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Abstract
We consider one-dimensional, reduced Noetherian rings R with finite normalization. We assume that there exists a positive integer NR such that, for every indecomposable finitely generated torsion-free R-module M and for every minimal prime ideal P of R, the dimension of MP, as a vector space over the field RP, R localized at P, is less than or equal to NR. We call the set of vector-space dimensions the rank-set of the module. We often call the sequence of vector-space dimensions the rank of the module. We are interested in what rank-sets occur for indecomposable R-modules.
In Chapter 2, with Meral Arnavut and Sylvia Wiegand, many possible ranks are eliminated. In the constant rank case, that is, when we have a module M such that the vector-space dimension of MP is the same for every minimal prime P of R, the only ranks occurring for indecomposable modules are 1,2,3,4 and 6. In the non-constant rank case, we show that for n ≥ 8 there are no indecomposable modules with rank-sets between n and 2n − 8. On the other hand, for each n ≥ 8, we construct an indecomposable module with rank-set the set of consecutive integers from n to 2n − 7.
In Chapter 3, for each set of consecutive integers not ruled out in Chapter 2, we produce a semilocal ring and an indecomposable module over that ring having that set as its rank-set. Could other ranks occur for indecomposable modules? To answer this, we construct some indecomposable modules with ranks of non-consecutive integers. We then give some conditions to show some additional sets of non-consecutive integers never occur as the rank-sets of indecomposable modules.
In Chapter 4, with Nick Baeth, we prove the last case in a theorem listing the ranks that occur for indecomposable modules over a local ring. This result was previously known with an additional hypothesis on the characteristic of R and its residue field. This hypothesis was assumed in the work of Chapter 2, but now we have eliminated it, and so, the results in Chapter 2 hold in more generality.
