Matrix factorizations over projective schemes

TL;DR

This paper investigates matrix factorizations of sheaves on schemes, providing explicit computational methods and a new proof of Orlov's theorem relating matrix factorizations to singularity categories.

Contribution

It offers an explicit hypercohomology approach to compute morphisms and presents a new proof of Orlov's embedding theorem with a detailed description of the functor's image.

Findings

Homomorphisms computed via hypercohomology of a mapping complex

New proof of Orlov's theorem on matrix factorizations and singularity categories

Complete description of the embedding functor's image

Abstract

We study matrix factorizations of locally free coherent sheaves on a scheme. For a scheme that is projective over an affine scheme, we show that homomorphisms in the homotopy category of matrix factorizations may be computed as the hypercohomology of a certain mapping complex. Using this explicit description, we give another proof of Orlov's theorem that there is a fully faithful embedding of the homotopy category of matrix factorizations into the singularity category of the corresponding zero subscheme. We also give a complete description of the image of this functor.