Maximal Cohen-Macaulay modules over the cone of an elliptic curve
Let \(k\) be an algebraically closed field with \(\text{char\,} k \neq 3\) and \(R = k[Y_1, Y_2, Y_3]/(Y_1^3 + Y_2^3 + Y_3^3)\). In the present article, the authors compute all the indecomposable Cohen-Macaulay \(R\)-modules of dimension~\(2\). Since~\(R\) is normal, it is equivalent to computing vector bundles over a curve \(\text{Proj} R\). The curve \(\text{Proj} R\) is an elliptic curve and \textit{M. F. Atiyah} [Proc. Lond. Math. Soc., III. Ser. 7, 414--452 (1957; Zbl 0084.17305)] classified vector bundles over such a curve. The novelty of this article is having computed free resolutions of such modules explicitly. The authors use the computer algebra system SINGULAR. This article contains some programmes.
scientific article; zbMATH DE number 1836726
scientific article