Thom-Sebastiani & Duality for Matrix Factorizations

TL;DR

This paper proves a Thom-Sebastiani theorem for matrix factorizations, relating tensor products and functors to coherent complexes on zero loci, with applications to Hochschild invariants, smoothness, properness, and Calabi-Yau structures.

Contribution

It introduces a Thom-Sebastiani type theorem for matrix factorizations and identifies related dg categories with coherent complexes, advancing understanding of their structure and invariants.

Findings

Identifies tensor products of dg categories with coherent complexes on zero loci.

Establishes the smoothness and properness conditions for matrix factorizations.

Refines computation of Hochschild invariants and relates Calabi-Yau structures to volume forms.

Abstract

The derived category of a hypersurface has an action by "cohomology operations" k[t], deg t=-2, underlying the 2-periodic structure on its category of singularities (as matrix factorizations). We prove a Thom-Sebastiani type Theorem, identifying the k[t]-linear tensor products of these dg categories with coherent complexes on the zero locus of the sum potential on the product (with a support condition), and identify the dg category of colimit-preserving k[t]-linear functors between Ind-completions with Ind-coherent complexes on the zero locus of the difference potential (with a support condition). These results imply the analogous statements for the 2-periodic dg categories of matrix factorizations. Some applications include: we refine and establish the expected computation of 2-periodic Hochschild invariants of matrix factorizations; we show that the category of matrix factorizations is smooth, and is proper when the critical locus is proper; we show how Calabi-Yau structures on matrix factorizations arise from volume forms on the total space; we establish a version of Kn\"orrer Periodicity for eliminating metabolic quadratic bundles over a base.