Maximal Cohen-Macaulay modules over isolated singularities
This paper is divided in two sections. In the first one the following theorem is proved: Let \((R,{\mathfrak m}, K)\) be an excellent henselian Cohen-Macaulay local ring, with isolated singularity \(\text{Spec}(R)\) containing a field such that \(K\) is perfect or \([K:K^p] < \infty\) if \(\text{ch} (K)=p<0\). Let \(x=\{x_1, \dots, x_s\}\) be a system of parameters in \(R\) and \(R_t=R/(x^t_1, \dots, x_s^t)\), \(t\in\mathbb{N}\). Let \(L_{R_2}\) be the category of all finitely generated \(R_2\)-modules such that \(((x_1, \dots, x_{j-1}) P:x_j)_P=(x_1, \dots, x_j)P\), for all \(j=1, \dots, s\). Let \(\text{MCM} (R)\) be the category of maximal CM-modules of \(R\). Then the base change functor \(F_1: \text{MCM} (R)\to \text{Mod} (R_1)\), induces a bijection from the isomorphism classes of \(\text{MCM} (R)\) to the isomorphism classes of \(R_1\)-modules of the type \(x_1\dots x_sP\) for \(P\in L_{R_2}\). In the second section it is shown that the isomorphism classes of maximal Cohen-Macaulay \(R\)-modules and the isomorphism classes of modules of \(L_{R_2}\) can be seen as orbits of an affine group acting on a Grassmannian variety when the residue field of \(R\) is algebraically closed.
isomorphism classes of finitely generated modules
isomorphism classes of maximal Cohen-Macaulay modules
scientific article; zbMATH DE number 840726