Classifying dg-categories of matrix factorizations

TL;DR

This paper classifies the homotopy categories of matrix factorizations for isolated singularities, showing they are determined by hypersurface conditions and using categorical equivalences to complete the classification.

Contribution

It provides a complete classification of dg categories of matrix factorizations for isolated singularities, linking quasi-equivalences to hypersurface singularities via Knörrer's periodicity.

Findings

Quasi-equivalence implies hypersurface singularity condition.

Classification completes the categorical understanding of matrix factorizations.

Uses categorical Mather–Yau theorem to finalize classification.

Abstract

We give a complete classification of differential $\mathbb{Z}$-graded homotopy categories of matrix factorizations of isolated singularities up to quasi-equivalence. This answers a question of Bernhard Keller and Evgeny Shinder. More generally, we show that a quasi-equivalence between the dg singularity category of a Gorenstein isolated singularity $R$ and the dg singularity category of a complete local Noetherian $\mathbb{C}$-algebra $S$ of different Krull dimension can always be realized by Kn\"orrer's periodicity -- in particular, the existence of such an equivalence implies that $R$ and $S$ are hypersurface singularities. This uses and is complemented by a recent categorical version of the Mather--Yau theorem for hypersurfaces of the same Krull dimension due to Hua & Keller, which completes the classification mentioned above.