Building modules from the singular locus

TL;DR

This paper determines the exact number of steps needed to construct any finitely generated module over a noetherian ring from the singular locus, refining previous bounds and exploring the resulting module category stratification.

Contribution

It precisely calculates the iteration count for building modules from the singular locus, improving upon Schoutens' earlier bounds.

Findings

Exact iteration number for module construction determined

Stratification of module category analyzed for local rings with isolated singularity

Provides detailed understanding of module building process from singular locus

Abstract

A finitely generated module over a commutative noetherian ring of finite Krull dimension can be built from the prime ideals in the singular locus by iteration of three procedures: taking extensions, direct summands, and cosyzygies. In 2003 Schoutens gave a bound on the number of iterations required to build any module, and in this note we determine the exact number. This building process yields a stratification of the module category, which we study in detail for local rings that have an isolated singularity.