The Stacks project

72.7 Fppf topology

In this section we discuss the notion of an fppf covering of algebraic spaces, and we define the big fppf site of an algebraic space. Please compare with Topologies, Section 34.7.

Definition 72.7.1. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. An fppf covering of $X$ is a family of morphisms $\{ f_ i : X_ i \to X\} _{i \in I}$ of algebraic spaces over $S$ such that each $f_ i$ is flat and locally of finite presentation and such that

\[ |X| = \bigcup \nolimits _{i \in I} |f_ i|(|X_ i|), \]

i.e., the morphisms are jointly surjective.

This is exactly the same as Topologies, Definition 34.7.1. In particular, if $X$ and all the $X_ i$ are schemes, then we recover the usual notion of an fppf covering of schemes.

Lemma 72.7.2. Any syntomic covering is an fppf covering, and a fortiori, any smooth, ├ętale, or Zariski covering is an fppf covering.

Proof. This is clear from the definitions, the fact that a syntomic morphism is flat and locally of finite presentation (Morphisms of Spaces, Lemmas 66.36.5 and 66.36.6) and Lemma 72.6.2. $\square$

Lemma 72.7.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.

  1. If $X' \to X$ is an isomorphism then $\{ X' \to X\} $ is an fppf covering of $X$.

  2. If $\{ X_ i \to X\} _{i\in I}$ is an fppf covering and for each $i$ we have an fppf covering $\{ X_{ij} \to X_ i\} _{j\in J_ i}$, then $\{ X_{ij} \to X\} _{i \in I, j\in J_ i}$ is an fppf covering.

  3. If $\{ X_ i \to X\} _{i\in I}$ is an fppf covering and $X' \to X$ is a morphism of algebraic spaces then $\{ X' \times _ X X_ i \to X'\} _{i\in I}$ is an fppf covering.

Proof. Omitted. $\square$

Lemma 72.7.4. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Suppose that $\mathcal{U} = \{ f_ i : X_ i \to X\} _{i \in I}$ is an fppf covering of $X$. Then there exists a refinement $\mathcal{V} = \{ g_ i : T_ i \to X\} $ of $\mathcal{U}$ which is an fppf covering such that each $T_ i$ is a scheme.

Proof. Omitted. Hint: For each $i$ choose a scheme $T_ i$ and a surjective ├ętale morphism $T_ i \to X_ i$. Then check that $\{ T_ i \to X\} $ is an fppf covering. $\square$

Lemma 72.7.5. Let $S$ be a scheme. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fppf covering of algebraic spaces over $S$. Then the map of sheaves

\[ \coprod X_ i \longrightarrow X \]

is surjective.

Proof. This follows from Spaces, Lemma 64.5.9. See also Spaces, Remark 64.5.2 in case you are confused about the meaning of this lemma. $\square$

Definition 72.7.6. Let $S$ be a scheme. A big fppf site $(\textit{Spaces}/S)_{fppf}$ is any site constructed as follows:

  1. Choose a big fppf site $(\mathit{Sch}/S)_{fppf}$ as in Topologies, Section 34.7.

  2. As underlying category take the category $\textit{Spaces}/S$ of algebraic spaces over $S$ (see discussion in Section 72.2 why this is a set).

  3. Choose any set of coverings as in Sets, Lemma 3.11.1 starting with the category $\textit{Spaces}/S$ and the class of fppf coverings of Definition 72.7.1.

Having defined this, we can localize to get the fppf site of an algebraic space.

Definition 72.7.7. Let $S$ be a scheme. Let $(\textit{Spaces}/S)_{fppf}$ be as in Definition 72.7.6. Let $X$ be an algebraic space over $S$, i.e., an object of $(\textit{Spaces}/S)_{fppf}$. Then the big fppf site $(\textit{Spaces}/X)_{fppf}$ of $X$ is the localization of the site $(\textit{Spaces}/S)_{fppf}$ at $X$ introduced in Sites, Section 7.25.

Next, we establish some relationships between the topoi associated to these sites.

Lemma 72.7.8. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. The functor

\[ u : (\textit{Spaces}/Y)_{fppf} \longrightarrow (\textit{Spaces}/X)_{fppf}, \quad V/Y \longmapsto V/X \]

is cocontinuous, and has a continuous right adjoint

\[ v : (\textit{Spaces}/X)_{fppf} \longrightarrow (\textit{Spaces}/Y)_{fppf}, \quad (U \to Y) \longmapsto (U \times _ X Y \to Y). \]

They induce the same morphism of topoi

\[ f_{big} : \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{fppf}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{fppf}) \]

We have $f_{big}^{-1}(\mathcal{G})(U/Y) = \mathcal{G}(U/X)$. We have $f_{big, *}(\mathcal{F})(U/X) = \mathcal{F}(U \times _ X Y/Y)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.

Proof. The functor $u$ is cocontinuous, continuous, and commutes with fibre products and equalizers. Hence Sites, Lemmas 7.21.5 and 7.21.6 apply and we deduce the formula for $f_{big}^{-1}$ and the existence of $f_{big!}$. Moreover, the functor $v$ is a right adjoint because given $U/T$ and $V/X$ we have $\mathop{\mathrm{Mor}}\nolimits _ X(u(U), V) = \mathop{\mathrm{Mor}}\nolimits _ Y(U, V \times _ X Y)$ as desired. Thus we may apply Sites, Lemmas 7.22.1 and 7.22.2 to get the formula for $f_{big, *}$. $\square$

Lemma 72.7.9. Let $S$ be a scheme. Given morphisms $f : X \to Y$, $g : Y \to Z$ of algebraic spaces over $S$ we have $g_{big} \circ f_{big} = (g \circ f)_{big}$.

Proof. This follows from the simple description of pushforward and pullback for the functors on the big sites from Lemma 72.7.8. $\square$


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