0E` ⊗ id⊗i∈[k]\{`} C 0 . For every a : [k] → [n] we set Ei L E(a) := pr∗a(1) E1 ⊗ · · · ⊗ pr∗a(k) Ek . Then E10,0 = a : [k]→[n] E(a). For s a local section of E10,0 we denote its component in E(a) by s(a). Then s is in E20,0 = ker d0,0 1 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 every a : [k] → [n]

which says that s|∆ij is (i j)-invariant. This shows that every local section of [n]

[n]

0,0 p∗ q ∗ (E1 ⊗ · · · ⊗ Ek ) = E∞ ⊂ 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 objects E, F ∈ Db (X [n] ) by the formula (see corollary 2.11) Exti [n] (E, F) ∼ = Exti n (Φ(E), Φ(F))Sn . X

Setting (

)Sn

X

)Sn

◦ π∗ we can rewrite the right-hand side as
ExtiX n (Φ(E), Φ(F)))Sn ∼ = Hi S n X, R HomX n (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 term R HomX n (Φ(E), Φ(F))Sn . We abbreviate the occurring bifunctor by Hom( , ) := HomX n ( , )Sn and set Exti = Ri Hom. Because of the exactness of ( )Sn we have R Hom( , ) ∼ = [R HomX n ( , )]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 a ˆ : 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 a ˆ is also an isomorphism. We construct the inverse b : Hom(E, G) → Hom(F, G) of a. For a morphism ϕ : E → G the morphism b(ϕ) sends s ∈ F (V ) to the unique section in G(V ) which restricts to ϕ(s|V ∩U ) ∈ G(V ∩ U ). 

EXTENSION GROUPS OF TAUTOLOGICAL SHEAVES ON HILBERT SCHEMES

9

D the big diagonal in X n, i.e. the reduced subvariety [ ∆ij = {(x1 , . . . , xn ) | #{x1 , . . . , xn } < n} ⊂ X n , D=

We denote by

1≤i
D its open complement in X n. Proposition 3.2. Let k ∈ N and E1 , . . . , Ek , F ∈ Coh(X) be locally free sheaves. Then there

and by U =

Xn

\

are natural isomorphisms [n]

[n]

1

k

(i) Hom(Φ(E1 ⊗ · · · ⊗ Ek ), OX n ) ∼ = Hom(CE0 1 ⊗ · · · ⊗ CE0 k , OX n ), [n] [n] (ii) Hom(Φ(E ⊗ · · · ⊗ E ), Φ(F [n] )) ∼ = Hom(C 0 ⊗ · · · ⊗ C 0 , C 0 ). E1

Ek

F

[n] [n] [n] [n] Proof. By theorem 2.13 we have Φ(E1 ⊗ · · · ⊗ Ek ) ∼ = p∗ q ∗ (E1 ⊗ · · · ⊗ Ek ). Moreover, [n] 0,0 p∗ q ∗ (⊗ki=1 Ei ) can be identified with the subsheaf E∞ of E10,0 = CE0 1 ⊗ · · · ⊗ CE0 k . Since the p,q terms E1 of the spectral sequence are for p ≥ 1 all supported on , we have

D

[n]

[n]

p∗ q ∗ (E1 ⊗ · · · ⊗ Ek )|U = (CE0 1 ⊗ · · · ⊗ CE0 k )|U .

D

Since X n is normal and 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 ⊂ S n X and a Sn -equivariant morphism [n]

[n]

ϕ : p∗ q ∗ (E1 ⊗ · · · ⊗ Ek )|π−1 W → CF0 |π−1 W . [n]

By remark 2.14 the morphism ϕ factorises over p∗ q ∗ F|π−1 W . This yields [n] [n] [n] [n] Hom(p∗ q ∗ (E1 ⊗ · · · ⊗ Ek ), p∗ q ∗ F [n] ) ∼ = Hom(p∗ q ∗ (E1 ⊗ · · · ⊗ Ek ), CF0 ) ∼ = Hom(C 0 ⊗ · · · ⊗ C 0 , C 0 ) E1

Ek

F

where we have used Lemma 3.1 again for the second isomorphism.



Vanishing of the higher Exti (Φ(F [n] ), OX n ). Lemma 3.3. Let F be a locally free sheaf on X. Then [(CFp )v ]Sn ∼ = 0 for p > 0. Proof. Let I ⊂ [n] with |I| = p + 1. By equivariant Grothendieck duality −p FIv ∼ = (F ∨ ⊗ ωX )I [−2p] . = ιI∗ R Hom∆I (p∗I F, ι!I OX n ) ∼ = R HomX n (ιI∗ p∗I F, OX n ) ∼ 2.4

2.9

Note that by lemma 2.8 the SI -action on FIv is indeed the trivial one. Now, M 2.2 v ⊗ a[p+1] ]S([p+1]) . [(CFp )v ]Sn ∼ (FI ⊗ aI )v ]Sn ∼ =[ = [F[p+1] |I|=p+1

v ⊗ a[p+1] by −1 which makes the invariants Every transposition in S[p+1] is acting on F[p+1] vanish.  Proposition 3.4. For every locally free sheaf F ∈ Coh(X) there is a natural isomorphism [(Φ(F [n] ))v ]Sn ∼ = [(CF0 )∨ ]Sn in Db (S n X). In particular, Exti (Φ(F [n] ), OX n ) = 0 for i > 0. Proof. Indeed, [Φ(F [n] )v ]Sn ∼ = [(CF• )v ]Sn ∼ = [(CF0 )v ]Sn ∼ = [(CF0 )∨ ]Sn . 2.12

3.3

lf



10

ANDREAS KRUG

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 sequence A associated to the functor Hom( , p∗ q ∗ F [n] ) given by (see e.g. [Huy06, Remark 2.67]) −p q ∗ [n] m m • ∗ [n] ∼ m [n] [n] Ap,q 1 = Ext (CE , p∗ q F ) =⇒ A = Ext (CE , p∗ q F ) = Ext (Φ(E ), Φ(F )) .

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 p q k m m k • ∼ m k ∗ [n] B(k)p,q 1 = Ext (CE , CF ) =⇒ B(k) = Ext (CE , CF ) = Ext (CE , p∗ q F ) .

Here as direct summands terms of the form Extq (EI ⊗ aI , FJ ⊗ aJ ) for ∅ = 6 I, J ⊂ [n] occur. In the case that I = {i} or J = {j} consist of just one element we set EI = pr∗i E or FJ = pr∗j F , respectively. Furthermore, we set ∆{i} := X n =: ∆∅ and N{i} := 0 =: N∅ . We also set ∧0 0 := OX n . 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 on M (k, p) by σ · (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 (3.1)

9 ∆I GG ss GG ιI s b ss GG j GG ss s G  ss ιI∩J # / Xn . ∆I ∩ ∆KJ ∆I∩J O w; KKK ww KKK w w i ww ι a KKK ww J %

∆J

The subvarieties ∆I and ∆J intersect transversely inside ∆I∩J . We denote the inclusion of ∆I ∩ ∆J into X n by t. Lemma 3.5. For (I, J) ∈ M (k, p) we have an S(I, J)-equivariant isomorphism   ∨ (3.2) Extq (EI , FJ ) = t∗ ((E ∨ )I ⊗ FJ )|∆I ∩∆J ⊗ (∧2k NI )|∆I ∩∆J ⊗ (∧2k−q NI∩J )|∆I ∩∆J . Proof. We will prove (3.2) with the roles of q and 2k − q interchanged. In DbS(I,J) (X n ) we have the isomorphisms R HomX n (EI , FJ ) ∼ = E v ⊗L FJ I

2.4

∼ = ιI∗ (p∗I E ∨ ⊗ ∧2k NI ) ⊗L ιJ∗ p∗J F [−2k]   PF ∼ = ιI∗ p∗I E ∨ ⊗ ∧2k NI ⊗ Lι∗I ιJ∗ p∗J F [−2k] . Taking cohomology in degree 2k − q we get   (3.3) Ext2k−q (EI , FJ ) ∼ = ιI∗ p∗I E ∨ ⊗ ∧2k NI ⊗ L−q ι∗I ιJ∗ p∗J F .

EXTENSION GROUPS OF TAUTOLOGICAL SHEAVES ON HILBERT SCHEMES

11

By diagram (3.1), Lι∗I ιJ∗ p∗J F ∼ = Lj ∗ Lι∗I∩J ιI∩J∗ i∗ p∗J F and by lemma 2.5 PF
∨ ∨ ∼ . L−q ι∗I∩J ιI∩J∗ i∗ p∗J F ∼ = i∗ p∗J F ⊗ i∗ ∧q NI∩J = i∗ p∗J F ⊗ ∧q NI∩J ∗ ∗ ∼ By the base change formula [Kuz06, Corollary 2.27] we have Lj i∗ = b∗ La . This yields

lf ∨ ∨ ∼ . Lj ∗ L−q ι∗I∩J ιI∩J∗ i∗ p∗J F ∼ = b∗ a∗ p∗J F ⊗ i∗ ∧q NI∩J = b∗ La∗ p∗J F ⊗ i∗ ∧q NI∩J

In particular, Lj ∗ L−q ι∗I∩J ιI∩J∗ i∗ p∗J F is cohomologically concentrated in degree zero for every q ∈ . Using the spectral sequence associated to the functor j ∗ (see e.g. [Huy06, p. 81]) this gives
L−q ι∗ ιJ∗ p∗ F ∼ = j ∗ L−q ι∗ ιI∩J∗ i∗ p∗ F ∼ = b∗ (p∗ F )|∆ ∩∆ ⊗ (∧q N ∨ )|∆ ∩∆ .

Z

I

J

I∩J

J

J

I

J

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) = 0 for every q∈ .

Z

Proof. By the previous lemma we see that the action of SI\J × SJ\I on Extq (EI , FJ ) is trivial (see also corollary 2.8). Since ∼ Extq (EI , FJ ) ⊗ aI ⊗ aJ = ∼ Extq (EI , FJ ) ⊗ aI\J ⊗ aJ\I Extq (EI ⊗ aI , FJ ⊗ aJ ) = as S(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 terms B(k)p,q 1 vanish unless p ∈ {k − 1, k, k + 1} and q ∈ {0, 2, . . . , 2k} is even. Proof. Let O(k, p) be a system of representatives of the Sn -orbits of M (k, p). By lemma 2.2 M S p ∼ q k B(k)p,q Extq (EI ⊗ aI , FJ ⊗ aJ ) n 1 = Ext (CE , CF ) = (3.4)

M (k,p)

M S(I,J) ∼ Extq (EI , FJ ) ⊗ aI\J ⊗ aJ\I . = O(k,p)

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 ∨ . Thus, the vanishing for q odd follows by corollary 2.8. is ∧2k−q NI∩J  Lemma 3.8. For k = 0, . . . , n − 1 and q = 1, . . . , k all the sequences 0 → B(k)1k−1,2q → B(k)k,2q → B(k)1k+1,2q → 0 1 k+1,0 are exact. Furthermore B(k)k−1,0 vanishes and B(k)k,0 is surjective. 1 1 → B(k)1

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  S [k+1] S[k] ×S[k+1] ∼ −q 2q ∼ Hom(E, F ) ⊗ ω B(k)k−1,2q Ext (E , F )) = = [k+1] [k] 1 X [k+1]

12

ANDREAS KRUG

for q = 1, . . . , k and B(k)k−1,0 = 0. In M (k, k) there are two orbits such that |I \J|, |J \I| ≤ 1; 1 namely those represented by ([k + 1], [k + 1]) and ([k + 1], [k] ∪ {k + 2}). In M (k, k + 1) there is one such orbit represented by ([k + 1], [k + 2]). This gives  Hom(E, F )S[k+1] for q = 0, k,2q ∼ [k+1] B(k)1 =

S[k+1] S[k+2] −q −q  Hom(E, F ) ⊗ ω ⊕ Hom(E, F ) ⊗ ω for q = 1, . . . , k, X

B(k)k+1,2q 1

∼ = Hom(E, F ) ⊗

X

[k+1]

−q
S[k+2] ωX [k+2]

[k+2]

for q = 0, . . . , k,

We first consider the case that q = 1, . . . , k. We show that under the above isomorphisms the component −q
S[k+1] −q
S[k+1] ϕ : Hom(E, F ) ⊗ ωX [k+1] → Hom(E, F ) ⊗ ωX [k+1] of dk−1,2q is an isomorphism (already before taking S[k+1] -invariants). For this we can argue 1 locally and thus may assume that E = F = OX . We have the following commutative diagram where the vertical maps are isomorphisms and the horizontal maps are induced by ϕ L ϕ1 S[k+1] 2q −−−−→ Ext2q (O∆[k+1] , O∆[k+1] )S[k+1] [ k+1 i=1 Ext (O∆[k+1] , O∆[k+1]\{i} )]   ∼ ∼ = y y= L ϕ2 −k −k ∨ 2(k−q) N ∨ )[k+1] ⊗ ∧2(k−q) N∆ ]S[k+1] )]S[k+1] −−−−→ [(ωX [ k+1 i=1 (ωX )[k+1] ⊗ ∧ ∆ |∆ [k+1] [k+1]\{i}

[k+1]

dk−1,2q 1

k−1 k , the components → CO Since the differential is induced by the differential dk−1 : CO X X of ϕ1 are induced by the restriction maps O∆[k+1]\{i} → O∆[k+1] . By a local computation using Koszul complexes (see also [Sca09a, Corollary B.4]) it follows that the components of ϕ2 are ∨ ∨ given by the inclusions N∆ → N[k+1] . Under the isomorphism of lemma 2.6 these, [k+1]\{i} |∆[k+1] in turn, are given by the inclusions %[k+1]\{i} → %[k+1] . It follows by [Sca09a, Lemma B.6(4)] that ϕ is indeed an isomorphism. By the same arguments, the component −q
S[k+2] −q
S[k+2] Hom(E, F ) ⊗ ωX [k+2] → Hom(E, F ) ⊗ ωX [k+2]

is an isomorphism for q = 1, . . . , k. These two facts together show that the short of dk,2q 1 exact sequences in the rows 2, 4, . . . , 2k are indeed exact. Consider now the case q = 0. Then, one is left with the component S

S

[k+1] [k+2] (Hom(E, F ))[k+1] → (Hom(E, F ))[k+2]

of dk,0 1 which is given by the restriction map. We have to show that it is still surjective after taking the invariants. Since S n X can be covered by open subsets S n U with U ⊂ X open and affine, it is sufficient that the map is surjective on the level of sections over S n U 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 R Hom(CEk , p∗ q ∗ F [n] ) is cohomologically concentrated in degree k, i.e. Extm (CEk , p∗ q ∗ F [n] ) = 0 for m 6= k. Proof. The previous two lemmas show that the only non-vanishing term on the 2-level of B(k) k,0 is B(k)k,0 2 . Thus, the only non-vanishing term on the ∞-level is B(k)∞ . It follows that for m 6= k we have 0 = B(k)m = Extm (CEk , p∗ q ∗ F [n] ). 

EXTENSION GROUPS OF TAUTOLOGICAL SHEAVES ON HILBERT SCHEMES

13

Proposition 3.10. Let E and F be locally free sheaves on X. Then R Hom(Φ(E [n] ), Φ(F [n] )) is cohomologically concentrated in degree zero and there is a natural isomorphism R Hom(Φ(E [n] ), Φ(F [n] )) ∼ = Hom(CE0 , CF0 )

in Db (S n X).

Proof. As mentioned above we use the spectral sequence −p q ∗ [n] m m • ∗ [n] ∼ m [n] [n] Ap,q 1 = Ext (CE , p∗ q F ) =⇒ A = Ext (CE , p∗ q F ) = Ext (Φ(E ), Φ(F )) .

By the above corollary the 1-sheet of A is concentrated on the diagonal p + q = 0. Thus Am = 0 for m 6= 0. This yields 3.2

[R Hom(Φ(E), Φ(F )) ∼ = Hom(CE0 , CF0 ) . = Hom(p∗ q ∗ (E [n] ), p∗ q ∗ (F [n] )) ∼  From tautological bundles to tautological objects. Remark 3.11. The functor C = C 0 = ⊕ni=1 p∗i : Coh(X) → CohSn (X n ) is exact, since the projections pi are flat. Thus, it descents to the functor C = ⊕ni=1 p∗i : 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
R Hom Φ((E • )[n] ), Φ((F • )[n] ) ∼ = R Hom(C(E • ), C(F • )) , R Hom(Φ((E • )[n] ), OX n ) ∼ = R Hom(C(E • ), OX n ) . Proof. Taking locally free resolutions A• ∼ = E • and B • ∼ = F • gives 2.12

Φ((E • )[n] ) ∼ = Φ((A• )[n] ) ∼ = p∗ q ∗ ((A• )[n] ) , Now, for every pair i, j ∈

2.12

Φ((F • )[n] ) ∼ = Φ((B • )[n] ) ∼ = p∗ q ∗ ((B • )[n] )

Z by proposition 3.10 we have
Extq p∗ q ∗ ((Ai )[n] ), p∗ q ∗ ((B i )[n] ) = 0

for q 6= 0. Thus, we can apply proposition 2.1 to the situation of the bifunctor Hom( , ) = [ ]Sn ◦ π∗ ◦ Hom( , ) : QCohSn (X n )◦ × QCohSn (X n ) → QCoh(S n X) and obtain using proposition 3.10 that

R Hom Φ((E • )[n] ), Φ((F • )[n] ) ∼ = R Hom 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

∼ = R Hom(C(A• ), C(B • )) 3.11

∼ = R Hom(C(E • ), C(F • )) .

The second isomorphism is shown similarly using proposition 3.4 instead of 3.10.



14

ANDREAS KRUG

Natural line bundles. At the level of Picard groups, there is a homomorphism which associates to any line bundle on X the natural line bundle on X [n] given by D : Pic X → Pic X [n]

,

L 7→ DL := µ∗ ((Ln )Sn ) .

Here the Sn -linearisation of Ln is given by the canonical isomorphisms p∗σ−1 (i) L ∼ = σ ∗ p∗i L, i.e. it is given by permutation of the factors. Remark 3.13. By [DN89, Theorem 2.3] the sheaf of invariants of Ln is the descent of Ln , i.e. Ln ∼ = π ∗ ((Ln )Sn ). Remark 3.14. The homomorphism D maps the trivial line bundle to the trivial line bundle and the canonical line bundle to the canonical line bundle, i.e. DOX ∼ = OX [n] and DωX ∼ = ωX [n] (see [NW04, Proposition 1.6]). Lemma 3.15. Let L be a line bundle on X. (i) For every F ∈ Db (X [n] ) there is a natural isomorphism Φ(F ⊗ DL ) ∼ = Φ(F) ⊗ Ln in n n ∼ DSn (X ). In particular, Φ(DL ) = L . (ii) For every G ∈ DbSn (X n ) 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
PF Rp∗ q ∗ (F ⊗ DL ) ∼ = Rp∗ (q ∗ F ⊗ q ∗ DL ) ∼ = Rp∗ q ∗ F ⊗ p∗ Ln ∼ = Rp∗ q ∗ (F) ⊗ Ln . Since Φ(OX [n] ) ∼ = OX n (see [Sca09a, Proposition 1.3.3]), we get Φ(DL ) ∼ = Ln as a special b b S n n n case. For (ii) we remember that the functor ( ) : DSn (X ) → D (S X) is an abbreviation of the composition ( )Sn ◦ π∗ . Then iSn PF h iSn  Sn h ∼ ∼ ∼ π∗ (G ⊗ Ln ) = (π∗ G)Sn ⊗ (Ln )Sn . = π∗ (G ⊗ π ∗ (Ln )Sn ) = π∗ (G) ⊗ (Ln )Sn  We call an object of the form E [n] ⊗ DL ∈ Db (X [n] ) with E ∈ Db (X) and L a line bundle on X a twisted tautological object. Proposition 3.16. Let E [n] ⊗ DL and F [n] ⊗ DM be twisted tautological objects. Then there are natural isomorphisms

R Hom Φ(E [n] ⊗ DL ), Φ(F [n] ⊗ DM ) ∼ = R Hom C(E) ⊗ Ln , C(F ) ⊗ M n ,

R Hom Φ(E [n] ⊗ DL ), Φ(DM ) ∼ = R Hom C(E) ⊗ Ln , M n . Proof. We will only show the first isomorphism since the proof of the second one is very similar. We have indeed h
iSn R Hom Φ(E [n] ⊗ DL ), Φ(F [n] ⊗ DM ) 3.15 h
iSn ∼ = R Hom Φ(E [n] ) ⊗ Ln , Φ(F [n] ) ⊗ M n h iSn ∼ = R Hom(Φ(E [n] ), Φ(F [n] )) ⊗ (Ln )∨ ⊗ M n

EXTENSION GROUPS OF TAUTOLOGICAL SHEAVES ON HILBERT SCHEMES

15

3.15

∼ = R Hom(Φ(E [n] ), Φ(F [n] ))Sn ⊗ ((Ln )∨ )Sn ⊗ (M n )Sn

3.12

∼ = R Hom(C(E), C(F ))Sn ⊗ ((Ln )∨ )Sn ⊗ (M n )Sn 3.15  Sn ∼ = R Hom(C(E), C(F )) ⊗ (Ln )∨ ⊗ M n
∼ = [R Hom C(E) ⊗ Ln , C(F ) ⊗ M n ]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 ∈ Pic X. 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 ) ⊗ S n−1 Ext∗ (L, M )⊕ Ext∗ (E ⊗ L, M ) ⊗ Ext∗ (L, F ⊗ M ) ⊗ S n−2 Ext∗ (L, M ),

Ext∗ (E [n] ⊗ DL , DM ) ∼ = Ext∗ (E ⊗ L, M ) ⊗ S n−1 Ext∗ (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
Sn ∼ = Ext∗ Φ(E [n] ⊗ DL ), Φ(F [n] ⊗ DM ) h iSn ∼ ) = H∗ (S n X, R Hom(Φ(E [n] ⊗ DL ), Φ(F [n] ⊗ DM ))  Sn ∼ ) = H∗ (S n X, R Hom(C(E) ⊗ Ln , C(F ) ⊗ M n )  ∗ n
 Sn ∼ . = H X , R Hom(C(E) ⊗ Ln , C(F ) ⊗ M n ) Applying lemma 2.2 we get  Sn R Hom(C(E) ⊗ Ln , C(F ) ⊗ M n )  Sn M ∼ R Hom(p∗i E ⊗ Ln , p∗j F ⊗ M n ) = i,j∈[n]

(∗)

∼ = R Hom(p∗1 E ⊗ Ln , p∗1 F ⊗ M n ) 

S[1]

 S ⊕ R Hom(p∗1 E ⊗ Ln , p∗2 F ⊗ M n ) [2] .

Using the compatibility of the derived sheaf-Hom with pullbacks gives ∼ R Hom(E ⊗ L, F ⊗ M )  Hom(L, M )n−1 R Hom(p∗ E ⊗ Ln , p∗ F ⊗ M n ) = 1

1

and also R Hom(p∗1 E ⊗ Ln , p∗2 F ⊗ M n ) ∼ Hom(E ⊗ L, M )  Hom(L, F ⊗ M )  Hom(L, M )n−2 . =R Now by the K¨ unneth formula  ∗
S[1] H X n , R Hom(E ⊗ L, F ⊗ M )  Hom(L, M )n−1

16

ANDREAS KRUG

 S ∼ = H∗ (R Hom(E ⊗ L, F ⊗ M )) ⊗ H∗ (Hom(L, M ))⊗n−1 [1]  S ∼ = Ext∗ (E ⊗ L, F ⊗ M ) ⊗ Ext∗ (L, M )⊗n−1 [1] ∼ = Ext∗ (E ⊗ L, F ⊗ M ) ⊗ S n−1 Ext∗ (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 automorphisms of X and pull-backs along open immersions U ⊂ X. Remark 3.19. By setting L = M = OX we get the following formulas for non-twisted tautological objects ∼ Ext∗ (E, F ) ⊗ S n−1 H∗ (OX ) ⊕ H∗ (E v ) ⊗ H∗ (F ) ⊗ S n−2 H∗ (OX ) Ext∗ (E [n] , F [n] ) = Ext∗ (E [n] , OX ) ∼ = H∗ (X [n] , (E [n] )v ) ∼ = H∗ (E v ) ⊗ S n−1 H∗ (OX ) .

Remark 3.20. Let F ∈ 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) ⊗ S n−1 H∗ (L) (see [Dan01] for the case of tautological bundles and [Sca09b] for the case of tautological ∨ ⊗ D ∼ sheaves). Moreover, the fact that DL M = DHom(L,M ) for L, M ∈ Pic(X) yields the formula Ext∗ (DL , F [n] ⊗ DM ) ∼ = Ext∗ (L, F ⊗ M ) ⊗ S n−1 Ext∗ (L, M ) . In the case that X is projective one can also directly deduce the formula of theorem 3.17 for Ext∗ (E [n] ⊗ DL , 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 ∈ Pic X natural isomorphisms Ext∗ (DL , DM ) ∼ = Ext∗ (Ln , M n )Sn ∼ = S n Ext∗ (L, M ) . Remark 3.22. Let for E1 , . . . , Ek , F be locally free sheaves on X. Using proposition 3.2 we [n] [n] [n] [n] also get formulas for Ext0 (E1 ⊗ · · · ⊗ Ek , OX [n] ) as well as for Ext0 (E1 ⊗ · · · ⊗ Ek , F [n] ). For ` ∈ and M a set let P (M, `) denote the set of partitions I = {I1 , . . . , I` } of M of length `. Then ! ` M O [n] [n] Hom(E1 ⊗ · · · ⊗ Ek , OX [n] ) ∼ H0 (⊗i∈Is Ei∨ ) ⊗ S n−` H0 (OX ) =

N

s=1

I∈P ([k],`),`≤n [n]

[n]

and Hom(E1 ⊗ · · · ⊗ Ek , F [n] ) is naturally isomorphic to M M ⊂[k] I∈P ([k]\M,`),`≤n−1

Hom(⊗i∈M Ei , F ) ⊗

` O

! H

∗

(⊗i∈Is Ei∨ )

⊗S

n−`−1

0

H (OX )

.

s=1

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 R Hom objects are in general not cohomologically concentrated in degree zero for k ≥ 2.

EXTENSION GROUPS OF TAUTOLOGICAL SHEAVES ON HILBERT SCHEMES

17

4. Consequences

P

Spherical and n -objects. Let M be a smooth projective variety with canonical bundle ωM . We call an object E ∈ Db (M ) an ωM -invariant object if E ⊗ ωM ∼ = E. A spherical object in Db (M ) is an ωM -invariant object E with the property ( if i = 0, dim M , Exti (E, E) = 0 else,

C

C

P

i.e. Ext∗ (E, E) ∼ = H∗ (S dim M , ), where S n denotes the real n-sphere. A n -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 ∗ ( n , ), where the multiplication on the left is given by the Yoneda product and on the right by the cup product. In particular, we have ( if 0 ≤ i ≤ 2n is even i Ext (E, E) = 0 if i is odd

P C

C

P

By Serre duality the dimension of M must be 2n as soon as Db (M ) contains a n -object. Spherical and n -objects are of interest because they induce automorphisms of Db (M ) (see [ST01] and [HT06]). For a smooth projective surface X with ωX = 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 n objects. But this is not the case by the following proposition.

P

P

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 n -objects.

P

Proof. Let E [n] ⊗ DL be a non-zero twisted tautological object on X [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) ⊗ S n−1 H∗ (OX ) ⊕ H∗ (E v ) ⊗ H∗ (E) ⊗ S n−2 H∗ (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) ⊗ S n−1 H∗ (OX ) contributes 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) ⊗ S n−1 H∗ (OX ) also contributes nontrivially to the degrees 2, 4, . . . , 2n. This yields exti (E [n] , E [n] ) ≥ 2 for i = 2, 4, 2n − 2 which shows that E [n] ⊗ DL is indeed neither a spherical nor a n -object. 

P

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 either twisted tautological objects or a natural line bundles. If E = E [n] ⊗ DL is a twisted tautological object we set Ψ(E) := C(E) ⊗ Ln and if E = DL is a natural line bundle we set Ψ(E) = Ln . Following the computations done in section 3 for E, F, G ∈ Db (X [n] ) twisted

18

ANDREAS KRUG

tautological objects or natural line bundles, we get the commutative diagram Ext∗ (F, G) × Ext∗ (E, F)  ∼ = y

Yon

−−−−→

Ext∗ (E, G)  ∼ y=

Yon

Ext∗ (Φ(F), Φ(G))Sn × Ext∗ (Φ(E), Φ(F))Sn −−−−→ Ext∗ (Φ(E), Φ(G))Sn   ∼ ∼ = y y= Yon

Ext∗ (Ψ(F), Ψ(G))Sn × Ext∗ (Ψ(E), Ψ(F))Sn −−−−→ Ext∗ (Ψ(E), Ψ(G))Sn . The formulas in terms of the extension groups on the surface X 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: let E1 , . . . , Ek ∈ Db (X). Then for σ ∈ Sk the map σ : H∗ (X k , E1  · · ·  Ek ) → H∗ (X k , Eσ−1 (1)  · · ·  Eσ−1 (k) ) induced by the action of σ on X k is given under the K¨ unneth isomorphism by (4.1)

σ(s1 ⊗ · · · ⊗ sk ) = εσ,p1 ,...,pk (sσ−1 (1) ⊗ · · · ⊗ sσ−1 (k) ) .

(compare (2.1)). Here, pi is the cohomological degree of si , i.e. si ∈ Hpi (X, Ei ). Let E, E 0 , E 00 ∈ Db (X) and F, F 0 , F 00 ∈ Db (Y ). Then the Yoneda pairing ◦ : Ext∗ (E 0  F 0 , E 00  F 00 ) × Ext∗ (E  F, E 0  F 0 ) → Ext∗ (E  F, E 00  F 00 ) is given under the K¨ unneth isomorphism for s1 ∈ Extp1 (E 0 , E 00 ), s2 ∈ Extp2 (F 0 , F 00 ), t3 ∈ Extp3 (E, E 0 ), and t4 ∈ Extp4 (F, F 0 ) by (4.2)

(s1 ⊗ s2 ) ◦ (t3 ⊗ t4 ) = (−1)p2 p3 (s1 ◦ t3 ) ⊗ (s2 ◦ t4 ) .

One way to see the need for the sign (−1)p2 p3 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 following notation: Let m ∈ 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 the si 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

N

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=1 Ei0 , ni=1 Ei00 ) × Ext∗ (ni=1 Ei , ni=1 Ei0 ) → Ext∗ (ni=1 Ei , ni=1 Ei00 ) is given under the K¨ unneth isomorphism by (s1 ⊗ · · · ⊗ sn ) ◦ (tn+1 ⊗ · · · ⊗ t2n ) = [(s1 ◦ tn+1 ) ⊗ · · · ⊗ (sn ◦ t2n )] .

EXTENSION GROUPS OF TAUTOLOGICAL SHEAVES ON HILBERT SCHEMES

19

P Furthermore, let i∈I Ti P be some finite P sum such that the Ti have the same property as T • from above. Then we set i∈I Ti := i∈I [Ti ]. For example, (2.2) reads • 1 X s1 · · · sn 7→ sσ−1 (1) ⊗ · · · ⊗ sσ−1 (n) . n! σ∈Sn

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 Ext∗ (E ⊗ L, F ⊗ M ) ⊗ S n−1 Ext∗ (L, M )⊕ P (E, L, F, M ) := Ext∗ (E ⊗ L, M ) ⊗ Ext∗ (L, F ⊗ M ) ⊗ S n−2 Ext∗ (L, M ) . We consider the elements   ϕ ⊗ s2 · · · sn ∈ P (F, M, G, N ) ∼ = Ext∗ (F [n] ⊗ DM , G[n] ⊗ DN ) , η ⊗ x ⊗ t3 · · · tn   ϕ0 ⊗ s02 · · · s0n ∼ Ext∗ (E [n] ⊗ DL , F [n] ⊗ DM ) . ∈ P (E, L, F, M ) = η 0 ⊗ x0 ⊗ t03 · · · t0n In order to use the sign convention from above we set s1 := ϕ , t1 := η , t2 := x , s01 := ϕ0 , t01 := η 0 , t02 := x0

,

sn+i := s0i , tn+i := t0i

and assume that all the si and ti are homogeneous. Now we can compute the Yoneda product
ϕ0 ⊗s02 ···s0n
ϕ⊗s2 ···sn ∗ [n] ⊗ D , G[n] ⊗ D ) and express it as an element in L N η⊗x⊗t3 ···tn ◦ η 0 ⊗x0 ⊗t03 ···t0n in Ext (E P (E, L, G, M ). Proposition 4.2. The Yoneda product is given by 1 (n − 1)!

• X σ∈S[2,n]

1 + (n − 2)! 1 ⊕ (n − 1)! +

+

(ϕϕ0 ) ⊗ (s2 s0σ−1 (2) ) · · · (sn s0σ−1 (n) )

1 (n − 1)! 1 (n − 2)!

• X

(xη 0 ) ⊗ (ηx0 ) · (t3 t0τ −1 (3) ) · · · (tn t0τ −1 (n) )

τ ∈S[3,n] • X

(ηϕ0 ) ⊗ (xs0β −1 (2) ) ⊗ (t3 s0β −1 (3) ) · · · (tn s0β −1 (n) )

β∈S[2,n] • X

(sγ −1 (2) η 0 ) ⊗ (ϕx0 ) ⊗ (sγ −1 (3) t03 ) · · · (sγ −1 (n) t0n )

γ∈S[2,n] • X i=3,...,n α∈S[3,...,n]

(ti η 0 ) ⊗ (xt0α−1 (i) ) ⊗ (ηx0 ) · (t3 t0α−1 (3) ) · · · (ti\ t0α−1 (i) ) · · · (tn t0α−1 (n) ) .

20

ANDREAS KRUG

Proof. The element

ϕ⊗s2 ···sn η⊗x⊗t3 ···tn




∈ P (F, M, G, N ) corresponds to the element

• • X X 1 1 sσ−1 (1) ⊗ · · · ⊗ sσ−1 (n) ⊕ tτ −1 (1) ⊗ · · · ⊗ tτ −1 (n) (n − 1)! (n − 2)! τ ∈Sn

σ∈Sn

0 ⊗ N n ). The coefficients are coming from the canonical isomorphism in Ext∗ (CF0 ⊗ M n , CG k S V → Sk V (see (2.2), note that the isomorphism of lemma 2.2 does not involve such coϕ0 ⊗s02 ···s0n
efficients). The analogue holds for η0 ⊗x0 ⊗t 0 ···t0 . The summand sσ −1 (1) ⊗ · · · ⊗ sσ −1 (n) is 3

n

an element of Ext∗ (pr∗σ(1) F ⊗ M n , pr∗σ(1) G ⊗ N n ) and tτ −1 (1) ⊗ · · · ⊗ tτ −1 (2) an element of Ext∗ (pr∗τ (1) F ⊗ M n , pr∗τ (2) G ⊗ N n ). There are five types of composable pairs of the 0 ⊗ N n ) × Ext∗ (C 0 ⊗ Ln , C 0 ⊗ M n ) , components of the classes in Ext∗ (CF0 ⊗ M n , CG E F namely pr∗i E ⊗ Ln → pr∗i F ⊗ M n → pr∗i G ⊗ N n , pr∗i E ⊗ Ln → pr∗j F ⊗ M n → pr∗i G ⊗ N n , pr∗i E ⊗ Ln → pr∗i F ⊗ M n → pr∗j G ⊗ N n , pr∗i E ⊗ Ln → pr∗j F ⊗ M n → pr∗j G ⊗ N n , pr∗i E ⊗ Ln → pr∗j F ⊗ M n → pr∗k G ⊗ N n 0 ⊗ N n ) with i, j, k ∈ [n] pairwise distinct. Thus, the Yoneda product in Ext∗ (CE0 ⊗ Ln , CG looks like this : • • X X 1 1 n 0 ⊗i=1 (sσ−1 (i) sσ0−1 (i) ) + ⊗ni=1 (tτ −1 (i) t0τ 0−1 (i) ) 2 (n − 1)!2 (n − 2)! 0 0 σ,σ ∈Sn σ(1)=σ 0 (1)

1 + (n − 1)!(n − 2)!

τ,τ ∈Sn τ (2)=τ 0 (1),τ (1)=τ 0 (2)

• X

⊗ni=1 (tτ −1 (i) s0σ0−1 (i) )

τ,σ 0 ∈Sn τ (1)=σ 0 (1)

1 + (n − 1)!(n − 2)!

• X

⊗ni=1 (sσ−1 (i) t0τ 0−1 (i) )

σ,τ 0 ∈Sn σ(1)=τ 0 (2)

(6) +

1 (n − 2)!2

• X

⊗ni=1 (tτ −1 (i) t0τ 0−1 (i) )

.

τ,τ 0 ∈Sn τ (1)=τ 0 (2),τ (2)6=τ 0 (1)

The first term of (6) is a Sn -invariant element of ⊕ni=1 Ext∗ (p∗i E ⊗ Ln , p∗i G ⊗ N n ). 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

⊗ni=1 (sσ−1 (i) s0σ0−1 (i) ) ∈ Ext∗ (p∗1 E ⊗ Ln , p∗1 G ⊗ N n )

σ,σ 0 ∈Sn σ(1)=σ 0 (1)=1 ∼ =

Under the isomorphism Sn−1 Ext∗ (L, N ) − → S n−1 Ext∗ (L, N ) this element is mapped to 1 (n − 1)!

• X

(ϕϕ0 ) ⊗ (s2 s0σ−1 (2) ) · · · (sn s0σ−1 (n) ) ∈ Ext∗ (E ⊗ L, G ⊗ N ) ⊗ S n−1 Ext∗ (L, N )

σ∈S[2,n]

which is exactly the first term of the formula we want to prove. Doing the same for the other four terms in (6) yields the desired element in P (E, L, G, M ). 

EXTENSION GROUPS OF TAUTOLOGICAL SHEAVES ON HILBERT SCHEMES

21

Remark 4.3. Let E, F ∈ Db (X) and M ∈ Pic X. For every morphism ∼ Exti (E, F ) ϕ ∈ Hom b (E, F [i]) = D (X)

the induced morphism ϕ[n] ⊗ idDM ∈ Exti (E [n] ⊗ DM , F [n] ⊗ DM ) corresponds to   ϕ ⊗ idM · · · idM ∈ P (E, M, F, M ) . 0 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 Mathematical 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 Monographs. 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]. SpringerVerlag, 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 of Lecture 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. Duke Math. J., 108(1):37–108, 2001.

22

ANDREAS KRUG

¨ t Bonn, Institut fu ¨ r Mathematik Universita E-mail address: [email protected]