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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2017: 1.686

CiteScore 2017: 0.96

SCImago Journal Rank (SJR) 2017: 2.585
Source Normalized Impact per Paper (SNIP) 2017: 1.203

Mathematical Citation Quotient (MCQ) 2016: 1.28

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Volume 2015, Issue 708


Height of exceptional collections and Hochschild cohomology of quasiphantom categories

Alexander Kuznetsov
Published Online: 2013-09-03 | DOI: https://doi.org/10.1515/crelle-2013-0077


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.

About the article

Received: 2013-04-10

Published Online: 2013-09-03

Published in Print: 2015-11-01

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

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2015, Issue 708, Pages 213–243, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2013-0077.

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