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

TL;DR

This paper refines existing theorems on the regularity of ideal powers, proving a conjecture, generalizing prior results, and comparing cohomology of stalks and fibers in projective morphisms.

Contribution

It provides a new, concise proof of a regularity refinement, proves a conjecture, and compares cohomology of stalks and fibers in projective morphisms.

Findings

Refined theorem on regularity of ideal powers

Proved a conjecture of H",

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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 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.