Ind-étale vs Formally étale

We show that when $A$ is a reduced algebra over a characteristic zero field $k$ and the module of Kähler differentials ${\mathrm{\Omega }}_{A\mathrm{/}k}=0$, then $A$ is ind-étale, 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.