Ind-\'etale vs Formally \'etale

TL;DR

This paper proves that reduced algebras over characteristic zero fields with trivial Kähler differentials are ind-étale, leading to new rigidity results and special cases of conjectures without noetherian assumptions.

Contribution

It establishes a link between vanishing Kähler differentials and ind-étaleness, partially answering Bhatt's question and deriving several algebraic rigidity results.

Findings

Reduced algebras with zero Kähler differentials are ind-étale.

Derived rigidity properties of Hochschild homology.

Proved special cases of Weibel's and Vorst's conjectures without noetherian assumptions.

Abstract

We show that when $A$ is a reduced algebra over a characteristic zero field $k$ and the module of K\"ahler differentials $\Omega_{A/k}=0$, then $A$ is ind-\'etale, partially answering a question of Bhatt. As further applications of this result, we deduce a rigidity property of Hochschild homology and special instances of Weibel's conjecture and Vorst's conjecture without any noetherian assumptions.