Maximal orders of global dimension and Krull dimension two
Let R be a (commutative) complete Noetherian two-dimensional local ring, with an algebraically closed residue field k of characteristic zero. The object of this paper is to classify the algebras A which are maximal R- orders and have global dimension 2. The case where \(A=R\) (so R is normal) is known - R is then a quotient singularity, so \(R=ks,t^ G\) where G is a small subgroup of \(GL_ 2(k)\), [\textit{J. Herzog}, Math. Ann. 233, 21-34 (1978; Zbl 0358.13009), \textit{H. Esnault}, J. Reine Angew. Math. 362, 63-71 (1985; Zbl 0553.14016), \textit{M. Auslander}, Almost split sequences and rational singularities (preprint)]. By a theorem of M. Auslander, this classification problem is equivalent to classifying the maximal R-orders in a division ring which have only finitely many isomorphism classes of indecomposable Cohen- Macaulay modules. The author classifies these orders, first for the case where R is a power series ring, and finally for the general case where R is a quotient singularity.
scientific article; zbMATH DE number 3948515
scientific article