The Stacks project

Proof. You can deduce this for the statements quasi-compact, quasi-separated, separated, and proper by using Lemmas 88.28.1 88.28.2, 88.28.3, 88.28.2, and 88.28.4 to translate the corresponding property into a property of the formal completion and using the argument of the proof of Lemma 88.30.1. However, there is a direct argument using fpqc descent as follows. First, you can reduce to proving the lemma for $A \to A^\wedge $ and $B \to B^\wedge $ since $A^\wedge \to B^\wedge $ is an isomorphism. Then note that $\{ U \to \mathop{\mathrm{Spec}}(A), \mathop{\mathrm{Spec}}(A^\wedge ) \to \mathop{\mathrm{Spec}}(A)\} $ is an fpqc covering with $U = \mathop{\mathrm{Spec}}(A) \setminus V(I)$ as before. The base change of $f$ by $U \to \mathop{\mathrm{Spec}}(A)$ is $\text{id}_ U$ by definition of our category (88.30.0.1). Let $P$ be a property of morphisms of algebraic spaces which is fpqc local on the base (Descent on Spaces, Definition 74.10.1) such that $P$ holds for identity morphisms. Then we see that $P$ holds for $f$ if and only if $P$ holds for $g$. This applies to $P$ equal to quasi-compact, quasi-separated, separated, proper, and finite by Descent on Spaces, Lemmas 74.11.1, 74.11.2, 74.11.18, 74.11.19, and 74.11.23. $\square$