Ranks of indecomposable modules over one-dimensional rings
Let R be an one-dimensional, reduced, commutative, Noetherian ring with finite dimension for which, locally, indecomposable torsionfree modules have all ranks at minimal primes \(\leq 1\). One may ask whether there exists a uniform global bound for the rank of these modules. Here, if \(P_ 1,...,P_ s\) are the minimal prime ideals of R, the rank of a module M is an s-tuple of non-negative integers \((a_ 1,...,a_ s)\) where \(a_ i\) is the dimension of \(M_{P_ i}\) as a vector space over \(R_{P_ i}.\) The answer is ``no''. \textit{J. Haefner} and \textit{L. S. Levy} [J. Pure Appl. Algebra 50, No.1, 1-20 (1988; Zbl 0653.13006)] have constructed, for every n, a ring R having an indecomposable module of rank \((1,1,1,2,...,2,4,...,2^ n)\). This paper shows that some upper and lower bounds on the ranks would force decomposition of torsionfree modules. Namely, if the ranks of a torsionfree module M at the various minimal primes are between n and 2n-2 for some integer \(n\geq 3\), then M decomposes. (This does not hold if the ranks are between n and 2n-1.) As a consequence, there are no torsionfree indecomposable modules of constant rank \(\geq 3\), although Levy has shown that such a ring can have an indecomposable torsionfree module of constant rank 2.
scientific article; zbMATH DE number 4071061