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

2010/09/07 by Chardin, Marc
#13A30 #13D02 #13D45 #14A15 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1009.1271

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 maximum over fibers and over stalks of the Castelnuovo-Mumford regularities of a family of projective schemes.

Related

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• Fiber invariants of projective morphisms and regularity of powers of ideals
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• Row Ideals and Fibers of Morphisms