In this section, we will prove Theorem6.1.1. Let X be a smooth projective surface over C
and 2≤n∈ N. We denote byδ:X → Xn the diagonal embedding. We want to show that
the composition
F:Db(X)−−→triv DbSn(X)−→δ∗ DbSn(Xn) is a Pn−1-functor. Here,S
nis considered to act trivially on X andtriv:Db(X)→DbSn(X)is
the functor which equips each object with the trivial linearisation. The right adjoint ofF is
given by the composition
R:DbSn(Xn)−→δ! DbSn(X) ( )
Sn
−−−−→Db(X)
of the usual right adjoint of the push-forward (see [LH09, Proposition 28.8] for equivari- ant Grothendieck duality for regular embeddings) and the functor of taking invariants. The approach is to compute the compositionδ!δ
∗ rst. Then taking Sn-invariants gives the com-
positionRF.
We consider the standard representation % of Sn as the quotient of the permutation representation Cn by the one-dimensional invariant subspace. The normal bundle sequence
0→TX →TXn|X →N →0, whereN :=Nδ=NX/Xn, is of the form
where the mapTX →TX⊕nis the diagonal embedding. When consideringTXn|X as aSn-sheaf
equipped with the natural linearisation it is given by TX ⊗Cn where Cn is the permutation
representation. Thus, as aSn-sheaf, the normal bundle N equals TX ⊗%. We also see that
the normal bundle sequence splits using e.g. the splitting
TX⊗Cn→TX , (v1, . . . , vn)7→
1
n(v1+· · ·+vn).
Theorem 6.3.1 ([AC12, Theorem 1.4]). Letι:Z ,→M be a regular embedding of codimension csuch that the normal bundle sequence splits. Then there is an isomorphism
ι∗ι∗( )'( )⊗ c M i=0 ∧iN∨ Z/M[i] (6.7) of endofunctors of Db(Z).
Recall that the right-adjoint ι! of ι∗ is given by ι!( ) = ι∗( )⊗ι!OM and ι!OM =
∧c:onderivedN
ι[−c]; see [Har66, Corollary III 7.3].
Corollary 6.3.2. Under the same assumptions, there is an isomorphism ι!ι∗( ) ' ( )⊗
Lc
i=0∧iNZ/M[−i]
.
Proof. Apply the tensor product withι!OM ' ∧cNZ/M[−c]on both sides of (6.7).
In the case thatι=δ from above this yields the isomorphism of functors
δ!δ∗( )'( )⊗ 2(n−1) M i=0 ∧i(T X⊗%)[−i] . (6.8)
Lemma 6.3.3. Letι:Z ,→M be as in Theorem6.3.1and letε:=ει∗:ι∗ι!→idbe the counit of adjunction. For0≤i, j≤c with i+j≤c, the component
( )⊗ ∧iNZ/M⊗ ∧jNZ/M[−(i+j)]→( )⊗ ∧i+jNZ/M[−(i+j)]
of the morphismι!ει∗:ι!ι∗ι!ι∗ →ι!ι∗ is given by the wedge pairing.
Proof. ForE ∈Db(M) the objectι!E can be identied with HomM(ι∗OZ, E), the latter con-
sidered as an object inDb(Z). Under this identication the counit mapε: HomM(ι∗OZ, E)→
E is given by the evaluation ϕ 7→ ϕ(1); see [Har66, Section III.6]. Corollary 6.3.2 says in particular that ι!ι∗(B) ' B ⊗OZ ι
!ι
∗(OZ) for B ∈ Db(Z). This gives the identications
ι!ι∗B ' HomM(ι∗OZ, ι∗OZ)⊗OZB and
ι!ι∗ι!ι∗B ' HomM(ι∗OZ, ι∗OZ)⊗OZHomM(ι∗OZ, ι∗OZ)⊗OZB .
Under these identications, the component
ExtiM(ι∗OZ, ι∗OZ)⊗OZExt
j
M(ι∗OZ, ι∗OZ)⊗OZ B→ Ext i+j
M (ι∗OZ, ι∗OZ)⊗OZB
of the monad multiplication equals the Yoneda product. The Yoneda product corresponds to the wedge product under the isomorphismExti
M(OZ,OZ)' ∧iNZ/M; see [LH09, Proposition
Lemma 6.3.4 ([Sca09a, Lemma B.5]). Let V be a two-dimensional vector space with a basis
consisting of vectorsu andv. Then the space of invariants[∧i(V ⊗%)]Sn is one-dimensional
if 0≤i≤2(n−1)is even and zero if it is odd. In the even casei= 2`the space of invariants
is spanned by the image of the vector ω`, where ω =
n X
i=1
uei∧vei∈ ∧2(V ⊗Cn),
under the projection induced by the quotient Cn→%.
Corollary 6.3.5. For a vector bundle E on X of rank two and 0 ≤ ` ≤ n−1 there is an isomorphism of line bundles[∧2`(E⊗%)]Sn '(∧2E)⊗`.
Proof. The isomorphism is given by composing the morphism (∧2E)⊗` → ∧2`(E⊗Cn) , x
1⊗ · · · ⊗x` 7→
X
1≤i1<···<i`≤n
x1ei1 ∧ · · · ∧x`ei`
with the projection induced by the quotient Cn→%.
We setD:= ( )⊗ ∧2T
X[−2]'Mω∨
X[−2]'S
−1
X as the inverse of the Serre functor on X.
Corollary 6.3.6. There is the isomorphism of functors RF 'id⊕D⊕D2⊕ · · · ⊕Dn−1.
Proof. We have RF = ( )Snδ!δ
∗triv. The assertion follows by formula (6.8) and Corollary 6.3.5.
Lemma 6.3.7. The functor F fulls condition (ii) of a Pn−1-functor with cotwistD=SX−1.
Proof. All the components cij:DDi−1 = Di → Dj of the morphism (6.3) are induced by
morphisms between the FourierMukai kernels. The FM kernel of Di = SX−i is ι∗ω−Xi[−2i] withι:X →X×X being the diagonal embedding. Fori < j we have
Hom(ι∗ωX−i[−2i], ι∗ω−Xj[−2j]) = Ext2(i−j)(ι∗ωX−i, ι∗ωX−j) = 0
hencecij = 0. The generatorsω` from Lemma 6.3.4 are mapped to each other by the wedge
product. For i = 1, . . . , n−1 the components cii: DDi−1 → Di are given by the wedge
product; see Lemma6.3.3. Hence, they are isomorphisms. Lemma 6.3.8. There is an isomorphismSXnF Dn−1 'F SX.
Proof. For E ∈Db(X) there are natural isomorphisms
SXnF Dn−1(E)'ωXn[2n]⊗δ∗(E ⊗ω−(n−1) X [−2(n−1)])'ω n X ⊗δ∗(E ⊗ω −(n−1) X )[2] 'δ∗(E ⊗ωX[2])'F SX(E)
where the second-to-last isomorphism is the projection formula.
All this together shows Theorem 6.1.1. This means that F = δ∗triv is indeed a Pn−1- functor with P-cotwistD=SX−1.