Matrix factorizations and semi-orthogonal decompositions for blowing-ups

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 [23] which defines a Landau–Ginzburg motivic measure using categories of matrix factorizations. Finally we explain some applications.