The Stacks project

Proof. Assume (1). Then for every finitely generated field extension $k'/k$ and every point $x' \in X_{k'}$ the local ring of $X_{k'}$ at $x'$ is regular. By Properties, Lemma 28.9.2 this means that $X_{k'}$ is regular. Hence (2).

It is clear that (2) implies (3).

Assume (3) and let $U \subset X$ be an affine open subscheme. Then $U_{k'}$ is a regular scheme for any finite purely inseparable extension $k'/k$ (including $k = k'$). This means that $k' \otimes _ k \mathcal{O}(U)$ is a regular ring for all finite purely inseparable extensions $k'/k$. Hence $\mathcal{O}(U)$ is a geometrically regular $k$-algebra and we see that (4) holds.

It is clear that (4) implies (5). Let $X = \bigcup U_ i$ be an affine open covering as in (5). For any field extension $k'/k$ the base change $X_{k'}$ is gotten by gluing the spectra of the rings $\mathcal{O}_ X(U_ i) \otimes _ k k'$ (see Schemes, Section 26.17). Hence $X_{k'}$ is regular. So (1) holds. $\square$