## 99.3 Properties of morphisms representable by algebraic spaces

We will study properties of (arbitrary) morphisms of algebraic stacks in its own chapter. For morphisms representable by algebraic spaces we know what it means to be surjective, smooth, or étale, etc. This applies in particular to morphisms $X \to \mathcal{Y}$ from algebraic spaces to algebraic stacks. In this section, we recall how this works, we list the properties to which this applies, and we prove a few easy lemmas.

Our first lemma says a morphism is representable by algebraic spaces if it is so after a base change by a flat, locally finitely presented, surjective morphism.

Lemma 99.3.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $W$ be an algebraic space and let $W \to \mathcal{Y}$ be surjective, locally of finite presentation, and flat. The following are equivalent

1. $f$ is representable by algebraic spaces, and

2. $W \times _\mathcal {Y} \mathcal{X}$ is an algebraic space.

Proof. The implication (1) $\Rightarrow$ (2) is Algebraic Stacks, Lemma 93.9.8. Conversely, let $W \to \mathcal{Y}$ be as in (2). To prove (1) it suffices to show that $f$ is faithful on fibre categories, see Algebraic Stacks, Lemma 93.15.2. Assumption (2) implies in particular that $W \times _\mathcal {Y} \mathcal{X} \to W$ is faithful. Hence the faithfulness of $f$ follows from Stacks, Lemma 8.6.9. $\square$

Let $P$ be a property of morphisms of algebraic spaces which is fppf local on the target and preserved by arbitrary base change. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Then we say $f$ has property $P$ if and only if for every scheme $T$ and morphism $T \to \mathcal{Y}$ the morphism of algebraic spaces $T \times _\mathcal {Y} \mathcal{X} \to T$ has property $P$, see Algebraic Stacks, Definition 93.10.1.

It turns out that if $f : \mathcal{X} \to \mathcal{Y}$ is representable by algebraic spaces and has property $P$, then for any morphism of algebraic stacks $\mathcal{Y}' \to \mathcal{Y}$ the base change $\mathcal{Y}' \times _\mathcal {Y} \mathcal{X} \to \mathcal{Y}'$ has property $P$, see Algebraic Stacks, Lemmas 93.9.7 and 93.10.6. If the property $P$ is preserved under compositions, then this holds also in the setting of morphisms of algebraic stacks representable by algebraic spaces, see Algebraic Stacks, Lemmas 93.9.9 and 93.10.5. Moreover, in this case products $\mathcal{X}_1 \times \mathcal{X}_2 \to \mathcal{Y}_1 \times \mathcal{Y}_2$ of morphisms representable by algebraic spaces having property $\mathcal{P}$ have property $\mathcal{P}$, see Algebraic Stacks, Lemma 93.10.8.

Finally, if we have two properties $P, P'$ of morphisms of algebraic spaces which are fppf local on the target and preserved by arbitrary base change and if $P(f) \Rightarrow P'(f)$ for every morphism $f$, then the same implication holds for the corresponding property of morphisms of algebraic stacks representable by algebraic spaces, see Algebraic Stacks, Lemma 93.10.9. We will use this without further mention in the following and in the following chapters.

The discussion above applies to each of the following properties of morphisms of algebraic spaces

1. quasi-compact, see Morphisms of Spaces, Lemma 66.8.4 and Descent on Spaces, Lemma 73.11.1,

2. quasi-separated, see Morphisms of Spaces, Lemma 66.4.4 and Descent on Spaces, Lemma 73.11.2,

3. universally closed, see Morphisms of Spaces, Lemma 66.9.3 and Descent on Spaces, Lemma 73.11.3,

4. universally open, see Morphisms of Spaces, Lemma 66.6.3 and Descent on Spaces, Lemma 73.11.4,

5. universally submersive, see Morphisms of Spaces, Lemma 66.7.3 and Descent on Spaces, Lemma 73.11.5,

6. universal homeomorphism, see Morphisms of Spaces, Lemma 66.53.4 and Descent on Spaces, Lemma 73.11.8,

7. surjective, see Morphisms of Spaces, Lemma 66.5.5 and Descent on Spaces, Lemma 73.11.6,

8. universally injective, see Morphisms of Spaces, Lemma 66.19.5 and Descent on Spaces, Lemma 73.11.7,

9. locally of finite type, see Morphisms of Spaces, Lemma 66.23.3 and Descent on Spaces, Lemma 73.11.9,

10. locally of finite presentation, see Morphisms of Spaces, Lemma 66.28.3 and Descent on Spaces, Lemma 73.11.10,

11. finite type, see Morphisms of Spaces, Lemma 66.23.3 and Descent on Spaces, Lemma 73.11.11,

12. finite presentation, see Morphisms of Spaces, Lemma 66.28.3 and Descent on Spaces, Lemma 73.11.12,

13. flat, see Morphisms of Spaces, Lemma 66.30.4 and Descent on Spaces, Lemma 73.11.13,

14. open immersion, see Morphisms of Spaces, Section 66.12 and Descent on Spaces, Lemma 73.11.14,

15. isomorphism, see Descent on Spaces, Lemma 73.11.15,

16. affine, see Morphisms of Spaces, Lemma 66.20.5 and Descent on Spaces, Lemma 73.11.16,

17. closed immersion, see Morphisms of Spaces, Section 66.12 and Descent on Spaces, Lemma 73.11.17,

18. separated, see Morphisms of Spaces, Lemma 66.4.4 and Descent on Spaces, Lemma 73.11.18,

19. proper, see Morphisms of Spaces, Lemma 66.40.3 and Descent on Spaces, Lemma 73.11.19,

20. quasi-affine, see Morphisms of Spaces, Lemma 66.21.5 and Descent on Spaces, Lemma 73.11.20,

21. integral, see Morphisms of Spaces, Lemma 66.45.5 and Descent on Spaces, Lemma 73.11.22,

22. finite, see Morphisms of Spaces, Lemma 66.45.5 and Descent on Spaces, Lemma 73.11.23,

23. (locally) quasi-finite, see Morphisms of Spaces, Lemma 66.27.4 and Descent on Spaces, Lemma 73.11.24,

24. syntomic, see Morphisms of Spaces, Lemma 66.36.3 and Descent on Spaces, Lemma 73.11.25,

25. smooth, see Morphisms of Spaces, Lemma 66.37.3 and Descent on Spaces, Lemma 73.11.26,

26. unramified, see Morphisms of Spaces, Lemma 66.38.4 and Descent on Spaces, Lemma 73.11.27,

27. étale, see Morphisms of Spaces, Lemma 66.39.4 and Descent on Spaces, Lemma 73.11.28,

28. finite locally free, see Morphisms of Spaces, Lemma 66.46.5 and Descent on Spaces, Lemma 73.11.29,

29. monomorphism, see Morphisms of Spaces, Lemma 66.10.5 and Descent on Spaces, Lemma 73.11.30,

30. immersion, see Morphisms of Spaces, Section 66.12 and Descent on Spaces, Lemma 73.12.1,

31. locally separated, see Morphisms of Spaces, Lemma 66.4.4 and Descent on Spaces, Lemma 73.12.2,

Lemma 99.3.2. Let $P$ be a property of morphisms of algebraic spaces as above. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. The following are equivalent:

1. $f$ has $P$,

2. for every algebraic space $Z$ and morphism $Z \to \mathcal{Y}$ the morphism $Z \times _\mathcal {Y} \mathcal{X} \to Z$ has $P$.

Proof. The implication (2) $\Rightarrow$ (1) is immediate. Assume (1). Let $Z \to \mathcal{Y}$ be as in (2). Choose a scheme $U$ and a surjective étale morphism $U \to Z$. By assumption the morphism $U \times _\mathcal {Y} \mathcal{X} \to U$ has $P$. But the diagram

$\xymatrix{ U \times _\mathcal {Y} \mathcal{X} \ar[d] \ar[r] & Z \times _\mathcal {Y} \mathcal{X} \ar[d] \\ U \ar[r] & Z }$

is cartesian, hence the right vertical arrow has $P$ as $\{ U \to Z\}$ is an fppf covering. $\square$

The following lemma tells us it suffices to check $P$ after a base change by a surjective, flat, locally finitely presented morphism.

Lemma 99.3.3. Let $P$ be a property of morphisms of algebraic spaces as above. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Let $W$ be an algebraic space and let $W \to \mathcal{Y}$ be surjective, locally of finite presentation, and flat. Set $V = W \times _\mathcal {Y} \mathcal{X}$. Then

$(f\text{ has }P) \Leftrightarrow (\text{the projection }V \to W\text{ has }P).$

Proof. The implication from left to right follows from Lemma 99.3.2. Assume $V \to W$ has $P$. Let $T$ be a scheme, and let $T \to \mathcal{Y}$ be a morphism. Consider the commutative diagram

$\xymatrix{ T \times _\mathcal {Y} \mathcal{X} \ar[d] & T \times _\mathcal {Y} V \ar[d] \ar[l] \ar[r] & V \ar[d] \\ T & T \times _\mathcal {Y} W \ar[l] \ar[r] & W }$

of algebraic spaces. The squares are cartesian. The bottom left morphism is a surjective, flat morphism which is locally of finite presentation, hence $\{ T \times _\mathcal {Y} V \to T\}$ is an fppf covering. Hence the fact that the right vertical arrow has property $P$ implies that the left vertical arrow has property $P$. $\square$

Lemma 99.3.4. Let $P$ be a property of morphisms of algebraic spaces as above. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks representable by algebraic spaces. Let $\mathcal{Z} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Set $\mathcal{W} = \mathcal{Z} \times _\mathcal {Y} \mathcal{X}$. Then

$(f\text{ has }P) \Leftrightarrow (\text{the projection }\mathcal{W} \to \mathcal{Z}\text{ has }P).$

Proof. Choose an algebraic space $W$ and a morphism $W \to \mathcal{Z}$ which is surjective, flat, and locally of finite presentation. By the discussion above the composition $W \to \mathcal{Y}$ is also surjective, flat, and locally of finite presentation. Denote $V = W \times _\mathcal {Z} \mathcal{W} = V \times _\mathcal {Y} \mathcal{X}$. By Lemma 99.3.3 we see that $f$ has $\mathcal{P}$ if and only if $V \to W$ does and that $\mathcal{W} \to \mathcal{Z}$ has $\mathcal{P}$ if and only if $V \to W$ does. The lemma follows. $\square$

Lemma 99.3.5. Let $P$ be a property of morphisms of algebraic spaces as above. Let $\tau \in \{ {\acute{e}tale}, smooth, syntomic, fppf\}$. Let $\mathcal{X} \to \mathcal{Y}$ and $\mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks representable by algebraic spaces. Assume

1. $\mathcal{X} \to \mathcal{Y}$ is surjective and étale, smooth, syntomic, or flat and locally of finite presentation,

2. the composition has $P$, and

3. $P$ is local on the source in the $\tau$ topology.

Then $\mathcal{Y} \to \mathcal{Z}$ has property $P$.

Proof. Let $Z$ be a scheme and let $Z \to \mathcal{Z}$ be a morphism. Set $X = \mathcal{X} \times _\mathcal {Z} Z$, $Y = \mathcal{Y} \times _\mathcal {Z} Z$. By (1) $\{ X \to Y\}$ is a $\tau$ covering of algebraic spaces and by (2) $X \to Z$ has property $P$. By (3) this implies that $Y \to Z$ has property $P$ and we win. $\square$

Lemma 99.3.6. Let $g : \mathcal{X}' \to \mathcal{X}$ be a morphism of algebraic stacks which is representable by algebraic spaces. Let $[U/R] \to \mathcal{X}$ be a presentation. Set $U' = U \times _\mathcal {X} \mathcal{X}'$, and $R' = R \times _\mathcal {X} \mathcal{X}'$. Then there exists a groupoid in algebraic spaces of the form $(U', R', s', t', c')$, a presentation $[U'/R'] \to \mathcal{X}'$, and the diagram

$\xymatrix{ [U'/R'] \ar[d]_{[\text{pr}]} \ar[r] & \mathcal{X}' \ar[d]^ g \\ [U/R] \ar[r] & \mathcal{X} }$

is $2$-commutative where the morphism $[\text{pr}]$ comes from a morphism of groupoids $\text{pr} : (U', R', s', t', c') \to (U, R, s, t, c)$.

Proof. Since $U \to \mathcal{Y}$ is surjective and smooth, see Algebraic Stacks, Lemma 93.17.2 the base change $U' \to \mathcal{X}'$ is also surjective and smooth. Hence, by Algebraic Stacks, Lemma 93.16.2 it suffices to show that $R' = U' \times _{\mathcal{X}'} U'$ in order to get a smooth groupoid $(U', R', s', t', c')$ and a presentation $[U'/R'] \to \mathcal{X}'$. Using that $R = V \times _\mathcal {Y} V$ (see Groupoids in Spaces, Lemma 77.22.2) this follows from

$R' = U \times _\mathcal {X} U \times _\mathcal {X} \mathcal{X}' = (U \times _\mathcal {X} \mathcal{X}') \times _{\mathcal{X}'} (U \times _\mathcal {X} \mathcal{X}')$

see Categories, Lemmas 4.31.8 and 4.31.10. Clearly the projection morphisms $U' \to U$ and $R' \to R$ give the desired morphism of groupoids $\text{pr} : (U', R', s', t', c') \to (U, R, s, t, c)$. Hence the morphism $[\text{pr}]$ of quotient stacks by Groupoids in Spaces, Lemma 77.21.1.

We still have to show that the diagram $2$-commutes. It is clear that the diagram

$\xymatrix{ U' \ar[d]_{\text{pr}_ U} \ar[r]_{f'} & \mathcal{X}' \ar[d]^ g \\ U \ar[r]^ f & \mathcal{X} }$

$2$-commutes where $\text{pr}_ U : U' \to U$ is the projection. There is a canonical $2$-arrow $\tau : f \circ t \to f \circ s$ in $\mathop{\mathrm{Mor}}\nolimits (R, \mathcal{X})$ coming from $R = U \times _\mathcal {X} U$, $t = \text{pr}_0$, and $s = \text{pr}_1$. Using the isomorphism $R' \to U' \times _{\mathcal{X}'} U'$ we get similarly an isomorphism $\tau ' : f' \circ t' \to f' \circ s'$. Note that $g \circ f' \circ t' = f \circ t \circ \text{pr}_ R$ and $g \circ f' \circ s' = f \circ s \circ \text{pr}_ R$, where $\text{pr}_ R : R' \to R$ is the projection. Thus it makes sense to ask if

99.3.6.1
\begin{equation} \label{stacks-properties-equation-verify} \tau \star \text{id}_{\text{pr}_ R} = \text{id}_ g \star \tau '. \end{equation}

Now we make two claims: (1) if Equation (99.3.6.1) holds, then the diagram $2$-commutes, and (2) Equation (99.3.6.1) holds. We omit the proof of both claims. Hints: part (1) follows from the construction of $f = f_{can}$ and $f' = f'_{can}$ in Algebraic Stacks, Lemma 93.16.1. Part (2) follows by carefuly working through the definitions. $\square$

Remark 99.3.7. Let $\mathcal{Y}$ be an algebraic stack. Consider the following $2$-category:

1. An object is a morphism $f : \mathcal{X} \to \mathcal{Y}$ which is representable by algebraic spaces,

2. a $1$-morphism $(g, \beta ) : (f_1 : \mathcal{X}_1 \to \mathcal{Y}) \to (f_2 : \mathcal{X}_2 \to \mathcal{Y})$ consists of a morphism $g : \mathcal{X}_1 \to \mathcal{X}_2$ and a $2$-morphism $\beta : f_1 \to f_2 \circ g$, and

3. a $2$-morphism between $(g, \beta ), (g', \beta ') : (f_1 : \mathcal{X}_1 \to \mathcal{Y}) \to (f_2 : \mathcal{X}_2 \to \mathcal{Y})$ is a $2$-morphism $\alpha : g \to g'$ such that $(\text{id}_{f_2} \star \alpha ) \circ \beta = \beta '$.

Let us denote this $2$-category $\textit{Spaces}/\mathcal{Y}$ by analogy with the notation of Topologies on Spaces, Section 72.2. Now we claim that in this $2$-category the morphism categories

$\mathop{\mathrm{Mor}}\nolimits _{\textit{Spaces}/\mathcal{Y}}( (f_1 : \mathcal{X}_1 \to \mathcal{Y}), (f_2 : \mathcal{X}_2 \to \mathcal{Y}))$

are all setoids. Namely, a $2$-morphism $\alpha$ is a rule which to each object $x_1$ of $\mathcal{X}_1$ assigns an isomorphism $\alpha _{x_1} : g(x_1) \longrightarrow g'(x_1)$ in the relevant fibre category of $\mathcal{X}_2$ such that the diagram

$\xymatrix{ & f_2(x_1) \ar[ld]_{\beta _{x_1}} \ar[rd]^{\beta '_{x_1}} \\ f_2(g(x_1)) \ar[rr]^{f_2(\alpha _{x_1})} & & f_2(g'(x_1)) }$

commutes. But since $f_2$ is faithful (see Algebraic Stacks, Lemma 93.15.2) this means that if $\alpha _{x_1}$ exists, then it is unique! In other words the $2$-category $\textit{Spaces}/\mathcal{Y}$ is very close to being a category. Namely, if we replace $1$-morphisms by isomorphism classes of $1$-morphisms we obtain a category. We will often perform this replacement without further mention.

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