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# Homology, Homotopy and Applications

## Volume 14 (2012)

### Number 2

### Matrix factorizations over projective schemes

Pages: 37 – 61

DOI: http://dx.doi.org/10.4310/HHA.2012.v14.n2.a3

#### Authors

#### Abstract

We study matrix factorizations of regular global sections of line bundles on schemes. If the line bundle is very ample relative to a Noetherian affine scheme we show that morphisms in the homotopy category of matrix factorizations may be computed as the hypercohomology of a certain mapping complex. Using this explicit description, we prove an analogue 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. Moreover, we give a complete description of the image of this functor.

#### Keywords

matrix factorization, singularity category

#### 2010 Mathematics Subject Classification

13D02, 13D09, 14F05