American Mathematical Society

The McKay correspondence as an equivalence of derived categories

By Tom Bridgeland, Alastair King, Miles Reid

Abstract

Let upper G be a finite group of automorphisms of a nonsingular three-dimensional complex variety upper M , whose canonical bundle omega Subscript upper M is locally trivial as a upper G -sheaf. We prove that the Hilbert scheme upper Y equals upper G hyphen normal upper H normal i normal l normal b upper M parametrising upper G -clusters in upper M is a crepant resolution of upper X equals upper M slash upper G and that there is a derived equivalence (Fourier–Mukai transform) between coherent sheaves on upper Y and coherent upper G -sheaves on upper M . This identifies the K theory of upper Y with the equivariant K theory of upper M , 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 upper G subset-of upper S upper L left-parenthesis 2 comma double-struck upper C right-parenthesis to the cohomology of the well-known minimal resolution of the Kleinian singularity double-struck upper C squared slash upper G . Gonzalez-Sprinberg and Verdier Reference10 interpreted the McKay correspondence as an isomorphism on K theory, observing that the representation ring of upper G is equal to the upper G -equivariant K theory of double-struck upper C squared . 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 upper G .

It is natural to ask what happens when double-struck upper C squared is replaced by an arbitrary nonsingular quasiprojective complex variety upper M of dimension n and upper G by a finite group of automorphisms of upper M , with the property that the stabiliser subgroup of any point x element-of upper M acts on the tangent space upper T Subscript x Baseline upper M as a subgroup of upper S upper L left-parenthesis upper T Subscript x Baseline upper M right-parenthesis . Thus the canonical bundle omega Subscript upper M is locally trivial as a upper G -sheaf, in the sense that every point of upper M has a upper G -invariant open neighbourhood on which there is a nonvanishing upper G -invariant n -form. This implies that the quotient variety upper X equals upper M slash upper G has only Gorenstein singularities.

A natural generalisation of the McKay correspondence would then be an isomorphism between the upper G -equivariant K theory of upper M and the ordinary K theory of a crepant resolution upper Y of upper X , that is, a resolution of singularities tau colon upper Y right-arrow upper X such that tau Superscript asterisk Baseline left-parenthesis omega Subscript upper X Baseline right-parenthesis equals omega Subscript upper Y . 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 n equals 3 , 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 4 , 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 upper G -equivariant objects on upper M , 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 upper G -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 upper X equals upper M slash upper G 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 upper G -sheaves on upper M , which may provide different crepant resolutions to the one considered here.

The upper G -Hilbert scheme upper G hyphen normal upper H normal i normal l normal b upper M was introduced by Nakamura as a good candidate for a crepant resolution of upper M slash upper G . It parametrises upper G -clusters or ‘scheme theoretic upper G -orbits’ on upper M : recall that a cluster upper Z subset-of upper M is a zero-dimensional subscheme, and a upper G -cluster is a upper G -invariant cluster whose global sections normal upper Gamma left-parenthesis script upper O Subscript upper Z Baseline right-parenthesis are isomorphic to the regular representation double-struck upper C left-bracket upper G right-bracket of upper G . Clearly, a upper G -cluster has length StartAbsoluteValue upper G EndAbsoluteValue and a free upper G -orbit is a upper G -cluster. There is a Hilbert–Chow morphism

tau colon upper G hyphen normal upper H normal i normal l normal b upper M long right-arrow upper X comma

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

When upper M equals double-struck upper C cubed and upper G subset-of upper S upper L left-parenthesis 3 comma double-struck upper C right-parenthesis is Abelian, Nakamura Reference18 proved that upper G hyphen normal upper H normal i normal l normal b upper M is irreducible and is a crepant resolution of upper X (compare also Reid Reference20 and Craw and Reid Reference8). He conjectured that the same result holds for an arbitrary finite subgroup upper G subset-of upper S upper L left-parenthesis 3 comma double-struck upper C right-parenthesis . Ito and Nakajima Reference12 observed that the construction of Gonzalez-Sprinberg and Verdier Reference10 is the upper M equals double-struck upper C squared case of a natural correspondence between the equivariant K theory of upper M and the ordinary K theory of upper G hyphen normal upper H normal i normal l normal b upper M . They proved that this correspondence is an isomorphism when upper M equals double-struck upper C cubed and upper G subset-of upper S upper L left-parenthesis 3 comma double-struck upper C right-parenthesis 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 script upper Q of normal upper D left-parenthesis upper Y times upper Y right-parenthesis 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 upper S upper L left-parenthesis 3 comma double-struck upper C right-parenthesis .

Since it is not known whether upper G hyphen normal upper H normal i normal l normal b upper M is irreducible or even connected in general, we actually take as our initial candidate for a resolution upper Y the irreducible component of upper G hyphen normal upper H normal i normal l normal b upper M containing the free upper G -orbits, that is, the component mapping birationally to upper X . The aim is to show that upper Y is a crepant resolution, and to construct an equivalence between the derived categories normal upper D left-parenthesis upper Y right-parenthesis of coherent sheaves on upper Y and normal upper D Superscript upper G Baseline left-parenthesis upper M right-parenthesis of coherent upper G -sheaves on upper M . A more detailed analysis of the equivalence shows that upper Y equals upper G hyphen normal upper H normal i normal l normal b upper M when upper M has dimension 3.

We now describe the correspondence and our results in more detail. Let upper M be a nonsingular quasiprojective complex variety of dimension n and let upper G subset-of upper A u t left-parenthesis upper M right-parenthesis be a finite group of automorphisms of upper M such that omega Subscript upper M is locally trivial as a upper G -sheaf. Put upper X equals upper M slash upper G and let upper Y subset-of upper G hyphen normal upper H normal i normal l normal b upper M be the irreducible component containing the free orbits, as described above. Write script upper Z for the universal closed subscheme script upper Z subset-of upper Y times upper M and p and q for its projections to upper Y and upper M . There is a commutative diagram of schemes

in which q and tau are birational, p and pi are finite, and p is flat. Let upper G act trivially on upper Y and upper X , so that all morphisms in the diagram are equivariant.

Define the functor

normal upper Phi equals bold upper R q Subscript asterisk Baseline ring p Superscript asterisk Baseline colon normal upper D left-parenthesis upper Y right-parenthesis long right-arrow normal upper D Superscript upper G Baseline left-parenthesis upper M right-parenthesis comma

where a sheaf upper E on upper Y is viewed as a upper G -sheaf by giving it the trivial action. Note that p Superscript asterisk is already exact, so we do not need to write bold upper L p Superscript asterisk . Our main result is the following.

Theorem 1.1

Suppose that the fibre product

upper Y times Subscript upper X Baseline upper Y equals StartSet left-parenthesis y 1 comma y 2 right-parenthesis element-of upper Y times upper Y vertical-bar tau left-parenthesis y 1 right-parenthesis equals tau left-parenthesis y 2 right-parenthesis EndSet subset-of upper Y times upper Y

has dimension n plus 1 . Then upper Y is a crepant resolution of upper X and normal upper Phi is an equivalence of categories.

When n less-than-or-equal-to 3 the condition of the theorem always holds because the exceptional locus of upper Y right-arrow upper X has dimension 2 . In this case we can also show that upper G hyphen normal upper H normal i normal l normal b upper M is irreducible, so we obtain

Theorem 1.2

Suppose n less-than-or-equal-to 3 . Then upper G hyphen normal upper H normal i normal l normal b upper M is irreducible and is a crepant resolution of upper X , and normal upper Phi is an equivalence of categories.

The condition of Theorem 1.1 also holds whenever upper G preserves a complex symplectic form on upper M and upper Y is a crepant resolution of upper X , because such a resolution is symplectic and hence semi-small (see Verbitsky Reference24, Theorem 2.8 and compare Kaledin Reference14).

Corollary 1.3

Suppose upper M is a complex symplectic variety and upper G acts by symplectic automorphisms. Assume that upper Y is a crepant resolution of upper X . Then normal upper Phi is an equivalence of categories.

Note that the condition of Theorem 1.1 certainly fails in dimension 4 whenever upper Y right-arrow upper X has an exceptional divisor over a point. This is to be expected since there are many examples of finite subgroups upper G subset-of upper S upper L left-parenthesis 4 comma double-struck upper C right-parenthesis for which the quotient singularity double-struck upper C Superscript 4 Baseline slash upper G has no crepant resolution and also examples where, although crepant resolutions do exist, upper G hyphen normal upper H normal i normal l normal b left-parenthesis double-struck upper C Superscript 4 Baseline right-parenthesis is not one.

Conventions

We work throughout in the category of schemes over double-struck upper C . 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 script upper A equipped with a shift automorphism upper T Subscript script upper A Baseline colon script upper A right-arrow script upper A colon a right-arrow from bar a left-bracket 1 right-bracket and a collection of distinguished triangles

a 1 right-arrow Overscript f 1 Endscripts a 2 right-arrow Overscript f 2 Endscripts a 3 right-arrow Overscript f 3 Endscripts a 1 left-bracket 1 right-bracket

of morphisms of script upper A satisfying certain axioms (see Verdier Reference25). We write a left-bracket i right-bracket for upper T Subscript script upper A Superscript i Baseline left-parenthesis a right-parenthesis and

upper H o m Subscript script upper A Superscript i Baseline left-parenthesis a 1 comma a 2 right-parenthesis equals upper H o m Subscript script upper A Baseline left-parenthesis a 1 comma a 2 left-bracket i right-bracket right-parenthesis period

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

The principal example of a triangulated category is the derived category normal upper D left-parenthesis upper A right-parenthesis of an Abelian category upper A . An object of normal upper D left-parenthesis upper A right-parenthesis is a bounded complex of objects of upper A 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 a 1 comma a 2 element-of upper A , one has upper H o m Subscript normal upper D left-parenthesis upper A right-parenthesis Superscript i Baseline left-parenthesis a 1 comma a 2 right-parenthesis equals upper E x t Subscript upper A Superscript i Baseline left-parenthesis a 1 comma a 2 right-parenthesis .

A functor upper F colon script upper A right-arrow script upper B between triangulated categories is exact if it commutes with the shift automorphisms and takes distinguished triangles of script upper A to distinguished triangles of script upper B . For example, derived functors between derived categories are exact.

2.2. Adjoint functors

Let upper F colon script upper A right-arrow script upper B and upper G colon script upper B right-arrow script upper A be functors. An adjunction for left-parenthesis upper G comma upper F right-parenthesis is a bifunctorial isomorphism

upper H o m Subscript script upper A Baseline left-parenthesis upper G minus comma minus right-parenthesis approximately-equals upper H o m Subscript script upper B Baseline left-parenthesis minus comma upper F minus right-parenthesis period

In this case, we say that upper G is left adjoint to upper F or that upper F is right adjoint to upper G . 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 epsilon colon upper G ring upper F right-arrow i d Subscript script upper A Baseline and eta colon i d Subscript script upper B Baseline right-arrow upper F ring upper G that come from applying the adjunction to 1 Subscript upper F a and 1 Subscript upper G b 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 upper F colon script upper A right-arrow script upper B is fully faithful if for any pair of objects a 1 , a 2 of script upper A , the map

upper F colon upper H o m Subscript script upper A Baseline left-parenthesis a 1 comma a 2 right-parenthesis right-arrow upper H o m Subscript script upper B Baseline left-parenthesis upper F a 1 comma upper F a 2 right-parenthesis

is an isomorphism. One should think of upper F as an ‘injective’ functor. This is clearer when upper F has a left adjoint upper G colon script upper B right-arrow script upper A (or a right adjoint upper H colon script upper B right-arrow script upper A ), in which case upper F is fully faithful if and only if the natural transformation upper G ring upper F right-arrow i d Subscript script upper A (or i d Subscript script upper A Baseline right-arrow upper H ring upper F ) is an isomorphism.

A functor upper F is an equivalence if there is an ‘inverse’ functor upper G colon script upper B right-arrow script upper A such that upper G ring upper F approximately-equals i d Subscript script upper A and upper F ring upper G approximately-equals i d Subscript script upper B . In this case upper G is both a left and right adjoint to upper F (see Mac Lane Reference16, IV.4). In practice, we show that upper F 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 script upper A and script upper B be triangulated categories and upper F colon script upper A right-arrow script upper B a fully faithful exact functor with a right adjoint upper H colon script upper B right-arrow script upper A . Then upper F is an equivalence if and only if upper H c approximately-equals 0 implies c approximately-equals 0 for all objects c element-of script upper B .

Proof.

By assumption eta colon i d Subscript script upper A Baseline right-arrow upper H ring upper F is an isomorphism, so upper F is an equivalence if and only if epsilon colon upper F ring upper H right-arrow i d Subscript script upper B Baseline is an isomorphism. Thus the ‘only if’ part of the lemma is immediate, since c approximately-equals upper F upper H c .

For the ‘if’ part, take any object b element-of script upper B and embed the natural adjunction map epsilon Subscript b in a triangle

StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel c right-arrow upper F upper H b right-arrow Overscript epsilon Subscript b Baseline Endscripts b right-arrow c left-bracket 1 right-bracket period EndLayout

If we apply upper H to this triangle, then upper H left-parenthesis epsilon Subscript b Baseline right-parenthesis is an isomorphism, because eta Subscript upper H b is an isomorphism and upper H left-parenthesis epsilon Subscript b Baseline right-parenthesis ring eta Subscript upper H b Baseline equals 1 Subscript upper H b (Reference16, IV.1, Theorem 1). Hence upper H c approximately-equals 0 and so c approximately-equals 0 by hypothesis. Thus epsilon Subscript b is an isomorphism, as required.

One may understand this lemma in a broader context as follows. The triangle (Equation1) shows that, when upper F is fully faithful with right adjoint upper H , there is a ‘semi-orthogonal’ decomposition script upper B equals left-parenthesis upper I m upper F comma upper K e r upper H right-parenthesis , where

StartLayout 1st Row 1st Column upper I m upper F 2nd Column equals StartSet b element-of script upper B colon b approximately-equals upper F a for some a element-of script upper A EndSet comma 2nd Row 1st Column upper K e r upper H 2nd Column equals StartSet c element-of script upper B colon upper H c approximately-equals 0 EndSet period EndLayout

Since upper F is fully faithful, the fact that b approximately-equals upper F a for some object a element-of script upper A necessarily means that b approximately-equals upper F upper H b , so only zero objects are in both subcategories. The semi-orthogonality condition also requires that upper H o m Subscript script upper B Baseline left-parenthesis b comma c right-parenthesis equals 0 for all b element-of upper I m upper F and c element-of upper K e r upper H , which is immediate from the adjunction. The lemma then has the very reasonable interpretation that if upper K e r upper H is trivial, then upper I m upper F equals script upper B and upper F is an equivalence. Note that if upper G is a left adjoint for upper F , then there is a similar semi-orthogonal decomposition on the other side script upper B equals left-parenthesis upper K e r upper G comma upper I m upper F right-parenthesis 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 script upper A is a subclass normal upper Omega of the objects of script upper A such that for any object a element-of script upper A

upper H o m Subscript script upper A Superscript i Baseline left-parenthesis a comma omega right-parenthesis equals 0 for all omega element-of normal upper Omega comma i element-of double-struck upper Z implies a approximately-equals 0

and

upper H o m Subscript script upper A Superscript i Baseline left-parenthesis omega comma a right-parenthesis equals 0 for all omega element-of normal upper Omega comma i element-of double-struck upper Z implies a approximately-equals 0 period

The following easy lemma is Reference6, Example 2.2.

Lemma 2.2

The set of skyscraper sheaves StartSet script upper O Subscript x Baseline colon x element-of upper X EndSet on a nonsingular projective variety upper X is a spanning class for normal upper D left-parenthesis upper X right-parenthesis .

A triangulated category script upper A is decomposable as an orthogonal direct sum of two full subcategories script upper A 1 and script upper A 2 if every object of script upper A is isomorphic to a direct sum a 1 circled-plus a 2 with a Subscript j Baseline element-of script upper A Subscript j , and if

upper H o m Subscript script upper A Superscript i Baseline left-parenthesis a 1 comma a 2 right-parenthesis equals upper H o m Subscript script upper A Superscript i Baseline left-parenthesis a 2 comma a 1 right-parenthesis equals 0

for any pair of objects a Subscript j Baseline element-of script upper A Subscript j and all integers i . The category script upper A is indecomposable if for any such decomposition one of the two subcategories script upper A Subscript i is trivial. For example, if upper X is a scheme, normal upper D left-parenthesis upper X right-parenthesis is indecomposable precisely when upper X 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 script upper A be a triangulated category in which all the upper H o m sets are finite dimensional vector spaces. A Serre functor for script upper A is an exact equivalence upper S colon script upper A right-arrow script upper A inducing bifunctorial isomorphisms

upper H o m Subscript script upper A Baseline left-parenthesis a comma b right-parenthesis right-arrow upper H o m Subscript script upper A Baseline left-parenthesis b comma upper S left-parenthesis a right-parenthesis right-parenthesis Superscript logical-or Baseline for all a comma b element-of script upper A

that satisfy a simple compatibility condition (see Reference5). When a Serre functor exists, it is unique up to isomorphism of functors. We say that script upper A has trivial Serre functor if for some integer i the shift functor left-bracket i right-bracket is a Serre functor for script upper A .

The main example is the bounded derived category of coherent sheaves normal upper D left-parenthesis upper X right-parenthesis on a nonsingular projective variety upper X , having the Serre functor

upper S Subscript upper X Baseline left-parenthesis minus right-parenthesis equals left-parenthesis minus circled-times omega Subscript upper X Baseline right-parenthesis left-bracket dimension upper X right-bracket period

Thus normal upper D left-parenthesis upper X right-parenthesis has trivial Serre functor if and only if the canonical bundle of upper X is trivial.

2.6. A criterion for equivalence

Let upper F colon script upper A right-arrow script upper B be an exact functor between triangulated categories with Serre functors upper S Subscript script upper A and upper S Subscript script upper B . Assume that upper F has a left adjoint upper G colon script upper B right-arrow script upper A . Then upper F also has a right adjoint upper H equals upper S Subscript script upper A Baseline ring upper G ring upper S Subscript script upper B Superscript negative 1 .

Theorem 2.3

With assumptions as above, suppose also that there is a spanning class normal upper Omega for script upper A such that

upper F colon upper H o m Subscript script upper A Superscript i Baseline left-parenthesis omega 1 comma omega 2 right-parenthesis right-arrow upper H o m Subscript script upper B Superscript i Baseline left-parenthesis upper F omega 1 comma upper F omega 2 right-parenthesis

is an isomorphism for all i element-of double-struck upper Z and all omega 1 comma omega 2 element-of normal upper Omega . Then upper F is fully faithful.

Proof.

See Reference6, Theorem 2.3.

Theorem 2.4

Suppose further that script upper A is nontrivial, that script upper B is indecomposable and that upper F upper S Subscript script upper A Baseline left-parenthesis omega right-parenthesis approximately-equals upper S Subscript script upper B Baseline upper F left-parenthesis omega right-parenthesis for all omega element-of normal upper Omega . Then upper F is an equivalence of categories.

Proof.

Consider an object b element-of script upper B . For any omega element-of normal upper Omega and i element-of double-struck upper Z we have isomorphisms

StartLayout 1st Row 1st Column Blank 2nd Column upper H o m Subscript script upper A Superscript i Baseline left-parenthesis omega comma upper G b right-parenthesis equals upper H o m Subscript script upper A Superscript i Baseline left-parenthesis upper G b comma upper S Subscript script upper A Baseline omega right-parenthesis Superscript logical-or Baseline equals upper H o m Subscript script upper B Superscript i Baseline left-parenthesis b comma upper F upper S Subscript script upper A Baseline omega right-parenthesis Superscript logical-or Baseline 2nd Row 1st Column Blank 2nd Column upper H o m Subscript script upper B Superscript i Baseline left-parenthesis b comma upper S Subscript script upper B Baseline upper F omega right-parenthesis Superscript logical-or Baseline equals upper H o m Subscript script upper B Superscript i Baseline left-parenthesis upper F omega comma b right-parenthesis equals upper H o m Subscript script upper A Superscript i Baseline left-parenthesis omega comma upper H b right-parenthesis comma EndLayout

using Serre duality and the adjunctions for left-parenthesis upper G comma upper F right-parenthesis and left-parenthesis upper F comma upper H right-parenthesis . Since normal upper Omega is a spanning class we can conclude that upper G b approximately-equals 0 precisely when upper H b approximately-equals 0 . Then the result follows from Reference6, Theorem 3.3.

The proof of Theorem 3.3 in Reference6 may be understood as follows. If upper K e r upper H subset-of upper K e r upper G , then the semi-orthogonal decomposition described at the end of Section 2.3 becomes an orthogonal decomposition. Hence upper K e r upper H must be trivial, because script upper B is indecomposable and script upper A , and hence upper I m upper F , is nontrivial. Thus upper I m upper F equals script upper B and upper F is an equivalence.

3. Derived categories of sheaves

This section is concerned with various general properties of complexes of script upper O Subscript upper X -modules on a scheme upper X . Note that all our schemes are of finite type over double-struck upper C . Given a scheme upper X , define normal upper D Superscript normal q normal c Baseline left-parenthesis upper X right-parenthesis to be the (unbounded) derived category of the Abelian category upper Q c o h left-parenthesis upper X right-parenthesis of quasicoherent sheaves on upper X . Also define normal upper D left-parenthesis upper X right-parenthesis to be the full subcategory of normal upper D Superscript normal q normal c Baseline left-parenthesis upper X right-parenthesis 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 upper X be a scheme and upper E element-of normal upper D left-parenthesis upper X right-parenthesis an object of finite homological dimension. Then the derived dual

upper E Superscript logical-or Baseline equals bold upper R script upper H times o m Subscript script upper O Sub Subscript upper X Baseline left-parenthesis upper E comma script upper O Subscript upper X Baseline right-parenthesis

also has finite homological dimension, and the functor minus circled-times upper E is both left and right adjoint to the functor minus circled-times upper E Superscript logical-or .

For the second example take a morphism of schemes f colon upper X right-arrow upper Y . The functor

bold upper R f Subscript asterisk Baseline colon normal upper D Superscript normal q normal c Baseline left-parenthesis upper X right-parenthesis long right-arrow normal upper D Superscript normal q normal c Baseline left-parenthesis upper Y right-parenthesis

has the left adjoint

bold upper L f Superscript asterisk Baseline colon normal upper D Superscript normal q normal c Baseline left-parenthesis upper Y right-parenthesis long right-arrow normal upper D Superscript normal q normal c Baseline left-parenthesis upper X right-parenthesis period

If f is proper, then bold upper R f Subscript asterisk takes normal upper D left-parenthesis upper X right-parenthesis into normal upper D left-parenthesis upper Y right-parenthesis . If f has finite Tor dimension (for example if f is flat, or upper Y is nonsingular), then bold upper L f Superscript asterisk takes normal upper D left-parenthesis upper Y right-parenthesis into normal upper D left-parenthesis upper X right-parenthesis .

The third example is Grothendieck duality. Again take a morphism of schemes f colon upper X right-arrow upper Y . The functor bold upper R f Subscript asterisk has a right adjoint

f Superscript factorial Baseline colon normal upper D Superscript normal q normal c Baseline left-parenthesis upper Y right-parenthesis long right-arrow normal upper D Superscript normal q normal c Baseline left-parenthesis upper X right-parenthesis

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

StartLayout 1st Row with Label left-parenthesis 2 right-parenthesis EndLabel f Superscript factorial Baseline left-parenthesis minus right-parenthesis approximately-equals bold upper L f Superscript asterisk Baseline left-parenthesis minus right-parenthesis circled-times Overscript bold upper L Endscripts f Superscript factorial Baseline left-parenthesis script upper O Subscript upper Y Baseline right-parenthesis period EndLayout

Neeman Reference19 has recently given a completely formal proof of these statements in terms of the Brown representability theorem.

Let upper X be a nonsingular projective variety of dimension n and write f colon upper X right-arrow upper Y equals upper S p e c left-parenthesis double-struck upper C right-parenthesis for the projection to a point. In this case f Superscript factorial Baseline left-parenthesis script upper O Subscript upper Y Baseline right-parenthesis equals omega Subscript upper X Baseline left-bracket n right-bracket . The above statement of Grothendieck duality implies that the functor

StartLayout 1st Row with Label left-parenthesis 3 right-parenthesis EndLabel upper S Subscript upper X Baseline left-parenthesis minus right-parenthesis equals left-parenthesis minus circled-times omega Subscript upper X Baseline right-parenthesis left-bracket n right-bracket EndLayout

is a Serre functor on normal upper D left-parenthesis upper X right-parenthesis .

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 upper E element-of normal upper D left-parenthesis upper X right-parenthesis is the locus of upper X where upper E is not exact, that is, the union of the supports of the cohomology sheaves of upper E . It is always a closed subset of upper X .

Given a scheme upper X , define the category normal upper D Subscript normal c Baseline left-parenthesis upper X right-parenthesis to be the full subcategory of normal upper D left-parenthesis upper X right-parenthesis consisting of complexes whose support is proper. Note that when upper X itself is proper, normal upper D Subscript normal c Baseline left-parenthesis upper X right-parenthesis is just the usual derived category normal upper D left-parenthesis upper X right-parenthesis .

If upper X is a quasiprojective variety and i colon upper X right-arrow with hook upper X overbar is some projective closure, then the functor i Subscript asterisk embeds normal upper D Subscript normal c Baseline left-parenthesis upper X right-parenthesis as a full triangulated subcategory of normal upper D left-parenthesis upper X overbar right-parenthesis . By resolution of singularities, if upper X is nonsingular we can assume that upper X overbar is too. Then the Serre functor on normal upper D left-parenthesis upper X overbar right-parenthesis restricts to give a Serre functor on normal upper D left-parenthesis upper X right-parenthesis . Thus if upper X is a nonsingular quasiprojective variety of dimension n , the category normal upper D Subscript normal c Baseline left-parenthesis upper X right-parenthesis 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 StartSet script upper O Subscript x Baseline colon x element-of upper X EndSet on a nonsingular quasiprojective variety upper X is a spanning class for normal upper D Subscript normal c Baseline left-parenthesis upper X right-parenthesis .

3.3. Crepant resolutions

Let upper X be a variety and f colon upper Y right-arrow upper X a resolution of singularities. Given a point x element-of upper X define normal upper D Subscript x Baseline left-parenthesis upper Y right-parenthesis to be the full subcategory of normal upper D Subscript normal c Baseline left-parenthesis upper Y right-parenthesis consisting of objects whose support is contained in the fibre f Superscript negative 1 Baseline left-parenthesis x right-parenthesis . We have the following categorical criterion for f to be crepant.

Lemma 3.1

Assume that upper X has rational singularities, that is, bold upper R f Subscript asterisk Baseline script upper O Subscript upper Y Baseline equals script upper O Subscript upper X . Suppose normal upper D Subscript x Baseline left-parenthesis upper Y right-parenthesis has trivial Serre functor for each x element-of upper X . Then upper X is Gorenstein and f colon upper Y right-arrow upper X is a crepant resolution.

Proof.

The Serre functor on normal upper D Subscript x Baseline left-parenthesis upper Y right-parenthesis is the restriction of the Serre functor on normal upper D Subscript normal c Baseline left-parenthesis upper Y right-parenthesis . Hence, by Section 3.2, the condition implies that for each x element-of upper X the restriction of the functor left-parenthesis minus circled-times omega Subscript upper Y Baseline right-parenthesis to the category normal upper D Subscript x Baseline left-parenthesis upper Y right-parenthesis is isomorphic to the identity. Since normal upper D Subscript x Baseline left-parenthesis upper Y right-parenthesis contains the structure sheaves of all fattened neighbourhoods of the fibre f Superscript negative 1 Baseline left-parenthesis x right-parenthesis this implies that the restriction of omega Subscript upper Y to each formal fibre of f is trivial. To get the result, we must show that omega Subscript upper X is a line bundle and that f Superscript asterisk Baseline omega Subscript upper X Baseline equals omega Subscript upper Y . Since omega Subscript upper X Baseline equals f Subscript asterisk Baseline omega Subscript upper Y , this is achieved by the following lemma.

Lemma 3.2

Assume that upper X has rational singularities. Then a line bundle upper L on upper Y is the pullback f Superscript asterisk Baseline upper M of some line bundle upper M on upper X if and only if the restriction of upper L to each formal fibre of f is trivial. Moreover, when this holds, upper M equals f Subscript asterisk Baseline upper L .

Proof.

For each point x element-of upper X , the formal fibre of f over x is the fibre product

upper Y times Subscript upper X Baseline upper S p e c left-parenthesis ModifyingAbove script upper O With caret Subscript upper X comma x Baseline right-parenthesis period

The restriction of the pullback of a line bundle from upper X 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 upper L to each of these formal fibres is trivial. The theorem on formal functions shows that the completions of the stalks of the sheaves bold upper R Superscript i Baseline f Subscript asterisk Baseline script upper O Subscript upper Y and bold upper R Superscript i Baseline f Subscript asterisk Baseline upper L at any point x element-of upper X are isomorphic for each i . Since upper X has rational singularities it follows that bold upper R Superscript i Baseline f Subscript asterisk Baseline upper L equals 0 for all i greater-than 0 , and upper M equals f Subscript asterisk Baseline upper L is a line bundle on upper X .

Since f Superscript asterisk Baseline upper M is torsion free, the natural adjunction map eta colon f Superscript asterisk Baseline f Subscript asterisk Baseline upper L right-arrow upper L is injective, so there is a short exact sequence

StartLayout 1st Row with Label left-parenthesis 4 right-parenthesis EndLabel 0 right-arrow f Superscript asterisk Baseline f Subscript asterisk Baseline upper L right-arrow Overscript eta Endscripts upper L right-arrow upper Q right-arrow 0 period EndLayout

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

bold upper R Superscript i Baseline f Subscript asterisk Baseline left-parenthesis f Superscript asterisk Baseline upper M right-parenthesis equals upper M circled-times bold upper R Superscript i Baseline f Subscript asterisk Baseline script upper O Subscript upper Y Baseline equals 0 for all i greater-than 0 period

The fact that eta is the unit of the adjunction for left-parenthesis f Superscript asterisk Baseline comma f Subscript asterisk Baseline right-parenthesis implies that f Subscript asterisk Baseline eta has a left inverse, and in particular is surjective. Applying f Subscript asterisk to (Equation4) we conclude that f Subscript asterisk Baseline upper Q equals 0 .

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

f Subscript asterisk Baseline left-parenthesis upper Q circled-times upper L Superscript negative 1 Baseline right-parenthesis equals 0 period

In particular, upper Q circled-times upper L Superscript negative 1 has no nonzero global sections. Tensoring (Equation4) with upper L Superscript negative 1 gives a contradiction unless upper Q equals 0 . Hence eta is an isomorphism and we are done.

4. upper G -sheaves

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

4.1. Sheaves and functors

A upper G -sheaf upper E on upper X is a quasicoherent sheaf of script upper O Subscript upper X -modules together with a lift of the upper G action to upper E . More precisely, for each g element-of upper G , there is a lift lamda Subscript g Superscript upper E Baseline colon upper E right-arrow g Superscript asterisk Baseline upper E satisfying lamda 1 Superscript upper E Baseline equals i d Subscript upper E and lamda Subscript h g Superscript upper E Baseline equals g Superscript asterisk Baseline left-parenthesis lamda Subscript h Superscript upper E Baseline right-parenthesis ring lamda Subscript g Superscript upper E .

If upper E and upper F are upper G -sheaves, then there is a (right) action of upper G on upper H o m Subscript upper X Baseline left-parenthesis upper E comma upper F right-parenthesis given by theta Superscript g Baseline equals left-parenthesis lamda Subscript g Superscript upper F Baseline right-parenthesis Superscript negative 1 Baseline ring g Superscript asterisk Baseline theta ring lamda Subscript g Superscript upper E and the spaces upper G hyphen normal upper H normal o normal m Subscript upper X Baseline left-parenthesis upper E comma upper F right-parenthesis of upper G -invariant maps give the morphisms in the Abelian categories upper Q c o h Superscript upper G Baseline left-parenthesis upper X right-parenthesis and upper C o h Superscript upper G Baseline left-parenthesis upper X right-parenthesis of upper G -sheaves.

The category upper Q c o h Superscript upper G Baseline left-parenthesis upper X right-parenthesis has enough injectives (Grothendieck Reference9, Proposition 5.1.2) so we may take upper G -equivariant injective resolutions. Since upper G is finite, if upper X is a quasiprojective scheme there is an ample invertible upper G -sheaf on upper X and so we may also take upper G -equivariant locally free resolutions. The functors upper G hyphen normal upper E normal x normal t Subscript upper X Superscript i Baseline left-parenthesis minus comma minus right-parenthesis are the upper G -invariant parts of upper E x t Subscript upper X Superscript i Baseline left-parenthesis minus comma minus right-parenthesis and are the derived functors of upper G hyphen normal upper H normal o normal m Subscript upper X Baseline left-parenthesis minus comma minus right-parenthesis . Thus if upper X is nonsingular of dimension n , so that upper Q c o h left-parenthesis upper X right-parenthesis has global dimension n , then the category upper Q c o h Superscript upper G Baseline left-parenthesis upper X right-parenthesis also has global dimension n .

The local functors script upper H o m and circled-times are defined in the obvious way on upper Q c o h Superscript upper G Baseline left-parenthesis upper X right-parenthesis , as are pullback f Superscript asterisk and pushforward f Subscript asterisk for any upper G -equivariant morphism of schemes f colon upper X right-arrow upper Y . Thus, for example, lamda Subscript g Superscript f Super Superscript asterisk Superscript upper E Baseline equals f Superscript asterisk Baseline lamda Subscript g Superscript upper E . Natural isomorphisms such as upper H o m Subscript upper X Baseline left-parenthesis f Superscript asterisk Baseline upper E comma upper F right-parenthesis approximately-equals upper H o m Subscript upper Y Baseline left-parenthesis upper E comma f Subscript asterisk Baseline upper F right-parenthesis are canonical, that is, commute with isomorphisms of the base, and hence are upper G -equivariant. Therefore they restrict to natural isomorphisms

upper G hyphen normal upper H normal o normal m Subscript upper X Baseline left-parenthesis f Superscript asterisk Baseline upper E comma upper F right-parenthesis approximately-equals upper G hyphen normal upper H normal o normal m Subscript upper Y Baseline left-parenthesis upper E comma f Subscript asterisk Baseline upper F right-parenthesis period

In other words, f Superscript asterisk and f Subscript asterisk are also adjoint functors between the categories upper Q c o h Superscript upper G Baseline left-parenthesis upper X right-parenthesis and upper Q c o h Superscript upper G Baseline left-parenthesis upper Y right-parenthesis .

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

It seems worthwhile to single out the following point:

Lemma 4.1

Let upper E and upper F be upper G -sheaves on upper X . Then, as a representation of upper G , we have a direct sum decomposition

upper H o m Subscript upper X Baseline left-parenthesis upper E comma upper F right-parenthesis equals circled-plus Underscript i equals 0 Overscript k Endscripts upper G hyphen normal upper H normal o normal m Subscript upper X Baseline left-parenthesis upper E circled-times rho Subscript i Baseline comma upper F right-parenthesis circled-times rho Subscript i

over the irreducible representations StartSet rho 0 comma period period period comma rho Subscript k Baseline EndSet .

Proof.

The result amounts to showing that

upper G hyphen normal upper H normal o normal m left-parenthesis rho Subscript i Baseline comma upper H o m Subscript upper X Baseline left-parenthesis upper E comma upper F right-parenthesis right-parenthesis equals upper G hyphen normal upper H normal o normal m Subscript upper X Baseline left-parenthesis upper E circled-times rho Subscript i Baseline comma upper F right-parenthesis period

Let f colon upper X right-arrow upper Y equals upper S p e c left-parenthesis double-struck upper C right-parenthesis be projection to a point, with upper G acting trivially on upper Y so that the map is equivariant. Then upper Q c o h Superscript upper G Baseline left-parenthesis upper Y right-parenthesis is just the category of double-struck upper C left-bracket upper G right-bracket -modules. Note that upper H o m Subscript upper X Baseline left-parenthesis upper E comma upper F right-parenthesis equals f Subscript asterisk Baseline script upper H times o m Subscript script upper O Sub Subscript upper X Baseline left-parenthesis upper E comma upper F right-parenthesis and f Superscript asterisk Baseline rho Subscript i Baseline equals script upper O Subscript upper X Baseline circled-times rho Subscript i , so that the adjunction between f Superscript asterisk and f Subscript asterisk gives

StartLayout 1st Row 1st Column upper G hyphen normal upper H normal o normal m Subscript upper Y Baseline left-parenthesis rho Subscript i Baseline comma f Subscript asterisk Baseline script upper H times o m Subscript script upper O Sub Subscript upper X Subscript Baseline left-parenthesis upper E comma upper F right-parenthesis right-parenthesis 2nd Column equals 3rd Column upper G hyphen normal upper H normal o normal m Subscript upper X Baseline left-parenthesis script upper O Subscript upper X Baseline circled-times rho Subscript i Baseline comma script upper H times o m Subscript script upper O Sub Subscript upper X Subscript Baseline left-parenthesis upper E comma upper F right-parenthesis right-parenthesis 2nd Row 1st Column Blank 2nd Column equals 3rd Column upper G hyphen normal upper H normal o normal m Subscript upper X Baseline left-parenthesis upper E circled-times rho Subscript i Baseline comma upper F right-parenthesis comma EndLayout

as required.

4.2. Trivial actions

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

upper E equals circled-plus Underscript i Endscripts upper E Subscript i Baseline circled-times rho Subscript i

over the irreducible representations StartSet rho 0 comma rho 1 comma ellipsis comma rho Subscript k Baseline EndSet of upper G (where rho 0 equals bold 1 is the trivial representation). The sheaves upper E Subscript i are just ordinary sheaves on upper X . Furthermore, upper G hyphen normal upper H normal o normal m Subscript upper X Baseline left-parenthesis upper E Subscript i Baseline circled-times rho Subscript i Baseline comma upper E Subscript j Baseline circled-times rho Subscript j Baseline right-parenthesis equals 0 for i not-equals j . Thus the category upper Q c o h Superscript upper G Baseline left-parenthesis upper X right-parenthesis decomposes as a direct sum circled-plus Underscript i Endscripts upper Q c o h Superscript i Baseline left-parenthesis upper X right-parenthesis and each summand is equivalent to upper Q c o h left-parenthesis upper X right-parenthesis .

In particular, every upper G -sheaf upper E has a fixed part left-bracket upper E right-bracket Superscript upper G and the functor

left-bracket minus right-bracket Superscript upper G Baseline colon upper Q c o h Superscript upper G Baseline left-parenthesis upper X right-parenthesis right-arrow upper Q c o h left-parenthesis upper X right-parenthesis

is the left and right adjoint to the functor

minus circled-times rho 0 colon upper Q c o h left-parenthesis upper X right-parenthesis right-arrow upper Q c o h Superscript upper G Baseline left-parenthesis upper X right-parenthesis comma

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

4.3. Derived categories

The upper G -equivariant derived category normal upper D Superscript upper G Baseline left-parenthesis upper X right-parenthesis is defined to be the full subcategory of the (unbounded) derived category of upper Q c o h Superscript upper G Baseline left-parenthesis upper X right-parenthesis consisting of complexes with bounded and coherent cohomology.

The usual derived functors bold upper R script upper H o m , circled-times Overscript bold upper L Endscripts , bold upper L f Superscript asterisk and bold upper R f Subscript asterisk 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 f colon upper X right-arrow upper Y be an equivariant morphism of schemes. The only thing to check is that equivariant pushdown bold upper R f Subscript asterisk commutes with small coproducts. This is proved exactly as in Reference19. Then the functor bold upper R f Subscript asterisk has a right adjoint f Superscript factorial , and (Equation2) holds when f is proper and of finite Tor dimension.

As in the nonequivariant case this implies that if upper X is a nonsingular quasiprojective variety of dimension n , the full subcategory normal upper D Subscript normal c Superscript upper G Baseline left-parenthesis upper X right-parenthesis subset-of normal upper D Superscript upper G Baseline left-parenthesis upper X right-parenthesis consisting of objects with compact supports has a Serre functor

upper S Subscript upper X Baseline left-parenthesis minus right-parenthesis equals left-parenthesis minus circled-times omega Subscript upper X Baseline right-parenthesis left-bracket n right-bracket comma

where omega Subscript upper X is the canonical bundle of upper X with its induced upper G -structure.

4.4. Indecomposability

If upper G acts trivially on upper X , then the results of Section 4.2 show that normal upper D Superscript upper G Baseline left-parenthesis upper X right-parenthesis decomposes as a direct sum of orthogonal subcategories indexed by the irreducible representations of upper G . More generally it is easy to see that normal upper D Superscript upper G Baseline left-parenthesis upper X right-parenthesis is decomposable unless upper G acts faithfully. We need the converse of this statement.

Lemma 4.2

Suppose a finite group upper G acts faithfully on a quasiprojective variety upper X . Then normal upper D Superscript upper G Baseline left-parenthesis upper X right-parenthesis is indecomposable.

Proof.

Suppose that normal upper D Superscript upper G Baseline left-parenthesis upper X right-parenthesis decomposes as an orthogonal direct sum of two subcategories script upper A 1 and script upper A 2 . Any indecomposable object of normal upper D Superscript upper G Baseline left-parenthesis upper X right-parenthesis lies in either script upper A 1 or script upper A 2 and

upper H o m Subscript normal upper D Sub Superscript upper G Subscript left-parenthesis upper X right-parenthesis Baseline left-parenthesis a 1 comma a 2 right-parenthesis equals 0 for all a 1 element-of script upper A 1 comma a 2 element-of script upper A 2 period

Since the action of upper G is faithful, the general orbit is free. Let upper D equals upper G dot x be a free orbit. Then script upper O Subscript upper D is indecomposable as a upper G -sheaf. Suppose without loss of generality that script upper O Subscript upper D lies in script upper A 1 .

Let rho Subscript i be an irreducible representation of upper G . The sheaf script upper O Subscript upper X Baseline circled-times rho Subscript i is indecomposable in normal upper D Superscript upper G Baseline left-parenthesis upper X right-parenthesis and there exists an equivariant map script upper O Subscript upper X Baseline circled-times rho Subscript i Baseline right-arrow script upper O Subscript upper D so script upper O Subscript upper X Baseline circled-times rho Subscript i also lies in script upper A 1 . Any indecomposable upper G -sheaf upper E supported in dimension 0 has a section, so by Lemma 4.1 there is an equivariant map script upper O Subscript upper X Baseline circled-times rho Subscript i Baseline right-arrow upper E , and thus upper E lies in script upper A 1 .

Finally given an indecomposable upper G -sheaf upper F , take an orbit upper G dot x contained in upper S u p p left-parenthesis upper F right-parenthesis and let i colon upper G dot x right-arrow with hook upper X be the inclusion. Then i Subscript asterisk Baseline i Superscript asterisk Baseline left-parenthesis upper F right-parenthesis is supported in dimension 0 and there is an equivariant map upper F right-arrow i Subscript asterisk Baseline i Superscript asterisk Baseline left-parenthesis upper F right-parenthesis , so upper F also lies in script upper A 1 . Now script upper A 2 is orthogonal to all sheaves, hence is trivial.

5. The intersection theorem

Our proof that upper G hyphen normal upper H normal i normal l normal b upper M 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 left-parenthesis upper A comma m right-parenthesis be a local double-struck upper C -algebra of dimension d . Suppose that

0 right-arrow upper M Subscript s Baseline right-arrow upper M Subscript s minus 1 Baseline right-arrow ellipsis right-arrow upper M 0 right-arrow 0

is a complex of finitely generated free upper A -modules with each homology module upper H Subscript i Baseline left-parenthesis upper M Subscript bullet Baseline right-parenthesis an upper A -module of finite length. Then s greater-than-or-equal-to d . Moreover, if s equals d and upper H 0 left-parenthesis upper M Subscript bullet Baseline right-parenthesis approximately-equals upper A slash m , then

upper H Subscript i Baseline left-parenthesis upper M Subscript bullet Baseline right-parenthesis equals 0 for all i not-equals 0 comma

and upper A is regular.

The basic idea is as follows. Serre’s criterion states that any finite length upper A -module has homological dimension d and that upper A is regular precisely if there is a finite length upper A -module which has homological dimension exactly d . The intersection theorem gives corresponding statements for complexes of upper A -modules 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 Reference22, Reference23; for the final clause, see Reference7.

We may rephrase the intersection theorem using the language of support and homological dimension. If upper X is a scheme and upper E an object in normal upper D left-parenthesis upper X right-parenthesis , then it is easy to check Reference7 that, for any point x element-of upper X ,

x element-of upper S u p p upper E long left right double arrow upper H o m Subscript normal upper D left-parenthesis upper X right-parenthesis Superscript i Baseline left-parenthesis upper E comma script upper O Subscript x Baseline right-parenthesis not-equals 0 for some i element-of double-struck upper Z period

The homological dimension of a nonzero object upper E element-of normal upper D left-parenthesis upper X right-parenthesis , written normal h times normal o times normal m times normal d times normal i times normal m upper E , is the smallest nonnegative integer s such that upper E is isomorphic in normal upper D left-parenthesis upper X right-parenthesis to a complex of locally free sheaves on upper X of length s (that is, having s plus 1 terms). If no such integer exists we put normal h times normal o times normal m times normal d times normal i times normal m upper E equals normal infinity . One can prove Reference7 that if upper X is quasiprojective, and n is a nonnegative integer, then normal h times normal o times normal m times normal d times normal i times normal m upper E less-than-or-equal-to n if and only if there is an integer j such that for all points x element-of upper X

upper H o m Subscript normal upper D left-parenthesis upper X right-parenthesis Superscript i Baseline left-parenthesis upper E comma script upper O Subscript x Baseline right-parenthesis equals 0 unless j less-than-or-equal-to i less-than-or-equal-to j plus n period

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

Corollary 5.2

Let upper X be a scheme and upper E a nonzero object of normal upper D left-parenthesis upper X right-parenthesis . Then

c o d i m left-parenthesis upper S u p p upper E right-parenthesis less-than-or-equal-to normal h times normal o times normal m times normal d times normal i times normal m upper E period

Corollary 5.3

Let upper X be a scheme, and fix a point x element-of upper X of codimension n . Suppose that there is an object upper E of normal upper D left-parenthesis upper X right-parenthesis such that for all points z element-of upper X , and any integer i ,

upper H o m Subscript normal upper D left-parenthesis upper X right-parenthesis Superscript i Baseline left-parenthesis upper E comma script upper O Subscript z Baseline right-parenthesis equals 0 unless z equals x and 0 less-than-or-equal-to i less-than-or-equal-to n period

Suppose also that upper H Superscript 0 Baseline left-parenthesis upper E right-parenthesis approximately-equals script upper O Subscript x . Then upper X is nonsingular at x and upper E approximately-equals script upper O Subscript x .

6. The projective case

The aim of this section is to prove Theorem 1.1 under the additional assumption that upper M 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 pi Subscript upper Y Baseline colon upper Y times upper M right-arrow upper Y and pi Subscript upper M Baseline colon upper Y times upper M right-arrow upper M denote the projections. The functor normal upper Phi may be rewritten

normal upper Phi left-parenthesis minus right-parenthesis approximately-equals bold upper R pi Subscript upper M Sub Subscript asterisk Subscript Baseline left-parenthesis script upper O Subscript script upper Z Baseline circled-times pi Subscript upper Y Superscript asterisk Baseline left-parenthesis minus circled-times rho 0 right-parenthesis right-parenthesis period

Note that script upper O Subscript script upper Z has finite homological dimension, because script upper Z is flat over upper Y and upper M is nonsingular. Hence the derived dual script upper O Subscript script upper Z Superscript logical-or Baseline equals bold upper R script upper H times o m Subscript script upper O Sub Subscript upper Y times upper M Baseline left-parenthesis script upper O Subscript script upper Z Baseline comma script upper O Subscript upper Y times upper M Baseline right-parenthesis also has finite homological dimension and we may define another functor normal upper Psi colon normal upper D Superscript upper G Baseline left-parenthesis upper M right-parenthesis right-arrow normal upper D left-parenthesis upper Y right-parenthesis , by the formula

normal upper Psi left-parenthesis minus right-parenthesis equals left-bracket bold upper R pi Subscript upper Y Sub Subscript asterisk Subscript Baseline left-parenthesis script upper P circled-times Overscript bold upper L Endscripts pi Subscript upper M Superscript asterisk Baseline left-parenthesis minus right-parenthesis right-parenthesis right-bracket Superscript upper G Baseline comma

where script upper P equals script upper O Subscript script upper Z Superscript logical-or Baseline circled-times pi Subscript upper M Superscript asterisk Baseline left-parenthesis omega Subscript upper M Baseline right-parenthesis left-bracket n right-bracket .

Now normal upper Psi is left adjoint to normal upper Phi because of the three standard adjunctions described in Section 3.1. The functor pi Subscript upper M Superscript asterisk is the left adjoint to bold upper R pi Subscript upper M comma Sub Superscript asterisk Subscript . The functor minus circled-times script upper O Subscript script upper Z has the (left and right) adjoint minus circled-times script upper O Subscript script upper Z Superscript logical-or . Finally the functor pi Subscript upper Y Superscript factorial has the left adjoint bold upper R pi Subscript upper Y Sub Subscript asterisk and

pi Subscript upper Y Superscript factorial Baseline left-parenthesis minus right-parenthesis equals pi Subscript upper Y Superscript asterisk Baseline left-parenthesis minus right-parenthesis circled-times pi Subscript upper M Superscript asterisk Baseline left-parenthesis omega Subscript upper M Baseline right-parenthesis left-bracket n right-bracket period

Step 2. The composite functor normal upper Psi ring normal upper Phi is given by

bold upper R pi 2 Subscript asterisk Baseline left-parenthesis script upper Q circled-times Overscript bold upper L Endscripts pi 1 Superscript asterisk Baseline left-parenthesis minus right-parenthesis right-parenthesis comma

where pi 1 and pi 2 are the projections of upper Y times upper Y onto its factors, and script upper Q is some object of normal upper D left-parenthesis upper Y times upper Y right-parenthesis . This is just composition of correspondences (see Mukai Reference17, Proposition 1.3).

If i Subscript y Baseline colon StartSet y EndSet times upper Y right-arrow with hook upper Y times upper Y is the closed embedding, then bold upper L i Subscript y Superscript asterisk Baseline left-parenthesis script upper Q right-parenthesis equals normal upper Psi normal upper Phi script upper O Subscript y . For any pair of points y 1 comma y 2 , one has script upper O Subscript left-parenthesis y 1 comma y 2 right-parenthesis Baseline equals i Subscript y 1 comma asterisk Baseline script upper O Subscript y 2 so that

StartLayout 1st Row with Label left-parenthesis 5 right-parenthesis EndLabel upper H o m Subscript normal upper D left-parenthesis upper Y times upper Y right-parenthesis Superscript i Baseline left-parenthesis script upper Q comma script upper O Subscript left-parenthesis y 1 comma y 2 right-parenthesis Baseline right-parenthesis equals upper H o m Subscript normal upper D left-parenthesis upper Y right-parenthesis Superscript i Baseline left-parenthesis normal upper Psi normal upper Phi script upper O Subscript y 1 Baseline comma script upper O Subscript y 2 Baseline right-parenthesis equals upper G hyphen normal upper E normal x normal t Subscript upper M Superscript i Baseline left-parenthesis script upper O Subscript upper Z Sub Subscript y 1 Subscript Baseline comma script upper O Subscript upper Z Sub Subscript y 2 Subscript Baseline right-parenthesis comma EndLayout

using the adjunctions for left-parenthesis bold upper L i Superscript asterisk Baseline comma i Subscript asterisk Baseline right-parenthesis and left-parenthesis normal upper Psi comma normal upper Phi right-parenthesis . Our first objective is to show that script upper Q is supported on the diagonal normal upper Delta subset-of upper Y times upper Y , or equivalently that the groups in (Equation5) vanish unless y 1 equals y 2 . When n equals 3 this plays the rôle of assumption (4.8) of Ito and Nakajima Reference12.

Step 3. Let upper Z 1 comma upper Z 2 subset-of upper M be upper G -clusters. Then

upper G hyphen normal upper H normal o normal m Subscript upper M Baseline left-parenthesis script upper O Subscript upper Z 1 Baseline comma script upper O Subscript upper Z 2 Baseline right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column double-struck upper C 2nd Column if upper Z 1 equals upper Z 2 comma 2nd Row 1st Column 0 2nd Column otherwise period EndLayout

To see this note that script upper O Subscript upper Z is generated as an script upper O Subscript upper M -module by any nonzero constant section. But, since the global sections normal upper Gamma left-parenthesis script upper O Subscript upper Z Baseline right-parenthesis form the regular representation of upper G , the constant sections are precisely the upper G -invariant sections. Hence any equivariant morphism maps a generator to a scalar multiple of a generator and so is determined by that scalar.

Let y 1 and y 2 be distinct points of upper Y . Serre duality, together with our assumption that omega Subscript upper M is locally trivial as a upper G -sheaf, implies that

upper G hyphen normal upper E normal x normal t Subscript upper M Superscript n Baseline left-parenthesis script upper O Subscript upper Z Sub Subscript y 1 Subscript Baseline comma script upper O Subscript upper Z Sub Subscript y 2 Subscript Baseline right-parenthesis equals upper G hyphen normal upper H normal o normal m Subscript upper M Baseline left-parenthesis script upper O Subscript upper Z Sub Subscript y 2 Subscript Baseline comma script upper O Subscript upper Z Sub Subscript y 1 Subscript Baseline right-parenthesis equals 0 comma

so that

upper G hyphen normal upper E normal x normal t Subscript upper M Superscript p Baseline left-parenthesis script upper O Subscript upper Z Sub Subscript y 1 Subscript Baseline comma script upper O Subscript upper Z Sub Subscript y 2 Subscript Baseline right-parenthesis equals 0 unless 1 less-than-or-equal-to p less-than-or-equal-to n minus 1 period

Hence script upper Q restricted to left-parenthesis upper Y times upper Y right-parenthesis minus normal upper Delta has homological dimension n minus 2 .

Step 4. Now we apply the intersection theorem. If y 1 and y 2 are points of upper Y such that tau left-parenthesis y 1 right-parenthesis not-equals tau left-parenthesis y 2 right-parenthesis , then the corresponding clusters upper Z Subscript y 1 and upper Z Subscript y 2 are disjoint, so that the groups in (Equation5) vanish. Thus the support of script upper Q vertical-bar Subscript left-parenthesis upper Y times upper Y right-parenthesis minus normal upper Delta Baseline is contained in the subscheme upper Y times Subscript upper X Baseline upper Y . By assumption this has codimension n minus 2 so Corollary 5.2 implies that

script upper Q vertical-bar Subscript left-parenthesis upper Y times upper Y right-parenthesis minus normal upper Delta Baseline approximately-equals 0 comma

that is, script upper Q is supported on the diagonal.

Step 5. Fix a point y element-of upper Y , and put upper E equals normal upper Psi normal upper Phi left-parenthesis script upper O Subscript y Baseline right-parenthesis . We proved above that upper E is supported at the point y . We claim that upper H Superscript 0 Baseline left-parenthesis upper E right-parenthesis equals script upper O Subscript y . Note that Corollary 5.3 then implies that upper Y is nonsingular at y and upper E approximately-equals script upper O Subscript y .

To prove the claim, note that there is a canonical map upper E right-arrow script upper O Subscript y , so we obtain a triangle

upper C right-arrow upper E right-arrow script upper O Subscript y Baseline right-arrow upper C left-bracket 1 right-bracket

for some object upper C of normal upper D left-parenthesis upper Y right-parenthesis . Using the adjoint pair left-parenthesis normal upper Psi comma normal upper Phi right-parenthesis , this gives a long exact sequence

StartLayout 1st Row ellipsis right-arrow upper H o m Subscript normal upper D left-parenthesis upper Y right-parenthesis Superscript 0 Baseline left-parenthesis script upper O Subscript y Baseline comma script upper O Subscript y Baseline right-parenthesis right-arrow upper H o m Subscript normal upper D Sub Superscript upper G Subscript left-parenthesis upper M right-parenthesis Superscript 0 Baseline left-parenthesis normal upper Phi script upper O Subscript y Baseline comma normal upper Phi script upper O Subscript y Baseline right-parenthesis right-arrow upper H o m Subscript normal upper D left-parenthesis upper Y right-parenthesis Superscript 0 Baseline left-parenthesis upper C comma script upper O Subscript y Baseline right-parenthesis 2nd Row upper H o m Subscript normal upper D left-parenthesis upper Y right-parenthesis Superscript 1 Baseline left-parenthesis script upper O Subscript y Baseline comma script upper O Subscript y Baseline right-parenthesis right-arrow upper H o m Subscript normal upper D Sub Superscript upper G Subscript left-parenthesis upper M right-parenthesis Superscript 1 Baseline left-parenthesis normal upper Phi script upper O Subscript y Baseline comma normal upper Phi script upper O Subscript y Baseline right-parenthesis right-arrow Overscript epsilon Endscripts ellipsis period EndLayout

The homomorphism epsilon is the Kodaira–Spencer map for the family of clusters StartSet script upper O Subscript upper Z Sub Subscript y Subscript Baseline colon y element-of upper Y EndSet (Bridgeland Reference6, Lemma 4.4). This is injective because upper G hyphen normal upper H normal i normal l normal b upper M is a fine moduli space for upper G -clusters on upper M . It follows that

upper H o m Subscript normal upper D left-parenthesis upper Y right-parenthesis Superscript i Baseline left-parenthesis upper C comma script upper O Subscript y Baseline right-parenthesis equals 0 for all i less-than-or-equal-to 0 period

An easy spectral sequence argument (see Reference6, Example 2.2) shows that upper H Superscript 0 Baseline left-parenthesis upper C right-parenthesis equals 0 . Taking cohomology sheaves of the above triangle gives upper H Superscript 0 Baseline left-parenthesis upper E right-parenthesis equals script upper O Subscript y , which proves the claim.

Step 6. We have now proved that upper Y is nonsingular, and that for any pair of points y 1 comma y 2 element-of upper Y , the homomorphisms

normal upper Phi colon upper E x t Subscript upper Y Superscript i Baseline left-parenthesis script upper O Subscript y 1 Baseline comma script upper O Subscript y 2 Baseline right-parenthesis right-arrow upper G hyphen normal upper E normal x normal t Subscript upper M Superscript i Baseline left-parenthesis script upper O Subscript upper Z Sub Subscript y 1 Subscript Baseline comma script upper O Subscript upper Z Sub Subscript y 2 Subscript Baseline right-parenthesis

are isomorphisms. By assumption, the action of upper G on upper M is such that omega Subscript upper M is trivial as a upper G -sheaf on an open neighbourhood of each orbit upper G dot x subset-of upper M . This implies that

script upper O Subscript upper Z Sub Subscript y Baseline circled-times omega Subscript upper M Baseline approximately-equals script upper O Subscript upper Z Sub Subscript y

in upper C o h Superscript upper G Baseline left-parenthesis upper M right-parenthesis , for each y element-of upper Y . Applying Theorem 2.4 shows that normal upper Phi is an equivalence of categories.

Step 7. It remains to show that tau colon upper Y right-arrow upper X is crepant. Take a point x element-of upper X equals upper M slash upper G . The equivalence normal upper Phi restricts to give an equivalence between the full subcategories normal upper D Subscript x Baseline left-parenthesis upper Y right-parenthesis subset-of normal upper D left-parenthesis upper Y right-parenthesis and normal upper D Subscript x Superscript upper G Baseline left-parenthesis upper M right-parenthesis subset-of normal upper D Superscript upper G Baseline left-parenthesis upper M right-parenthesis consisting of objects supported on the fibre tau Superscript negative 1 Baseline left-parenthesis x right-parenthesis and the orbit pi Superscript negative 1 Baseline left-parenthesis x right-parenthesis respectively.

The category normal upper D Subscript x Superscript upper G Baseline left-parenthesis upper M right-parenthesis has trivial Serre functor because omega Subscript upper M is trivial as a upper G -sheaf on a neighbourhood of pi Superscript negative 1 Baseline left-parenthesis x right-parenthesis . Thus normal upper D Subscript x Baseline left-parenthesis upper Y right-parenthesis also has trivial Serre functor and Lemma 3.1 gives the result.

This completes the proof of Theorem 1.1 in the case that upper Y 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 upper M 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 upper M overbar of upper M and define adjoint functors as before. Then we restrict normal upper Phi to a functor

normal upper Phi Subscript normal c Baseline colon normal upper D Subscript normal c Baseline left-parenthesis upper Y right-parenthesis long right-arrow normal upper D Subscript normal c Superscript upper G Baseline left-parenthesis upper M right-parenthesis period

The argument of the last section carries through to show that upper Y is nonsingular and crepant and that normal upper Phi Subscript normal c is an equivalence. It remains for us to show that normal upper Phi colon normal upper D left-parenthesis upper Y right-parenthesis right-arrow normal upper D Superscript upper G Baseline left-parenthesis upper M right-parenthesis is also an equivalence.

Step 8. The functor normal upper Phi has a right adjoint

normal upper Upsilon left-parenthesis minus right-parenthesis equals left-bracket p Subscript asterisk Baseline ring q Superscript factorial Baseline left-parenthesis minus right-parenthesis right-bracket Superscript upper G Baseline equals left-bracket bold upper R pi Subscript upper Y Subscript asterisk Baseline left-parenthesis omega Subscript upper Z slash upper M Baseline circled-times Overscript bold upper L Endscripts pi Subscript upper M Superscript asterisk Baseline left-parenthesis minus right-parenthesis right-parenthesis right-bracket Superscript upper G Baseline period

As before, the composition normal upper Upsilon ring normal upper Phi is given by

bold upper R pi 2 Subscript asterisk Baseline left-parenthesis script upper Q circled-times Overscript bold upper L Endscripts pi 1 Superscript asterisk Baseline left-parenthesis minus right-parenthesis right-parenthesis comma

where pi 1 and pi 2 are the projections of upper Y times upper Y onto its factors, and script upper Q is some object of normal upper D left-parenthesis upper Y times upper Y right-parenthesis .

Since normal upper Phi Subscript normal c is an equivalence, normal upper Upsilon normal upper Phi script upper O Subscript y Baseline equals script upper O Subscript y for any point y element-of upper Y , and it follows that script upper Q is actually the pushforward of a line bundle upper L on upper Y to the diagonal in upper Y times upper Y . The functor normal upper Upsilon ring normal upper Phi is then just twisting by upper L , and to show that normal upper Phi is fully faithful we must show that upper L is trivial.

There is a morphism of functors epsilon colon i d right-arrow normal upper Upsilon ring normal upper Phi , which for any point y element-of upper Y gives a commutative diagram

StartLayout 1st Row 1st Column script upper O Subscript upper Y 2nd Column right-arrow Overscript epsilon left-parenthesis script upper O Subscript upper Y Baseline right-parenthesis Endscripts 3rd Column upper L 2nd Row 1st Column f down-arrow 2nd Column Blank 3rd Column upper L circled-times f 4th Column Blank 3rd Row 1st Column script upper O Subscript y 2nd Column right-arrow Overscript epsilon left-parenthesis script upper O Subscript y Baseline right-parenthesis Endscripts 3rd Column script upper O Subscript y EndLayout

where f is nonzero. Since epsilon is an isomorphism on the subcategory normal upper D Subscript normal c Baseline left-parenthesis upper Y right-parenthesis , the maps epsilon left-parenthesis script upper O Subscript y Baseline right-parenthesis are all isomorphisms, so the section epsilon left-parenthesis script upper O Subscript upper Y Baseline right-parenthesis is an isomorphism.

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

normal upper Upsilon left-parenthesis upper E right-parenthesis approximately-equals 0 long right double arrow upper E approximately-equals 0 for any object upper E of normal upper D Superscript upper G Baseline left-parenthesis upper M right-parenthesis period

Suppose normal upper Upsilon left-parenthesis upper E right-parenthesis approximately-equals 0 . Using the adjunction for left-parenthesis normal upper Phi comma normal upper Upsilon right-parenthesis ,

upper H o m Subscript normal upper D Sub Superscript upper G Subscript left-parenthesis upper M right-parenthesis Superscript i Baseline left-parenthesis upper B comma upper E right-parenthesis equals 0 for all i comma

whenever upper B approximately-equals normal upper Phi left-parenthesis upper A right-parenthesis for some object upper A element-of normal upper D left-parenthesis upper Y right-parenthesis . In particular, this holds for any upper B with compact support.

If upper E is nonzero, let upper D equals upper G dot x be an orbit of upper G contained in the support of upper E . Let i colon upper D right-arrow with hook upper M denote the inclusion, a projective equivariant morphism of schemes. Then the adjunction morphism i Subscript asterisk Baseline i Superscript factorial Baseline left-parenthesis upper E right-parenthesis right-arrow upper E 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 upper Y to be an irreducible component of upper G hyphen normal upper H normal i normal l normal b upper M . Note that when upper Y is nonsingular and normal upper Phi is an equivalence, upper Y is actually a connected component. This is simply because for any point y element-of upper Y , the bijection

normal upper Phi colon upper E x t Subscript upper Y Superscript 1 Baseline left-parenthesis script upper O Subscript y Baseline comma script upper O Subscript y Baseline right-parenthesis right-arrow upper G hyphen normal upper E normal x normal t Subscript upper M Superscript 1 Baseline left-parenthesis script upper O Subscript upper Z Sub Subscript y Subscript Baseline comma script upper O Subscript upper Z Sub Subscript y Subscript Baseline right-parenthesis

identifies the tangent space of upper Y at y with the tangent space of upper G hyphen normal upper H normal i normal l normal b upper M at y . In this section we wish to go further and prove that when upper M has dimension 3, upper G hyphen normal upper H normal i normal l normal b upper M is in fact connected.

Proof of Nakamura’s conjecture.

Suppose for contradiction that there exists a upper G -cluster upper Z subset-of upper M not contained among the StartSet upper Z Subscript y Baseline colon y element-of upper Y EndSet . Since normal upper Phi is an equivalence we can take an object upper E element-of normal upper D Subscript normal c Baseline left-parenthesis upper Y right-parenthesis such that normal upper Phi left-parenthesis upper E right-parenthesis equals script upper O Subscript upper Z . The argument of Section Equation6, Step 3, shows that for any point y element-of upper Y

upper H o m Subscript normal upper D left-parenthesis upper Y right-parenthesis Superscript i Baseline left-parenthesis upper E comma script upper O Subscript y Baseline right-parenthesis equals upper G hyphen normal upper E normal x normal t Subscript upper M Superscript i Baseline left-parenthesis script upper O Subscript upper Z Baseline comma script upper O Subscript upper Z Sub Subscript y Subscript Baseline right-parenthesis equals 0 unless 1 less-than-or-equal-to i less-than-or-equal-to 2 period

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

StartLayout 1st Row with Label left-parenthesis 6 right-parenthesis EndLabel 0 right-arrow upper L 2 right-arrow Overscript f Endscripts upper L 1 right-arrow 0 period EndLayout

But script upper O Subscript upper Z is supported on some upper G -orbit in upper M , so upper E is supported on a fibre of upper Y , and hence in codimension 1 . It follows that the complex (Equation6) is exact on the left, so upper E approximately-equals c o k e r f left-bracket 1 right-bracket . In particular left-bracket upper E right-bracket equals minus left-bracket c o k e r f right-bracket in the Grothendieck group upper K Subscript normal c Baseline left-parenthesis upper Y right-parenthesis of normal upper D Subscript normal c Baseline left-parenthesis upper Y right-parenthesis .

Let y be a point of the fibre that is the support of upper E . By Lemma 8.1 below, left-bracket script upper O Subscript upper Z Sub Subscript y Subscript Baseline right-bracket equals left-bracket script upper O Subscript upper Z Baseline right-bracket in upper K Subscript normal c Superscript upper G Baseline left-parenthesis upper M right-parenthesis , so that left-bracket script upper O Subscript y Baseline right-bracket equals left-bracket upper E right-bracket in upper K Subscript normal c Baseline left-parenthesis upper Y right-parenthesis , since the equivalence normal upper Phi gives an isomorphism of Grothendieck groups.

Let upper Y overbar be a nonsingular projective variety with an open inclusion i colon upper Y right-arrow with hook upper Y overbar . The functor i Subscript asterisk Baseline colon normal upper D Subscript normal c Baseline left-parenthesis upper Y right-parenthesis right-arrow normal upper D left-parenthesis upper Y overbar right-parenthesis induces a map on K groups, so left-bracket c o k e r f right-bracket equals minus left-bracket script upper O Subscript y Baseline right-bracket in upper K Subscript normal c Baseline left-parenthesis upper Y overbar right-parenthesis . But this contradicts Riemann–Roch, because if upper L is a sufficiently ample line bundle on upper Y overbar , then chi left-parenthesis c o k e r f circled-times upper L right-parenthesis and chi left-parenthesis script upper O Subscript y Baseline circled-times upper L right-parenthesis are both positive.

Lemma 8.1

If upper Z 1 and upper Z 2 are two upper G -clusters on upper M supported on the same orbit, then the corresponding elements left-bracket script upper O Subscript upper Z 1 Baseline right-bracket and left-bracket script upper O Subscript upper Z 2 Baseline right-bracket in the Grothendieck group upper K Subscript normal c Superscript upper G Baseline left-parenthesis upper M right-parenthesis of normal upper D Subscript normal c Superscript upper G Baseline left-parenthesis upper M right-parenthesis are equal.

Proof.

We need to show that, as upper G -sheaves, script upper O Subscript upper Z 1 and script upper O Subscript upper Z 2 have composition series with the same simple factors. Suppose that they are both supported on the upper G -orbit upper D equals upper G dot x subset-of upper M and let upper H be the stabiliser subgroup of x in upper G . The restriction functor is an equivalence of categories from finite length upper G -sheaves supported on upper D to finite length upper H -sheaves supported at x . The reverse equivalence is the induction functor left-parenthesis minus circled-times double-struck upper C left-bracket upper G right-bracket right-parenthesis . Since the restriction of a upper G -cluster supported on upper D is an upper H -cluster supported at x , it is sufficient to prove the result for upper H -clusters supported at x .

If StartSet rho 0 comma period period period comma rho Subscript k Baseline EndSet are the irreducible representations of upper H , then we claim that the simple upper H -sheaves supported at x are precisely

StartSet upper S Subscript i Baseline equals script upper O Subscript x Baseline circled-times rho Subscript i Baseline colon 0 less-than-or-equal-to i less-than-or-equal-to k EndSet period

These sheaves are certainly simple, since they are simple as double-struck upper C left-bracket upper H right-bracket -modules. On the other hand, any upper H -sheaf upper E supported at x has a nonzero ordinary sheaf morphism script upper O Subscript x Baseline right-arrow upper E . By Lemma 4.1 there must be a n