The Stacks project

Proof. The problem is local on $X$, hence we may assume $X$ is quasi-compact. Let $X = X_1 \cup \ldots \cup X_ n$ be the irreducible components of $X$. First suppose that $X$ contains a dense open which is smooth over $k$. Then $X$ is geometrically reduced at the generic point of each $X_ i$ (for example by Lemma 33.25.4). By Lemma 33.6.8, it follows that $X$ is geometrically reduced.

Conversely, suppose that $X$ is geometrically reduced. Since $Z = \bigcup _{i \not= j} X_ i \cap X_ j$ is nowhere dense in $X$, we may replace $X$ by $X \setminus Z$. As $X \setminus Z$ is a disjoint union of irreducible schemes, this reduces us to the case where $X$ is irreducible. As $X$ is irreducible and reduced, it is integral, see Properties, Lemma 28.3.4. Let $\eta \in X$ be its generic point. Then the function field $K = k(X) = \kappa (\eta ) = \mathcal{O}_{X, \eta }$ is geometrically reduced over $k$, hence separable over $k$, see Algebra, Lemma 10.44.2. Let $U = \mathop{\mathrm{Spec}}(A) \subset X$ be any nonempty affine open so that $K = A_{(0)}$ is the fraction field of $A$. Apply Algebra, Lemma 10.140.5 to conclude that $A$ is smooth at $(0)$ over $k$. By definition this means that some principal localization of $A$ is smooth over $k$ and we win. $\square$