The Stacks project

Lemma 86.26.2. Notation and assumptions as in Lemma 86.26.1. Let $f : X' \to \mathop{\mathrm{Spec}}(A)$ correspond to $g : Y' \to \mathop{\mathrm{Spec}}(B)$ via the equivalence. Then $f$ is quasi-compact, quasi-separated, separated, proper, finite, and add more here if and only if $g$ is so.

Proof. You can deduce this for the statements quasi-compact, quasi-separated, separated, and proper by using Lemmas 86.24.1 86.24.2, 86.24.3, 86.24.2, and 86.24.4 to translate the corresponding property into a property of the formal completion and using the argument of the proof of Lemma 86.26.1. 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 (86.26.0.1). Let $P$ be a property of morphisms of algebraic spaces which is fpqc local on the base (Descent on Spaces, Definition 72.9.1) 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 72.10.1, 72.10.2, 72.10.18, 72.10.19, and 72.10.23. $\square$


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