On a question of Swan. With an Appendix by K\k{e}stutis \v{C}esnavi\v{c}ius

TL;DR

This paper demonstrates that regular local rings can be expressed as filtered inductive limits of similar rings over integers, simplifying the proof of the cohomological purity conjecture by reducing it to the complete case.

Contribution

It establishes that regular local rings are filtered inductive limits of regular local rings of finite type over Z, providing a new approach to the cohomological purity conjecture.

Findings

Regular local rings are filtered inductive limits of rings of finite type over Z.

The cohomological purity conjecture reduces to the complete case.

Provides a new structural understanding of regular local rings.

Abstract

We show that a regular local ring is a filtered inductive limit of regular local rings, essentially of finite type over $\bf Z$. As an application the cohomological purity conjecture is reduced to the complete case.