Derived categories and algebraic geometry.

*(English)*Zbl 1208.14006
Holm, Thorsten (ed.) et al., Triangulated categories. Based on a workshop, Leeds, UK, August 2006. Cambridge: Cambridge University Press (ISBN 978-0-521-74431-7/pbk). London Mathematical Society Lecture Note Series 375, 351-370 (2010).

The article under review presents some basic properties of \(D^{b}(X)\), the bounded derived category of coherent sheaves on a variety \(X\). The approach is to show some similarities to the case of the abelian category of coherent sheaves. There are two parts.

The first part deals with localisation. The abelian case is considered first: It is known that given a closed subscheme \(Z\) of the variety \(X\), one can describe the abelian category of coherent sheaves on \(U=X\setminus Z\) as a quotient \({\text{Coh}}(X)/ {\text{Coh}}_Z(X)\), where \({\text{Coh}}_Z(X)\) denotes the category of coherent sheaves supported on \(Z\). This fact can also be expressed as an exact sequence of abelian categories. Going to triangulated categories we have a similar result when one substitutes coherent sheaves by the derived category of quasi-coherent sheaves. However, one usually works with another triangulated category. Assume for simplicity that \(X\) is quasi-projective and recall that a complex is perfect if it is quasi-isomorphic to a bounded complex of free sheaves of finite rank. The triangulated category of perfect complexes is denoted by \({\text{Perf}}(X)\) and it coincides with \({D^{b}}(X)\) if \(X\) is smooth. With notation as above the inclusion of \(U\) into \(X\) induces a fully faithful functor from \({\text{Perf}}(X)/{\text{Perf}}_Z(X)\) to \({\text{Perf}}(U)\) and a perfect complex on \(U\) is the restriction of a perfect complex on \(X\) if and only if its class in the Grothendieck group \(K_0(U)\) is the restriction of an element in \(K_0(X)\). The first part ends with a decription of some applications to \(K\)-theory.

In the second part the author considers reconstruction. A classical result of Gabriel states that a variety \(X\) is determined by its abelian category of coherent sheaves \({\text{Coh}}(X)\). This result relies on the description of so called Serre subcategories of \({\text{Coh}}(X)\). In the triangulated setting P. Balmer introduced a spectrum of a tensor triangulated category in [J. Reine Angew. Math. 588, 149–168 (2005; Zbl 1080.18007)] and, using the classification of thick triangulated subcategories of \({\text{Perf}}(X)\), showed that the category \({\text{Perf}}(X)\), viewed as a tensor triangulated category, determines the variety \(X\).

In the last section the author recalls the well-known results due to A. Bondal and D. Orlov [Compos. Math. 125, No. 3, 327–344 (2001; Zbl 0994.18007)] stating that \({D^b}(X)\) (without the tensor structure) determines \(X\) if \(X\) is a smooth and projective variety whose canonical (or anticanonical) bundle is ample. The description of the group of autoequivalences of \({D^b}(X)\) in this case is also presented.

For the entire collection see [Zbl 1195.18001].

The first part deals with localisation. The abelian case is considered first: It is known that given a closed subscheme \(Z\) of the variety \(X\), one can describe the abelian category of coherent sheaves on \(U=X\setminus Z\) as a quotient \({\text{Coh}}(X)/ {\text{Coh}}_Z(X)\), where \({\text{Coh}}_Z(X)\) denotes the category of coherent sheaves supported on \(Z\). This fact can also be expressed as an exact sequence of abelian categories. Going to triangulated categories we have a similar result when one substitutes coherent sheaves by the derived category of quasi-coherent sheaves. However, one usually works with another triangulated category. Assume for simplicity that \(X\) is quasi-projective and recall that a complex is perfect if it is quasi-isomorphic to a bounded complex of free sheaves of finite rank. The triangulated category of perfect complexes is denoted by \({\text{Perf}}(X)\) and it coincides with \({D^{b}}(X)\) if \(X\) is smooth. With notation as above the inclusion of \(U\) into \(X\) induces a fully faithful functor from \({\text{Perf}}(X)/{\text{Perf}}_Z(X)\) to \({\text{Perf}}(U)\) and a perfect complex on \(U\) is the restriction of a perfect complex on \(X\) if and only if its class in the Grothendieck group \(K_0(U)\) is the restriction of an element in \(K_0(X)\). The first part ends with a decription of some applications to \(K\)-theory.

In the second part the author considers reconstruction. A classical result of Gabriel states that a variety \(X\) is determined by its abelian category of coherent sheaves \({\text{Coh}}(X)\). This result relies on the description of so called Serre subcategories of \({\text{Coh}}(X)\). In the triangulated setting P. Balmer introduced a spectrum of a tensor triangulated category in [J. Reine Angew. Math. 588, 149–168 (2005; Zbl 1080.18007)] and, using the classification of thick triangulated subcategories of \({\text{Perf}}(X)\), showed that the category \({\text{Perf}}(X)\), viewed as a tensor triangulated category, determines the variety \(X\).

In the last section the author recalls the well-known results due to A. Bondal and D. Orlov [Compos. Math. 125, No. 3, 327–344 (2001; Zbl 0994.18007)] stating that \({D^b}(X)\) (without the tensor structure) determines \(X\) if \(X\) is a smooth and projective variety whose canonical (or anticanonical) bundle is ample. The description of the group of autoequivalences of \({D^b}(X)\) in this case is also presented.

For the entire collection see [Zbl 1195.18001].

Reviewer: Pawel Sosna (Milano)

##### MSC:

14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |

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

18E10 | Abelian categories, Grothendieck categories |