Equivalences of derived categories of sheaves on smooth stacks.

*(English)*Zbl 1076.14023A well known theorem due to D. O. Orlov [J. Math. Sci., New York 84, 1361–1381 (1997; Zbl 0938.14019)] asserts that any equivalence \(F\) of derived categories of sheaves on smooth projective varieties \(X\) and \(Y\) is representable, i.e. there exists an object \(e\) of \(D(X \times Y)\) such that \(F\) is the integral functor \(\Phi^e\) associated to \(e\). In the paper under review the author extends this result to derived categories of sheaves on smooth stacks. More precisely he proves that, given stacks \(\mathcal{X}\) and \(\mathcal{Y}\) associated to normal projective varieties with quotient singularities \(X\) and \(Y\), any exact fully faithful functor \(F\) admitting a left adjoint is representable by a unique object (up to isomorphism).

The first step is the construction of a (possibly infinite) Beilinson-type resolution of the diagonal over \(X \times X\). It is pointed out here that this construction agrees with A. Canonaco’s resolution for weighted projective spaces [J. Algebra 225, 28–46 (2000; Zbl 0963.14007)]. After proving some boundedness results, the author uses this resolution to define the representing object \(e\). The isomorphism between the integral functor \(\Phi^e\) and the original functor \(F\) is first constructed on a spanning class of locally free sheaves.

Some applications are also considered. The first one is a comparison of numerical invariants for varieties with quotient singularities having equivalent derived categories. The second one is an extension to orbifolds of A. Bondal and D. Orlov’s reconstruction theorem for varieties with ample (anti)canonical bundle [Compos. Math. 125, 327–344 (2001; Zbl 0994.18007)].

The first step is the construction of a (possibly infinite) Beilinson-type resolution of the diagonal over \(X \times X\). It is pointed out here that this construction agrees with A. Canonaco’s resolution for weighted projective spaces [J. Algebra 225, 28–46 (2000; Zbl 0963.14007)]. After proving some boundedness results, the author uses this resolution to define the representing object \(e\). The isomorphism between the integral functor \(\Phi^e\) and the original functor \(F\) is first constructed on a spanning class of locally free sheaves.

Some applications are also considered. The first one is a comparison of numerical invariants for varieties with quotient singularities having equivalent derived categories. The second one is an extension to orbifolds of A. Bondal and D. Orlov’s reconstruction theorem for varieties with ample (anti)canonical bundle [Compos. Math. 125, 327–344 (2001; Zbl 0994.18007)].

Reviewer: Daniele Faenzi (Firenze)

##### MSC:

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |

14A20 | Generalizations (algebraic spaces, stacks) |

18E30 | Derived categories, triangulated categories (MSC2010) |