The McKay correspondence as an equivalence of derived categories

By Tom Bridgeland, Alastair King, and Miles Reid

To Andrei Tyurin on his 60th birthday

Abstract

Let be a finite group of automorphisms of a nonsingular three-dimensional complex variety , whose canonical bundle is locally trivial as a -sheaf. We prove that the Hilbert scheme parametrising -clusters in is a crepant resolution of and that there is a derived equivalence (Fourier–Mukai transform) between coherent sheaves on and coherent -sheaves on . This identifies the K theory of with the equivariant K theory of , and thus generalises the classical McKay correspondence. Some higher-dimensional extensions are possible.

1. Introduction

The classical McKay correspondence relates representations of a finite subgroup to the cohomology of the well-known minimal resolution of the Kleinian singularity . Gonzalez-Sprinberg and Verdier Reference 10 interpreted the McKay correspondence as an isomorphism on K theory, observing that the representation ring of is equal to the -equivariant K theory of . More precisely, they identify a basis of the K theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of .

It is natural to ask what happens when is replaced by an arbitrary nonsingular quasiprojective complex variety of dimension and by a finite group of automorphisms of , with the property that the stabiliser subgroup of any point acts on the tangent space as a subgroup of . Thus the canonical bundle is locally trivial as a -sheaf, in the sense that every point of has a -invariant open neighbourhood on which there is a nonvanishing -invariant -form. This implies that the quotient variety has only Gorenstein singularities.

A natural generalisation of the McKay correspondence would then be an isomorphism between the -equivariant K theory of and the ordinary K theory of a crepant resolution of , that is, a resolution of singularities such that . In the classical McKay case, the minimal resolution is crepant, but in higher dimensions crepant resolutions do not necessarily exist and, even when they do, they are not usually unique. However, it is now known that crepant resolutions of Gorenstein quotient singularities do exist in dimension , through a case by case analysis of the local linear actions by Ito, Markushevich and Roan (see Roan Reference 21 and references given there). In dimension , even such quotient singularities only have crepant resolutions in rather special cases.

In this paper, we take the point of view that the appropriate way to formulate and prove the McKay correspondence on K theory is to lift it to an equivalence of derived categories. In itself, this is not a new observation and it turns out that it was actually known to Gonzalez-Sprinberg and Verdier (see also Reid Reference 20, Conjecture 4.1). Furthermore, if the resolution is constructed as a moduli space of -equivariant objects on , then the correspondence should be given by a Fourier-Mukai transform determined by the universal object. This is the natural analogue of the classical statement that the tautological sheaves are a basis of the K theory. Both points of view are taken by Kapranov and Vasserot Reference 15 in proving the derived category version of the classical two-dimensional McKay correspondence.

The new and remarkable feature is that, by using the derived category and Fourier-Mukai transforms and, in particular, techniques developed in Reference 6 and Reference 7, the process of proving the equivalence of derived categories—when it works—also yields a proof that the moduli space is a crepant resolution. More specifically, we will give a sufficient condition for a certain natural moduli space, namely Nakamura’s -Hilbert scheme, to be a crepant resolution for which the McKay correspondence holds as an equivalence of derived categories. This condition is automatically satisfied in dimensions 2 and 3. Thus we simultaneously prove the existence of one crepant resolution of in three dimensions, without a case by case analysis, and verify the McKay correspondence for this resolution. We do not prove the McKay correspondence for an arbitrary crepant resolution although our methods should easily adapt to more general moduli spaces of -sheaves on , which may provide different crepant resolutions to the one considered here.

The -Hilbert scheme was introduced by Nakamura as a good candidate for a crepant resolution of . It parametrises -clusters or ‘scheme theoretic -orbits’ on : recall that a cluster is a zero-dimensional subscheme, and a -cluster is a -invariant cluster whose global sections are isomorphic to the regular representation of . Clearly, a -cluster has length and a free -orbit is a -cluster. There is a Hilbert–Chow morphism

which, on closed points, sends a -cluster to the orbit supporting it. Note that is a projective morphism, is onto and is birational on one component.

When and is Abelian, Nakamura Reference 18 proved that is irreducible and is a crepant resolution of (compare also Reid Reference 20 and Craw and Reid Reference 8). He conjectured that the same result holds for an arbitrary finite subgroup . Ito and Nakajima Reference 12 observed that the construction of Gonzalez-Sprinberg and Verdier Reference 10 is the case of a natural correspondence between the equivariant K theory of and the ordinary K theory of . They proved that this correspondence is an isomorphism when and is Abelian by constructing an explicit resolution of the diagonal in Beilinson style. Our approach via Fourier–Mukai transforms leaves this resolution of the diagonal implicit (it appears as the object of in Section 6), and seems to give a more direct argument. Two of the main consequences of the results of this paper are that Nakamura’s conjecture is true and that the natural correspondence on K theory is an isomorphism for all finite subgroups of .

Since it is not known whether is irreducible or even connected in general, we actually take as our initial candidate for a resolution the irreducible component of containing the free -orbits, that is, the component mapping birationally to . The aim is to show that is a crepant resolution, and to construct an equivalence between the derived categories of coherent sheaves on and of coherent -sheaves on . A more detailed analysis of the equivalence shows that when has dimension 3.

We now describe the correspondence and our results in more detail. Let be a nonsingular quasiprojective complex variety of dimension and let be a finite group of automorphisms of such that is locally trivial as a -sheaf. Put and let be the irreducible component containing the free orbits, as described above. Write for the universal closed subscheme and and for its projections to and . There is a commutative diagram of schemes

in which and are birational, and are finite, and is flat. Let act trivially on and , so that all morphisms in the diagram are equivariant.

Define the functor

where a sheaf on is viewed as a -sheaf by giving it the trivial action. Note that is already exact, so we do not need to write . Our main result is the following.

Theorem 1.1.

Suppose that the fibre product

has dimension . Then is a crepant resolution of and is an equivalence of categories.

When the condition of the theorem always holds because the exceptional locus of has dimension . In this case we can also show that is irreducible, so we obtain

Theorem 1.2.

Suppose . Then is irreducible and is a crepant resolution of , and is an equivalence of categories.

The condition of Theorem 1.1 also holds whenever preserves a complex symplectic form on and is a crepant resolution of , because such a resolution is symplectic and hence semi-small (see Verbitsky Reference 24, Theorem 2.8 and compare Kaledin Reference 14).

Corollary 1.3.

Suppose is a complex symplectic variety and acts by symplectic automorphisms. Assume that is a crepant resolution of . Then is an equivalence of categories.

Note that the condition of Theorem 1.1 certainly fails in dimension whenever has an exceptional divisor over a point. This is to be expected since there are many examples of finite subgroups for which the quotient singularity has no crepant resolution and also examples where, although crepant resolutions do exist, is not one.

Conventions

We work throughout in the category of schemes over . A point of a scheme always means a closed point.

2. Category theory

This section contains some basic category theory, most of which is well known. The only nontrivial part is Section 2.6 where we state a condition for an exact functor between triangulated categories to be an equivalence.

2.1. Triangulated categories

A triangulated category is an additive category equipped with a shift automorphism and a collection of distinguished triangles

of morphisms of satisfying certain axioms (see Verdier Reference 25). We write for and

A triangulated category is trivial if every object is a zero object.

The principal example of a triangulated category is the derived category of an Abelian category . An object of is a bounded complex of objects of up to quasi-isomorphism, the shift functor moves a complex to the left by one place and a distinguished triangle is the mapping cone of a morphism of complexes. In this case, for objects , one has .

A functor between triangulated categories is exact if it commutes with the shift automorphisms and takes distinguished triangles of to distinguished triangles of . For example, derived functors between derived categories are exact.

2.2. Adjoint functors

Let and be functors. An adjunction for is a bifunctorial isomorphism

In this case, we say that is left adjoint to or that is right adjoint to . When it exists, a left or right adjoint to a given functor is unique up to isomorphism of functors. The adjoint of a composite functor is the composite of the adjoints. An adjunction determines and is determined by two natural transformations and that come from applying the adjunction to and respectively (see Mac Lane Reference 16, IV.1 for more details).

The basic adjunctions we use in this paper are described in Section 3.1 below.

2.3. Fully faithful functors and equivalences

A functor is fully faithful if for any pair of objects , of , the map

is an isomorphism. One should think of as an ‘injective’ functor. This is clearer when has a left adjoint (or a right adjoint ), in which case is fully faithful if and only if the natural transformation (or ) is an isomorphism.

A functor is an equivalence if there is an ‘inverse’ functor such that and . In this case is both a left and right adjoint to (see Mac Lane Reference 16, IV.4). In practice, we show that is an equivalence by writing down an adjoint (a priori, one-sided) and proving that it is an inverse. One simple example of this is the following.

Lemma 2.1.

Let and be triangulated categories and a fully faithful exact functor with a right adjoint . Then is an equivalence if and only if implies for all objects .

Proof.

By assumption is an isomorphism, so is an equivalence if and only if is an isomorphism. Thus the ‘only if’ part of the lemma is immediate, since .

For the ‘if’ part, take any object and embed the natural adjunction map in a triangle

If we apply to this triangle, then is an isomorphism, because is an isomorphism and (Reference 16, IV.1, Theorem 1). Hence and so by hypothesis. Thus is an isomorphism, as required.

One may understand this lemma in a broader context as follows. The triangle (Equation 1) shows that, when is fully faithful with right adjoint , there is a ‘semi-orthogonal’ decomposition , where

Since is fully faithful, the fact that for some object necessarily means that , so only zero objects are in both subcategories. The semi-orthogonality condition also requires that for all and , which is immediate from the adjunction. The lemma then has the very reasonable interpretation that if is trivial, then and is an equivalence. Note that if is a left adjoint for , then there is a similar semi-orthogonal decomposition on the other side and a corresponding version of the lemma. For more details on semi-orthogonal decompositions see Bondal Reference 4.

2.4. Spanning classes and orthogonal decomposition

A spanning class for a triangulated category is a subclass of the objects of such that for any object

and

The following easy lemma is Reference 6, Example 2.2.

Lemma 2.2.

The set of skyscraper sheaves on a nonsingular projective variety is a spanning class for .

A triangulated category is decomposable as an orthogonal direct sum of two full subcategories and if every object of is isomorphic to a direct sum with , and if

for any pair of objects and all integers . The category is indecomposable if for any such decomposition one of the two subcategories is trivial. For example, if is a scheme, is indecomposable precisely when is connected. For more details see Bridgeland Reference 6.

2.5. Serre functors

The properties of Serre duality on a nonsingular projective variety were abstracted by Bondal and Kapranov Reference 5 into the notion of a Serre functor on a triangulated category. Let be a triangulated category in which all the sets are finite dimensional vector spaces. A Serre functor for is an exact equivalence inducing bifunctorial isomorphisms

that satisfy a simple compatibility condition (see Reference 5). When a Serre functor exists, it is unique up to isomorphism of functors. We say that has trivial Serre functor if for some integer the shift functor is a Serre functor for .

The main example is the bounded derived category of coherent sheaves on a nonsingular projective variety , having the Serre functor

Thus has trivial Serre functor if and only if the canonical bundle of is trivial.

2.6. A criterion for equivalence

Let be an exact functor between triangulated categories with Serre functors and . Assume that has a left adjoint . Then also has a right adjoint .

Theorem 2.3.

With assumptions as above, suppose also that there is a spanning class for such that

is an isomorphism for all and all . Then is fully faithful.

Proof.

See Reference 6, Theorem 2.3.

Theorem 2.4.

Suppose further that is nontrivial, that is indecomposable and that for all . Then is an equivalence of categories.

Proof.

Consider an object . For any and we have isomorphisms

using Serre duality and the adjunctions for and . Since is a spanning class we can conclude that precisely when . Then the result follows from Reference 6, Theorem 3.3.

The proof of Theorem 3.3 in Reference 6 may be understood as follows. If , then the semi-orthogonal decomposition described at the end of Section 2.3 becomes an orthogonal decomposition. Hence must be trivial, because is indecomposable and , and hence , is nontrivial. Thus and is an equivalence.

3. Derived categories of sheaves

This section is concerned with various general properties of complexes of -modules on a scheme . Note that all our schemes are of finite type over . Given a scheme , define to be the (unbounded) derived category of the Abelian category of quasicoherent sheaves on . Also define to be the full subcategory of consisting of complexes with bounded and coherent cohomology.

3.1. Geometric adjunctions

Here we describe three standard adjunctions that arise in algebraic geometry and are used frequently in what follows. For the first example, let be a scheme and an object of finite homological dimension. Then the derived dual

also has finite homological dimension, and the functor is both left and right adjoint to the functor .

For the second example take a morphism of schemes . The functor

has the left adjoint

If is proper, then takes into . If has finite Tor dimension (for example if is flat, or is nonsingular), then takes into .

The third example is Grothendieck duality. Again take a morphism of schemes . The functor has a right adjoint

and moreover, if is proper and of finite Tor dimension, there is an isomorphism of functors

Neeman Reference 19 has recently given a completely formal proof of these statements in terms of the Brown representability theorem.

Let be a nonsingular projective variety of dimension and write for the projection to a point. In this case . The above statement of Grothendieck duality implies that the functor

is a Serre functor on .

3.2. Duality for quasiprojective schemes

In order to apply Grothendieck duality on quasiprojective schemes, we need to restrict attention to sheaves with compact support. The support of an object is the locus of where is not exact, that is, the union of the supports of the cohomology sheaves of . It is always a closed subset of .

Given a scheme , define the category to be the full subcategory of consisting of complexes whose support is proper. Note that when itself is proper, is just the usual derived category .

If is a quasiprojective variety and is some projective closure, then the functor embeds as a full triangulated subcategory of . By resolution of singularities, if is nonsingular we can assume that is too. Then the Serre functor on restricts to give a Serre functor on . Thus if is a nonsingular quasiprojective variety of dimension , the category has a Serre functor given by (Equation 3).

The argument used to prove Lemma 2.2 is easily generalised to give the statement that the set of skyscraper sheaves on a nonsingular quasiprojective variety is a spanning class for .

3.3. Crepant resolutions

Let be a variety and a resolution of singularities. Given a point define to be the full subcategory of consisting of objects whose support is contained in the fibre . We have the following categorical criterion for to be crepant.

Lemma 3.1.

Assume that has rational singularities, that is, . Suppose has trivial Serre functor for each . Then is Gorenstein and is a crepant resolution.

Proof.

The Serre functor on is the restriction of the Serre functor on . Hence, by Section 3.2, the condition implies that for each the restriction of the functor to the category is isomorphic to the identity. Since contains the structure sheaves of all fattened neighbourhoods of the fibre this implies that the restriction of to each formal fibre of is trivial. To get the result, we must show that is a line bundle and that . Since , this is achieved by the following lemma.

Lemma 3.2.

Assume that has rational singularities. Then a line bundle on is the pullback of some line bundle on if and only if the restriction of to each formal fibre of is trivial. Moreover, when this holds, .

Proof.

For each point , the formal fibre of over is the fibre product

The restriction of the pullback of a line bundle from to each of these schemes is trivial because a line bundle has trivial formal stalks at points.

For the converse suppose that the restriction of to each of these formal fibres is trivial. The theorem on formal functions shows that the completions of the stalks of the sheaves and at any point are isomorphic for each . Since has rational singularities it follows that for all , and is a line bundle on .

Since is torsion free, the natural adjunction map is injective, so there is a short exact sequence

By the projection formula and the fact that has rational singularities,

The fact that is the unit of the adjunction for implies that has a left inverse, and in particular is surjective. Applying to (Equation 4) we conclude that .

Using the theorem on formal functions again, we can deduce that