Rings with only finitely many isomorphism classes of indecomposable maximal Buchsbaum modules
The main theorem of this article is as follows: Let \(P=kX_ 1,X_ 2,...,X_ n\) be a formal power series ring over an algebraically closed field k of characteristic \(k\neq 2.\) Let \(R=P/I\), where I is an ideal of P and suppose that \(\dim (R)=d\geq 2\). Then the following two conditions are equivalent: (1) R is a regular local ring. (2) R is a Cohen-Macaulay ring that possesses only finitely many isomorphism classes of indecomposable maximal Buchsbaum modules. When this is the case, the syzygy modules of the residue class field k of R are the representatives of indecomposable maximal Buchsbaum modules and so there are exactly d non-isomorphic indecomposable maximal Buchsbaum modules over R. The implication \((1) \Rightarrow (2)\) and the last assertion has been proved by the first author [in Commutative algebra and combinatorics, US- Jap. joint Semin., Kyoto 1985, Adv. Stud. Pure Math. 11, 39-64 (1987; Zbl 0649.13009)] and this article is proving the implication \((2) \Rightarrow (1)\). Furthermore, the authors are showing that if we drop the assumption \(\dim(R)\geq 2\), then \(R=kX,Y/(X^ 3+Y^ 2)\) gives us a one-dimensional counterexample to the assertion.
scientific article; zbMATH DE number 4073140