The Stacks project

Proof. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is stable under base change and composition and holds for closed immersions. Assume $f : X \to Y$ has $\mathcal{P}$ and is universally injective. Then, in the situation of Definition 76.3.1 the morphism $(\mathop{\mathrm{Spec}}(K) \times _ Y X)_{red} \to \mathop{\mathrm{Spec}}(K)$ is universally injective and has $\mathcal{P}$. This reduces the problem of proving

to the problem of proving that any nonempty reduced algebraic space $X$ over field whose structure morphism $X \to \mathop{\mathrm{Spec}}(K)$ is universally injective and $\mathcal{P}$ is representable by the spectrum of a field. Namely, then $X \to \mathop{\mathrm{Spec}}(K)$ will be a morphism of schemes and we conclude by the equivalence of radicial and universally injective for morphisms of schemes, see Morphisms, Lemma 29.10.2.

Let us prove (1). Assume $f$ is decent and universally injective. By Decent Spaces, Lemmas 68.17.4, 68.17.6, and 68.17.2 (to see that an immersion is decent) we see that the discussion in the first paragraph applies. Let $X$ be a nonempty decent reduced algebraic space universally injective over a field $K$. In particular we see that $|X|$ is a singleton. By Decent Spaces, Lemma 68.14.2 we conclude that $X \cong \mathop{\mathrm{Spec}}(L)$ for some extension $K \subset L$ as desired.

A quasi-separated morphism is decent, see Decent Spaces, Lemma 68.17.2. Hence (1) implies (2).

Let us prove (3). Recall that the separation axioms are stable under base change and composition and that closed immersions are separated, see Morphisms of Spaces, Lemmas 67.4.4, 67.4.8, and 67.10.7. Thus the discussion in the first paragraph of the proof applies. Let $X$ be a reduced algebraic space universally injective and locally separated over a field $K$. In particular $|X|$ is a singleton hence $X$ is quasi-compact, see Properties of Spaces, Lemma 66.5.2. We can find a surjective étale morphism $U \to X$ with $U$ affine, see Properties of Spaces, Lemma 66.6.3. Consider the morphism of schemes

As $X \to \mathop{\mathrm{Spec}}(K)$ is universally injective $j$ is surjective, and as $X \to \mathop{\mathrm{Spec}}(K)$ is locally separated $j$ is an immersion. A surjective immersion is a closed immersion, see Schemes, Lemma 26.10.4. Hence $R = U \times _ X U$ is affine as a closed subscheme of an affine scheme. In particular $R$ is quasi-compact. It follows that $X = U/R$ is quasi-separated, and the result follows from (2). $\square$