Matrix factorizations in higher codimension

TL;DR

This paper establishes an equivalence between the singularity category of affine complete intersections and matrix factorizations, leading to new geometric and homological insights, including stable support sets and module resolutions.

Contribution

It introduces a geometric construction of cohomology rings, generalizes support varieties to stable support sets, and constructs projective resolutions from matrix factorizations for complete intersections.

Findings

Equivalence between singularity categories and matrix factorizations

Construction of cohomology rings via geometric methods

Resolution of modules using matrix factorizations in complete intersections

Abstract

We observe that there is an equivalence between the singularity category of an affine complete intersection and the homotopy category of matrix factorizations over a related scheme. This relies in part on a theorem of Orlov. Using this equivalence, we give a geometric construction of the ring of cohomology operators, and a generalization of the theory of support varieties, which we call stable support sets. We settle a question of Avramov about which stable support sets can arise for a given complete intersection ring. We also use the equivalence to construct a projective resolution of a module over a complete intersection ring from a matrix factorization, generalizing the well-known result in the hypersurface case.