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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Indecomposable modules of large rank over Cohen-Macaulay local rings


Authors: Wolfgang Hassler, Ryan Karr, Lee Klingler and Roger Wiegand
Journal: Trans. Amer. Math. Soc. 360 (2008), 1391-1406
MSC (2000): Primary 13C05, 13E05, 13H10
Published electronically: October 3, 2007
MathSciNet review: 2357700
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Abstract: A commutative Noetherian local ring $ (R,\mathfrak{m},k)$ is called Dedekind-like provided $ R$ is one-dimensional and reduced, the integral closure $ \overline{R}$ is generated by at most 2 elements as an $ R$-module, and $ \mathfrak{m}$ is the Jacobson radical of $ \overline{R}$. If $ M$ is an indecomposable finitely generated module over a Dedekind-like ring $ R$, and if $ P$ is a minimal prime ideal of $ R$, it follows from a classification theorem due to L. Klingler and L. Levy that $ M_P$ must be free of rank 0, 1 or 2.

Now suppose $ (R,\mathfrak{m},k)$ is a one-dimensional Cohen-Macaulay local ring that is not Dedekind-like, and let $ P_1,\dotsc,P_t$ be the minimal prime ideals of $ R$. The main theorem in the paper asserts that, for each non-zero $ t$-tuple $ (n_1,\dotsc,n_t)$ of non-negative integers, there is an infinite family of pairwise non-isomorphic indecomposable finitely generated $ R$-modules $ M$ satisfying $ M_{P_i}\cong(R_{P_i})^{(n_i)}$ for each $ i$.


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Additional Information

Wolfgang Hassler
Affiliation: Institut für Mathematik und wissenschaftliches Rechnen, Karl-Franzens-Universi- tät Graz, Heinrichstraße 36/IV, A-8010 Graz, Austria

Ryan Karr
Affiliation: Honors College, Florida Atlantic University, Jupiter, Florida 33458

Lee Klingler
Affiliation: Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, Florida 33431-6498

Roger Wiegand
Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0130

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04226-2
PII: S 0002-9947(07)04226-2
Received by editor(s): November 2, 2004
Received by editor(s) in revised form: October 14, 2005
Published electronically: October 3, 2007
Additional Notes: The first author’s research was supported by a grant from the Fonds zur Förderung der wissenschaftlichen Forschung, project number P16770–N12. The fourth author was partially supported by a grant from the National Science Foundation. The third author thanks the University of Nebraska-Lincoln, where much of the research was completed.
Article copyright: © Copyright 2007 American Mathematical Society