Matrix factorization for quasi-homogeneous singularities

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 \(\mathbb{C}^*\)-divisors, modulo linear equivalence. When \(\dim(X)=2\) we reduce the problem of finding...

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Bibliographic Details
Published in:arXiv.org 2023-01
Main Authors: Ananyo Dan, Romano-Velázquez, Agustín
Format: Article
Language:English
Subjects:
Group theory
Isomorphism
Modules
Sheaves
Singularities
Online Access:Get full text
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Summary: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 \(\mathbb{C}^*\)-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.
ISSN:2331-8422