Compact generators in categories of matrix factorizations. (English) Zbl 1252.18026

Dyckerhoff, Tobias

Duke Math. J. 159, No. 2, 223-274 (2011).

The author studies the category of matrix factorizations

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associated to the germ of an isolated hypersurface singularity. This category is shown to admit a compact generator which is given by the stabilization of the residue field

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. The author deduces a quasi-equivalence between the category of matrix factorizations and the differential graded derived category of an explicitly computable differential graded algebra

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. Building on this result, the author employs a variant of Toën’s derived Morita theory to identify continuous functors between matrix factorization categories as integral transforms. This enables the author to calculate the Hochschild chain and cochain complexes

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of these categories. Finally, the author gives interpretations of the results of this work in terms of noncommutative geometry

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based on differential graded categories.

Reviewer: Tong Wenting (Nanjing)

MSC:

18E30	Derived categories, triangulated categories (MSC2010)
14B05	Singularities in algebraic geometry

Keywords:

category of matrix factorizations; compact generator; cohomology; differential graded derived category; differential graded algebra; noncommutative geometry

References:

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