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

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