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

Research paper by Lennart Galinat

Indexed on: 06 Feb '13Published on: 06 Feb '13Published in: Mathematics - Algebraic Geometry



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.