Compact generators in categories of matrix factorizations

Abstract

We study the category of matrix factorizations 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. We deduce a quasi-equivalence between the category of matrix factorizations and the dg derived category of an explicitly computable dg algebra. Building on this result, we employ a variant of Toen's derived Morita theory to identify continuous functors between matrix factorization categories as integral transforms. This enables us to calculate the Hochschild chain and cochain complexes of these categories. Finally, we give interpretations of the results of this work in terms of noncommutative geometry based on dg categories.

43 pages, revised version after referee report: corrected a mistake in the proof of Theorem 4.7, slightly stronger assumptions are needed to make the Morita theory work (see new Section 3), added discussion of Knoerrer periodicity (5.3), general reorganization; to appear in Duke Mathematical Journal

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