Derived categories of sheaves on singular schemes with an application to reconstruction.(English)Zbl 1213.14031
If \(X\) is a (quasi-)projective variety, there are several triangulated categories naturally associated to it: Probably the most common ones are the bounded derived category of coherent sheaves \({\text D}^{\text b}(X)\) and its full triangulated subcategory of perfect complexes \({\text{Perf}}(X)\), which is roughly the smallest triangulated subcategory of \({\text D}^{\text b}(X)\) containing all finite-rank locally free sheaves. These categories coincide if and only if \(X\) is smooth.
In recent years it has become commonplace to investigate the geometry of a smooth projective variety through its bounded derived category of coherent sheaves, so most of the results are formulated, and the proofs usually only work, under the smoothness assumption. In the article under review the author extends two such results to the singular case.
Firstly,A.Bondal andM.van den Bergh proved [Mosc.Math.J.3, No.1, 1–36 (2003;Zbl 1135.18302)] that for a smooth and proper variety \(X\) any covariant or contravariant locally-finite (a certain boundedness condition) cohomological functor from \({\text D}^{\text b}(X)\) to the category of vector spaces is representable. Furthermore, \({\text D}^{\text b}(X)\) is equivalent to either of these categories of functors. The first main result of this paper states that dropping the smoothness one has that \({\text D}^{\text b}(X)\) is equivalent to the category of locally-finite, cohomological functors on \({\text{Perf}}(X)\). The proof uses the machinery of compactly-generated triangulated categories.
Secondly, a result ofA.Bondal andD.Orlov [Compos.Math.125, No.3, 327–344 (2001;Zbl 0994.18007)] says that if a smooth and projective variety \(X\) has ample or anti-ample canonical bundle, then any variety \(Y\) satisfying \({\text D}^{\text b}(X)\cong{\text D}^{\text b}(Y)\) is isomorphic to \(X\). The author extends this result to the case where \(X\) is projective Gorenstein using, in particular, a relativization of the notion of a Serre functor.
Besides the above mentioned results the author also proves that two projective schemes have equivalent bounded derived categories if and only if the respective categories of perfect complexes are equivalent. Furthermore, for a projective scheme \(X\) the groups of autoequivalences of \({\text D}^{\text b}(X)\) and of \({\text{Perf}}(X)\) coincide. This result is proved using so-called pseudo-adjoint functors.
In recent years it has become commonplace to investigate the geometry of a smooth projective variety through its bounded derived category of coherent sheaves, so most of the results are formulated, and the proofs usually only work, under the smoothness assumption. In the article under review the author extends two such results to the singular case.
Firstly,A.Bondal andM.van den Bergh proved [Mosc.Math.J.3, No.1, 1–36 (2003;Zbl 1135.18302)] that for a smooth and proper variety \(X\) any covariant or contravariant locally-finite (a certain boundedness condition) cohomological functor from \({\text D}^{\text b}(X)\) to the category of vector spaces is representable. Furthermore, \({\text D}^{\text b}(X)\) is equivalent to either of these categories of functors. The first main result of this paper states that dropping the smoothness one has that \({\text D}^{\text b}(X)\) is equivalent to the category of locally-finite, cohomological functors on \({\text{Perf}}(X)\). The proof uses the machinery of compactly-generated triangulated categories.
Secondly, a result ofA.Bondal andD.Orlov [Compos.Math.125, No.3, 327–344 (2001;Zbl 0994.18007)] says that if a smooth and projective variety \(X\) has ample or anti-ample canonical bundle, then any variety \(Y\) satisfying \({\text D}^{\text b}(X)\cong{\text D}^{\text b}(Y)\) is isomorphic to \(X\). The author extends this result to the case where \(X\) is projective Gorenstein using, in particular, a relativization of the notion of a Serre functor.
Besides the above mentioned results the author also proves that two projective schemes have equivalent bounded derived categories if and only if the respective categories of perfect complexes are equivalent. Furthermore, for a projective scheme \(X\) the groups of autoequivalences of \({\text D}^{\text b}(X)\) and of \({\text{Perf}}(X)\) coincide. This result is proved using so-called pseudo-adjoint functors.
Reviewer: Pawel Sosna (Milano)
MSC:
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
Keywords:
derived categories;compactly-generated triangulated categories;perfect complexes;saturated triangulated categories;pseudo-adjoint functorsCite
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