The McKay correspondence as an equivalence of derived categories
By Tom Bridgeland, Alastair King, and Miles Reid
To Andrei Tyurin on his 60th birthday
Abstract
Let $G$ be a finite group of automorphisms of a nonsingular three-dimensional complex variety $M$, whose canonical bundle $\omega _M$ is locally trivial as a $G$-sheaf. We prove that the Hilbert scheme $Y=\operatorname {\text{$G$}-Hilb}{M}$ parametrising $G$-clusters in $M$ is a crepant resolution of $X=M/G$ and that there is a derived equivalence (Fourier–Mukai transform) between coherent sheaves on $Y$ and coherent $G$-sheaves on $M$. This identifies the K theory of $Y$ with the equivariant K theory of $M$, 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 $G\subset \operatorname {SL}(2,\mathbb{C})$ to the cohomology of the well-known minimal resolution of the Kleinian singularity $\mathbb{C}^2/G$. Gonzalez-Sprinberg and Verdier Reference 10 interpreted the McKay correspondence as an isomorphism on K theory, observing that the representation ring of $G$ is equal to the $G$-equivariant K theory of $\mathbb{C}^2$. 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 $G$.
It is natural to ask what happens when $\mathbb{C}^2$ is replaced by an arbitrary nonsingular quasiprojective complex variety $M$ of dimension $n$ and $G$ by a finite group of automorphisms of $M$, with the property that the stabiliser subgroup of any point $x\in M$ acts on the tangent space $T_xM$ as a subgroup of $\operatorname {SL}(T_xM)$. Thus the canonical bundle $\omega _M$ is locally trivial as a $G$-sheaf, in the sense that every point of $M$ has a $G$-invariant open neighbourhood on which there is a nonvanishing $G$-invariant$n$-form. This implies that the quotient variety $X=M/G$ has only Gorenstein singularities.
A natural generalisation of the McKay correspondence would then be an isomorphism between the $G$-equivariant K theory of $M$ and the ordinary K theory of a crepant resolution $Y$ of $X$, that is, a resolution of singularities $\tau \colon Y\to X$ such that $\tau ^*(\omega _X)=\omega _Y$. 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 $n=3$, through a case by case analysis of the local linear actions by Ito, Markushevich and Roan (see Roan Reference 21 and references given there). In dimension $\ge 4$, 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 Reference 20, Conjecture 4.1). Furthermore, if the resolution is constructed as a moduli space of $G$-equivariant objects on $M$, 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 Reference 15 in proving the derived category version of the classical two-dimensional McKay correspondence.
The new and remarkable feature is that, by using the derived category and Fourier-Mukai transforms and, in particular, techniques developed in Reference 6 and Reference 7, the process of proving the equivalence of derived categories—when it works—also yields a proof that the moduli space is a crepant resolution. More specifically, we will give a sufficient condition for a certain natural moduli space, namely Nakamura’s $G$-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 $X=M/G$ 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 $G$-sheaves on $M$, which may provide different crepant resolutions to the one considered here.
The $G$-Hilbert scheme $\operatorname {\text{$G$}-Hilb}{M}$ was introduced by Nakamura as a good candidate for a crepant resolution of $M/G$. It parametrises $G$-clusters or ‘scheme theoretic $G$-orbits’ on $M$: recall that a cluster$Z\subset M$ is a zero-dimensional subscheme, and a $G$-cluster is a $G$-invariant cluster whose global sections $\Gamma (\mathcal{O}_Z)$ are isomorphic to the regular representation $\mathbb{C}[G]$ of $G$. Clearly, a $G$-cluster has length $|G|$ and a free $G$-orbit is a $G$-cluster. There is a Hilbert–Chow morphism
which, on closed points, sends a $G$-cluster to the orbit supporting it. Note that $\tau$ is a projective morphism, is onto and is birational on one component.
When $M=\mathbb{C}^3$ and $G\subset \operatorname {SL}(3,\mathbb{C})$ is Abelian, Nakamura Reference 18 proved that $\operatorname {\text{$G$}-Hilb}{M}$ is irreducible and is a crepant resolution of $X$ (compare also Reid Reference 20 and Craw and Reid Reference 8). He conjectured that the same result holds for an arbitrary finite subgroup $G\subset \operatorname {SL}(3,\mathbb{C})$. Ito and Nakajima Reference 12 observed that the construction of Gonzalez-Sprinberg and Verdier Reference 10 is the $M=\mathbb{C}^2$ case of a natural correspondence between the equivariant K theory of $M$ and the ordinary K theory of $\operatorname {\text{$G$}-Hilb}{M}$. They proved that this correspondence is an isomorphism when $M=\mathbb{C}^3$ and $G\subset \operatorname {SL}(3,\mathbb{C})$ 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 $\operatorname {\mathcal{Q}}$ of $\operatorname {D}(Y\times Y)$ 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 $\operatorname {SL}(3,\mathbb{C})$.
Since it is not known whether $\operatorname {\text{$G$}-Hilb}{M}$ is irreducible or even connected in general, we actually take as our initial candidate for a resolution $Y$ the irreducible component of $\operatorname {\text{$G$}-Hilb}{M}$ containing the free $G$-orbits, that is, the component mapping birationally to $X$. The aim is to show that $Y$ is a crepant resolution, and to construct an equivalence between the derived categories $\operatorname {D}(Y)$ of coherent sheaves on $Y$ and $\operatorname {D}^G(M)$ of coherent $G$-sheaves on $M$. A more detailed analysis of the equivalence shows that $Y=\operatorname {\text{$G$}-Hilb}{M}$ when $M$ has dimension 3.
We now describe the correspondence and our results in more detail. Let $M$ be a nonsingular quasiprojective complex variety of dimension $n$ and let $G\subset \operatorname {Aut}(M)$ be a finite group of automorphisms of $M$ such that $\omega _M$ is locally trivial as a $G$-sheaf. Put $X=M/G$ and let $Y\subset \operatorname {\text{$G$}-Hilb}{M}$ be the irreducible component containing the free orbits, as described above. Write $\mathcal{Z}$ for the universal closed subscheme $\mathcal{Z}\subset Y\times M$ and $p$ and $q$ for its projections to $Y$ and $M$. There is a commutative diagram of schemes
$$\begin{equation*} \vcenter{\img[][84pt][81pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{xy} \xymatrix{ {} &\mathcal Z \ar[dl]_{\textstyle{p}} \ar[dr]^{\textstyle{q}} & {}\\ Y \ar[dr]_{\textstyle{\tau}} & {} & M \ar[dl]^{\textstyle{\pi}}\\ {} & X & } \end{xy}}]{Images/imgfd0e4ffb971170b7f42b8dabaf96022f.svg}} \end{equation*}$$
in which $q$ and $\tau$ are birational, $p$ and $\pi$ are finite, and $p$ is flat. Let $G$ act trivially on $Y$ and $X$, so that all morphisms in the diagram are equivariant.
where a sheaf $E$ on $Y$ is viewed as a $G$-sheaf by giving it the trivial action. Note that $p^*$ is already exact, so we do not need to write $\mathbf{L}p^*$. Our main result is the following.
Theorem 1.1.
Suppose that the fibre product
$$\begin{equation*} Y\times _X Y= \Bigl \{(y_1, y_2)\in Y\times Y \Bigm | \tau (y_1)=\tau (y_2)\Bigr \} \subset Y\times Y \end{equation*}$$
has dimension $\le n+1$. Then $Y$ is a crepant resolution of $X$ and $\Phi$ is an equivalence of categories.
When $n\le 3$ the condition of the theorem always holds because the exceptional locus of $Y\to X$ has dimension $\le 2$. In this case we can also show that $\operatorname {\text{$G$}-Hilb}{M}$ is irreducible, so we obtain
Theorem 1.2.
Suppose $n\le 3$. Then $\operatorname {\text{$G$}-Hilb}{M}$ is irreducible and is a crepant resolution of $X$, and $\Phi$ is an equivalence of categories.
The condition of Theorem 1.1 also holds whenever $G$ preserves a complex symplectic form on $M$ and $Y$ is a crepant resolution of $X$, because such a resolution is symplectic and hence semi-small (see Verbitsky Reference 24, Theorem 2.8 and compare Kaledin Reference 14).
Corollary 1.3.
Suppose $M$ is a complex symplectic variety and $G$ acts by symplectic automorphisms. Assume that $Y$ is a crepant resolution of $X$. Then $\Phi$ is an equivalence of categories.
Note that the condition of Theorem 1.1 certainly fails in dimension $\ge 4$ whenever $Y\to X$ has an exceptional divisor over a point. This is to be expected since there are many examples of finite subgroups $G\subset \operatorname {SL}(4,\mathbb{C})$ for which the quotient singularity $\mathbb{C}^4/G$ has no crepant resolution and also examples where, although crepant resolutions do exist, $\operatorname {\text{$G$}-Hilb}{\bigl (\mathbb{C}^4\bigr )}$ is not one.
Conventions
We work throughout in the category of schemes over $\mathbb{C}$. 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 $\mathcal{A}$ equipped with a shift automorphism$T_{\mathcal{A}}\colon \mathcal{A}\to \mathcal{A}\colon a\mapsto a[1]$ and a collection of distinguished triangles
A triangulated category $\mathcal{A}$ is trivial if every object is a zero object.
The principal example of a triangulated category is the derived category $\operatorname {D}(A)$ of an Abelian category $A$. An object of $\operatorname {D}(A)$ is a bounded complex of objects of $A$ 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 $a_1,a_2\in A$, one has $\operatorname {Hom}^i_{\operatorname {D}(A)}(a_1,a_2)=\operatorname {Ext}^i_A(a_1,a_2)$.
A functor $F\colon \mathcal{A}\to \mathcal{B}$ between triangulated categories is exact if it commutes with the shift automorphisms and takes distinguished triangles of $\mathcal{A}$ to distinguished triangles of $\mathcal{B}$. For example, derived functors between derived categories are exact.
2.2. Adjoint functors
Let $F\colon \mathcal{A}\to \mathcal{B}$ and $G\colon \mathcal{B}\to \mathcal{A}$ be functors. An adjunction for $(G,F)$ is a bifunctorial isomorphism
In this case, we say that $G$ is left adjoint to $F$ or that $F$ is right adjoint to $G$. 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 $\varepsilon \colon G\circ F\to \operatorname {id}_\mathcal{A}$ and $\eta \colon \operatorname {id}_\mathcal{B}\to F\circ G$ that come from applying the adjunction to $1_{Fa}$ and $1_{Gb}$ respectively (see Mac Lane Reference 16, IV.1 for more details).
The basic adjunctions we use in this paper are described in Section 3.1 below.
2.3. Fully faithful functors and equivalences
A functor $F\colon \mathcal{A}\to \mathcal{B}$ is fully faithful if for any pair of objects $a_1$,$a_2$ of $\mathcal{A}$, the map
is an isomorphism. One should think of $F$ as an ‘injective’ functor. This is clearer when $F$ has a left adjoint $G\colon \mathcal{B}\to \mathcal{A}$ (or a right adjoint $H\colon \mathcal{B}\to \mathcal{A}$), in which case $F$ is fully faithful if and only if the natural transformation $G\circ F\to \operatorname {id}_{\mathcal{A}}$ (or $\operatorname {id}_{\mathcal{A}}\to H\circ F$) is an isomorphism.
A functor $F$ is an equivalence if there is an ‘inverse’ functor $G\colon \mathcal{B}\to \mathcal{A}$ such that $G\circ F\cong \operatorname {id}_{\mathcal{A}}$ and $F\circ G\cong \operatorname {id}_{\mathcal{B}}$. In this case $G$ is both a left and right adjoint to $F$ (see Mac Lane Reference 16, IV.4). In practice, we show that $F$ 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 $\mathcal{A}$ and $\mathcal{B}$ be triangulated categories and $F\colon \mathcal{A}\to \mathcal{B}$ a fully faithful exact functor with a right adjoint $H\colon \mathcal{B}\to \mathcal{A}$. Then $F$ is an equivalence if and only if $Hc\cong 0$ implies $c\cong 0$ for all objects $c\in \mathcal{B}$.
Proof.
By assumption $\eta \colon \operatorname {id}_\mathcal{A}\to H\circ F$ is an isomorphism, so $F$ is an equivalence if and only if $\varepsilon \colon F\circ H\to \operatorname {id}_\mathcal{B}$ is an isomorphism. Thus the ‘only if’ part of the lemma is immediate, since $c\cong FHc$.
For the ‘if’ part, take any object $b\in \mathcal{B}$ and embed the natural adjunction map $\varepsilon _b$ in a triangle
If we apply $H$ to this triangle, then $H(\varepsilon _b)$ is an isomorphism, because $\eta _{Hb}$ is an isomorphism and $H(\varepsilon _b)\circ \eta _{Hb}=1_{Hb}$ (Reference 16, IV.1, Theorem 1). Hence $Hc\cong 0$ and so $c\cong 0$ by hypothesis. Thus $\varepsilon _b$ is an isomorphism, as required.
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One may understand this lemma in a broader context as follows. The triangle (Equation 1) shows that, when $F$ is fully faithful with right adjoint $H$, there is a ‘semi-orthogonal’ decomposition $\mathcal{B}=(\operatorname {Im}F,\operatorname {Ker}H)$, where
Since $F$ is fully faithful, the fact that $b\cong Fa$ for some object $a\in \mathcal{A}$ necessarily means that $b\cong FHb$, so only zero objects are in both subcategories. The semi-orthogonality condition also requires that $\operatorname {Hom}_{\mathcal{B}}(b,c)=0$ for all $b\in \operatorname {Im}F$ and $c\in \operatorname {Ker}H$, which is immediate from the adjunction. The lemma then has the very reasonable interpretation that if $\operatorname {Ker}H$ is trivial, then $\operatorname {Im}F=\mathcal{B}$ and $F$ is an equivalence. Note that if $G$ is a left adjoint for $F$, then there is a similar semi-orthogonal decomposition on the other side $\mathcal{B}=(\operatorname {Ker}G,\operatorname {Im}F)$ and a corresponding version of the lemma. For more details on semi-orthogonal decompositions see Bondal Reference 4.
2.4. Spanning classes and orthogonal decomposition
A spanning class for a triangulated category $\mathcal{A}$ is a subclass $\Omega$ of the objects of $\mathcal{A}$ such that for any object $a\in \mathcal{A}$
The set of skyscraper sheaves $\{\mathcal{O}_x\colon x\in X\}$ on a nonsingular projective variety $X$ is a spanning class for $\operatorname {D}(X)$.
A triangulated category $\mathcal{A}$ is decomposable as an orthogonal direct sum of two full subcategories $\mathcal{A}_1$ and $\mathcal{A}_2$ if every object of $\mathcal{A}$ is isomorphic to a direct sum $a_1\oplus a_2$ with $a_j\in \mathcal{A}_j$, and if
for any pair of objects $a_j\in \mathcal{A}_j$ and all integers $i$. The category $\mathcal{A}$ is indecomposable if for any such decomposition one of the two subcategories $\mathcal{A}_i$ is trivial. For example, if $X$ is a scheme, $\operatorname {D}(X)$ is indecomposable precisely when $X$ is connected. For more details see Bridgeland Reference 6.
2.5. Serre functors
The properties of Serre duality on a nonsingular projective variety were abstracted by Bondal and Kapranov Reference 5 into the notion of a Serre functor on a triangulated category. Let $\mathcal{A}$ be a triangulated category in which all the $\operatorname {Hom}$ sets are finite dimensional vector spaces. A Serre functor for $\mathcal{A}$ is an exact equivalence $S\colon \mathcal{A}\to \mathcal{A}$ inducing bifunctorial isomorphisms
that satisfy a simple compatibility condition (see Reference 5). When a Serre functor exists, it is unique up to isomorphism of functors. We say that $\mathcal{A}$ has trivial Serre functor if for some integer $i$ the shift functor $[i]$ is a Serre functor for $\mathcal{A}$.
The main example is the bounded derived category of coherent sheaves $\operatorname {D}(X)$ on a nonsingular projective variety $X$, having the Serre functor
Thus $\operatorname {D}(X)$ has trivial Serre functor if and only if the canonical bundle of $X$ is trivial.
2.6. A criterion for equivalence
Let $F\colon \mathcal{A}\to \mathcal{B}$ be an exact functor between triangulated categories with Serre functors $S_{\mathcal{A}}$ and $S_{\mathcal{B}}$. Assume that $F$ has a left adjoint $G\colon \mathcal{B}\to \mathcal{A}$. Then $F$ also has a right adjoint $H=S_{\mathcal{A}}\circ G\circ S_{\mathcal{B}}^{-1}$.
Theorem 2.3.
With assumptions as above, suppose also that there is a spanning class $\Omega$ for $\mathcal{A}$ such that
Suppose further that $\mathcal{A}$ is nontrivial, that $\mathcal{B}$ is indecomposable and that $FS_{\mathcal{A}}(\omega )\cong S_{\mathcal{B}}F(\omega )$ for all $\omega \in \Omega$. Then $F$ is an equivalence of categories.
Proof.
Consider an object $b\in \mathcal{B}$. For any $\omega \in \Omega$ and $i\in \mathbb{Z}$ we have isomorphisms
using Serre duality and the adjunctions for $(G,F)$ and $(F,H)$. Since $\Omega$ is a spanning class we can conclude that $Gb\cong 0$ precisely when $Hb\cong 0$. Then the result follows from Reference 6, Theorem 3.3.
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The proof of Theorem 3.3 in Reference 6 may be understood as follows. If $\operatorname {Ker}H\subset \operatorname {Ker}G$, then the semi-orthogonal decomposition described at the end of Section 2.3 becomes an orthogonal decomposition. Hence $\operatorname {Ker}H$ must be trivial, because $\mathcal{B}$ is indecomposable and $\mathcal{A}$, and hence $\operatorname {Im}F$, is nontrivial. Thus $\operatorname {Im}F=\mathcal{B}$ and $F$ is an equivalence.
3. Derived categories of sheaves
This section is concerned with various general properties of complexes of $\mathcal{O}_X$-modules on a scheme $X$. Note that all our schemes are of finite type over $\mathbb{C}$. Given a scheme $X$, define $\operatorname {D}^{\mathrm{qc}}(X)$ to be the (unbounded) derived category of the Abelian category $\operatorname {Qcoh}(X)$ of quasicoherent sheaves on $X$. Also define $\operatorname {D}(X)$ to be the full subcategory of $\operatorname {D}^{\mathrm{qc}}(X)$ 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 $X$ be a scheme and $E\in \operatorname {D}(X)$ an object of finite homological dimension. Then the derived dual
also has finite homological dimension, and the functor $-\stackrel{\mathbf{L}}{\otimes }E$ is both left and right adjoint to the functor $-\stackrel{\mathbf{L}}{\otimes }E^{\vee }$.
For the second example take a morphism of schemes $f\colon X\to Y$. The functor
If $f$ is proper, then $\mathbf{R}f_*$ takes $\operatorname {D}(X)$ into $\operatorname {D}(Y)$. If $f$ has finite Tor dimension (for example if $f$ is flat, or $Y$ is nonsingular), then $\mathbf{L}f^*$ takes $\operatorname {D}(Y)$ into $\operatorname {D}(X)$.
The third example is Grothendieck duality. Again take a morphism of schemes $f\colon X\to Y$. The functor $\mathbf{R}f_*$ has a right adjoint
Neeman Reference 19 has recently given a completely formal proof of these statements in terms of the Brown representability theorem.
Let $X$ be a nonsingular projective variety of dimension $n$ and write $f\colon X\to Y=\operatorname {Spec}(\mathbb{C})$ for the projection to a point. In this case $f^!(\mathcal{O}_Y)=\omega _X[n]$. The above statement of Grothendieck duality implies that the functor
In order to apply Grothendieck duality on quasiprojective schemes, we need to restrict attention to sheaves with compact support. The support of an object $E\in \operatorname {D}(X)$ is the locus of $X$ where $E$ is not exact, that is, the union of the supports of the cohomology sheaves of $E$. It is always a closed subset of $X$.
Given a scheme $X$, define the category $\operatorname {D_{\mathrm{c}}}(X)$ to be the full subcategory of $\operatorname {D}(X)$ consisting of complexes whose support is proper. Note that when $X$ itself is proper, $\operatorname {D_{\mathrm{c}}}(X)$ is just the usual derived category $\operatorname {D}(X)$.
If $X$ is a quasiprojective variety and $i\colon X\hookrightarrow \overline{X}$ is some projective closure, then the functor $i_*$ embeds $\operatorname {D_{\mathrm{c}}}(X)$ as a full triangulated subcategory of $\operatorname {D}(\overline{X})$. By resolution of singularities, if $X$ is nonsingular we can assume that $\overline{X}$ is too. Then the Serre functor on $\operatorname {D}(\overline{X})$ restricts to give a Serre functor on $\operatorname {D}(X)$. Thus if $X$ is a nonsingular quasiprojective variety of dimension $n$, the category $\operatorname {D_{\mathrm{c}}}(X)$ has a Serre functor given by (Equation 3).
The argument used to prove Lemma 2.2 is easily generalised to give the statement that the set of skyscraper sheaves $\{\mathcal{O}_x\colon x\in X\}$ on a nonsingular quasiprojective variety $X$ is a spanning class for $\operatorname {D_{\mathrm{c}}}(X)$.
3.3. Crepant resolutions
Let $X$ be a variety and $f\colon Y\to X$ a resolution of singularities. Given a point $x\in X$ define $\operatorname {D}_x(Y)$ to be the full subcategory of $\operatorname {D_{\mathrm{c}}}(Y)$ consisting of objects whose support is contained in the fibre $f^{-1}(x)$. We have the following categorical criterion for $f$ to be crepant.
Lemma 3.1.
Assume that $X$ has rational singularities, that is, $\mathbf{R}f_* \mathcal{O}_Y=\mathcal{O}_X$. Suppose $\operatorname {D}_x(Y)$ has trivial Serre functor for each $x\in X$. Then $X$ is Gorenstein and $f\colon Y\to X$ is a crepant resolution.
Proof.
The Serre functor on $\operatorname {D}_x(Y)$ is the restriction of the Serre functor on $\operatorname {D_{\mathrm{c}}}(Y)$. Hence, by Section 3.2, the condition implies that for each $x\in X$ the restriction of the functor $(-\otimes \omega _Y)$ to the category $\operatorname {D}_x(Y)$ is isomorphic to the identity. Since $\operatorname {D}_x(Y)$ contains the structure sheaves of all fattened neighbourhoods of the fibre $f^{-1}(x)$ this implies that the restriction of $\omega _Y$ to each formal fibre of $f$ is trivial. To get the result, we must show that $\omega _X$ is a line bundle and that $f^*\omega _X=\omega _Y$. Since $\omega _X=f_*\omega _Y$, this is achieved by the following lemma.
Lemma 3.2.
Assume that $X$ has rational singularities. Then a line bundle $L$ on $Y$ is the pullback $f^*M$ of some line bundle $M$ on $X$ if and only if the restriction of $L$ to each formal fibre of $f$ is trivial. Moreover, when this holds, $M=f_*L$.
Proof.
For each point $x\in X$, the formal fibre of $f$ over $x$ is the fibre product
The restriction of the pullback of a line bundle from $X$ 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 $L$ to each of these formal fibres is trivial. The theorem on formal functions shows that the completions of the stalks of the sheaves $\mathbf{R}^i f_*\mathcal{O}_Y$ and $\mathbf{R}^i f_*L$ at any point $x\in X$ are isomorphic for each $i$. Since $X$ has rational singularities it follows that $\mathbf{R}^i f_*L=0$ for all $i>0$, and $M=f_*L$ is a line bundle on $X$.
Since $f^*M$ is torsion free, the natural adjunction map $\eta \colon f^* f_*L \to L$ is injective, so there is a short exact sequence
The fact that $\eta$ is the unit of the adjunction for $(f^*,f_*)$ implies that $f_*\eta$ has a left inverse, and in particular is surjective. Applying $f_*$ to (Equation 4) we conclude that $f_* Q=0$.
Using the theorem on formal functions again, we can deduce that