This section is the analogue of Morphisms of Spaces, Section 67.4 for morphisms of formal algebraic spaces.
Definition 87.30.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. Let $\Delta _{X/Y} : X \to X \times _ Y X$ be the diagonal morphism.
We say $f$ is separated if $\Delta _{X/Y}$ is a closed immersion.
We say $f$ is quasi-separated if $\Delta _{X/Y}$ is quasi-compact.
Since $\Delta _{X/Y}$ is representable (by schemes) by Lemma 87.15.5 we can test this by considering morphisms $T \to X \times _ Y X$ from affine schemes $T$ and checking whether
is quasi-compact or a closed immersion, see Lemma 87.17.5 or Definition 87.27.1. Note that the scheme $E$ is the equalizer of two morphisms $a, b : T \to X$ which agree as morphisms into $Y$ and that $E \to T$ is a monomorphism and locally of finite type.
Lemma 87.30.2. All of the separation axioms listed in Definition 87.30.1 are stable under base change.
Proof. Let $f : X \to Y$ and $Y' \to Y$ be morphisms of formal algebraic spaces. Let $f' : X' \to Y'$ be the base change of $f$ by $Y' \to Y$. Then $\Delta _{X'/Y'}$ is the base change of $\Delta _{X/Y}$ by the morphism $X' \times _{Y'} X' \to X \times _ Y X$. Each of the properties of the diagonal used in Definition 87.30.1 is stable under base change. Hence the lemma is true. $\square$
Lemma 87.30.3. Let $S$ be a scheme. Let $f : X \to Z$, $g : Y \to Z$ and $Z \to T$ be morphisms of formal algebraic spaces over $S$. Consider the induced morphism $i : X \times _ Z Y \to X \times _ T Y$. Then
$i$ is representable (by schemes), locally of finite type, locally quasi-finite, separated, and a monomorphism,
if $Z \to T$ is separated, then $i$ is a closed immersion, and
if $Z \to T$ is quasi-separated, then $i$ is quasi-compact.
Proof. By general category theory the following diagram
is a fibre product diagram. Hence $i$ is the base change of the diagonal morphism $\Delta _{Z/T}$. Thus the lemma follows from Lemma 87.15.5. $\square$
Lemma 87.30.4. All of the separation axioms listed in Definition 87.30.1 are stable under composition of morphisms.
Proof. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of formal algebraic spaces to which the axiom in question applies. The diagonal $\Delta _{X/Z}$ is the composition
Our separation axiom is defined by requiring the diagonal to have some property $\mathcal{P}$. By Lemma 87.30.3 above we see that the second arrow also has this property. Hence the lemma follows since the composition of (representable) morphisms with property $\mathcal{P}$ also is a morphism with property $\mathcal{P}$. $\square$
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
$f$ is $\mathcal{P}$,
for every scheme $Z$ and morphism $Z \to Y$ the base change $Z \times _ Y X \to Z$ of $f$ is $\mathcal{P}$,
for every affine scheme $Z$ and every morphism $Z \to Y$ the base change $Z \times _ Y X \to Z$ of $f$ is $\mathcal{P}$,
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),
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
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
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
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
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. $\square$
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