Ideal-adic completion of quasi-excellent rings (after Gabber)

TL;DR

This paper provides a detailed proof of Gabber's result on lifting quasi-excellent rings, showing that their ideal-adic completions preserve excellence and quasi-excellence, thereby advancing the understanding of ring properties in algebraic geometry.

Contribution

It offers a comprehensive proof of Gabber's unpublished result, extending the class of rings known to maintain excellence under ideal-adic completion.

Findings

Ideal-adic completion of an excellent ring is excellent.

Ideal-adic completion of a quasi-excellent ring is quasi-excellent.

The paper extends previous work on the lifting problem for these rings.

Abstract

In this paper, we give a detailed proof to a result of Gabber (unpublished) on the lifting problem of quasi-excellent rings, extending the previous work on Nishimura-Nishimura. As a corollary, we establish that an ideal-adic completion of an excellent (resp. quasi-excellent) ring is excellent (resp. quasi-excellent).