A descent theorem for formal smoothness

TL;DR

This paper establishes a descent theorem for formal smoothness in noetherian local rings, unifying and improving various known results in commutative algebra, and also explores similar descent results for the complete intersection property.

Contribution

It provides a new descent theorem for formal smoothness under specific conditions, enhancing understanding of related properties like regularity and quasi-excellence.

Findings

Conditions on commutative squares imply formal smoothness descent

Unifies and improves results like Greco's theorem and Kunz's characterization

Extends the descent concept to the complete intersection property

Abstract

Let u be a local homomorphism of noetherian local rings forming part of a commutative square vf=gu. We give some conditions on the square which imply that u is formally smooth. This result encapsulates a variety of (apparently unrelated) results in commutative algebra greatly improving some of them: Greco's theorem on descent of quasi-excellence property by finite surjective morphisms, Kunz's characterization of regular local rings in positive characteristic by means of the Frobenius homomorphism (and in fact the relative version obtained by Andre and Radu), etc. In the second part of the paper, we study a similar question for the complete intersection property instead of formal smoothness, giving also some applications.