Vol. 7, No. 1, 2013

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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
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