The Stacks project

Remark 76.3.5. Let $X \to Y$ be a morphism of algebraic spaces. For some applications (of radicial morphisms) it is enough to require that for every $\mathop{\mathrm{Spec}}(K) \to Y$ where $K$ is a field

1. the space $|\mathop{\mathrm{Spec}}(K) \times _ Y X|$ is a singleton,

2. there exists a monomorphism $\mathop{\mathrm{Spec}}(L) \to \mathop{\mathrm{Spec}}(K) \times _ Y X$, and

3. $K \subset L$ is purely inseparable.

If needed later we will may call such a morphism weakly radicial. For example if $X \to Y$ is a surjective weakly radicial morphism then $X(k) \to Y(k)$ is surjective for every algebraically closed field $k$. Note that the base change $X_{\overline{\mathbf{Q}}} \to \mathop{\mathrm{Spec}}(\overline{\mathbf{Q}})$ of the morphism in Example 76.3.3 is weakly radicial, but not radicial. The analogue of Lemma 76.3.4 is that if $X \to Y$ has property ($\beta$) and is universally injective, then it is weakly radicial (proof omitted).