Ind-étale versus formally étale

We show that when A$A$ is a reduced algebra over a characteristic zero field k$k$ and the module of K & auml;hler differentials Omega A/k=0$\Omega _{A/k}=0$, then A$A$ is ind-& eacute;tale, 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.