Hochschild homology and semiorthogonal decompositions

By Alexander Kuznetsov. 2009

112 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7ba086742c2120814a184a9a0b620d7f77e08331

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In Algebra and Number Theory

1. Nicolas Addington
2. Evgeny Shinder
3. Sergey Galkin

In Geometry and Topology

1. Nicolas Addington
2. Evgeny Shinder
3. Sergey Galkin
4. Alexander Perry
5. Alexander Kuznetsov
6. D. Huybrechts

In Mathematical Physics

1. Nicolas Addington
2. Evgeny Shinder
3. Sergey Galkin
4. Alexander Kuznetsov
5. Alexander Perry
6. D. Huybrechts

Citations in the corpus

• New derived symmetries of some hyperkähler varieties (2011, background): “gle (4.4) is split because Ext1([−2],idA) = Hom(idA,[−3]) = HH3(A) = 0. This argument is valid because we are working with Fourier–Mukai kernels, not just functors; everything is proved rigorously in [23]. The first map of (4.4) agrees with the unit η: idA →RFup to a scalar multiple, because Hom(idA,”
• Exceptional collections of line bundles on the Beauville surface (2012, methodology): “lgebraic varieties, their admissible subcategories and semiorthogonal decompositions have been studied intensively by Bondal, Kapranov, Kuznetsov, Orlov, and others [Bon], [BK], [BO], [Kap], [Kuz06], [Kuz09]. It has been questioned which additive invariants of admissible geometric triangulated categ”