Definition 34.6.1. Let $T$ be a scheme. An syntomic 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 syntomic and such that $T = \bigcup f_ i(T_ i)$.
34.6 The syntomic topology
In this section we define the syntomic topology. This topology is quite interesting in that it often has the same cohomology groups as the fppf topology but is technically easier to deal with.
Lemma 34.6.2. Any smooth covering is a syntomic covering, and a fortiori, any étale or Zariski covering is a syntomic covering.
Proof. This is clear from the definitions and the fact that a smooth morphism is syntomic, see Morphisms, Lemma 29.34.7 and Lemma 34.5.2. $\square$
Next, we show that this notion satisfies the conditions of Sites, Definition 7.6.2.
Lemma 34.6.3. Let $T$ be a scheme.
If $T' \to T$ is an isomorphism then $\{ T' \to T\} $ is a syntomic covering of $T$.
If $\{ T_ i \to T\} _{i\in I}$ is a syntomic covering and for each $i$ we have a syntomic covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a syntomic covering.
If $\{ T_ i \to T\} _{i\in I}$ is a syntomic covering and $T' \to T$ is a morphism of schemes then $\{ T' \times _ T T_ i \to T'\} _{i\in I}$ is a syntomic covering.
Proof. Omitted. $\square$
Lemma 34.6.4. Let $T$ be an affine scheme. Let $\{ T_ i \to T\} _{i \in I}$ be a syntomic covering of $T$. Then there exists a syntomic 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, and such that each morphism $U_ j \to T$ is standard syntomic, see Morphisms, Definition 29.30.1. Moreover, we may choose each $U_ j$ to be open affine in one of the $T_ i$.
Proof. Omitted, but see Algebra, Lemma 10.136.15. $\square$
Thus we define the corresponding standard coverings of affines as follows.
Definition 34.6.5. Let $T$ be an affine scheme. A standard syntomic covering of $T$ is a family $\{ f_ j : U_ j \to T\} _{j = 1, \ldots , m}$ with each $U_ j$ is affine, $U_ j \to T$ standard syntomic and $T = \bigcup f_ j(U_ j)$.
Definition 34.6.6. A big syntomic site is any site $\mathit{Sch}_{syntomic}$ as in Sites, Definition 7.6.2 constructed as follows:
Choose any set of schemes $S_0$, and any set of syntomic 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 syntomic 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 syntomic site of a scheme $S$, let us point out that the topology on a big syntomic site $\mathit{Sch}_{syntomic}$ is in some sense induced from the syntomic topology on the category of all schemes.
Lemma 34.6.7. Let $\mathit{Sch}_{syntomic}$ be a big syntomic site as in Definition 34.6.6. Let $T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{syntomic})$. Let $\{ T_ i \to T\} _{i \in I}$ be an arbitrary syntomic covering of $T$.
There exists a covering $\{ U_ j \to T\} _{j \in J}$ of $T$ in the site $\mathit{Sch}_{syntomic}$ which refines $\{ T_ i \to T\} _{i \in I}$.
If $\{ T_ i \to T\} _{i \in I}$ is a standard syntomic covering, then it is tautologically equivalent to a covering in $\mathit{Sch}_{syntomic}$.
If $\{ T_ i \to T\} _{i \in I}$ is a Zariski covering, then it is tautologically equivalent to a covering in $\mathit{Sch}_{syntomic}$.
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 34.6.3 the refinement $\{ T_{ij} \to T\} _{i \in I, j \in J_ i}$ is a syntomic 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 $\mathit{Sch}_{syntomic}$. But then $T_ i$ as a finite type scheme over $W_ i$ is isomorphic to an object $V_ i$ of $\mathit{Sch}_{syntomic}$ 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 $\mathit{Sch}_{syntomic}$ by Sets, Lemma 3.9.9. The covering $\{ U_ j \to T\} _{j \in J}$ is a covering as in (1). In the situation of (2), (3) each of the schemes $T_ i$ is isomorphic to an object of $\mathit{Sch}_{syntomic}$ by Sets, Lemma 3.9.9, and another application of Sets, Lemma 3.11.1 gives what we want. $\square$
Definition 34.6.8. Let $S$ be a scheme. Let $\mathit{Sch}_{syntomic}$ be a big syntomic site containing $S$.
The big syntomic site of $S$, denoted $(\mathit{Sch}/S)_{syntomic}$, is the site $\mathit{Sch}_{syntomic}/S$ introduced in Sites, Section 7.25.
The big affine syntomic site of $S$, denoted $(\textit{Aff}/S)_{syntomic}$, is the full subcategory of $(\mathit{Sch}/S)_{syntomic}$ whose objects are affine $U/S$. A covering of $(\textit{Aff}/S)_{syntomic}$ is any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{syntomic}$ which is a standard syntomic covering.
Next, we check that the big affine site defines the same topos as the big site.
Lemma 34.6.9. Let $S$ be a scheme. Let $\mathit{Sch}_{syntomic}$ be a big syntomic site containing $S$. The functor $(\textit{Aff}/S)_{syntomic} \to (\mathit{Sch}/S)_{syntomic}$ is special cocontinuous and induces an equivalence of topoi from $\mathop{\mathit{Sh}}\nolimits ((\textit{Aff}/S)_{syntomic})$ to $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{syntomic})$.
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 $u : (\textit{Aff}/S)_{syntomic} \to (\mathit{Sch}/S)_{syntomic}$. Being cocontinuous just means that any syntomic covering of $T/S$, $T$ affine, can be refined by a standard syntomic covering of $T$. This is the content of Lemma 34.6.4. Hence (1) holds. We see $u$ is continuous simply because a standard syntomic covering is a syntomic 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. $\square$
To be continued...
Lemma 34.6.10. Let $\mathit{Sch}_{syntomic}$ be a big syntomic site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{syntomic}$. 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.
\[ u : (\mathit{Sch}/T)_{syntomic} \longrightarrow (\mathit{Sch}/S)_{syntomic}, \quad V/T \longmapsto V/S \]
\[ v : (\mathit{Sch}/S)_{syntomic} \longrightarrow (\mathit{Sch}/T)_{syntomic}, \quad (U \to S) \longmapsto (U \times _ S T \to T). \]
\[ f_{big} : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/T)_{syntomic}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{syntomic}) \]
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 $f_{big, *}$. $\square$
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