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Orlov’s equivalence and maximal Cohen-Macaulay modules over the cone of an elliptic curve. (English) Zbl 1329.14034
Let \(K\) be a field and \(f\in K[x_0, \ldots, x_n]\) be a homogeneous polynomial of degree \(n+1\) and \(X=\mathrm{Proj}\;K[x_0, \ldots, x_n]/f\). D. Orlov [Prog. Math. 270, 503–531 (2009; Zbl 1200.18007)] proved that there is an equivalence between the bounded derived category of coherent sheaves on \(X\) and the homotopy category of graded matrix factorisations of \(f\). It is described how to compute this equivalence in case of a smooth elliptic curve over an algebraically closed field. The indecomposable graded matrix factorisations of rank one are described. Every indecomposable maximal Cohen-Macaulay module over the completion of a smooth curve is gradable. This gives explicit descriptions of all indecomposable rank one matrix factorisations. It is explained how to compute all indecomposable matrix factorisations of higher rank using a computer algebra system as Singular.

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
16G50 Cohen-Macaulay modules in associative algebras
14H52 Elliptic curves
Software:
SINGULAR
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