Compact generators in categories of matrix factorizations

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.

💡 Research Summary

The paper investigates the differential‑graded (dg) category of matrix factorizations attached to an isolated hypersurface singularity ((R,w)). The authors first prove that the stabilization of the residue field (k=R/\mathfrak m), denoted (k^{\mathrm{stab}}), is a compact generator of (\operatorname{MF}(R,w)). This result is striking because it shows that a single, very concrete object generates the entire category regardless of the dimension of the singularity.

Using this generator, they define the endomorphism dg‑algebra
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