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 $i$.References
- Hyman Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8–28. MR 153708, DOI 10.1007/BF01112819
- Nuri Cimen, Roger Wiegand, and Sylvia Wiegand, One-dimensional rings of finite representation type, Abelian groups and modules (Padova, 1994) Math. Appl., vol. 343, Kluwer Acad. Publ., Dordrecht, 1995, pp. 95–121. MR 1378192
- Ju. A. Drozd and A. V. Roĭter, Commutative rings with a finite number of indecomposable integral representations, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 783–798 (Russian). MR 0220716
- W. Hassler, R. Karr, L. Klingler, and R. Wiegand, Large indecomposable modules over local rings, J. Algebra 303 (2006), no. 1, 202–215. MR 2253659, DOI 10.1016/j.jalgebra.2006.05.016
- Wolfgang Hassler and Roger Wiegand, Direct sum cancellation for modules over one-dimensional rings, J. Algebra 283 (2005), no. 1, 93–124. MR 2102074, DOI 10.1016/j.jalgebra.2004.05.011
- Lee Klingler and Lawrence S. Levy, Representation type of commutative Noetherian rings. I. Local wildness, Pacific J. Math. 200 (2001), no. 2, 345–386. MR 1868696, DOI 10.2140/pjm.2001.200.345
- Lee Klingler and Lawrence S. Levy, Representation type of commutative Noetherian rings. I. Local wildness, Pacific J. Math. 200 (2001), no. 2, 345–386. MR 1868696, DOI 10.2140/pjm.2001.200.345
- Lee Klingler and Lawrence S. Levy, Representation type of commutative Noetherian rings. III. Global wildness and tameness, Mem. Amer. Math. Soc. 176 (2005), no. 832, viii+170. MR 2147090, DOI 10.1090/memo/0832
- Lawrence S. Levy and Roger Wiegand, Dedekind-like behavior of rings with $2$-generated ideals, J. Pure Appl. Algebra 37 (1985), no. 1, 41–58. MR 794792, DOI 10.1016/0022-4049(85)90086-6
- Graham J. Leuschke and Roger Wiegand, Hypersurfaces of bounded Cohen-Macaulay type, J. Pure Appl. Algebra 201 (2005), no. 1-3, 204–217. MR 2158755, DOI 10.1016/j.jpaa.2004.12.028
- Graham J. Leuschke and Roger Wiegand, Local rings of bounded Cohen-Macaulay type, Algebr. Represent. Theory 8 (2005), no. 2, 225–238. MR 2162283, DOI 10.1007/s10468-004-8319-5
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
- R. B. Warfield Jr., Decomposability of finitely presented modules, Proc. Amer. Math. Soc. 25 (1970), 167–172. MR 254030, DOI 10.1090/S0002-9939-1970-0254030-4
- Roger Wiegand, Noetherian rings of bounded representation type, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 497–516. MR 1015536, DOI 10.1007/978-1-4612-3660-3_{2}7
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