Bounds for indecomposable torsion-free modules
Let \(M\) be a finitely generated torsion-free module over a one-dimensional reduced noetherian ring \(R\) with finitely generated normalization \(\widetilde{R}\) (i.e., \(R\) is a ring-order). The rank of \(M\) is the tuple of vector-space dimensions \(\dim_{R_P}M_P\), where \(P\) ranges over the minimal primes of \(R\). A ring-order \(R\) is said to have bounded representation type if there exists a bound \(N_R\) on the ranks of all indecomposable finitely generated torsion-free \(R\)-modules. \(M\) is said to have constant rank if all the dimensions \(\dim_{R_P}M_P\) are the same. Following a long list of results (see loc. cit.) this article studies what bounds and ranks occur for indecomposable torsion-free \(R\)-modules, when \(R\) is of bounded representation type. The main result (Main Theorem 1.2) of this paper states the following: If \(R\) is a local ring-order of bounded representation type, then \(R\) has at most three minimal primes. Moreover, if \(M\) is an indecomposable torsion-free finitely generated \(R\)-module then: (1) if \(R\) is a domain, then \(\text{rank}(M)\in\{1,2,3\}\); (2) if \(R\) has exactly two minimal primes then \(\text{rank}(M)\in\{(0,1),(1,0),(1,1),(1,2),(2,1), (2,2)\}\); (3) if \(R\) has exactly three minimal primes then, with a suitable ordering of the minimal primes, \[ \text{rank}(M)\in\{(0,0,1),(0,1,0),(1,0,0),(0,1,1),(1,0,1),(1,1,0),(1,1,1),(2,1,1)\}. \] Note that \(R\) is not necessarily equicharacteristic with perfect residue field and characteristic not 2,3 or 5 as previous results required [see \textit{N. R. Baeth}, J. Pure Appl. Algebra 208, No. 3, 923--940 (2007; Zbl 1106.13010); \textit{M. Arnavut, M. Luckas} and \textit{S. Wiegand} [J. Pure Appl. Algebra 208, No. 2, 739--760 (2007; Zbl 1111.13007)]. Essential to the argument is to observe (Theorem 2.2 and [\textit{N. Cimen, R. Wiegand} and \textit{S. Wiegand}, One-dimensional rings of finite Cohen-Macaulay type. in: Abelian groups and modules, Kluwer Academic, Doldrecht (1995)]) that the local ring-order of bounded representation type are Cohen-Macaulay local rings with finite Cohen-Macaulay type and therefore they satisfy the Drozd-Roiter conditions: \(R\) has multiplicity at most 3 and \((\underline{m}\widetilde{R}+R)/R\) is cyclic as an \(R\)-module, where \(\underline{m}\) is the maximal ideal of \(R\). With this, the proof of the main result relies on representing \(R\) as an Artinian pair [see \textit{E. L. Green, I. Reiner}, Mich. Math. J. 25, 53--84 (1978; Zbl 0365.16015); \textit{R. Wiegand}, in: Algebraic geometry and Application, Coll. Pap. Abhyankar's 60th birthday Conf., Purdue Univ. 1990, 381--389 (1994; Zbl 0813.13022); \textit{R. Wiegand}, J. Algebra 203, No. 1, 156--168 (1998; Zbl 0921.13015); \textit{R. Wiegand, S. Wiegand}, J. Pure Appl. Algebra 93, No. 3, 311--342 (1994; Zbl 0813.13013)]: consider \(\underline{c}=\{r\in R|r\widetilde{R}\subset R\}\) the conductor of \(R\) in \(\widetilde{R}\) and consider \(A:=R/\underline{c}\rightarrow\widetilde{R}/\underline{c}=: B\) the pullback of \(R\rightarrow\widetilde{R}\). \((A\rightarrow B)\) is a module-finite extension of Artinian rings, called Artinian pair. A module over an Artinian pair is a pair of modules \((V\rightarrow W)\), where \(W\) is a finitely generated projective \(B-\)module and \(V\) is an \(A\)-submodule of \(W\) with \(BV=W\). The module \((V\rightarrow W)\) has a matrix representation and the rank of this module matches the rank of its representation matrix. Using elementary row and column operations on this matrix (without changing the module) one can decompose the module it represents into direct sum of submodules of smaller ranks. This nice technique is used by the authors to prove the main result. For example, a module of rank \((4,2)\) is decomposable and from [\textit{R. Wiegand, S. Wiegand}, J. Pure Appl. Algebra 93, No. 3, 311--342 (1994; Zbl 0813.13013)] one can obtain part (2) in the main result (Proposition 3.4). At the end the authors study (Corollaries 5.3 and 5.4) the direct summands of constant rank maximal Cohen-Macauly \(R\)-modules; and also discuss what happens if \(R\) is not necessarily a local ring (Theorem 5.5). The article is very well written. We can find a very useful explanatory section (Section 2) with the definitions and background needed. It also describes previously known results about the problem studied, and the proofs of the results and the techniques used are presented carefully and with full details.
scientific article; zbMATH DE number 5544379
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