The Stacks project

79.4 Properties of maps of presheaves representable by algebraic spaces

Here is the definition that makes this work.

Definition 79.4.1. Let $S$ be a scheme. Let $a : F \to G$ be a map of presheaves on $(\mathit{Sch}/S)_{fppf}$ which is representable by algebraic spaces. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which

  1. is preserved under any base change, and

  2. is fppf local on the base, see Descent on Spaces, Definition 73.10.1.

In this case we say that $a$ has property $\mathcal{P}$ if for every scheme $U$ and $\xi : U \to G$ the resulting morphism of algebraic spaces $U \times _ G F \to U$ has property $\mathcal{P}$.

It is important to note that we will only use this definition for properties of morphisms that are stable under base change, and local in the fppf topology on the base. This is not because the definition doesn't make sense otherwise; rather it is because we may want to give a different definition which is better suited to the property we have in mind.

The definition above applies1 for example to the properties of being “surjective”, “quasi-compact”, “étale”, “smooth”, “flat”, “separated”, “(locally) of finite type”, “(locally) quasi-finite”, “(locally) of finite presentation”, “affine”, “proper”, and “a closed immersion”. In other words, $a$ is surjective (resp. quasi-compact, étale, smooth, flat, separated, (locally) of finite type, (locally) quasi-finite, (locally) of finite presentation, proper, a closed immersion) if for every scheme $T$ and map $\xi : T \to G$ the morphism of algebraic spaces $T \times _{\xi , G} F \to T$ is surjective (resp. quasi-compact, étale, flat, separated, (locally) of finite type, (locally) quasi-finite, (locally) of finite presentation, proper, a closed immersion).

Next, we check consistency with the already existing notions. By Lemma 79.3.2 any morphism between algebraic spaces over $S$ is representable by algebraic spaces. And by Morphisms of Spaces, Lemma 66.5.3 (resp. 66.8.8, 66.39.2, 66.37.4, 66.30.5, 66.4.12, 66.23.4, 66.27.6, 66.28.4, 66.20.3, 66.40.2, 66.12.1) the definition of surjective (resp. quasi-compact, étale, smooth, flat, separated, (locally) of finite type, (locally) quasi-finite, (locally) of finite presentation, affine, proper, closed immersion) above agrees with the already existing definition of morphisms of algebraic spaces.

Some formal lemmas follow.

Lemma 79.4.2. Let $S$ be a scheme. Let $\mathcal{P}$ be a property as in Definition 79.4.1. Let

\[ \xymatrix{ G' \times _ G F \ar[r] \ar[d]^{a'} & F \ar[d]^ a \\ G' \ar[r] & G } \]

be a fibre square of presheaves on $(\mathit{Sch}/S)_{fppf}$. If $a$ is representable by algebraic spaces and has $\mathcal{P}$ so does $a'$.

Proof. Omitted. Hint: This is formal. $\square$

Lemma 79.4.3. Let $S$ be a scheme. Let $\mathcal{P}$ be a property as in Definition 79.4.1, and assume $\mathcal{P}$ is stable under composition. Let

\[ \xymatrix{ F \ar[r]^ a & G \ar[r]^ b & H } \]

be maps of presheaves on $(\mathit{Sch}/S)_{fppf}$. If $a$, $b$ are representable by algebraic spaces and has $\mathcal{P}$ so does $b \circ a$.

Proof. Omitted. Hint: See Lemma 79.3.8 and use stability under composition. $\square$

Lemma 79.4.4. Let $S$ be a scheme. Let $F_ i, G_ i : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$, $i = 1, 2$. Let $a_ i : F_ i \to G_ i$, $i = 1, 2$ be representable by algebraic spaces. Let $\mathcal{P}$ be a property as in Definition 79.4.1 which is stable under composition. If $a_1$ and $a_2$ have property $\mathcal{P}$ so does $a_1 \times a_2 : F_1 \times F_2 \longrightarrow G_1 \times G_2$.

Proof. Note that the lemma makes sense by Lemma 79.3.9. Proof omitted. $\square$

Lemma 79.4.5. Let $S$ be a scheme. Let $F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$ be a transformation of functors representable by algebraic spaces. Let $\mathcal{P}$, $\mathcal{P}'$ be properties as in Definition 79.4.1. Suppose that for any morphism $f : X \to Y$ of algebraic spaces over $S$ we have $\mathcal{P}(f) \Rightarrow \mathcal{P}'(f)$. If $a$ has property $\mathcal{P}$, then $a$ has property $\mathcal{P}'$.

Proof. Formal. $\square$

Lemma 79.4.6. Let $S$ be a scheme. Let $F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be sheaves. Let $a : F \to G$ be representable by algebraic spaces, flat, locally of finite presentation, and surjective. Then $a : F \to G$ is surjective as a map of sheaves.

Proof. Let $T$ be a scheme over $S$ and let $g : T \to G$ be a $T$-valued point of $G$. By assumption $T' = F \times _ G T$ is an algebraic space and the morphism $T' \to T$ is a flat, locally of finite presentation, and surjective morphism of algebraic spaces. Let $U \to T'$ be a surjective étale morphism, where $U$ is a scheme. Then by the definition of flat morphisms of algebraic spaces the morphism of schemes $U \to T$ is flat. Similarly for “locally of finite presentation”. The morphism $U \to T$ is surjective also, see Morphisms of Spaces, Lemma 66.5.3. Hence we see that $\{ U \to T\} $ is an fppf covering such that $g|_ U \in G(U)$ comes from an element of $F(U)$, namely the map $U \to T' \to F$. This proves the map is surjective as a map of sheaves, see Sites, Definition 7.11.1. $\square$

[1] Being preserved under base change holds by Morphisms of Spaces, Lemmas 66.5.5, 66.8.4, 66.39.4, 66.37.3, 66.30.4, 66.4.4, 66.23.3, 66.27.4, 66.28.3, 66.20.5, 66.40.3, and Spaces, Lemma 64.12.3. Being fppf local on the base holds by Descent on Spaces, Lemmas 73.11.6, 73.11.1, 73.11.28, 73.11.26, 73.11.13, 73.11.18, 73.11.11, 73.11.24, 73.11.10, 73.11.16, 73.11.19, and 73.11.17.

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