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On the regularity of products and intersections of complete intersections


Authors: Marc Chardin, Nguyen Cong Minh and Ngo Viet Trung
Journal: Proc. Amer. Math. Soc. 135 (2007), 1597-1606
MSC (2000): Primary 13D02
DOI: https://doi.org/10.1090/S0002-9939-06-08842-3
Published electronically: December 27, 2006
MathSciNet review: 2286067
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper proves the formulae

$\displaystyle \operatorname{reg}(IJ)$ $\displaystyle \le \operatorname{reg}(I) + \operatorname{reg}(J),$    
$\displaystyle \operatorname{reg}(I \cap J)$ $\displaystyle \le \operatorname{reg}(I) + \operatorname{reg}(J)$    

for arbitrary monomial complete intersections $ I$ and $ J$, and provides examples showing that these inequalities do not hold for general complete intersections.


References [Enhancements On Off] (What's this?)

  • [BaM] D. Bayer and D. Mumford, What can be computed in algebraic geometry? In: D. Eisenbud and L. Robbiano (eds.), Computational Algebraic Geometry and Commutative Algebra, Proceedings, Cortona 1991, Cambridge University Press, 1993, 1-48. MR 1253986 (95d:13032)
  • [BCFH] H. Bresinsky, F. Curtis, M. Fiorentini, L. T. Hoa, On the structure of local cohomology modules for projective monomial curves in $ {\mathbb{P}}^3$. Nagoya Math. J. 136 (1994), 81-114. MR 1309382 (96b:14040)
  • [BrH] W. Bruns and J. Herzog, On multigraded resolutions, Math. Proc. Camb. Phil. Soc. 118 (1995), 245-257. MR 1341789 (96g:13013)
  • [Ch] M. Chardin, Regularity of ideals and their powers, Prépublication 364, Institut de Mathematiques de Jussieu, 2004.
  • [CD] M. Chardin, C. D'Cruz, Castelnuovo-Mumford regularity: examples of curves and surfaces, J. Algebra 270 (2003), 347-360. MR 2016666 (2004m:13036)
  • [CU] M. Chardin, B. Ulrich, Liaison and Castelnuovo-Mumford regularity, Amer. J. Math. 124 (2002), 1103-1124. MR 1939782 (2004c:14095)
  • [CoH] A. Conca and J. Herzog, Castelnuovo-Mumford regularity of products of ideals, Collect. Math. 54 (2003), 137-152. MR 1995137 (2004k:13020)
  • [DS] H. Derksen and J. Sidman, A sharp bound for the Castelnuovo-Mumford regularity of subspace arrangements, Adv. Math. 172 (2002), no. 2, 151-157. MR 1942401 (2003m:13013)
  • [E] D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer, 1994. MR 1322960 (97a:13001)
  • [HT] L. T. Hoa and N. V. Trung, On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals, Math. Zeits. 229 (1998), 519-537. MR 1658557 (99k:13034)
  • [M] N. C. Minh, On Castelnuovo-Mumford regularity of products of monomial ideals, Acta Math. Vietnam 30 (2005), 203-209. MR 2192521 (2006i:13034)
  • [Si] J. Sidman, On the Castelnuovo-Mumford regularity of products of ideal sheaves, Adv. Geom. 2 (2002), 219-229. MR 1924756 (2003f:13021)
  • [St] B. Sturmfels, Four counter examples in combinatorial algebraic geometry, J. Algebra 230 (2000), 282-294. MR 1774768 (2001g:13047)

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Additional Information

Marc Chardin
Affiliation: Institut de Mathématiques de Jussieu, CNRS & Université Paris VI, Paris, France
Email: chardin@math.jussieu.fr

Nguyen Cong Minh
Affiliation: Department of Mathematics, University of Education, 136 Xuân Thuy, Hanoi, Vietnam
Email: ngcminh@gmail.com

Ngo Viet Trung
Affiliation: Institute of Mathematics, Viên Toán Hoc, 18 Hoàng Quôc Viêt, 1037 Hanoi, Vietnam
Email: nvtrung@math.ac.vn

DOI: https://doi.org/10.1090/S0002-9939-06-08842-3
Received by editor(s): March 7, 2005
Received by editor(s) in revised form: February 6, 2006
Published electronically: December 27, 2006
Additional Notes: The second author was partially supported by the National Basic Research Program of Vietnam
Communicated by: Michael Stillman
Article copyright: © Copyright 2006 American Mathematical Society

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