Compact generators in categories of matrix factorizations
Tobias Dyckerhoff
TL;DR
This work develops a comprehensive framework for matrix factorizations MF$(R,w)$ of isolated hypersurface singularities, showing the associated noncommutative space $\mathcal{X}$ is dg affine, homologically smooth, proper, and Calabi–Yau. It identifies a compact generator given by the stabilization of the residue field, establishing a quasi-equivalence between MF$^{\infty}(R,w)$ and the dg derived category of a concrete endomorphism dg algebra $A$. Using Toën’s derived Morita theory, it expresses functors as integral transforms, enabling explicit computations of Hochschild cohomology (as the Jacobian algebra) and Hochschild homology, and proving Hodge-to-de Rham degeneration. The paper further develops stabilization techniques, Nakayama-type lemmas, Knörrer periodicity, formal completions, and minimal $A_\infty$ structures, situating matrix factorizations within a robust noncommutative geometric framework with Calabi–Yau interpretations and consequences for deformation theory.”
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.