NOTES ON PROCESI BUNDLES AND THE SYMPLECTIC MCKAY EQUIVALENCE

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1 NOTES ON PROCESI BUNDLES AND THE SYMPLECTIC MCKAY EQUIVALENCE GUFANG ZHAO Contents 1. Introduction 1 2. What is a Procesi bundle 2 3. Derived equivalences from exceptional objects 4 4. Splitting of the Azumaya algebras 6 Appendix A. The Brauer groups 9 Appendix B. Uniqueness of the G m -equivariant structure of the Procesi bundle 9 References 1 1. Introduction We work over a separably closed field k. Let G be GL n and g its Lie algebra. Let A 2n be endowed with the usual S n -action. Recall that for a suitable choice of stability condition θ, we have T (g A n )/// θ G = Hilb n (A 2 ) =: X T (g A n )///G = V/S n V η Note that the map π is equivariant under the G m -action, where the G m -action on A 2 is by dilation. When k is a field of positive characteristic, for simplicity, we denote A 2n(1) simply by V and denote Hilb n (A 2 ) (1) by X. As has been explained in [Vain14], for any integral weight λ of G, on X there is a quantization O λ X, coming from quantum Hamiltonian reduction of D(g A n ), the sheaf of differential operators on g A n. For simplicity, we will denote O X by O X. Let W be the Weyl algebra on the vector space A n. The ring of global sections of O X is isomorphic to W Sn. We have already seen in [Vain14] that O X is an Azumaya algebra on X. We will show that RΓ : D b (Coh O X ) D b (Mod W Sn ) π Date: April 1,

2 2 G. ZHAO is an equivalence of derived categories. Let (V/S n ) be the formal neighborhood of in (V/S n ). Let X be π 1 ((V/S n ) ). Note that X is a projective algebraic scheme over Spec k [V ] Sn. Let W Sn be the restriction of the Azumaya algebra W Sn to (V/S n ). The above equivalence restricts to an equivalence D b (Coh O X ) = D b (Mod W Sn ). When k has large enough characteristic, then W Sn is Morita equivalent to W#S n. It is almost evident that W viewed as an Azumaya algebra on V (1) admits an S n -equivariant splitting when restricted to the formal neighborhood of the origin. Let W be the restriction of W to the formal neighborhood. Therefore, there is an equivalence of categories Mod W #S n = Mod k [V ]#Sn. We will also show that O X splits on X. The splitting of the Azumaya algebra yields a Morita equivalence Coh O X = Coh X. Putting all the equivalences together, we have D b (Coh X ) = D b (Coh O X ) = D b (Mod W Sn ) = D b (Mod W #S n ) = D b (Coh Sn V ). The objects Ê Db (Coh X ) corresponding to O V is a vector bundle on X, as will be proved. We will show that this vector bundle Ê extends to a vector bundle E on the entire X. The extension E is called the Procesi bundle. Also we will show that R Hom X (E, ) induces an equivalence D b (Coh X) = D b (Mod O[V ]#S n ), which is called the symplectic McKay equivalence. The main goal of this notes it to show the existence of a Procesi bundle following the arguments in [BK4]. The classification has been carried out in [Los13]. 2. What is a Procesi bundle Definition 2.1. A Procesi bundle on X is an G m -equivariant vector bundle E, satisfying (1) End X (E) = k[v ]#S n as G m -equivariant k[v ]-algebras; (2) H i (X, E nd X (E)) = for all i > ; (3) E Sn = O X. Lemma 2.2. By (1) every fiber of E carries an action of S n, which is isomorphic to the regular representation. Proof. Let V be the open subset of V on which S n acts freely. The map η is a Galois cover when restricted to V. By (1) we have a morphism of sheaves O(η V )#S n E nd X (E X ). This implies E X is a module over the sheaf of algebras η V, which is equivariant with respect to the S n -action. Therefore, there is a vector bundle Ē on V, equivariant with respect to S n, such that E X = η Ē. Counting the rank we know that Ē is a line bundle on V. Since

3 THE PROCESI BUNDLES 3 V is obtained from V by removing a codimension-2 locus, the Piccard group of V is trivial. Hence, Ē = O V. So, E X = η O V. In particular, on X, for any irreducible S n -representation N, the component Hom Sn (N, E) has rank the same as the dimension of N. Then by upper-semi-continuity, and the fact that E is a vector bundle, we know that Hom Sn (N, E) has rank the same as the dimension of N everywhere on X. Note that by this Lemma, (3) makes sense, as for any x in X, the fiber of E at x is isomorphic to ks n as S n -modules, hence E Sn is a line bundle. Let e = 1 n! g S n g. Then the global sections of E can be calculated as H (X, E) = H (X, E nd X (E)e) = H (X, E nd X (E))e = (k[v ]#S n )e = k[v ]. It worth mentioning that the requirement for E to be equivariant under the G m - action is automatic, given the vanishing of its self-extension and the structure of its endomorphism ring. Also this G m -action can be modified such that the induced action on End X (E) is compatible with the one on k[v ]#S n. Remark 2.3. Let X be the locus where π is an isomorphism. Conditions (1) and (2) in the definition can be reformulated as a) E X = π η O X ; b) Ext i (E, E) = for all i >, and End(E) = End(E X ) as G m -equivariant Γ(O X )-algebras. Proposition 2.4. Suppose there is a vector bundle Ê on X for all i >, and H i (X, E nd X (Ê)) = End X (Ê) = k [V ]#S n such that as k [V ] Sn -algebras. Then this Ê extends to a Procesi bundle on X. We need some lemmas to prove this Proposition. Lemma 2.5. Let Z be an affine variety over k, carrying a G m -action which contracts Z to Z. Let π : Y Z be a projective morphism which is equivariant with respect to G m. Let Z be the formal neighborhood of Z, and let Y be the preimage of Z in Y. Assume there is a vector bundle Ê on Y with 1 Ext (Ê, Ê) =. Then, there exist a G m -equivariant vector bundle E on Y whose restriction to is isomorphic to Ê. Y We need a standard fact the proof of which can be found in, e.g., [BFG6, Proposition ]. Let Y Z be a proper morphism. Let Z be the formal neighborhood of Z, and let Y be the preimage of Z in Y. Let E be a vector bundle on Y such that Ext1 (E, E) =. If Y is endowed with a G m-action, then this action can be lifted to E. If E is indecomposable, then the G m -equivariant structure on E is unique up to a twisting.

4 4 G. ZHAO Proof of Lemma 2.5. As Y is projective over Z, there is a graded algebra B, such that Z = Spec B and Y = Proj B. The fact that π : Y Z is equivariant with respect to the G m -action implies that B, as a module over the graded ring B with a non-trivial grading defined by the weights of the G m -action, is also a graded module. Define B to be the completion of B at the maximal ideal corresponding to Z. Define B := B B B, which is a graded algebra over B. The projective scheme Y is Proj B. The vector bundle Ê on Y corresponds to a graded G m -equivariant locally free module over B. Taking the G m -finite vectors, we get a graded locally free module over B. This module corresponds to a vector bundle E on Proj B which is Y, whose restriction to Y is Ê. Proof of Proposition 2.4. By assumption we have a vector bundle Ê on X with Ext 1 (Ê, Ê) =. By Lemma 2.5, Ê extends to a G m-equivariant vector bundle E on X. We only need to check that E satisfies (1) and (2) in Definition 2.1. Lemma B.1 implies that the G m -action on Ê can be modified so that End X (Ê) = k [V ]#S n as G m -equivariant k [V ] Sn -algebras. Not to interrupt the flow of this notes, we put the proof of Lemma B.1 in to an Appendix. We need to verify that R i π E nd(e) = for all i >. As E is equivariant with respect to G m, the support of R i π E nd(e) need to contain a G m -stable closed point. The only such point is. We only need to show that origin is not in the support of R i π E nd(e). Indeed, as base change to (V/S n ) is flat, we have R i π E nd(e) =. Again by flatness of the base change, R π E nd(e) = k [V ]#S n. As E is equivariant with respect to G m, so is R π E nd(e). By taking G m -finite vectors, we get the desired isomorphism R π E nd(e) = k[v ]#S n. 3. Derived equivalences from exceptional objects Definition 3.1. An object M in an abelian category is said to be almost exceptional if it is nonzero and Ext i (M, M) = for all i >, and the algebra End(M) has finite homological dimension. The goal of this section is to show the following proposition. Proposition 3.2. Let Y be a smooth irreducible variety, projective over some affine variety. Let A be an Azumaya algebra on Y. If E is an almost exceptional object in Coh A, then the functor is an equivalence. R Hom A (E, ) : D b (Coh A) D b (Mod End(E)) 3.1. Exceptional objects on smooth varieties. Lemma 3.3. For any exceptional E in Coh A, the functor L End A (E) E is fullyfaithfull, i.e., it identifies D b (Mod End A (E)) with an admissible subcategory of D b (Coh A).

5 THE PROCESI BUNDLES 5 Note that the assumption End(E) having finite global dimension makes the functor L End A (E) E well-defined on the level of bounded derived categories, i.e., sends bounded complexes to bounded ones. Recall that for a triangulated category C, a subcategory I is said to be admissible if any object F in C fits into an exact triangle F 1 F F 2 F 1 [1] with F 1 I and F 2 I. Any triangulated subcategory ι : I C, whose embedding functor ι admits a right adjoint τ is admissible. For any F in C, define F 1 := τι(f), and F 1 F the adjunction morphism. One can easily check that the cone of F 1 F lies in I by adjointness. Proof. The functor L End A (E) E is the right adjoint of R Hom A(E, ). We would like to show that for any M D b (Mod(E)), we have R Hom A (E, M L End(E) E) = M. But this follows from the fact that Ext i (E, E) = for all i In the Calabi-Yau setting. We say a triangulated category D is indecomposable if it can not be written as D 1 D 2 for non-zero D 1 and D 2. Lemma 3.4. Let Y be a connected quasi-projective variety over k, and let A be an Azumaya algebra on Y. Then the category D b (Coh A) is indecomposable. Proof. For any A-module P and any ample line bundle L on Y, the set {P L k } generates D b (Coh A). Assume D b (Coh A) = D 1 D 2, then take P to be an indecomposable direct summand of a free A-module. Take L to be an ample line bundle such that H (Y, L H om A (P, P )). For any k, the module P L k is also indecomposable, hence either in D 1 or in D 2. Assuming the former one. Note that Hom(P L n, P L m ) for all n m, therefore for any k the object P L k lies in D 2. So, D 1 is zero as for any M D 1 we have Ext i (P L k, M) = for any i and any k. Recall that on any smooth variety Y, proper over Spec R, for any Azumaya algebra A on Y, there is a relative Serre functor S : D b (Coh A) D b (Coh A) given by K Y [dim Y ]. Let D be the dualizing complex in D b (Mod R), then the Grothendieck-Serre duality yields, for any complexes of coherent sheaves F and G on Y, R Hom R (R Hom Y (F, G), D) = R Hom Y (G, F K Y [dim Y ]). Therefore, for any subcategory I of D b (Coh A), the relative Serre functor S sends I into I. Proposition 3.5. Let Y be a connected smooth quasi-projective variety over k with trivial canonical bundle, and A an Azumaya algebra on Y. Let F : D b (Coh A) D be a non-zero triangulated functor what admits a fully-faithful right adjoint F. Then F is an equivalence of categories.

6 6 G. ZHAO Proof. As the Serre functor intertwines left and right perpendiculars of any subcategory, if K Y = OY, then the left and right perpendiculars of any subcategory coincide. In particular, under the assumptions we have D b (Coh A) = D D. But we know D b (Coh A) is indecomposable since Y is connected, and D therefore D =. Corollary 3.6. Let X = Hilb n (A 2 ) (1), and let O X be the quantization of X constructed in [Vain14], then there is an equivalence of derived categories D b (Coh O X ) = D b (Mod Γ(O X )) The only thing need to show is that O has no higher cohomology. But this follows from the Grauert-Riemenschneider vanishing argument as in [Vain14]. Corollary 3.7. Let X be Hilb n (A 2 ) and let E be any Procesi bundle on X, then R Hom X (E, ) induces an equivalence D b (Coh X) = D b (Coh Sn V ). 4. Splitting of the Azumaya algebras 4.1. Existence of Procesi bundles in positive characteristic. Proposition 4.1. The Azumaya algebra O X splits on X. We want show that there is an Azumaya algebra A on (V/S n ) such that F r O X is Morita equivalent to π A. Or equivalently, we want to construct a class β Het((V/S 2 n ), G m ) tor such that [F r O X ] = π β. As is explained in Appendix A, there is a norm map η : Het(V, 2 G m ) Sn tor H2 et((v/s n ), G m ) tor. Then, the class β Het((V/S 2 n ), G m ) tor can be chosen to be the η (W), where W is viewed as a S n -invariant class on V. Let us verify that π β = [F r O X ]. As the restriction to any open subset induces an injection on Brauer groups, it suffices to show that after restricted to X, we have π β = [F r O X ]. Let V be the open subset of V on which S n acts freely. In particular, η is Galois on V. Therefore we have η W V = W Sn V /S n. Recall that when restricted to X F r O X is also isomorphic to π W Sn. Therefore, we have π β = [F r O X ]. In other words, the Azumaya algebra F r O X is the pull-back of an Azumaya algebra on (V/S n ) (1). It follows from the Hensel s Lemma that on the formal neighborhood of the origin, any Azumaya algebra on (V/S n ) splits. Our F r O X X is the pull-back of a splitting Azumaya algebra, hence also splits. Corollary 4.2. There is an equivalence of derived categories D b (Coh X ) = D b (Mod W Sn ). Theorem 4.3. There exists a Procesi bundle on X. Proof. We have D b (Coh X ) = D b (Coh O X ) = D b (Mod W Sn ) = D b (Coh Sn W ) = D b (Coh Sn V ).

7 THE PROCESI BUNDLES 7 Let Ê be the complex of coherent sheaves on X corresponding to O V. We would like to show that Ê is actually a vector bundle, with no higher self-extensions, whose endomorphism ring is isomorphic to k [V ]#S n. Let S be the splitting bundle of O on X, and let S be the equivariant splitting bundle of W. Tracing back all the equivalences above, one can see that S is send to S. ALso S is the bimodule induces the equivalence between Mod W Sn and Coh Sn V. By standard Morita theory, S is a projective generator in Coh Sn V. In particular, every indecomposable projective objet is a direct summand of S. Note that k [V ]#S n is a direct sum of indecomposable projective objects. Therefore, every indecomposable direct summand is send to a direct summand of S via the derived equivalence, which is a vector bundle. So, k [V ]#S n itself is send to a vector bundle on X. The other claims follow from the fact that R Hom(Ê, Ê) = k [V ]#S n. Corollary 4.4. There is an equivalence of derived categories 4.2. Lifting to characteristic zero. D b (Coh X) = D b (Coh Sn V ). Proposition 4.5. (1) Let R be a local Noetherian ring with maximal ideal m and residue field k. Let X R be a variety smooth over R, whose fiber over k is denoted by X k. Then every almost exceptional object E on X k extends uniquely to an almost exceptional object Ê on X, the formal completion of X R at X k. (2) If moreover R is complete with respect to the m-adic topology, and X R is equipped with a positive weight G m -action such that E is equivariant, then E extends uniquely to a G m -equivariant vector bundle E R on X R. Proof of (1). Denote R/(m n ) by R n, and denote the base change of X R to R n by X n. Iteratively, we need to extend the vector bundle E n on X n to a vector bundle E n+1 on X n+1. By standard deformation theory, the obstruction of the extension lies in Ext 2 (E, E) (m n /m n+1 ), and the set of equivalence classes of extensions is a torsor under Ext 1 (E, E) (m n /m n+1 ). By assumption, both groups are trivial. Therefore, the extension in each step exists and is unique. As Ext i i (E, E) = for i >, we also have Ext (Ê, Ê) = for i >. The fact that End(Ê) has finite homological dimension can be checked by calculating the Ext groups using the spectral sequence from the m-adic filtration. We need to recall the formal function theorem of Grothendieck. Let p : X Y be a proper morphism. Let Yy be the formal neighborhood of y Y, and let Xy be the preimage of Yy in X. Let E be a vector bundle on Xy. Then (R i p E) y = R i p X y (E X y ). Proof of (2). 1 Denote R/(m n ) by R n. Note that both X n and E n are flat over R n. As a projective variety over an affine R n -variety, X n can be written as Proj B n for 1 The proof will be skipped in the talk

8 8 G. ZHAO some graded ring B n. The vector bundle E n corresponds to a graded module M n over B n, flat a module over R n. The system M n for n satisfies the property that M n/(m n 1 ) = M n 1. The fact that X n carries a contracting action of G m (R n ), with respect to which E n is equivariant, decomposes B n into weight spaces B n = i= B n[i], and M n = M n[i]. This weight space decomposition is compatible with respect to the grading. Each M k n[i] and Bn[i] k are finitely generated flat (hence free) modules over R n. The formal scheme X R is lim Proj B n. Please note that lim B n is different from i lim B n[i]; but the space of finite vectors with respect to lim G m (R n ) in the former is the later. Denote i lim B n[i] by B. We have Proj B = X R. Similarly, define M := i lim M n[i]. We want to show that M is a finitely generated graded B -module, and corresponds to a vector bundle E R on X R with the property that E R XR = Ê (or equivalently E R Xn = En ). Let {b n s s I B } be a finite set of homogeneous generators of B n as R n -algebra, and let {fh n h I M} be a finite set of homogeneous generators of M n as B n-module. In particular, monomials in b n s and fh n of degree i generates M n[i] as R n -module. Take any homogeneous lifting of b n s in B n+1, called bn+1 s, and homogeneous liftings of fh n n+1 called fh. Then the Nakayama s Lemma implies that monomials of b n s and f n+1 h of degree i generates M n+1 as B n+1 -module. Iterate this process we get elements b s B and f h M whose monomials of degree i generates M [i]. This shows the finite-generation of M. By construction each M k [i] is free as module over R, therefore E is flat over R. Such an E is automatically locally free. Being locally free is equivalent to being projective on every members of an affine cover of X R. Let U R be an affine subset, flat over R. We want to show that Ext i U R (E UR, M) = for any coherent sheaf M on U R and any i >. For this it suffices to show that Ext i U R (E UR, M)/m =. By the formal function theorem, Ext i U R (E UR, M) = lim Ext i U n (E Un, M Un ). But E Un is isomorphic to E n restricted to an affine open set, hence is projective. Each Ext i U n (E Un, M Un ) vanishes. Remark 4.6. Note that by the formal function theorem, H i (X R, E nd(e)) m = lim Hi (X n, E nd(e n )). Let R be a finitely generated Z-algebra on which X is well-defined, whose field of fractions is denoted by K. Let k be a geometric point in Spec R with large enough characteristic, and R the formal completion of R at k, whose field of fractions is denoted by K. On X k there is a Procesi bundle by Theorem 4.3. Then the Proposition implies that E K exists and satisfies the axioms of a Procesi bundle. Let R be a finitely generated K-subalgebra in K, so that E R is welldefined and there is an isomorphism End XR (E R ) = R [V ]#S n as G m -equivariant R [V ] Sn -algebras. Then the field of fractions K of R is a field of characteristic zero on which a Procesi bundle exists.

9 THE PROCESI BUNDLES 9 Appendix A. The Brauer groups Let us recall some facts about Brauer groups. For any scheme Y, the set of Azumaya algebras, up to Morita equivalence, is called the Brauer group of this scheme, denoted by Br(Y ). The group structure is given by tensor of Azumaya algebras. Clearly it is an abelian group with identity given by the structure sheaf. We list some basic properties without proof that are used in the body of the notes. All of them can be found in [M8]. Proposition A.1. (1) For any Zarisky open subset U of Y, restricting to U induces an injection on Brauer groups Br(Y ) Br(U). (2) For any affine scheme Y, let Het(Y, 2 G m ) tor be the torsion subgroup of the second étale cohomology. We have an isomorphism Br(Y ) = Het(Y, 2 G m ) tor. (3) Let B be a complete local ring with separably closed residue field, and let B be a finite étale (flat unramified) extension of B. Then B = n B for some n. In particular, any connected finite étale map Y Spec B is an isomorphism, hence any étale cohomology of Spec B vanishes. (4) Let B be a complete local ring with separably closed residue field, and let B be a finite extension of B, then any étale cohomology of Spec B vanishes. For any finite group Γ acting on an affine variety S over a separably closed field, the norm morphism η : H i (S, G m ) H i (S/Γ, G m ) is well-defined. In general for any geometric point x in S/Γ, the stalk (R i η G m ) x = H i (S S/Γ Spec O S/Γ, x, G m ). The later vanishes since S S/Γ Spec O S/Γ, x, as a finite extension of a complete local ring (over a separably closed field), has no étale cohomology. Therefore, R i η G m = for i >. Then, Het(S/Γ, 2 η G m ) = Het(S, 2 G m ) by the Leray spectral sequence. Therefore, the norm map η : G ms G ms/γ induces a morphism on the level of étale cohomology. Appendix B. Uniqueness of the G m -equivariant structure of the Procesi bundle For any irreducible representation N of S n, the bundle ÊN := Hom Sn (N, Ê) is indecomposable. Indeed, let e N ks n be the indecomposable idempotent corresponding to N. Then End(ÊN ) = e N (k [V ]#S n )e N, Were ÊN decomposable, the algebra e N (k [V ]#S n )e N would have more than one non-trivial idempotents. This is a contradiction. So ÊN is indecomposable. Lemma B.1. Up to a twist, H (X, Ê) and k [V ] are isomorphic as G m -equivariant k [V ] Sn -modules. Proof. For any irreducible S n -representation N, let k [V ] N := Hom Sn (N, k [V ]). We only need to show that k [V ] N and Γ(ÊN ) are isomorphic as G m -equivariant k [V ] Sn -modules. In turn, it suffices to show that the G m -equivariant structure on k [V ] N is unique.

10 1 G. ZHAO Let ˆV be V V, where V is the open subscheme on which S n -acts freely. Note that k [V ] N ˆV /S n is a vector bundle. We only need to show that this bundle is indecomposable and admits a unique G m -equivariant structure up to twisting. This will follow if we show η (k [V ] N ˆV /S n ) is indecomposable and admits a unique G m -equivariant structure up to twisting. But η (k [V ] N ˆV /S n ) = k [V ] N ˆV /S n O ˆV = O ˆV k N. The global section of O ˆV k N is k [V ] N, which is clearly indecomposable as k [V ]#S n -module and admits a unique G m -equivariant structure up to a grading. As its restriction to an open subset, O ˆV k N also admits a unique G m -equivariant structure. Therefore, so does k [V ] N. This Lemma implies that we can endow Ê with suitable G m-equivariant structure, so that the isomorphism Γ(Ê) = k [V ] is G m -equivariant. Then using the fact that the natural morphism k [V ]#S n End k[[v ]] Sn (k [V ]) is a G m -equivariant isomorphism, we conclude that k [V ]#S n and End(Ê) are isomorphic as G m- equivariant algebras. References [BK4] R. Bezrukavnikov, D. Kaledin, McKay equivalence for symplectic resolutions of singularities, preprints, arxiv:412, [BFG6] R. Bezrukavnikov, M. Finkelberg, and V. Ginzburg, Cherednik algebras and Hilbert schemes in characteristic p, Represent. Theory 1 (26), MR [Los13] I. Losev, On Procesi bundles, preprint, 213. arxiv: [M8] J. Milne, Etale cohomology, Princeton Mathematical Series, 33. Princeton University Press, Princeton, N.J., 198. MR A [Sim14] J. Simental Rodriguez, Rational Cherednik algebras of type A, this seminar. [Sun14] Y. Sun, Quantum Hamiltonian reduction, this seminar. [Vain14] D. Vaintrob, Hamiltonian reduction in characteristic p, this seminar. 1, 3.6, 3.2 Department of Mathematics, Northeastern University, Boston, MA 2115, USA address: zhao.g@husky.neu.edu