Compact generators in categories of matrix factorizations

TL;DR

This paper establishes a compact generator for the category of matrix factorizations of isolated hypersurface singularities, providing explicit algebraic models and functorial characterizations with applications in noncommutative geometry.

Contribution

It introduces a compact generator for matrix factorization categories and relates them to dg derived categories, enabling explicit computations and functor classifications.

Findings

Existence of a compact generator given by the stabilization of the residue field.

Quasi-equivalence between matrix factorizations and dg derived categories of explicit dg algebras.

Identification of continuous functors as integral transforms and applications to Hochschild complexes.

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.