Periodic free resolutions from twisted matrix factorizations

TL;DR

This paper extends matrix factorizations to regular normal elements in graded algebras, relating them to Zhang twists and establishing their connection to syzygy modules and singularity categories in the AS-regular setting.

Contribution

It introduces twisted matrix factorizations for regular normal elements and links them to Zhang twists and singularity categories, generalizing Eisenbud's original concept.

Findings

High syzygy modules are cokernels of twisted matrix factorizations.

Equivalence between homotopy category of twisted matrix factorizations and singularity category.

Extension of matrix factorizations to graded algebra setting.

Abstract

The notion of a matrix factorization was introduced by Eisenbud in the commutative case in his study of bounded (periodic) free resolutions over complete intersections. In this work, we extend the notion of (homogeneous) matrix factorizations to regular normal elements of connected graded algebras over a field. Next, we relate the category of twisted matrix factorizations to an element over a ring and certain Zhang twists. We also show that in the AS-regular setting, every sufficiently high syzygy module is the cokernel of some twisted matrix factorization. Furthermore, we show that in this setting there is an equivalence of categories between the homotopy category of twisted matrix factorizations and the singularity category of the hypersurface, following work of Orlov.