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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotic behavior of dimensions of syzygies


Authors: Kristen A. Beck and Micah J. Leamer
Journal: Proc. Amer. Math. Soc. 141 (2013), 2245-2252
MSC (2010): Primary 13C15, 13D02, 13E05; Secondary 13D45
Published electronically: March 8, 2013
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ R$ be a commutative noetherian local ring and $ M$ be a finitely generated $ R$-module of infinite projective dimension. It is well-known that the depths of the syzygy modules of $ M$ eventually stabilize to the depth of $ R$. In this paper, we investigate the conditions under which a similar statement can be made regarding dimension. In particular, we show that if $ R$ is equidimensional and the Betti numbers of $ M$ are eventually non-decreasing, then the dimension of any sufficiently high syzygy module of $ M$ coincides with the dimension of $ R$.


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Additional Information

Kristen A. Beck
Affiliation: Department of Mathematics, University of Texas at Arlington, P. O. Box 19408, Arlington, Texas 76019-0408
Email: kbeck@uta.edu

Micah J. Leamer
Affiliation: Department of Mathematics, University of Nebraska-Lincoln, P. O. Box 880130, Lincoln, Nebraska 68588-0130
Email: s-mleamer1@math.unl.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11510-8
PII: S 0002-9939(2013)11510-8
Received by editor(s): June 22, 2011
Received by editor(s) in revised form: October 14, 2011
Published electronically: March 8, 2013
Additional Notes: This material is based on work that began at the 2011 Mathematical Research Community in Commutative Algebra, located in Snowbird, UT. The MRC was funded by the American Mathematical Society and the National Science Foundation.
The first author was partially supported by NSA Grant H98230-07-1-0197.
The second author was funded in part by a GAANN grant from the Department of Education. Part of this work also appears in the second author’s Ph.D. thesis.
Communicated by: Irena Peeva
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.