Reduction of isolated singularities
Let \(f\in {\mathbb{C}}[X_ 0,...,X_ d]\) be a polynomial such that the hypersurface defined by f has an isolated singularity. Denote by A the complete local ring \({\mathbb{C}}X_ 0,...,X_ d/(f)\). It is known that A has a simple singularity iff A has only finitely many isomorphism classes of indecomposable Cohen-Macaulay modules (Knörrer, Buchweitz- Greuel-Schreyer). In this case the Auslander-Reiten quiver Q(A) of A is determined and turns out to be closely related to the Dynkin diagram of A. The author begins the study of the category of Cohen-Macaulay A-modules in the case when A has an isolated but not a simple singularity, proving the following results: (1) Let C be the connected component of Q(A) containing A. Then \(Q(A)- C={\dot \cup}_{i\in I}A_{\infty}/<x^{n(i)}>,\) where I is an index set and \(n(i)\in\{1,2\}\). Moreover, if d is even, \(n(i)=1\) for all i. (2) There exists an arithmetic sequence of positive integers r, 2r, 3r,... such that for each mr \((m\geq 1)\) there exists an infinite sequence \((M_{m,n})_{n\geq 1}\) of indecomposable pairwise non-isomorphic Cohen- Macaulay A-modules all of which of rank mr.
scientific article; zbMATH DE number 4067075