Algebra & Number Theory

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

Marc Chardin

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
Revised: 10 January 2012
Accepted: 7 February 2012
First available in Project Euclid: 20 December 2017

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

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

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.