On a construction of indecomposable modules and applications
The main purpose of this paper is to introduce a new method of getting a family \(\{M_ n| n\in {\mathbb{N}}^*\}\) of indecomposable modules over a commutative Noetherian local ring \((R,{\mathfrak m})\), when R possesses a finitely generated R-module C of depth greater than 1 (or equal) such that \(C\otimes_ R\hat R\) (\(\hat R\) is the completion of R with respect to the \({\mathfrak m}\)-adic topology) is indecomposable and the initial part of a minimal free resolution of C satisfies certain conditions: Each \(M_ n\) is a finitely generated R-module with \(\dim_ RM_ n=\dim_ RC\) and depth\(_ RM_ n=0\) and if C is Cohen-Macaulay, then \(M_ n\) is Buchsbaum. Furthermore \(M_ n/H_{{\mathfrak m}}(M_ n)\) is isomorphic to the direct sum of n copies of C (here \(H_{{\mathfrak m}}(M_ n)=\cup_{i\geq 1}[(0):{\mathfrak m}^ i]_ M ).\) The author applies his results to the Buchsbaum-representation theory in one-dimensional case. (A Noetherian local ring R has finite Buchsbaum- representation type if there are only finitely many isomorphism classes of indecomposable Buchsbaum R-modules M which are maximal, i.e. \(\dim_ RM=\dim (R).)\)
scientific article; zbMATH DE number 4130523