# The McKay correspondence as an equivalence of derived categories

## Abstract

Let be a finite group of automorphisms of a nonsingular three-dimensional complex variety whose canonical bundle , is locally trivial as a We prove that the Hilbert scheme -sheaf. parametrising in -clusters is a crepant resolution of and that there is a derived equivalence (Fourier–Mukai transform) between coherent sheaves on and coherent on -sheaves 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 .Reference10 interpreted the McKay correspondence as an isomorphism on K theory, observing that the representation ring of is equal to the K theory of -equivariant 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 in the sense that every point of -sheaf, has a open neighbourhood on which there is a nonvanishing -invariant -invariant This implies that the quotient variety -form. has only Gorenstein singularities.

A natural generalisation of the McKay correspondence would then be an isomorphism between the K theory of -equivariant 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 ,Reference21 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 Reference20, Conjecture 4.1). Furthermore, if the resolution is constructed as a moduli space of objects on -equivariant 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 ,Reference15 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 Reference6 and Reference7, 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 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 -Hilbert 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 on -sheaves which may provide different crepant resolutions to the one considered here. ,

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

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

When and is Abelian, Nakamura Reference18 proved that is irreducible and is a crepant resolution of (compare also Reid Reference20 and Craw and Reid Reference8). He conjectured that the same result holds for an arbitrary finite subgroup Ito and Nakajima .Reference12 observed that the construction of Gonzalez-Sprinberg and Verdier Reference10 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 that is, the component mapping birationally to -orbits, 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 on -sheaves 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 Put -sheaf. 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 by giving it the trivial action. Note that -sheaf 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 ,Reference24, Theorem 2.8 and compare Kaledin Reference14).

### 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 Reference25). 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 Reference16, 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 Reference16, 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 (Reference16, 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 (Equation1) 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 Reference4.

### 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 Reference6, 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 Reference6.

### 2.5. Serre functors

The properties of Serre duality on a nonsingular projective variety were abstracted by Bondal and Kapranov Reference5 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 Reference5). 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 Reference6, 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 .Reference6, Theorem 3.3.

The proof of Theorem 3.3 in Reference6 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 on a scheme -modules 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 Reference19 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 (Equation3).

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 (Equation4) we conclude that .

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

In particular, has no nonzero global sections. Tensoring (Equation4) with gives a contradiction unless Hence . is an isomorphism and we are done.

## 4. -sheaves

Throughout this section is a finite group acting on a scheme (on the left) by automorphisms. As in the last section, all schemes are of finite type over We list some results we need concerning the category of sheaves on . equipped with a compatible action, or ‘ for short. Since -sheaves’ is finite, most of the proofs are trivial and are left to the reader. The main point is that natural constructions involving sheaves on are canonical, so commute with automorphisms of .

### 4.1. Sheaves and functors

A -sheaf on is a quasicoherent sheaf of together with a lift of the -modules action to More precisely, for each . there is a lift , satisfying and .

If and are then there is a (right) action of -sheaves, on given by and the spaces of maps give the morphisms in the Abelian categories -invariant and of -sheaves.

The category has enough injectives (Grothendieck Reference9, Proposition 5.1.2) so we may take injective resolutions. Since -equivariant is finite, if is a quasiprojective scheme there is an ample invertible on -sheaf and so we may also take locally free resolutions. The functors -equivariant are the parts of -invariant and are the derived functors of Thus if . is nonsingular of dimension so that , has global dimension then the category , also has global dimension .

The local functors and are defined in the obvious way on as are pullback , and pushforward for any morphism of schemes -equivariant Thus, for example, . Natural isomorphisms such as . are canonical, that is, commute with isomorphisms of the base, and hence are Therefore they restrict to natural isomorphisms -equivariant.

In other words, and are also adjoint functors between the categories and .

Similarly, the natural isomorphisms implicit in the projection formula, flat base change, etc. are canonical and hence -equivariant.

It seems worthwhile to single out the following point:

#### Lemma 4.1

Let and be on -sheaves Then, as a representation of . we have a direct sum decomposition ,

over the irreducible representations .

#### Proof.

The result amounts to showing that

Let be projection to a point, with acting trivially on so that the map is equivariant. Then is just the category of Note that -modules. and so that the adjunction between , and gives

as required.

### 4.2. Trivial actions

If the group acts trivially on then any , -sheaf decomposes as a direct sum

over the irreducible representations of (where is the trivial representation). The sheaves are just ordinary sheaves on Furthermore, . for Thus the category . decomposes as a direct sum and each summand is equivalent to .

In particular, every -sheaf has a fixed part and the functor

is the left and right adjoint to the functor

that is, ‘let act trivially’. Both functors are exact.

### 4.3. Derived categories

The derived category -equivariant is defined to be the full subcategory of the (unbounded) derived category of consisting of complexes with bounded and coherent cohomology.

The usual derived functors , , and may be defined on the equivariant derived category, and, as for sheaves, the standard properties of adjunctions, projection formula and flat base change then hold because the implicit natural isomorphisms are sufficiently canonical.

One way to obtain an equivariant Grothendieck duality is to refer to Neeman’s results Reference19. Let be an equivariant morphism of schemes. The only thing to check is that equivariant pushdown commutes with small coproducts. This is proved exactly as in Reference19. Then the functor has a right adjoint and ( ,Equation2) holds when is proper and of finite Tor dimension.

As in the nonequivariant case this implies that if is a nonsingular quasiprojective variety of dimension the full subcategory , consisting of objects with compact supports has a Serre functor

where is the canonical bundle of with its induced -structure.

### 4.4. Indecomposability

If acts trivially on then the results of Section ,4.2 show that decomposes as a direct sum of orthogonal subcategories indexed by the irreducible representations of More generally it is easy to see that . is decomposable unless acts faithfully. We need the converse of this statement.

#### Lemma 4.2

Suppose a finite group acts faithfully on a quasiprojective variety Then . is indecomposable.

#### Proof.

Suppose that decomposes as an orthogonal direct sum of two subcategories and Any indecomposable object of . lies in either or and

Since the action of is faithful, the general orbit is free. Let be a free orbit. Then is indecomposable as a Suppose without loss of generality that -sheaf. lies in .

Let be an irreducible representation of The sheaf . is indecomposable in and there exists an equivariant map so also lies in Any indecomposable . -sheaf supported in dimension 0 has a section, so by Lemma 4.1 there is an equivariant map and thus , lies in .

Finally given an indecomposable -sheaf take an orbit , contained in and let be the inclusion. Then is supported in dimension 0 and there is an equivariant map so , also lies in Now . is orthogonal to all sheaves, hence is trivial.

## 5. The intersection theorem

Our proof that is nonsingular follows an idea developed in Bridgeland and Maciocia Reference7 for moduli spaces over K3 fibrations, and uses the following famous and difficult result of commutative algebra:

### Theorem 5.1 (Intersection theorem).

Let be a local of dimension -algebra Suppose that .

is a complex of finitely generated free with each homology module -modules an of finite length. Then -module Moreover, if . and then ,

and is regular.

The basic idea is as follows. Serre’s criterion states that any finite length has homological dimension -module and that is regular precisely if there is a finite length which has homological dimension exactly -module The intersection theorem gives corresponding statements for complexes of . with finite length homology. As a rough slogan, “regularity is a property of the derived category”. For the main part of the proof, see Roberts -modulesReference22, Reference23; for the final clause, see Reference7.

We may rephrase the intersection theorem using the language of support and homological dimension. If is a scheme and an object in then it is easy to check ,Reference7 that, for any point ,

The *homological dimension* of a nonzero object written , is the smallest nonnegative integer , such that is isomorphic in to a complex of locally free sheaves on of length (that is, having terms). If no such integer exists we put One can prove .Reference7 that if is quasiprojective, and is a nonnegative integer, then if and only if there is an integer such that for all points

The two parts of Theorem 5.1 now become the following (cf. Reference7).

### Corollary 5.2

Let be a scheme and a nonzero object of Then .

### Corollary 5.3

Let be a scheme, and fix a point of codimension Suppose that there is an object . of such that for all points and any integer , ,

Suppose also that Then . is nonsingular at and .

## 6. The projective case

The aim of this section is to prove Theorem 1.1 under the additional assumption that is projective. The quasiprojective case involves some further technical difficulties that we deal with in the next section. Take notation as in the Introduction. We break the proof up into seven steps.

Step 1. Let and denote the projections. The functor may be rewritten

Note that has finite homological dimension, because is flat over and is nonsingular. Hence the derived dual also has finite homological dimension and we may define another functor by the formula ,

where .

Now is left adjoint to because of the three standard adjunctions described in Section 3.1. The functor is the left adjoint to The functor . has the (left and right) adjoint Finally the functor . has the left adjoint and

Step 2. The composite functor is given by

where and are the projections of onto its factors, and is some object of This is just composition of correspondences (see Mukai .Reference17, Proposition 1.3).

If is the closed embedding, then For any pair of points . one has , so that

using the adjunctions for and Our first objective is to show that . is supported on the diagonal or equivalently that the groups in ( ,Equation5) vanish unless When . this plays the rôle of assumption (4.8) of Ito and Nakajima Reference12.

Step 3. Let be Then -clusters.

To see this note that is generated as an by any nonzero constant section. But, since the global sections -module form the regular representation of the constant sections are precisely the , sections. Hence any equivariant morphism maps a generator to a scalar multiple of a generator and so is determined by that scalar. -invariant

Let and be distinct points of Serre duality, together with our assumption that . is locally trivial as a implies that -sheaf,

so that

Hence restricted to has homological dimension .

Step 4. Now we apply the intersection theorem. If and are points of such that then the corresponding clusters , and are disjoint, so that the groups in (Equation5) vanish. Thus the support of is contained in the subscheme By assumption this has codimension . so Corollary 5.2 implies that

that is, is supported on the diagonal.

Step 5. Fix a point and put , We proved above that . is supported at the point We claim that . Note that Corollary .5.3 then implies that is nonsingular at and .

To prove the claim, note that there is a canonical map so we obtain a triangle ,

for some object of Using the adjoint pair . this gives a long exact sequence ,

The homomorphism is the Kodaira–Spencer map for the family of clusters (Bridgeland Reference6, Lemma 4.4). This is injective because is a fine moduli space for on -clusters It follows that .

An easy spectral sequence argument (see Reference6, Example 2.2) shows that Taking cohomology sheaves of the above triangle gives . which proves the claim. ,

Step 6. We have now proved that is nonsingular, and that for any pair of points the homomorphisms ,

are isomorphisms. By assumption, the action of on is such that is trivial as a on an open neighbourhood of each orbit -sheaf This implies that .

in for each , Applying Theorem .2.4 shows that is an equivalence of categories.

Step 7. It remains to show that is crepant. Take a point The equivalence . restricts to give an equivalence between the full subcategories and consisting of objects supported on the fibre and the orbit respectively.

The category has trivial Serre functor because is trivial as a on a neighbourhood of -sheaf Thus . also has trivial Serre functor and Lemma 3.1 gives the result.

This completes the proof of Theorem 1.1 in the case that is projective.

## 7. The quasiprojective case

In this section we complete the proof of Theorem 1.1. Once again, take notation as in the Introduction. The problem with the argument of the last section is that when is not projective, Grothendieck duality in the form we need only applies to objects with compact support. To get round this we first take a projective closure of and define adjoint functors as before. Then we restrict to a functor

The argument of the last section carries through to show that is nonsingular and crepant and that is an equivalence. It remains for us to show that is also an equivalence.

Step 8. The functor has a right adjoint

As before, the composition is given by

where and are the projections of onto its factors, and is some object of .

Since is an equivalence, for any point and it follows that , is actually the pushforward of a line bundle on to the diagonal in The functor . is then just twisting by and to show that , is fully faithful we must show that is trivial.

There is a morphism of functors which for any point , gives a commutative diagram

where is nonzero. Since is an isomorphism on the subcategory the maps , are all isomorphisms, so the section is an isomorphism.

Step 9. The fact that is an equivalence follows from Lemma 2.1 once we show that

Suppose Using the adjunction for . ,

whenever for some object In particular, this holds for any . with compact support.

If is nonzero, let be an orbit of contained in the support of Let . denote the inclusion, a projective equivariant morphism of schemes. Then the adjunction morphism is nonzero, which gives a contradiction.

This completes the proof of Theorem 1.1.

## 8. Nakamura’s conjecture

Recall that in Theorem 1.1 we took the space to be an irreducible component of Note that when . is nonsingular and is an equivalence, is actually a connected component. This is simply because for any point the bijection ,

identifies the tangent space of at with the tangent space of at In this section we wish to go further and prove that when . has dimension 3, is in fact connected.

### Proof of Nakamura’s conjecture.

Suppose for contradiction that there exists a -cluster not contained among the Since . is an equivalence we can take an object such that The argument of Section .Equation6, Step 3, shows that for any point

This implies that has homological dimension 1, or more precisely, that is quasi-isomorphic to a complex of locally free sheaves of the form

But is supported on some in -orbit so , is supported on a fibre of and hence in codimension , It follows that the complex ( .Equation6) is exact on the left, so In particular . in the Grothendieck group of .

Let be a point of the fibre that is the support of By Lemma .8.1 below, in so that , in since the equivalence , gives an isomorphism of Grothendieck groups.

Let be a nonsingular projective variety with an open inclusion The functor . induces a map on K groups, so in But this contradicts Riemann–Roch, because if . is a sufficiently ample line bundle on then , and are both positive.

### Lemma 8.1

If and are two on -clusters supported on the same orbit, then the corresponding elements and in the Grothendieck group of are equal.

### Proof.

We need to show that, as -sheaves, and have composition series with the same simple factors. Suppose that they are both supported on the -orbit and let be the stabiliser subgroup of in The restriction functor is an equivalence of categories from finite length . supported on -sheaves to finite length supported at -sheaves The reverse equivalence is the induction functor . Since the restriction of a . supported on -cluster is an supported at -cluster it is sufficient to prove the result for , supported at -clusters .

If are the irreducible representations of then we claim that the simple , supported at -sheaves are precisely

These sheaves are certainly simple, since they are simple as On the other hand, any -modules. -sheaf supported at has a nonzero ordinary sheaf morphism