Asymptotic behavior of dimensions of syzygies
Authors:
Kristen A. Beck and Micah J. Leamer
Journal:
Proc. Amer. Math. Soc. 141 (2013), 22452252
MSC (2010):
Primary 13C15, 13D02, 13E05; Secondary 13D45
Published electronically:
March 8, 2013
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Abstract 
References 
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Additional Information
Abstract: Let be a commutative noetherian local ring and be a finitely generated module of infinite projective dimension. It is wellknown 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 nondecreasing, then the dimension of any sufficiently high syzygy module of coincides with the dimension of .
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S. Okiyama, A local ring is CM if and only if its residue field has a CM syzygy, Tokyo J. Math. 14 (1991), 489500. MR 1138183 (92m:13017)
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P. Roberts, Le théorème d'intersection, C. R. Acad. Sci., Paris Sér. I 304 (1987), 177180. MR 880574 (89b:14008)
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L.C. Sun, Growth of Betti numbers of modules over rings of small embedding codimension or small linkage number, J. Pure Appl. Algebra 96 (1994), 5771. MR 1297441 (95j:13014)
 12.
L.C. Sun, Growth of Betti numbers of modules over generalized Golod rings, J. Algebra 199 (1998), 8893. MR 1489355 (98i:13023)
 1.
 L. L. Avramov, Local algebra and rational homotopy, Homotopie algébrique et algèbre locale; Luminy, 1982 (J.M. Lemaire, J.C. Thomas, eds.), Astérisque 113114, Soc. Math. France, Paris, 1984, 1543. MR 749041 (85j:55021)
 2.
 L. L. Avramov, Infinite free resolutions, Six lectures on commutative algebra (Bellaterra, 1996), Prog. Math. 166, Birkhäuser, Basel, 1998, 1118. MR 1648664 (99m:13022)
 3.
 L. L. Avramov, V. N. Gasharov, I. V. Peeva, Complete intersection dimension, Publ. Math. I.H.E.S. 86 (1997), 67114. MR 1608565 (99c:13033)
 4.
 S. Choi, Betti numbers and the integral closure of an ideal, Math. Scand. 66 (1990), 173184. MR 1075135 (91m:13007)
 5.
 A. Crabbe, D. Katz, J. Striuli, E. Theodorescu, HilbertSamuel polynomials for the contravariant extension functor, Nagoya Math. J. 198 (2010), 122. MR 2666575 (2011g:13033)
 6.
 D. R. Grayson, M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/
 7.
 S. B. Iyengar, G. J. Leuschke, A. Leykin, C. Miller, E. Miller, A. K. Singh, and U. Walther, Twentyfour hours of local cohomology, Graduate Studies in Mathematics, 87, American Mathematical Society, Providence, RI, 2007. MR 2355715 (2009a:13025)
 8.
 J. Lescot, Séries de Poincaré et modules inertes, J. Algebra 132 (1990), 2249. MR 1060830 (91k:13010)
 9.
 S. Okiyama, A local ring is CM if and only if its residue field has a CM syzygy, Tokyo J. Math. 14 (1991), 489500. MR 1138183 (92m:13017)
 10.
 P. Roberts, Le théorème d'intersection, C. R. Acad. Sci., Paris Sér. I 304 (1987), 177180. MR 880574 (89b:14008)
 11.
 L.C. Sun, Growth of Betti numbers of modules over rings of small embedding codimension or small linkage number, J. Pure Appl. Algebra 96 (1994), 5771. MR 1297441 (95j:13014)
 12.
 L.C. Sun, Growth of Betti numbers of modules over generalized Golod rings, J. Algebra 199 (1998), 8893. MR 1489355 (98i:13023)
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Additional Information
Kristen A. Beck
Affiliation:
Department of Mathematics, University of Texas at Arlington, P. O. Box 19408, Arlington, Texas 760190408
Email:
kbeck@uta.edu
Micah J. Leamer
Affiliation:
Department of Mathematics, University of NebraskaLincoln, P. O. Box 880130, Lincoln, Nebraska 685880130
Email:
smleamer1@math.unl.edu
DOI:
http://dx.doi.org/10.1090/S000299392013115108
PII:
S 00029939(2013)115108
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 H982300710197.
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.
