Definition 59.20.1. (See Topologies, Definitions 34.7.1, 34.6.1, 34.5.1, 34.4.1, and 34.3.1.) Let $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}, Zariski\} $. A family of morphisms of schemes $\{ f_ i : T_ i \to T\} _{i \in I}$ with fixed target is called a $\tau $-covering if and only if each $f_ i$ is flat of finite presentation, syntomic, smooth, étale, resp. an open immersion, and we have $\bigcup f_ i(T_ i) = T$.
59.20 Big and small sites of schemes
Let $S$ be a scheme. Let $\tau $ be one of the topologies we will be discussing. Thus $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}, Zariski\} $. Of course if you are only interested in the étale topology, then you can simply assume $\tau = {\acute{e}tale}$ throughout. Moreover, we will discuss étale morphisms, étale coverings, and étale sites in more detail starting in Section 59.25. In order to proceed with the discussion of cohomology of quasi-coherent sheaves it is convenient to introduce the big $\tau $-site and in case $\tau \in \{ {\acute{e}tale}, Zariski\} $, the small $\tau $-site of $S$. In order to do this we first introduce the notion of a $\tau $-covering.
The class of all $\tau $-coverings satisfies the axioms (1), (2) and (3) of Definition 59.10.2 (our definition of a site), see Topologies, Lemmas 34.7.3, 34.6.3, 34.5.3, 34.4.3, and 34.3.2.
Let us introduce the sites we will be working with. Contrary to what happens in [SGA4], we do not want to choose a universe. Instead we pick a “partial universe” (which is a suitably large set as in Sets, Section 3.5), and consider all schemes contained in this set. Of course we make sure that our favorite base scheme $S$ is contained in the partial universe. Having picked the underlying category we pick a suitably large set of $\tau $-coverings which turns this into a site. The details are in the chapter on topologies on schemes; there is a lot of freedom in the choices made, but in the end the actual choices made will not affect the étale (or other) cohomology of $S$ (just as in [SGA4] the actual choice of universe doesn't matter at the end). Moreover, the way the material is written the reader who is happy using strongly inaccessible cardinals (i.e., universes) can do so as a substitute.
Definition 59.20.2. Let $S$ be a scheme. Let $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}, \linebreak[0] Zariski\} $.
A big $\tau $-site of $S$ is any of the sites $(\mathit{Sch}/S)_\tau $ constructed as explained above and in more detail in Topologies, Definitions 34.7.8, 34.6.8, 34.5.8, 34.4.8, and 34.3.7.
If $\tau \in \{ {\acute{e}tale}, Zariski\} $, then the small $\tau $-site of $S$ is the full subcategory $S_\tau $ of $(\mathit{Sch}/S)_\tau $ whose objects are schemes $T$ over $S$ whose structure morphism $T \to S$ is étale, resp. an open immersion. A covering in $S_\tau $ is a covering $\{ U_ i \to U\} $ in $(\mathit{Sch}/S)_\tau $ such that $U$ is an object of $S_\tau $.
The underlying category of the site $(\mathit{Sch}/S)_\tau $ has reasonable “closure” properties, i.e., given a scheme $T$ in it any locally closed subscheme of $T$ is isomorphic to an object of $(\mathit{Sch}/S)_\tau $. Other such closure properties are: closed under fibre products of schemes, taking countable disjoint unions, taking finite type schemes over a given scheme, given an affine scheme $\mathop{\mathrm{Spec}}(R)$ one can complete, localize, or take the quotient of $R$ by an ideal while staying inside the category, etc. On the other hand, for example arbitrary disjoint unions of schemes in $(\mathit{Sch}/S)_\tau $ will take you outside of it. Also note that, given an object $T$ of $(\mathit{Sch}/S)_\tau $ there will exist $\tau $-coverings $\{ T_ i \to T\} _{i \in I}$ (as in Definition 59.20.1) which are not coverings in $(\mathit{Sch}/S)_\tau $ for example because the schemes $T_ i$ are not objects of the category $(\mathit{Sch}/S)_\tau $. But our choice of the sites $(\mathit{Sch}/S)_\tau $ is such that there always does exist a covering $\{ U_ j \to T\} _{j \in J}$ of $(\mathit{Sch}/S)_\tau $ which refines the covering $\{ T_ i \to T\} _{i \in I}$, see Topologies, Lemmas 34.7.7, 34.6.7, 34.5.7, 34.4.7, and 34.3.6. We will mostly ignore these issues in this chapter.
If $\mathcal{F}$ is a sheaf on $(\mathit{Sch}/S)_\tau $ or $S_\tau $, then we denote
\[ H^ p_\tau (U, \mathcal{F}), \text{ in particular } H^ p_\tau (S, \mathcal{F}) \]
the cohomology groups of $\mathcal{F}$ over the object $U$ of the site, see Section 59.14. Thus we have $H^ p_{fppf}(S, \mathcal{F})$, $H^ p_{syntomic}(S, \mathcal{F})$, $H^ p_{smooth}(S, \mathcal{F})$, $H^ p_{\acute{e}tale}(S, \mathcal{F})$, and $H^ p_{Zar}(S, \mathcal{F})$. The last two are potentially ambiguous since they might refer to either the big or small étale or Zariski site. However, this ambiguity is harmless by the following lemma.
Lemma 59.20.3. Let $\tau \in \{ {\acute{e}tale}, Zariski\} $. If $\mathcal{F}$ is an abelian sheaf defined on $(\mathit{Sch}/S)_\tau $, then the cohomology groups of $\mathcal{F}$ over $S$ agree with the cohomology groups of $\mathcal{F}|_{S_\tau }$ over $S$.
Proof. By Topologies, Lemmas 34.3.14 and 34.4.14 the functors $S_\tau \to (\mathit{Sch}/S)_\tau $ satisfy the hypotheses of Sites, Lemma 7.21.8. Hence our lemma follows from Cohomology on Sites, Lemma 21.7.2. $\square$
The category of sheaves on the big or small étale site of $S$ depends only on the full subcategory of $(\mathit{Sch}/S)_{\acute{e}tale}$ or $S_{\acute{e}tale}$ consisting of affines and one only needs to consider the standard étale coverings between them (as defined below). This gives rise to sites $(\textit{Aff}/S)_{\acute{e}tale}$ and $S_{affine, {\acute{e}tale}}$, see Topologies, Definition 34.4.8. The comparison results are proven in Topologies, Lemmas 34.4.11 and 34.4.12. Here is our definition of standard coverings in some of the topologies we will consider in this chapter.
Definition 59.20.4. (See Topologies, Definitions 34.7.5, 34.6.5, 34.5.5, 34.4.5, and 34.3.4.) Let $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}, Zariski\} $. Let $T$ be an affine scheme. A standard $\tau $-covering of $T$ is a family $\{ f_ j : U_ j \to T\} _{j = 1, \ldots , m}$ with each $U_ j$ is affine, and each $f_ j$ flat and of finite presentation, standard syntomic, standard smooth, étale, resp. the immersion of a standard principal open in $T$ and $T = \bigcup f_ j(U_ j)$.
Lemma 59.20.5. Let $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}, Zariski\} $. Any $\tau $-covering of an affine scheme can be refined by a standard $\tau $-covering.
Proof. See Topologies, Lemmas 34.7.4, 34.6.4, 34.5.4, 34.4.4, and 34.3.3. $\square$
For completeness we state and prove the invariance under choice of partial universe of the cohomology groups we are considering. We will prove invariance of the small étale topos in Lemma 59.21.2 below. For notation and terminology used in this lemma we refer to Topologies, Section 34.12.
Lemma 59.20.6. Let $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}, Zariski\} $. Let $S$ be a scheme. Let $(\mathit{Sch}/S)_\tau $ and $(\mathit{Sch}'/S)_\tau $ be two big $\tau $-sites of $S$, and assume that the first is contained in the second. In this case
for any abelian sheaf $\mathcal{F}'$ defined on $(\mathit{Sch}'/S)_\tau $ and any object $U$ of $(\mathit{Sch}/S)_\tau $ we have
In words: the cohomology of $\mathcal{F}'$ over $U$ computed in the bigger site agrees with the cohomology of $\mathcal{F}'$ restricted to the smaller site over $U$.
for any abelian sheaf $\mathcal{F}$ on $(\mathit{Sch}/S)_\tau $ there is an abelian sheaf $\mathcal{F}'$ on $(\mathit{Sch}/S)_\tau '$ whose restriction to $(\mathit{Sch}/S)_\tau $ is isomorphic to $\mathcal{F}$.
\[ H^ p_\tau (U, \mathcal{F}'|_{(\mathit{Sch}/S)_\tau }) = H^ p_\tau (U, \mathcal{F}') \]
Proof. By Topologies, Lemma 34.12.2 the inclusion functor $(\mathit{Sch}/S)_\tau \to (\mathit{Sch}'/S)_\tau $ satisfies the assumptions of Sites, Lemma 7.21.8. This implies (2) and (1) follows from Cohomology on Sites, Lemma 21.7.2. $\square$
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