Definition 34.7.1. Let $T$ be a scheme. An fppf covering of $T$ is a family of morphisms $\{ f_ i : T_ i \to T\} _{i \in I}$ of schemes such that each $f_ i$ is flat, locally of finite presentation and such that $T = \bigcup f_ i(T_ i)$.
34.7 The fppf topology
Let $S$ be a scheme. We would like to define the fppf-topology1 on the category of schemes over $S$. According to our general principle we first introduce the notion of an fppf-covering.
Lemma 34.7.2. Any syntomic covering is an fppf covering, and a fortiori, any smooth, étale, 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, see Morphisms, Lemmas 29.30.6 and 29.30.7, and Lemma 34.6.2. $\square$
Next, we show that this notion satisfies the conditions of Sites, Definition 7.6.2.
Lemma 34.7.3. Let $T$ be a scheme.
If $T' \to T$ is an isomorphism then $\{ T' \to T\} $ is an fppf covering of $T$.
If $\{ T_ i \to T\} _{i\in I}$ is an fppf covering and for each $i$ we have an fppf covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is an fppf covering.
If $\{ T_ i \to T\} _{i\in I}$ is an fppf covering and $T' \to T$ is a morphism of schemes then $\{ T' \times _ T T_ i \to T'\} _{i\in I}$ is an fppf covering.
Proof. The first assertion is clear. The second follows as the composition of flat morphisms is flat (see Morphisms, Lemma 29.25.6) and the composition of morphisms of finite presentation is of finite presentation (see Morphisms, Lemma 29.21.3). The third follows as the base change of a flat morphism is flat (see Morphisms, Lemma 29.25.8) and the base change of a morphism of finite presentation is of finite presentation (see Morphisms, Lemma 29.21.4). Moreover, the base change of a surjective family of morphisms is surjective (proof omitted). $\square$
Lemma 34.7.4. Let $T$ be an affine scheme. Let $\{ T_ i \to T\} _{i \in I}$ be an fppf covering of $T$. Then there exists an fppf covering $\{ U_ j \to T\} _{j = 1, \ldots , m}$ which is a refinement of $\{ T_ i \to T\} _{i \in I}$ such that each $U_ j$ is an affine scheme. Moreover, we may choose each $U_ j$ to be open affine in one of the $T_ i$.
Proof. This follows directly from the definitions using that a morphism which is flat and locally of finite presentation is open, see Morphisms, Lemma 29.25.10. $\square$
Thus we define the corresponding standard coverings of affines as follows.
Definition 34.7.5. Let $T$ be an affine scheme. A standard fppf covering of $T$ is a family $\{ f_ j : U_ j \to T\} _{j = 1, \ldots , m}$ with each $U_ j$ is affine, flat and of finite presentation over $T$ and $T = \bigcup f_ j(U_ j)$.
Definition 34.7.6. A big fppf site is any site $\mathit{Sch}_{fppf}$ as in Sites, Definition 7.6.2 constructed as follows:
Choose any set of schemes $S_0$, and any set of fppf coverings $\text{Cov}_0$ among these schemes.
As underlying category take any category $\mathit{Sch}_\alpha $ constructed as in Sets, Lemma 3.9.2 starting with the set $S_0$.
Choose any set of coverings as in Sets, Lemma 3.11.1 starting with the category $\mathit{Sch}_\alpha $ and the class of fppf coverings, and the set $\text{Cov}_0$ chosen above.
See the remarks following Definition 34.3.5 for motivation and explanation regarding the definition of big sites.
Before we continue with the introduction of the big fppf site of a scheme $S$, let us point out that the topology on a big fppf site $\mathit{Sch}_{fppf}$ is in some sense induced from the fppf topology on the category of all schemes.
Lemma 34.7.7. Let $\mathit{Sch}_{fppf}$ be a big fppf site as in Definition 34.7.6. Let $T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{fppf})$. Let $\{ T_ i \to T\} _{i \in I}$ be an arbitrary fppf covering of $T$.
There exists a covering $\{ U_ j \to T\} _{j \in J}$ of $T$ in the site $\mathit{Sch}_{fppf}$ which refines $\{ T_ i \to T\} _{i \in I}$.
If $\{ T_ i \to T\} _{i \in I}$ is a standard fppf covering, then it is tautologically equivalent to a covering of $\mathit{Sch}_{fppf}$.
If $\{ T_ i \to T\} _{i \in I}$ is a Zariski covering, then it is tautologically equivalent to a covering of $\mathit{Sch}_{fppf}$.
Proof. For each $i$ choose an affine open covering $T_ i = \bigcup _{j \in J_ i} T_{ij}$ such that each $T_{ij}$ maps into an affine open subscheme of $T$. By Lemma 34.7.3 the refinement $\{ T_{ij} \to T\} _{i \in I, j \in J_ i}$ is an fppf covering of $T$ as well. Hence we may assume each $T_ i$ is affine, and maps into an affine open $W_ i$ of $T$. Applying Sets, Lemma 3.9.9 we see that $W_ i$ is isomorphic to an object of $\mathit{Sch}_{fppf}$. But then $T_ i$ as a finite type scheme over $W_ i$ is isomorphic to an object $V_ i$ of $\mathit{Sch}_{fppf}$ by a second application of Sets, Lemma 3.9.9. The covering $\{ V_ i \to T\} _{i \in I}$ refines $\{ T_ i \to T\} _{i \in I}$ (because they are isomorphic). Moreover, $\{ V_ i \to T\} _{i \in I}$ is combinatorially equivalent to a covering $\{ U_ j \to T\} _{j \in J}$ of $T$ in the site $\mathit{Sch}_{fppf}$ by Sets, Lemma 3.9.9. The covering $\{ U_ j \to T\} _{j \in J}$ is a refinement as in (1). In the situation of (2), (3) each of the schemes $T_ i$ is isomorphic to an object of $\mathit{Sch}_{fppf}$ by Sets, Lemma 3.9.9, and another application of Sets, Lemma 3.11.1 gives what we want. $\square$
Definition 34.7.8. Let $S$ be a scheme. Let $\mathit{Sch}_{fppf}$ be a big fppf site containing $S$.
The big fppf site of $S$, denoted $(\mathit{Sch}/S)_{fppf}$, is the site $\mathit{Sch}_{fppf}/S$ introduced in Sites, Section 7.25.
The big affine fppf site of $S$, denoted $(\textit{Aff}/S)_{fppf}$, is the full subcategory of $(\mathit{Sch}/S)_{fppf}$ whose objects are affine $U/S$. A covering of $(\textit{Aff}/S)_{fppf}$ is any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{fppf}$ which is a standard fppf covering.
It is not completely clear that the big affine fppf site is a site. We check this now.
Lemma 34.7.9. Let $S$ be a scheme. Let $\mathit{Sch}_{fppf}$ be a big fppf site containing $S$. Then $(\textit{Aff}/S)_{fppf}$ is a site.
Proof. Let us show that $(\textit{Aff}/S)_{fppf}$ is a site. Reasoning as in the proof of Lemma 34.4.9 it suffices to show that the collection of standard fppf coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 7.6.2. This is clear since for example, given a standard fppf covering $\{ T_ i \to T\} _{i\in I}$ and for each $i$ we have a standard fppf covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a standard fppf covering because $\bigcup _{i\in I} J_ i$ is finite and each $T_{ij}$ is affine. $\square$
Lemma 34.7.10. Let $S$ be a scheme. Let $\mathit{Sch}_{fppf}$ be a big fppf site containing $S$. The underlying categories of the sites $\mathit{Sch}_{fppf}$, $(\mathit{Sch}/S)_{fppf}$, and $(\textit{Aff}/S)_{fppf}$ have fibre products. In each case the obvious functor into the category $\mathit{Sch}$ of all schemes commutes with taking fibre products. The category $(\mathit{Sch}/S)_{fppf}$ has a final object, namely $S/S$.
Proof. For $\mathit{Sch}_{fppf}$ it is true by construction, see Sets, Lemma 3.9.9. Suppose we have $U \to S$, $V \to U$, $W \to U$ morphisms of schemes with $U, V, W \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{fppf})$. The fibre product $V \times _ U W$ in $\mathit{Sch}_{fppf}$ is a fibre product in $\mathit{Sch}$ and is the fibre product of $V/S$ with $W/S$ over $U/S$ in the category of all schemes over $S$, and hence also a fibre product in $(\mathit{Sch}/S)_{fppf}$. This proves the result for $(\mathit{Sch}/S)_{fppf}$. If $U, V, W$ are affine, so is $V \times _ U W$ and hence the result for $(\textit{Aff}/S)_{fppf}$. $\square$
Next, we check that the big affine site defines the same topos as the big site.
Lemma 34.7.11. Let $S$ be a scheme. Let $\mathit{Sch}_{fppf}$ be a big fppf site containing $S$. The functor $(\textit{Aff}/S)_{fppf} \to (\mathit{Sch}/S)_{fppf}$ is cocontinuous and induces an equivalence of topoi from $\mathop{\mathit{Sh}}\nolimits ((\textit{Aff}/S)_{fppf})$ to $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})$.
Proof. The notion of a special cocontinuous functor is introduced in Sites, Definition 7.29.2. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 7.29.1. Denote the inclusion functor $u : (\textit{Aff}/S)_{fppf} \to (\mathit{Sch}/S)_{fppf}$. Being cocontinuous just means that any fppf covering of $T/S$, $T$ affine, can be refined by a standard fppf covering of $T$. This is the content of Lemma 34.7.4. Hence (1) holds. We see $u$ is continuous simply because a standard fppf covering is a fppf covering. Hence (2) holds. Parts (3) and (4) follow immediately from the fact that $u$ is fully faithful. And finally condition (5) follows from the fact that every scheme has an affine open covering. $\square$
Next, we establish some relationships between the topoi associated to these sites.
Lemma 34.7.12. Let $\mathit{Sch}_{fppf}$ be a big fppf site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{fppf}$. The functor is cocontinuous, and has a continuous right adjoint They induce the same morphism of topoi We have $f_{big}^{-1}(\mathcal{G})(U/T) = \mathcal{G}(U/S)$. We have $f_{big, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times _ S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.
\[ u : (\mathit{Sch}/T)_{fppf} \longrightarrow (\mathit{Sch}/S)_{fppf}, \quad V/T \longmapsto V/S \]
\[ v : (\mathit{Sch}/S)_{fppf} \longrightarrow (\mathit{Sch}/T)_{fppf}, \quad (U \to S) \longmapsto (U \times _ S T \to T). \]
\[ f_{big} : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/T)_{fppf}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf}) \]
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 $V/S$ we have $\mathop{\mathrm{Mor}}\nolimits _ S(u(U), V) = \mathop{\mathrm{Mor}}\nolimits _ T(U, V \times _ S T)$ as desired. Thus we may apply Sites, Lemmas 7.22.1 and 7.22.2 to get the formula for $f_{big, *}$. $\square$
Lemma 34.7.13. Given schemes $X$, $Y$, $Y$ in $(\mathit{Sch}/S)_{fppf}$ and morphisms $f : X \to Y$, $g : Y \to Z$ 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 34.7.12. $\square$
[1] The letters fppf stand for “fidèlement plat de présentation finie”.
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