The Stacks project

64.4 Separation axioms

It makes sense to list some a priori properties of the diagonal of a morphism of algebraic spaces.

Lemma 64.4.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\Delta _{X/Y} : X \to X \times _ Y X$ be the diagonal morphism. Then

  1. $\Delta _{X/Y}$ is representable,

  2. $\Delta _{X/Y}$ is locally of finite type,

  3. $\Delta _{X/Y}$ is a monomorphism,

  4. $\Delta _{X/Y}$ is separated, and

  5. $\Delta _{X/Y}$ is locally quasi-finite.

Proof. We are going to use the fact that $\Delta _{X/S}$ is representable (by definition of an algebraic space) and that it satisfies properties (2) – (5), see Spaces, Lemma 62.13.1. Note that we have a factorization

\[ X \longrightarrow X \times _ Y X \longrightarrow X \times _ S X \]

of the diagonal $\Delta _{X/S} : X \to X \times _ S X$. Since $X \times _ Y X \to X \times _ S X$ is a monomorphism, and since $\Delta _{X/S}$ is representable, it follows formally that $\Delta _{X/Y}$ is representable. In particular, the rest of the statements now make sense, see Section 64.3.

Choose a surjective étale morphism $U \to X$, with $U$ a scheme. Consider the diagram

\[ \xymatrix{ R = U \times _ X U \ar[r] \ar[d] & U \times _ Y U \ar[d] \ar[r] & U \times _ S U \ar[d] \\ X \ar[r] & X \times _ Y X \ar[r] & X \times _ S X } \]

Both squares are cartesian, hence so is the outer rectangle. The top row consists of schemes, and the vertical arrows are surjective étale morphisms. By Spaces, Lemma 62.11.4 the properties (2) – (5) for $\Delta _{X/Y}$ are equivalent to those of $R \to U \times _ Y U$. In the proof of Spaces, Lemma 62.13.1 we have seen that $R \to U \times _ S U$ has properties (2) – (5). The morphism $U \times _ Y U \to U \times _ S U$ is a monomorphism of schemes. These facts imply that $R \to U \times _ Y U$ have properties (2) – (5).

Namely: For (3), note that $R \to U \times _ Y U$ is a monomorphism as the composition $R \to U \times _ S U$ is a monomorphism. For (2), note that $R \to U \times _ Y U$ is locally of finite type, as the composition $R \to U \times _ S U$ is locally of finite type (Morphisms, Lemma 29.14.8). A monomorphism which is locally of finite type is locally quasi-finite because it has finite fibres (Morphisms, Lemma 29.19.7), hence (5). A monomorphism is separated (Schemes, Lemma 26.23.3), hence (4). $\square$

Definition 64.4.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of 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 locally separated1 if $\Delta _{X/Y}$ is an immersion.

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

This definition makes sense since $\Delta _{X/Y}$ is representable, and hence we know what it means for it to have one of the properties described in the definition. We will see below (Lemma 64.4.13) that this definition matches the ones we already have for morphisms of schemes and representable morphisms.

Lemma 64.4.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is separated, then $f$ is locally separated and $f$ is quasi-separated.

Proof. This is true, via the general principle Spaces, Lemma 62.5.8, because a closed immersion of schemes is an immersion and is quasi-compact. $\square$

Lemma 64.4.4. All of the separation axioms listed in Definition 64.4.2 are stable under base change.

Proof. Let $f : X \to Y$ and $Y' \to Y$ be morphisms of 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 $X' \times _{Y'} X' \to X \times _ Y X$. By the results of Section 64.3 each of the properties of the diagonal used in Definition 64.4.2 is stable under base change. Hence the lemma is true. $\square$

slogan

Lemma 64.4.5. Let $S$ be a scheme. Let $f : X \to Z$, $g : Y \to Z$ and $Z \to T$ be morphisms of algebraic spaces over $S$. Consider the induced morphism $i : X \times _ Z Y \to X \times _ T Y$. Then

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

  2. if $Z \to T$ is locally separated, then $i$ is an immersion,

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

  4. 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 64.4.1, and the material in Section 64.3. $\square$

slogan

Lemma 64.4.6. Let $S$ be a scheme. Let $T$ be an algebraic space over $S$. Let $g : X \to Y$ be a morphism of algebraic spaces over $T$. Consider the graph $i : X \to X \times _ T Y$ of $g$. Then

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

  2. if $Y \to T$ is locally separated, then $i$ is an immersion,

  3. if $Y \to T$ is separated, then $i$ is a closed immersion, and

  4. if $Y \to T$ is quasi-separated, then $i$ is quasi-compact.

Proof. This is a special case of Lemma 64.4.5 applied to the morphism $X = X \times _ Y Y \to X \times _ T Y$. $\square$

Lemma 64.4.7. Let $S$ be a scheme. Let $f : X \to T$ be a morphism of algebraic spaces over $S$. Let $s : T \to X$ be a section of $f$ (in a formula $f \circ s = \text{id}_ T$). Then

  1. $s$ is representable, locally of finite type, locally quasi-finite, separated and a monomorphism,

  2. if $f$ is locally separated, then $s$ is an immersion,

  3. if $f$ is separated, then $s$ is a closed immersion, and

  4. if $f$ is quasi-separated, then $s$ is quasi-compact.

Proof. This is a special case of Lemma 64.4.6 applied to $g = s$ so the morphism $i = s : T \to T \times _ T X$. $\square$

Lemma 64.4.8. All of the separation axioms listed in Definition 64.4.2 are stable under composition of morphisms.

Proof. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of 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 64.4.5 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 $\mathcal{P}$, see Section 64.3. $\square$

Lemma 64.4.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

  1. If $Y$ is separated and $f$ is separated, then $X$ is separated.

  2. If $Y$ is quasi-separated and $f$ is quasi-separated, then $X$ is quasi-separated.

  3. If $Y$ is locally separated and $f$ is locally separated, then $X$ is locally separated.

  4. If $Y$ is separated over $S$ and $f$ is separated, then $X$ is separated over $S$.

  5. If $Y$ is quasi-separated over $S$ and $f$ is quasi-separated, then $X$ is quasi-separated over $S$.

  6. If $Y$ is locally separated over $S$ and $f$ is locally separated, then $X$ is locally separated over $S$.

Proof. Parts (4), (5), and (6) follow immediately from Lemma 64.4.8 and Spaces, Definition 62.13.2. Parts (1), (2), and (3) reduce to parts (4), (5), and (6) by thinking of $X$ and $Y$ as algebraic spaces over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Properties of Spaces, Definition 63.3.1. $\square$

Lemma 64.4.10. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of algebraic spaces over $S$.

  1. If $g \circ f$ is separated then so is $f$.

  2. If $g \circ f$ is locally separated then so is $f$.

  3. If $g \circ f$ is quasi-separated then so is $f$.

Proof. Consider the factorization

\[ X \to X \times _ Y X \to X \times _ Z X \]

of the diagonal morphism of $g \circ f$. In any case the last morphism is a monomorphism. Hence for any scheme $T$ and morphism $T \to X \times _ Y X$ we have the equality

\[ X \times _{(X \times _ Y X)} T = X \times _{(X \times _ Z X)} T. \]

Hence the result is clear. $\square$

Lemma 64.4.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.

  1. If $X$ is separated then $X$ is separated over $S$.

  2. If $X$ is locally separated then $X$ is locally separated over $S$.

  3. If $X$ is quasi-separated then $X$ is quasi-separated over $S$.

Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

  1. If $X$ is separated over $S$ then $f$ is separated.

  2. If $X$ is locally separated over $S$ then $f$ is locally separated.

  3. If $X$ is quasi-separated over $S$ then $f$ is quasi-separated.

Proof. Parts (4), (5), and (6) follow immediately from Lemma 64.4.10 and Spaces, Definition 62.13.2. Parts (1), (2), and (3) follow from parts (4), (5), and (6) by thinking of $X$ and $Y$ as algebraic spaces over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Properties of Spaces, Definition 63.3.1. $\square$

Lemma 64.4.12. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{P}$ be any of the separation axioms of Definition 64.4.2. 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 algebraic space $Z \times _ Y X$ is $\mathcal{P}$ (see Properties of Spaces, Definition 63.3.1),

  5. there exists a scheme $V$ and a surjective étale morphism $V \to Y$ such that the base change $V \times _ Y X \to V$ has $\mathcal{P}$, and

  6. there exists a Zariski covering $Y = \bigcup Y_ i$ such that each of the morphisms $f^{-1}(Y_ i) \to Y_ i$ has $\mathcal{P}$.

Proof. We will repeatedly use Lemma 64.4.4 without further mention. In particular, it is clear that (1) implies (2) and (2) implies (3).

Let us prove that (3) and (4) are equivalent. Note that if $Z$ is an affine scheme, then the morphism $Z \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is a separated morphism as a morphism of algebraic spaces over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. If $Z \times _ Y X \to Z$ is $\mathcal{P}$, then $Z \times _ Y X \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is $\mathcal{P}$ as a composition (see Lemma 64.4.8). Hence the algebraic space $Z \times _ Y X$ is $\mathcal{P}$. Conversely, if the algebraic space $Z \times _ Y X$ is $\mathcal{P}$, then $Z \times _ Y X \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is $\mathcal{P}$, and hence by Lemma 64.4.10 we see that $Z \times _ Y X \to Z$ is $\mathcal{P}$.

Let us prove that (3) implies (5). Assume (3). Let $V$ be a scheme and let $V \to Y$ be étale surjective. We have to show that $V \times _ Y X \to V$ has property $\mathcal{P}$. In other words, we have to show that the morphism

\[ V \times _ Y X \longrightarrow (V \times _ Y X) \times _ V (V \times _ Y X) = V \times _ Y X \times _ Y X \]

has the corresponding property (i.e., is a closed immersion, immersion, or quasi-compact). Let $V = \bigcup V_ j$ be an affine open covering of $V$. By assumption we know that each of the morphisms

\[ V_ j \times _ Y X \longrightarrow V_ j \times _ Y X \times _ Y X \]

does have the corresponding property. Since being a closed immersion, immersion, quasi-compact immersion, or quasi-compact is Zariski local on the target, and since the $V_ j$ cover $V$ we get the desired conclusion.

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

\[ \xymatrix{ V \times _ Y X \ar[r] \ar[d] & X \ar[d] \\ V \times _ Y X \times _ Y X \ar[r] & X \times _ Y X } \]

By assumption the left vertical arrow is a closed immersion, immersion, quasi-compact immersion, or quasi-compact. It follows from Spaces, Lemma 62.5.6 that also the right vertical arrow is a closed immersion, immersion, quasi-compact immersion, or quasi-compact.

It is clear that (1) implies (6) by taking the covering $Y = Y$. Assume $Y = \bigcup Y_ i$ is as in (6). Choose schemes $V_ i$ and surjective étale morphisms $V_ i \to Y_ i$. Note that the morphisms $V_ i \times _ Y X \to V_ i$ have $\mathcal{P}$ as they are base changes of the morphisms $f^{-1}(Y_ i) \to Y_ i$. Set $V = \coprod V_ i$. Then $V \to Y$ is a morphism as in (5) (details omitted). Hence (6) implies (5) and we are done. $\square$

Lemma 64.4.13. Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$.

  1. The morphism $f$ is locally separated.

  2. The morphism $f$ is (quasi-)separated in the sense of Definition 64.4.2 above if and only if $f$ is (quasi-)separated in the sense of Section 64.3.

In particular, if $f : X \to Y$ is a morphism of schemes over $S$, then $f$ is (quasi-)separated in the sense of Definition 64.4.2 if and only if $f$ is (quasi-)separated as a morphism of schemes.

Proof. This is the equivalence of (1) and (2) of Lemma 64.4.12 combined with the fact that any morphism of schemes is locally separated, see Schemes, Lemma 26.21.2. $\square$

[1] In the literature this term often refers to quasi-separated and locally separated morphisms.

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