Castelnuovo-Mumford regularity of Ext modules and homological degree

Authors:
Marc Chardin, Dao Thanh Ha and Lê Tuân Hoa

Journal:
Trans. Amer. Math. Soc. **363** (2011), 3439-3456

MSC (2000):
Primary 13D45

Published electronically:
February 8, 2011

MathSciNet review:
2775813

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Abstract | References | Similar Articles | Additional Information

Abstract: Bounds for the Castelnuovo-Mumford regularity of Ext modules, over a polynomial ring over a field, are given in terms of the initial degrees, Castelnuovo-Mumford regularities and the number of generators of the two graded modules involved. These general bounds are refined in the case where the second module is the ring. Other estimates, for instance on the size of graded pieces of these modules, are given. We also derive a bound on the homological degree in terms of the Castelnuovo-Mumford regularity. This answers positively a question raised by Vasconcelos.

**[CD]**Marc Chardin and Kamran Divaani-Aazar,*Generalized local cohomology and regularity of Ext modules*, J. Algebra**319**(2008), no. 11, 4780–4797. MR**2416743**, 10.1016/j.jalgebra.2007.11.032**[CFN]**Marc Chardin, Amadou Lamine Fall, and Uwe Nagel,*Bounds for the Castelnuovo-Mumford regularity of modules*, Math. Z.**258**(2008), no. 1, 69–80. MR**2350034**, 10.1007/s00209-007-0157-9**[DGV]**Luisa Rodrigues Doering, Tor Gunston, and Wolmer V. Vasconcelos,*Cohomological degrees and Hilbert functions of graded modules*, Amer. J. Math.**120**(1998), no. 3, 493–504. MR**1623400****[HaH]**Dao Thanh Ha and Lê Tuân Hoa,*Castelnuovo-Mumford regularity of some modules*, Comm. Algebra**36**(2008), no. 3, 992–1004. MR**2394265**, 10.1080/00927870701776748**[H]**Lê Tuân Hoa,*Finiteness of Hilbert functions and bounds for Castelnuovo-Mumford regularity of initial ideals*, Trans. Amer. Math. Soc.**360**(2008), no. 9, 4519–4540. MR**2403695**, 10.1090/S0002-9947-08-04424-3**[HHy1]**Lê Tuân Hoa and Eero Hyry,*Castelnuovo-Mumford regularity of initial ideals*, J. Symbolic Comput.**38**(2004), no. 5, 1327–1341. MR**2168718**, 10.1016/j.jsc.2004.04.001**[HHy2]**Lê Tuân Hoa and Eero Hyry,*Castelnuovo-Mumford regularity of canonical and deficiency modules*, J. Algebra**305**(2006), no. 2, 877–900. MR**2266858**, 10.1016/j.jalgebra.2006.05.001**[Na]**Uwe Nagel,*Comparing Castelnuovo-Mumford regularity and extended degree: the borderline cases*, Trans. Amer. Math. Soc.**357**(2005), no. 9, 3585–3603. MR**2146640**, 10.1090/S0002-9947-04-03595-0**[Sc]**Peter Schenzel,*Dualisierende Komplexe in der lokalen Algebra und Buchsbaum-Ringe*, Lecture Notes in Mathematics, vol. 907, Springer-Verlag, Berlin-New York, 1982 (German). With an English summary. MR**654151****[Va]**Wolmer V. Vasconcelos,*Computational methods in commutative algebra and algebraic geometry*, Algorithms and Computation in Mathematics, vol. 2, Springer-Verlag, Berlin, 1998. With chapters by David Eisenbud, Daniel R. Grayson, Jürgen Herzog and Michael Stillman. MR**1484973**

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Additional Information

**Marc Chardin**

Affiliation:
Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4, place Jussieu, F-75005 Paris, France

Email:
chardin@math.jussieu.fr

**Dao Thanh Ha**

Affiliation:
Department of Mathematics, University of Vinh, Vietnam

Email:
thahanh@yahoo.com

**Lê Tuân Hoa**

Affiliation:
Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam

Email:
lthoa@math.ac.vn

DOI:
http://dx.doi.org/10.1090/S0002-9947-2011-05062-2

Keywords:
Castelnuovo-Mumford regularity,
local cohomology,
canonical module,
deficiency module,
homological degree.

Received by editor(s):
February 6, 2009

Published electronically:
February 8, 2011

Additional Notes:
The second and third authors were supported in part by the National Basic Research Program (Vietnam). The third author would also like to thank University of Paris 6 for their financial support and hospitality during his visit in 2007 when this work was started.

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.