## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 14, Number 2 (2012), 37-61.

### Matrix factorizations over projective schemes

Jesse Burke and Mark E. Walker

#### 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.

#### Article information

**Source**

Homology Homotopy Appl., Volume 14, Number 2 (2012), 37-61.

**Dates**

First available in Project Euclid: 12 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1355321479

**Mathematical Reviews number (MathSciNet)**

MR3007084

**Zentralblatt MATH identifier**

1259.14015

**Subjects**

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 13D09: Derived categories 13D02: Syzygies, resolutions, complexes

**Keywords**

Matrix factorization singularity category

#### Citation

Burke, Jesse; Walker, Mark E. Matrix factorizations over projective schemes. Homology Homotopy Appl. 14 (2012), no. 2, 37--61. https://projecteuclid.org/euclid.hha/1355321479