Transactions of the American Mathematical Society

Comparing Castelnuovo-Mumford regularity and extended degree: The borderline cases

Author(s): Uwe Nagel
Journal: Trans. Amer. Math. Soc. 357 (2005), 3585-3603.
MSC (2000): Primary 13D40, 13D45; Secondary 13P10, 14M05
Posted: October 28, 2004
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Abstract: Castelnuovo-Mumford regularity and any extended degree function can be thought of as complexity measures for the structure of finitely generated graded modules. A recent result of Doering, Gunston, and Vasconcelos shows that both can be compared in the case of a graded algebra. We extend this result to modules and analyze when the estimate is in fact an equality. A complete classification is obtained if we choose as extended degree the homological or the smallest extended degree. The corresponding algebras are characterized in three ways: by relations among the algebra generators, by using generic initial ideals, and by their Hilbert series.

1.: R. Achilles, P. Schenzel, On bounds for Castelnuovo's index of regularity, J. Math. Kyoto Univ. 29 (1989), 91-104. MR 90i:14002
2.: M. Auslander, D. A. Buchsbaum, Codimension and multiplicity, Ann. Math. 68 (1958), 625-657. MR 20:6414
3.: D. Bayer, M. Stillman, A criterion for detecting -regularity, Invent. Math. 87 (1987), 1-11. MR 87k:13019
4.: M. Brodmann, U. Nagel, Bounding cohomological Hilbert functions by hyperplane sections, J. Algebra 174 (1995), 323-348. MR 96d:14017
5.: L. R. Doering, T. Gunston, W. V. Vasconcelos, Cohomological degrees and Hilbert functions of graded modules, Amer. J. Math. 120 (1998), 493-504. MR 99h:13019
6.: D. Eisenbud, ``Commutative algebra with a view toward algebraic geometry,'' Graduate Texts in Mathematics 150, Springer-Verlag, 1995.MR 97a:13001
7.: M. Green, Generic initial ideals, In: ``Six lectures on commmutative algebra'' (J. Elias, J. M. Giral, R. M. Miro-Roig, S. Zarzuela, Eds.), Progress in Mathematics 166, Birkhäuser, 1998, pp. 119-185. MR 99m:13040
8.: T. Gunston, Cohomological degrees, Dilworth numbers and linear resolution, Ph.D. Thesis, Rutgers University, 1998.
9.: J. Migliore, U. Nagel, C. Peterson, Bezout's theorem and Cohen-Macaulay modules, Math. Z. 237 (2001), 373-394. MR 2002e:13030
10.: U. Nagel, Castelnuovo-Regularität und Hilbertreihen, Math. Nachr. 142 (1989), 27-43. MR 90m:13025
11.: U. Nagel, On Castelnuovo's regularity and Hilbert functions, Compositio Math. 76 (1990), 265-275. MR 92c:14003
12.: K. Pardue, Nonstandard Borel-fixed ideals, Ph.D. Thesis, Brandeis University, 1994.
13.: M. E. Rossi, N. V. Trung, G. Valla, Castelnuovo-Mumford regularity and extended degree, Trans. Amer. Math. Soc. 355 (2003), 1773-1786. MR 2004b:13020
14.: J. Stückrad, W. Vogel, Castelnuovo's regularity and multiplicity, Math. Ann. 281 (1988), 355-368. MR 90b:14059a
15.: W. V. Vasconcelos, Cohomological degrees of graded modules, In: ``Six lectures on commutative algebra'' (J. Elias, J. M. Giral, R. M. Miro-Roig, S. Zarzuela, Eds.), Progress in Mathematics 166, Birkhäuser, Basel, 1998, pp. 345-392. MR 99j:13012
16.: W. V. Vasconcelos, The homological degree of a module, Trans. Amer. Math. Soc. 350 (1998), 1167-1179. MR 98i:13046
17.: O. Zarisky, P. Samuel, ``Commutative algebra,'' Vol. II, van Nostrand, Toronto, New York, London, 1960. MR 22:11006

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13D40, 13D45, 13P10, 14M05

Uwe Nagel
Affiliation: Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506-0027
Email: uwenagel@ms.uky.edu

DOI: 10.1090/S0002-9947-04-03595-0
PII: S 0002-9947(04)03595-0
Keywords: Castelnuovo-Mumford regularity, extended degree, generic initial ideal, Hilbert series, homological degree, smallest extended degree
Received by editor(s): April 2, 2003
Received by editor(s) in revised form: December 5, 2003
Posted: October 28, 2004
Copyright of article: Copyright 2004, American Mathematical Society

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