Algebra & Number Theory

Powers of ideals and the cohomology of stalks and fibers of morphisms

Marc Chardin

Full-text: Open access

Abstract

We first provide here a very short proof of a refinement of a theorem of Kodiyalam and Cutkosky, Herzog and Trung on the regularity of powers of ideals. This result implies a conjecture of Hà and generalizes a result of Eisenbud and Harris concerning the case of ideals primary for the graded maximal ideal in a standard graded algebra over a field. It also implies a new result on the regularities of powers of ideal sheaves. We then compare the cohomology of the stalks and the cohomology of the fibers of a projective morphism to the effect of comparing the maximums over fibers and over stalks of the Castelnuovo–Mumford regularities of a family of projective schemes.

Article information

Source
Algebra Number Theory, Volume 7, Number 1 (2013), 1-18.

Dates
Received: 22 June 2010
Revised: 10 January 2012
Accepted: 7 February 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729927

Digital Object Identifier
doi:10.2140/ant.2013.7.1

Mathematical Reviews number (MathSciNet)
MR3037888

Zentralblatt MATH identifier
1270.13008

Subjects
Primary: 13D02: Syzygies, resolutions, complexes
Secondary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13D45: Local cohomology [See also 14B15] 14A15: Schemes and morphisms

Keywords
cohomology stalks Rees algebras fibers of morphisms powers of ideals Castelnuovo–Mumford regularity

Citation

Chardin, Marc. Powers of ideals and the cohomology of stalks and fibers of morphisms. Algebra Number Theory 7 (2013), no. 1, 1--18. doi:10.2140/ant.2013.7.1. https://projecteuclid.org/euclid.ant/1513729927


Export citation

References

  • M. André, Homologie des algèbres commutatives, Die Grundlehren der mathematischen Wissenschaften 206, Springer, Berlin, 1974.
  • N. Bourbaki, Éléments de mathématique: Algèbre, Chapitre 10: Algèbre homologique, Masson, Paris, 1980.
  • N. Bourbaki, Éléments de mathématique: Algèbre, Chapitres 1 á 4, Masson, Paris, 1985.
  • M. Chardin, “Some results and questions on Castelnuovo–Mumford regularity”, pp. 1–40 in Syzygies and Hilbert functions, Lect. Notes Pure Appl. Math. 254, Chapman & Hall/CRC, Boca Raton, FL, 2007.
  • M. Chardin, A. L. Fall, and U. Nagel, “Bounds for the Castelnuovo–Mumford regularity of modules”, Math. Z. 258:1 (2008), 69–80.
  • S. D. Cutkosky, “Irrational asymptotic behaviour of Castelnuovo–Mumford regularity”, J. Reine Angew. Math. 522 (2000), 93–103.
  • S. D. Cutkosky, J. Herzog, and N. V. Trung, “Asymptotic behaviour of the Castelnuovo–Mumford regularity”, Compositio Math. 118:3 (1999), 243–261.
  • S. D. Cutkosky, L. Ein, and R. Lazarsfeld, “Positivity and complexity of ideal sheaves”, Math. Ann. 321:2 (2001), 213–234.
  • K. Divaani-Aazar, R. Naghipour, and M. Tousi, “Cohomological dimension of certain algebraic varieties”, Proc. Amer. Math. Soc. 130 (2002), 3537–3544.
  • D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Springer, New York, 1995.
  • D. Eisenbud and J. Harris, “Powers of ideals and fibers of morphisms”, Math. Res. Lett. 17:2 (2010), 267–273.
  • H. T. Hà, “Asymptotic linearity of regularity and $a\sp \ast$-invariant of powers of ideals”, Math. Res. Lett. 18:1 (2011), 1–9.
  • R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer, New York, 1977.
  • V. Kodiyalam, “Asymptotic behaviour of Castelnuovo–Mumford regularity”, Proc. Amer. Math. Soc. 128:2 (2000), 407–411.
  • M. Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics 13, Wiley, New York-London, 1962.
  • W. Niu, “Some results on asymptotic regularity of ideal sheaves”, J. Algebra 377 (2013), 157–172.
  • N. V. Trung and H.-J. Wang, “On the asymptotic linearity of Castelnuovo–Mumford regularity”, J. Pure Appl. Algebra 201 (2005), 42–48.