Height of exceptional collections and Hochschild cohomology of quasiphantom categories
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Alexander Kuznetsov
Abstract
We define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective variety X, a graded vector space which controls the restriction morphism from the Hochschild cohomology of X to the Hochschild cohomology of the orthogonal complement of this admissible subcategory. When the subcategory is generated by an exceptional collection, we define a new invariant (the height) and show that the orthogonal to an exceptional collection of height h in the derived category of a smooth projective variety X has the same Hochschild cohomology as X in degrees up to h - 2. We use this to describe the second Hochschild cohomology of quasiphantom categories in the derived categories of some surfaces of general type. We also give necessary and sufficient conditions for the fullness of an exceptional collection in terms of its height and of its normal Hochschild cohomology.
Funding source: RFFI
Award Identifier / Grant number: 11-01-00393, 11-01-00568, 12-01-33024, NSh-5139.2012.1
Funding source: Simons Foundation
Funding source: AG Laboratory SU-HSE, RF
Award Identifier / Grant number: ag.11.G34.31.0023
I would like to thank Dima Orlov for suggesting the problem and useful discussions, Roman Mikhailov for a help with references, and Olaf Schnürer for corrections to the first version of the paper. I am very grateful to Christian Böhning, Hans-Christian Graf von Bothmer and Pawel Sosna for the help with computing the height of their exceptional collection, careful reading of the first draft of the paper, and many useful comments. I am also very grateful to the anonymous referee for useful suggestions and references.
© 2015 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- The length metric on the set of orthogonal projections and new estimates in the subspace perturbation problem
- A Giambelli formula for even orthogonal Grassmannians
- Lifting in Frattini covers and a characterization of finite solvable groups
- Note on basic features of large time behaviour of heat kernels
- A decomposition theorem in II1-factors
- Wonderful resolutions and categorical crepant resolutions of singularities
- Tannaka–Kreĭn duality for compact quantum homogeneous spaces II. Classification of quantum homogeneous spaces for quantum SU(2)
- An approach of the minimal model program for horospherical varieties via moment polytopes
- Height of exceptional collections and Hochschild cohomology of quasiphantom categories
Artikel in diesem Heft
- Frontmatter
- The length metric on the set of orthogonal projections and new estimates in the subspace perturbation problem
- A Giambelli formula for even orthogonal Grassmannians
- Lifting in Frattini covers and a characterization of finite solvable groups
- Note on basic features of large time behaviour of heat kernels
- A decomposition theorem in II1-factors
- Wonderful resolutions and categorical crepant resolutions of singularities
- Tannaka–Kreĭn duality for compact quantum homogeneous spaces II. Classification of quantum homogeneous spaces for quantum SU(2)
- An approach of the minimal model program for horospherical varieties via moment polytopes
- Height of exceptional collections and Hochschild cohomology of quasiphantom categories
