Journal of Noncommutative Geometry


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

Matrix factorizations and semi-orthogonal decompositions for blowing-ups

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

(1) Department of Mathematics, Indiana University, Rawles Hall, 831 East 3rd St, IN 47405, Bloomington, USA
(2) Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115, 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