Effective generic freeness and applications to local cohomology

TL;DR

This paper investigates the conditions under which local cohomology modules are generically free over a Noetherian domain and introduces an effective Gr"obner basis method to explicitly determine generic freeness.

Contribution

It establishes generic freeness of local cohomology modules over smooth algebras and develops a Gr"obner basis approach for explicit computation of generic freeness.

Findings

Local cohomology modules are generically free over $A$ under certain smoothness conditions.

A Gr"obner basis method is provided for explicit determination of generic freeness.

The approach applies to finitely generated modules over Noetherian rings.

Abstract

Let $A$ be a Noetherian domain and $R$ be a finitely generated $A$-algebra. We study several features regarding the generic freeness over $A$ of an $R$-module. For an ideal $I \subset R$, we show that the local cohomology modules ${\rm H}_I^i(R)$ are generically free over $A$ under certain settings where $R$ is a smooth $A$-algebra. By utilizing the theory of Gr\"obner bases over arbitrary Noetherian rings, we provide an effective method to make explicit the generic freeness over $A$ of a finitely generated $R$-module.