Totally reflexive modules constructed from smooth projective curves of genus \(g \geq 2\) (Q2642248)

The purpose of the paper is to construct a non-Gorenstein Cohen-Macaulay domain with a non-free totally reflexive module from any given smooth projective curve of genus at least two. More precisely: Let \(C\) be a smooth projective curve of genus \(g\) over an algebraically closed field \(k.\) Then: (1) There is a Weil divisor \(D\) on \(C\) of degree \(g+1\) such that \({\mathcal O}_C(D)\) is generated by its global sections and satisfies \(\dim_k H^0(C, {\mathcal O}_C(D)) = 2\) and \(H^1({\mathcal O}_C(D)) = 0.\) (2) Set \(R = \bigoplus_{n \geq 0} H^0(C, {\mathcal O}_C(2nD))\) and \(M = \bigoplus_{n \geq 0} H^0(C, {\mathcal O}_C((2n+1)D)).\) Then \(R\) is a standard graded \(k\)-algebra of dimension 2 and Cohen-Macaulay type \(g.\) \(M\) is a finitely generated graded non-free totally reflexive \(R\)-module of rank \(1\) with a linear periodic free resolution (over \(R\)) defined by a \(2 \times 2\)-matrix with linear entries. Here totally reflexive means a module of Gorenstein dimension 0. Over a Gorenstein local ring totally reflexive modules are precisely the maximal Cohen-Macaulay modules. So the construction of the authors provide further examples of non-free totally reflexive modules over non-Gorenstein rings. Moreover this is the first application of \(G\)-dimensions in algebraic geometry.