Homology, Homotopy and Applications

Volume 14 (2012)

Number 2

Matrix factorizations over projective schemes

Pages: 37 – 61

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


Jesse Burke (Department of Mathematics, Universität Bielefeld, Germany)

Mark E. Walker (Department of Mathematics, University of Nebraska, Lincoln, Nebraska, U.S.A.)


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.


matrix factorization, singularity category

2010 Mathematics Subject Classification

13D02, 13D09, 14F05

Published 4 December 2012