Openness of the regular locus and generators for module categories

TL;DR

This paper explores how the openness of the regular locus in a commutative Noetherian ring influences the existence of generators in various module categories, linking geometric properties to algebraic structures.

Contribution

It establishes new connections between the geometric property of the regular locus and the algebraic existence of generators in module and derived categories.

Findings

Openness of the regular locus implies the existence of generators in module categories.

The relationship extends to the bounded derived and singularity categories.

Results provide a geometric criterion for algebraic generation properties.

Abstract

This work clarifies the relationship between the openness of the regular locus of a commutative Noetherian ring R and the existence of generators for the category of finitely generated R-modules, the corresponding bounded derived category, and for the singularity category of R.