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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Indecomposable modules of large rank over Cohen-Macaulay local rings
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by Wolfgang Hassler, Ryan Karr, Lee Klingler and Roger Wiegand PDF
Trans. Amer. Math. Soc. 360 (2008), 1391-1406 Request permission

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
  • MR Author ID: 228650
  • Roger Wiegand
  • Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0130
  • MR Author ID: 205253
  • 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.
  • © Copyright 2007 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 1391-1406
  • MSC (2000): Primary 13C05, 13E05, 13H10
  • DOI: https://doi.org/10.1090/S0002-9947-07-04226-2
  • MathSciNet review: 2357700