Height of exceptional collections and Hochschild cohomology of quasiphantom categories

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 its 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 of fullness of an exceptional collection in terms of its height and of its normal Hochschild cohomology.

23 pages, the construction of the Čech enhancement is corrected

References in corpus (5)

• Hochschild homology and semiorthogonal decompositions
• Determinantal Barlow surfaces and phantom categories
• On the derived category of the classical Godeaux surface
• Derived categories of Burniat surfaces and exceptional collections
• Exceptional collections of line bundles on the Beauville surface

Cited by in corpus (17)

• Semiorthogonal decompositions in algebraic geometry
• Derived categories view on rationality problems
• Geometric Phantom Categories
• Determinantal Barlow surfaces and phantom categories
• Exceptional collections of line bundles on the Beauville surface
• Derived categories of Gushel-Mukai varieties
• Derived noncommutative schemes, geometric realizations, and finite dimensional algebras
• Exceptional collections on 2-adically uniformised fake projective planes
• A Phantom on a Rational Surface
• Perverse sheaves of categories and some applications
• Exceptional collections on Dolgachev surfaces associated with degenerations
• T-structures and twisted complexes on derived injectives
• Polyvector fields for Fano 3-folds
• Quasiphantom categories on a family of surfaces isogenous to a higher product
• Derived categories of noncommutative quadrics and Hilbert squares
• Deformation of exceptional collections
• Categorical characterization of quadrics