Matrix factorization for quasi-homogeneous singularities

Abstract

Given an isolated, quasi-homogeneous singularity $X$ we prove that there is a group isomorphism between the group of rank one reflexive sheaves on $X$ and the free abelian group generated by ${\mathbb{C}}^{\ast }$-divisors, modulo linear equivalence. When $\dim (X)=2$ we reduce the problem of finding matrix factorizations of arbitrary reflexive ${\mathcal{O}}_X$-modules to the same question on rank one reflexive sheaves. We then enumerate the matrix factorizations of all rank one reflexive sheaves. As a consequence, we prove a conjecture of Etingof and Ginzburg on point modules.

18 pages