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

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

2010-09-07

arxiv.org/abs/1009.1271v2

Mathematics

Commutative Algebra

Scientific paper

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 H\`a 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.

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