Rank-sets of indecomposable modules over one-dimensional rings
A ring \(R\) is a \textit{ring-order}, if it is a reduced, one-dimensional, commutative noetherian ring with integral closure a finitely generated \(R-\)module. Let \(M\) be a finitely generated, torsion-free \(R-\)module. If \(P_1,\ldots,P_s\) are the minimal prime ideals of \(R\), define the \textit{rank} of \(M\) to be \(rank(M)=(r_1,\ldots,r_s),\) where \(r_i:=\dim_{R_{P_i}}(M_{P_i}), 1\leq i\leq s\). The set \(\text{Ranks}(M)=\{r_1,\ldots,r_s\}\) is called the rank-set of \(M\). \(M\) has \textit{constant rank} \(N\) if \(\text{rank}(M)=(N,\ldots,N)\). \(R\) has \textit{bounded representation type} if there exists an \(N\) such that for each indecomposable finitely generated torsion-free \(R-\)module \(M\), \(rank(M)\leq (N,\ldots,N)\). The article fits into the problem of describing the isomorphism classes of indecomposable modules over one-dimensional rings, by investigating what subsets of the integers are realizable as the rank-sets of indecomposable modules (Question 1.3). By \textit{M. Arnavut, M. Luckas} and \textit{S. Wiegand} [J. Pure Appl. Algebra 208, No. 2, 739--760 (2007; Zbl 1111.13007)] and [\textit{N. Baeth, M. Luckas}, Bounds for indecomposable torsion-free modules, J. Pure Appl. Algebra (submitted)], the rank-set of every indecomposable module is either a set or a proper subset of a set listed in Table 1.5. For example, from this table, there are no indecomposable modules of constant rank 5. The Main Theorem 1.6. guarantees that for every set in Table 1.5. there exist a semilocal ring-order \(R\) of bounded representation type and an indecomposable finitely generated torsion-free \(R-\)module with rank-set exactly that set. The key of the Main Theorem is Proposition 3.1 which enables to construct indecomposable modules of rank \((a_1,\ldots,a_s,t), t\leq 2a_{\ell}\) for some \(1\leq\ell\leq s\), from indecomposable modules of rank \((a_1,\ldots,a_s)\). The base for this construction are three local ring-orders of bounded representation type described in [\textit{R. Wiegand, S. Wiegand}, J. Pure Appl. Algebra 93, No. 3, 311--342 (1994; Zbl 0813.13013)]. The last section of the article studies the case of proper subsets of sets in Table 1.5., and generalizes Proposition 3.1. to some beautiful numerical results. The whole article is very carefully written, with lots of examples, clear explanations and ingenious detailed proofs. I found very useful Section 2 about the background results needed in the proofs and the comments and parenthesis in the text, which also give a more insight look to the problem treated.
scientific article; zbMATH DE number 5354972
scientific article