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

DOI:
https://doi.org/10.1090/S0002-9939-2013-11510-8

Published electronically:
March 8, 2013

MathSciNet review:
3043006

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Abstract | References | Similar Articles | Additional Information

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

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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:
https://doi.org/10.1090/S0002-9939-2013-11510-8

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.