Orlov's Equivalence and Maximal Cohen-Macaulay Modules over the Cone of an Elliptic Curve

TL;DR

This paper develops a computational approach to study Orlov's equivalence for hypersurfaces, specifically describing indecomposable matrix factorizations over elliptic curves and providing methods for higher rank cases.

Contribution

It introduces a method for explicit computation of indecomposable matrix factorizations over elliptic curves using Orlov's equivalence, including higher rank cases with computer algebra.

Findings

Explicit descriptions of indecomposable rank one matrix factorizations for elliptic curves.

A computational framework for higher rank matrix factorizations.

Connection between maximal Cohen-Macaulay modules and matrix factorizations.

Abstract

We describe a method for doing computations with Orlov's equivalence between the bounded derived category of certain hypersurfaces and the stable category of graded matrix factorisations of the polynomials describing these hypersurfaces. In the case of a smooth elliptic curve over an algebraically closed field we describe the indecomposable graded matrix factorisations of rank one. Since every indecomposable Maximal Cohen-Macaulay module over the completion of a smooth cubic curve is gradable, we obtain explicit descriptions of all indecomposable rank one matrix factorisations of such potentials. Finally, we explain how to compute all indecomposable matrix factorisations of higher rank with the help of a computer algebra system.