Hochschild homology and semiorthogonal decompositions

TL;DR

This paper explores Hochschild homology and cohomology of admissible subcategories in derived categories of smooth projective varieties, revealing their properties, additive behavior, and explicit computations for certain Fano threefolds and conic bundles.

Contribution

It establishes isomorphisms between Hochschild (co)homology and derived endomorphisms of kernels, and analyzes their behavior under semiorthogonal decompositions.

Findings

Hochschild homology is additive with respect to semiorthogonal decompositions.

Constructs long exact sequences relating Hochschild cohomology of categories and components.

Computes Hochschild (co)homology for specific Fano threefolds and conic bundle components.

Abstract

We investigate Hochschild cohomology and homology of admissible subcategories of derived categories of coherent sheaves on smooth projective varieties. We show that the Hochschild cohomology of an admissible subcategory is isomorphic to the derived endomorphisms of the kernel giving the corresponding projection functor, and the Hochschild homology is isomorphic to derived morphisms from this kernel to its convolution with the kernel of the Serre functor. We investigate some basic properties of Hochschild homology and cohomology of admissible subcategories. In particular, we check that the Hochschild homology is additive with respect to semiorthogonal decompositions and construct some long exact sequences relating the Hochschild cohomology of a category and its semiorthogonal components. We also compute Hochschild homology and cohomology of some interesting admissible subcategories, in particular of the nontrivial components of derived categories of some Fano threefolds and of the nontrivial components of the derived categories of conic bundles.