Journal of Noncommutative Geometry


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Volume 10, Issue 3, 2016, pp. 907–979
DOI: 10.4171/JNCG/252

Published online: 2016-09-28

Matrix factorizations and semi-orthogonal decompositions for blowing-ups

Valery A. Lunts[1] and Olaf M. Schnürer[2]

(1) Indiana University, Bloomington, USA
(2) Universität Bonn, Germany

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.

Keywords: Matrix factorization, semi-orthogonal decomposition, blowing-up, projective space bundle, dg enhancement

Lunts Valery, Schnürer Olaf: Matrix factorizations and semi-orthogonal decompositions for blowing-ups. J. Noncommut. Geom. 10 (2016), 907-979. doi: 10.4171/JNCG/252