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

Chardin, Marc

doi:10.2140/ant.2013.7.1

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

Marc Chardin

Vol. 7 (2013), No. 1, 1–18

DOI: 10.2140/ant.2013.7.1

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.

À Jean-Pierre Jouanolou, avec admiration et amitié

Keywords

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

Mathematical Subject Classification 2000

Primary: 13D02

Secondary: 13A30, 13D45, 14A15

Milestones

Received: 22 June 2010

Revised: 10 January 2012

Accepted: 7 February 2012

Published: 28 March 2013

Authors

Marc Chardin
Institut de Mathématiques de Jussieu CNRS & UPMC 4, place Jussieu F-75005 Paris France

Vol. 7, No. 1, 2013