The Stacks project

85.23 Separation axioms for morphisms

This section is the analogue of Morphisms of Spaces, Section 65.4 for morphisms of formal algebraic spaces.

Definition 85.23.1. 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.

  1. We say $f$ is separated if $\Delta _{X/Y}$ is a closed immersion.

  2. We say $f$ is quasi-separated if $\Delta _{X/Y}$ is quasi-compact.

Since $\Delta _{X/Y}$ is representable (by schemes) by Lemma 85.10.5 we can test this by considering morphisms $T \to X \times _ Y X$ from affine schemes $T$ and checking whether

\[ E = T \times _{X \times _ Y X} X \longrightarrow T \]

is quasi-compact or a closed immersion, see Lemma 85.12.5 or Definition 85.20.1. 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.

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 85.23.1 is stable under base change. Hence the lemma is true. $\square$

Lemma 85.23.3. 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

  1. $i$ is representable (by schemes), locally of finite type, locally quasi-finite, separated, and a monomorphism,

  2. if $Z \to T$ is separated, then $i$ is a closed immersion, and

  3. if $Z \to T$ is quasi-separated, then $i$ is quasi-compact.

Proof. By general category theory the following diagram

\[ \xymatrix{ X \times _ Z Y \ar[r]_ i \ar[d] & X \times _ T Y \ar[d] \\ Z \ar[r]^-{\Delta _{Z/T}} \ar[r] & Z \times _ T Z } \]

is a fibre product diagram. Hence $i$ is the base change of the diagonal morphism $\Delta _{Z/T}$. Thus the lemma follows from Lemma 85.10.5. $\square$

Lemma 85.23.4. All of the separation axioms listed in Definition 85.23.1 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

\[ X \longrightarrow X \times _ Y X \longrightarrow X \times _ Z X. \]

Our separation axiom is defined by requiring the diagonal to have some property $\mathcal{P}$. By Lemma 85.23.3 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 85.23.5. 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 85.23.1. 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 85.11.3),

  5. there exists a covering $\{ Y_ j \to Y\} $ as in Definition 85.7.1 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 85.23.2 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 85.7.1. 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 85.5.1. 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 85.5.4.

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 63.5.6 that also the right vertical arrow is quasi-compact or a closed immersion. $\square$


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