Matrix factorizations and semi-orthogonal decompositions for blowing-ups

TL;DR

This paper develops the theory of matrix factorizations on schemes, establishing foundational aspects and demonstrating a semi-orthogonal decomposition for blow-ups, which parallels classical results for derived categories and aids in Landau-Ginzburg motivic measures.

Contribution

It introduces foundational tools for matrix factorizations on schemes and proves a semi-orthogonal decomposition for blow-ups, extending classical derived category results to this setting.

Findings

Established derived functors and dg enhancements for matrix factorizations.

Proved semi-orthogonal decomposition of matrix factorizations on blow-ups.

Set the stage for defining Landau-Ginzburg motivic measures using these categories.

Abstract

We study categories of matrix factorizations. These categories are defined for any regular function on a suitable regular scheme. Our paper has two parts. In the first part we develop the foundations; for example we discuss derived direct and inverse image functors and dg enhancements. In the second part we prove that the category of matrix factorizations on the blowing-up of a suitable regular scheme X along a regular closed subscheme Y has a semi-orthogonal decomposition into admissible subcategories in terms of matrix factorizations on Y and X. This is the analog of a well-known theorem for bounded derived categories of coherent sheaves, and is an essential step in our forthcoming article which defines a Landau-Ginzburg motivic measure using categories of matrix factorizations. Finally we explain some applications.