Some genericity results over Noetherian rings

TL;DR

This paper investigates generic properties in filtered modules over Noetherian rings, establishing that regular sequences are generic and deriving various corollaries related to module grades, ideal heights, and homological properties.

Contribution

It proves that being a regular sequence is a generic property in filtered modules over Noetherian rings, with applications to several algebraic invariants and complexes.

Findings

Regular sequences are generic in filtered modules.

Corollaries on generic grades and heights of modules and ideals.

Results on acyclicity and vanishing of Tor/Ext.

Abstract

Let M be a filtered module. Some properties of elements of M are "generic" in the following sense: (being open/stable) if an element z of M has a property P then any approximation of z has P; (being dense) any element of M is approximated by an element that has P. (Here the approximation is taken in the filtered sense.) \\ Moreover, one can often ensure an approximation with further special properties, e.g. avoiding a prescribed set of submodules. We prove that being a regular sequence is a generic property. As immediate applications we get corollaries on the generic grades of modules, heights of ideals, properties of determinantal ideals, acyclicity of generalized Eagon-Northcott complexes and vanishing of Tor/Ext.