The Stacks project

Lemma 87.30.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. Let $\mathcal{P}$ be any of the separation axioms of Definition 87.30.1. The following are equivalent

  1. $f$ is $\mathcal{P}$,

  2. for every scheme $Z$ and morphism $Z \to Y$ the base change $Z \times _ Y X \to Z$ of $f$ is $\mathcal{P}$,

  3. for every affine scheme $Z$ and every morphism $Z \to Y$ the base change $Z \times _ Y X \to Z$ of $f$ is $\mathcal{P}$,

  4. for every affine scheme $Z$ and every morphism $Z \to Y$ the formal algebraic space $Z \times _ Y X$ is $\mathcal{P}$ (see Definition 87.16.3),

  5. there exists a covering $\{ Y_ j \to Y\} $ as in Definition 87.11.1 such that the base change $Y_ j \times _ Y X \to Y_ j$ has $\mathcal{P}$ for all $j$.

Proof. We will repeatedly use Lemma 87.30.2 without further mention. In particular, it is clear that (1) implies (2) and (2) implies (3).

Assume (3) and let $Z \to Y$ be a morphism where $Z$ is an affine scheme. Let $U$, $V$ be affine schemes and let $a : U \to Z \times _ Y X$ and $b : V \to Z \times _ Y X$ be morphisms. Then

\[ U \times _{Z \times _ Y X} V = (Z \times _ Y X) \times _{\Delta , (Z \times _ Y X) \times _ Z (Z \times _ Y X)} (U \times _ Z V) \]

and we see that this is quasi-compact if $\mathcal{P} =$“quasi-separated” or an affine scheme equipped with a closed immersion into $U \times _ Z V$ if $\mathcal{P} =$“separated”. Thus (4) holds.

Assume (4) and let $Z \to Y$ be a morphism where $Z$ is an affine scheme. Let $U$, $V$ be affine schemes and let $a : U \to Z \times _ Y X$ and $b : V \to Z \times _ Y X$ be morphisms. Reading the argument above backwards, we see that $U \times _{Z \times _ Y X} V \to U \times _ Z V$ is quasi-compact if $\mathcal{P} =$“quasi-separated” or a closed immersion if $\mathcal{P} =$“separated”. Since we can choose $U$ and $V$ as above such that $U$ varies through an étale covering of $Z \times _ Y X$, we find that the corresponding morphisms

\[ U \times _ Z V \to (Z \times _ Y X) \times _ Z (Z \times _ Y X) \]

form an étale covering by affines. Hence we conclude that $\Delta : (Z \times _ Y X) \to (Z \times _ Y X) \times _ Z (Z \times _ Y X)$ is quasi-compact, resp. a closed immersion. Thus (3) holds.

Let us prove that (3) implies (5). Assume (3) and let $\{ Y_ j \to Y\} $ be as in Definition 87.11.1. We have to show that the morphisms

\[ \Delta _ j : Y_ j \times _ Y X \longrightarrow (Y_ j \times _ Y X) \times _{Y_ j} (Y_ j \times _ Y X) = Y_ j \times _ Y X \times _ Y X \]

has the corresponding property (i.e., is quasi-compact or a closed immersion). Write $Y_ j = \mathop{\mathrm{colim}}\nolimits Y_{j, \lambda }$ as in Definition 87.9.1. Replacing $Y_ j$ by $Y_{j, \lambda }$ in the formula above, we have the property by our assumption that (3) holds. Since the displayed arrow is the colimit of the arrows $\Delta _{j, \lambda }$ and since we can test whether $\Delta _ j$ has the corresponding property by testing after base change by affine schemes mapping into $Y_ j \times _ Y X \times _ Y X$, we conclude by Lemma 87.9.4.

Let us prove that (5) implies (1). Let $\{ Y_ j \to Y\} $ be as in (5). Then we have the fibre product diagram

\[ \xymatrix{ \coprod Y_ j \times _ Y X \ar[r] \ar[d] & X \ar[d] \\ \coprod Y_ j \times _ Y X \times _ Y X \ar[r] & X \times _ Y X } \]

By assumption the left vertical arrow is quasi-compact or a closed immersion. It follows from Spaces, Lemma 65.5.6 that also the right vertical arrow is quasi-compact or a closed immersion. $\square$

Comments (2)

Comment #1967 by Brian Conrad on

In the proof that (4) implies (3), the phrase "find an etale covering of by and as above" might be misunderstood to refer to using a finite disjoint union of affines to make an affine etale cover, which however cannot be done in general (without quasi-compactness), thereby leading to reader confusion at first sight. So I suggest that phrase be replaced with "choose and to vary through a fixed etale covering of by affine schemes over ".

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