The Stacks project

Proof. Assume $\mathcal{X}$ is well-nigh affine. Choose an affine scheme $U$ and a surjective, flat, finite, and finitely presented morphism $U \to \mathcal{X}$. Set $R = U \times _\mathcal {X} U$. Then we obtain a groupoid $(U, R, s, t, c)$ in algebraic spaces and an isomorphism $[U/R] \to \mathcal{X}$, see Algebraic Stacks, Lemma 94.16.1 and Remark 94.16.3. Since $s, t : R \to U$ are flat, finite, and finitely presented morphisms (as base changes of $U \to \mathcal{X})$ we see that $s, t$ are finite locally free (Morphisms, Lemma 29.48.2). This implies that $R$ is affine (as finite morphisms are affine) and hence (2) holds.

Suppose that we have a groupoid scheme $(U, R, s, t, c)$ with $U$ and $R$ are affine and $s, t : R \to U$ finite locally free. Set $\mathcal{X} = [U/R]$. Then $\mathcal{X}$ is an algebraic stack by Criteria for Representability, Theorem 97.17.2 (strictly speaking we don't need this here, but it can't be stressed enough that this is true). The morphism $U \to \mathcal{X}$ is surjective, flat, and locally of finite presentation by Criteria for Representability, Lemma 97.17.1. Thus we can check whether $U \to \mathcal{X}$ is finite by checking whether the projection $U \times _\mathcal {X} U \to U$ has this property, see Properties of Stacks, Lemma 100.3.3. Since $U \times _\mathcal {X} U = R$ by Groupoids in Spaces, Lemma 78.22.2 we see that this is true. Thus $\mathcal{X}$ is well-nigh affine.

Proof of the final statement. We see that $\mathcal{X}$ is quasi-compact by Properties of Stacks, Lemma 100.6.2. We see that $\mathcal{X} = [U/R]$ is quasi-DM and separated by Morphisms of Stacks, Lemma 101.20.1. $\square$