Introduction
In this document we explain what the different topologies on the category of schemes are. Some references are [SGA1] and [Ner]. Before doing so we would like to point out that there are many different choices of sites (as defined in Sites, Definition 00VH) which give rise to the same notion of sheaf on the underlying category. Hence our choices may be slightly different from those in the references but ultimately lead to the same cohomology groups, etc.
The general procedure
In this section we explain a general procedure for producing the sites we will be working with. Suppose we want to study sheaves over schemes with respect to some topology \(\tau\). In order to get a site, as in Sites, Definition 00VH, of schemes with that topology we have to do some work. Namely, we cannot simply say “consider all schemes with the Zariski topology” since that would give a “big” category. Instead, in each section of this chapter we will proceed as follows:
We define a class \(\text{Cov}_\tau\) of coverings of schemes satisfying the axioms of Sites, Definition 00VH. It will always be the case that a Zariski open covering of a scheme is a covering for \(\tau\).
We single out a notion of standard \(\tau\)-covering within the category of affine schemes.
We define what is an “absolute” big \(\tau\)-site \(\Sch_\tau\). These are the sites one gets by appropriately choosing a set of schemes and a set of coverings.
For any object \(S\) of \(\Sch_\tau\) we define the big \(\tau\)-site \((\Sch/S)_\tau\) and for suitable \(\tau\) the small1 \(\tau\)-site \(S_\tau\).
In addition there is a site \((\textit{Aff}/S)_\tau\) using the notion of standard \(\tau\)-covering of affines2 whose category of sheaves is equivalent to the category of sheaves on \((\Sch/S)_\tau\).
The above is a little clumsy in that we do not end up with a canonical choice for the big \(\tau\)-site of a scheme, or even the small \(\tau\)-site of a scheme. If you are willing to ignore set theoretic difficulties, then you can work with classes and end up with canonical big and small sites...
The Zariski topology
Definition
Let \(T\) be a scheme. A Zariski 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 an open immersion and such that \(T = \bigcup f_i(T_i)\).
This defines a (proper) class of coverings. Next, we show that this notion satisfies the conditions of Sites, Definition 00VH.
Lemma
Let \(T\) be a scheme.
If \(T' \to T\) is an isomorphism then \(\{T' \to T\}\) is a Zariski covering of \(T\).
If \(\{T_i \to T\}_{i\in I}\) is a Zariski covering and for each \(i\) we have a Zariski covering \(\{T_{ij} \to T_i\}_{j\in J_i}\), then \(\{T_{ij} \to T\}_{i \in I, j\in J_i}\) is a Zariski covering.
If \(\{T_i \to T\}_{i\in I}\) is a Zariski covering and \(T' \to T\) is a morphism of schemes then \(\{T' \times_T T_i \to T'\}_{i\in I}\) is a Zariski covering.
Proof
Omitted.
Lemma
Let \(T\) be an affine scheme. Let \(\{T_i \to T\}_{i \in I}\) be a Zariski covering of \(T\). Then there exists a Zariski 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 a standard open of \(T\), see Schemes, Definition 01HT. Moreover, we may choose each \(U_j\) to be an open of one of the \(T_i\).
Proof
Follows as \(T\) is quasi-compact and standard opens form a basis for its topology. This is also proved in Schemes, Lemma 01HS.
Thus we define the corresponding standard coverings of affines as follows.
Definition
Compare Schemes, Definition 01HT. Let \(T\) be an affine scheme. A standard Zariski covering of \(T\) is a Zariski covering \(\{U_j \to T\}_{j = 1, \ldots, m}\) with each \(U_j \to T\) inducing an isomorphism with a standard affine open of \(T\).
Definition
A big Zariski site is any site \(\Sch_{Zar}\) as in Sites, Definition 00VH constructed as follows:
Choose any set of schemes \(S_0\), and any set of Zariski coverings \(\text{Cov}_0\) among these schemes.
As underlying category of \(\Sch_{Zar}\) take any category \(\Sch_\alpha\) constructed as in Sets, Lemma 000J starting with the set \(S_0\).
As coverings of \(\Sch_{Zar}\) choose any set of coverings as in Sets, Lemma 000X starting with the category \(\Sch_\alpha\) and the class of Zariski coverings, and the set \(\text{Cov}_0\) chosen above.
It is shown in Sites, Lemma 00VY that, after having chosen the category \(\Sch_\alpha\), the category of sheaves on \(\Sch_\alpha\) does not depend on the choice of coverings chosen in (3) above. In other words, the topos \(\Sh(\Sch_{Zar})\) only depends on the choice of the category \(\Sch_\alpha\). It is shown in Sets, Lemma 000R that these categories are closed under many constructions of algebraic geometry, e.g., fibre products and taking open and closed subschemes. We can also show that the exact choice of \(\Sch_\alpha\) does not matter too much, see Section 022I.
Another approach would be to assume the existence of a strongly inaccessible cardinal and to define \(\Sch_{Zar}\) to be the category of schemes contained in a chosen universe with set of coverings the Zariski coverings contained in that same universe.
Before we continue with the introduction of the big Zariski site of a scheme \(S\), let us point out that the topology on a big Zariski site \(\Sch_{Zar}\) is in some sense induced from the Zariski topology on the category of all schemes.
Lemma
Let \(\Sch_{Zar}\) be a big Zariski site as in Definition 020S. Let \(T \in \Ob(\Sch_{Zar})\). Let \(\{T_i \to T\}_{i \in I}\) be an arbitrary Zariski covering of \(T\). There exists a covering \(\{U_j \to T\}_{j \in J}\) of \(T\) in the site \(\Sch_{Zar}\) which is tautologically equivalent (see Sites, Definition 00VU) to \(\{T_i \to T\}_{i \in I}\).
Proof
Since each \(T_i \to T\) is an open immersion, we see by Sets, Lemma 000R that each \(T_i\) is isomorphic to an object \(V_i\) of \(\Sch_{Zar}\). The covering \(\{V_i \to T\}_{i \in I}\) is tautologically equivalent to \(\{T_i \to T\}_{i \in I}\) (using the identity map on \(I\) both ways). 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 \(\Sch_{Zar}\) by Sets, Lemma 000X.
Definition
Let \(S\) be a scheme. Let \(\Sch_{Zar}\) be a big Zariski site containing \(S\).
The big Zariski site of \(S\), denoted \((\Sch/S)_{Zar}\), is the site \(\Sch_{Zar}/S\) introduced in Sites, Section 00XZ.
The small Zariski site of \(S\), which we denote \(S_{Zar}\), is the full subcategory of \((\Sch/S)_{Zar}\) whose objects are those \(U/S\) such that \(U \to S\) is an open immersion. A covering of \(S_{Zar}\) is any covering \(\{U_i \to U\}\) of \((\Sch/S)_{Zar}\) with \(U \in \Ob(S_{Zar})\).
The big affine Zariski site of \(S\), denoted \((\textit{Aff}/S)_{Zar}\), is the full subcategory of \((\Sch/S)_{Zar}\) consisting of objects \(U/S\) such that \(U\) is an affine scheme. A covering of \((\textit{Aff}/S)_{Zar}\) is any covering \(\{U_i \to U\}\) of \((\Sch/S)_{Zar}\) with \(U \in \Ob((\textit{Aff}/S)_{Zar})\) which is a standard Zariski covering.
The small affine Zariski site of \(S\), denoted \(S_{affine, Zar}\), is the full subcategory of \(S_{Zar}\) whose objects are those \(U/S\) such that \(U\) is an affine scheme. A covering of \(S_{affine, Zar}\) is any covering \(\{U_i \to U\}\) of \(S_{Zar}\) with \(U \in \Ob(S_{affine, Zar})\) which is a standard Zariski covering.
It is not completely clear that the small Zariski site, the big affine Zariski site, and the small affine Zariski site are sites. We check this now.
Lemma
Let \(S\) be a scheme. Let \(\Sch_{Zar}\) be a big Zariski site containing \(S\). The structures \(S_{Zar}\), \((\textit{Aff}/S)_{Zar}\), and \(S_{affine, Zar}\) defined above are sites.
Proof
Let us show that \(S_{Zar}\) is a site. It is a category with a given set of families of morphisms with fixed target. Thus we have to show properties (1), (2) and (3) of Sites, Definition 00VH. Since \((\Sch/S)_{Zar}\) is a site, it suffices to prove that given any covering \(\{U_i \to U\}\) of \((\Sch/S)_{Zar}\) with \(U \in \Ob(S_{Zar})\) we also have \(U_i \in \Ob(S_{Zar})\). This follows from the definitions as the composition of open immersions is an open immersion.
Let us show that \((\textit{Aff}/S)_{Zar}\) is a site. Reasoning as above, it suffices to show that the collection of standard Zariski coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 00VH. Let \(R\) be a ring. Let \(f_1, \ldots, f_n \in R\) generate the unit ideal. For each \(i \in \{1, \ldots, n\}\) let \(g_{i1}, \ldots, g_{in_i} \in R_{f_i}\) be elements generating the unit ideal of \(R_{f_i}\). Write \(g_{ij} = f_{ij}/f_i^{e_{ij}}\) which is possible. After replacing \(f_{ij}\) by \(f_i f_{ij}\) if necessary, we have that \(D(f_{ij}) \subset D(f_i) \cong \Spec(R_{f_i})\) is equal to \(D(g_{ij}) \subset \Spec(R_{f_i})\). Hence we see that the family of morphisms \(\{D(g_{ij}) \to \Spec(R)\}\) is a standard Zariski covering. From these considerations it follows that (2) holds for standard Zariski coverings. We omit the verification of (1) and (3).
We omit the proof that \(S_{affine, Zar}\) is a site.
Lemma
Let \(S\) be a scheme. Let \(\Sch_{Zar}\) be a big Zariski site containing \(S\). The underlying categories of the sites \(\Sch_{Zar}\), \((\Sch/S)_{Zar}\), \(S_{Zar}\), \((\textit{Aff}/S)_{Zar}\), and \(S_{affine, Zar}\) have fibre products. In each case the obvious functor into the category \(\Sch\) of all schemes commutes with taking fibre products. The categories \((\Sch/S)_{Zar}\), and \(S_{Zar}\) both have a final object, namely \(S/S\).
Proof
For \(\Sch_{Zar}\) it is true by construction, see Sets, Lemma 000R. Suppose we have \(U \to S\), \(V \to U\), \(W \to U\) morphisms of schemes with \(U, V, W \in \Ob(\Sch_{Zar})\). The fibre product \(V \times_U W\) in \(\Sch_{Zar}\) is a fibre product in \(\Sch\) 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 \((\Sch/S)_{Zar}\). This proves the result for \((\Sch/S)_{Zar}\). If \(U \to S\), \(V \to U\) and \(W \to U\) are open immersions then so is \(V \times_U W \to S\) and hence we get the result for \(S_{Zar}\). If \(U, V, W\) are affine, so is \(V \times_U W\) and hence the result for \((\textit{Aff}/S)_{Zar}\) and \(S_{affine, Zar}\).
Next, we check that the big, resp. small affine site defines the same topos as the big, resp. small site.
Lemma
Let \(S\) be a scheme. Let \(\Sch_{Zar}\) be a big Zariski site containing \(S\). The functor \((\textit{Aff}/S)_{Zar} \to (\Sch/S)_{Zar}\) is a special cocontinuous functor. Hence it induces an equivalence of topoi from \(\Sh((\textit{Aff}/S)_{Zar})\) to \(\Sh((\Sch/S)_{Zar})\).
Proof
The notion of a special cocontinuous functor is introduced in Sites, Definition 03CG. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 03A0. Denote the inclusion functor \(u : (\textit{Aff}/S)_{Zar} \to (\Sch/S)_{Zar}\). Being cocontinuous just means that any Zariski covering of \(T/S\), \(T\) affine, can be refined by a standard Zariski covering of \(T\). This is the content of Lemma 020Q. Hence (1) holds. We see \(u\) is continuous simply because a standard Zariski covering is a Zariski 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.
Lemma
Let \(S\) be a scheme. Let \(\Sch_{Zar}\) be a big Zariski site containing \(S\). The functor \(S_{affine, Zar} \to S_{Zar}\) is a special cocontinuous functor. Hence it induces an equivalence of topoi from \(\Sh(S_{affine, Zar})\) to \(\Sh(S_{Zar})\).
Proof
Omitted. Hint: compare with the proof of Lemma 020W.
Let us check that the notion of a sheaf on the small Zariski site corresponds to notion of a sheaf on \(S\).
Lemma
The category of sheaves on \(S_{Zar}\) is equivalent to the category of sheaves on the underlying topological space of \(S\).
Proof
We will use repeatedly that for any object \(U/S\) of \(S_{Zar}\) the morphism \(U \to S\) is an isomorphism onto an open subscheme. Let \(\mathcal{F}\) be a sheaf on \(S\). Then we define a sheaf on \(S_{Zar}\) by the rule \(\mathcal{F}'(U/S) = \mathcal{F}(\Im(U \to S))\). For the converse, we choose for every open subscheme \(U \subset S\) an object \(U'/S \in \Ob(S_{Zar})\) with \(\Im(U' \to S) = U\) (here you have to use Sets, Lemma 000R). Given a sheaf \(\mathcal{G}\) on \(S_{Zar}\) we define a sheaf on \(S\) by setting \(\mathcal{G}'(U) = \mathcal{G}(U'/S)\). To see that \(\mathcal{G}'\) is a sheaf we use that for any open covering \(U = \bigcup_{i \in I} U_i\) the covering \(\{U_i \to U\}_{i \in I}\) is combinatorially equivalent to a covering \(\{U_j' \to U'\}_{j \in J}\) in \(S_{Zar}\) by Sets, Lemma 000X, and we use Sites, Lemma 00VW. Details omitted.
From now on we will not make any distinction between a sheaf on \(S_{Zar}\) or a sheaf on \(S\). We will always use the procedures of the proof of the lemma to go between the two notions. Next, we establish some relationships between the topoi associated to these sites.
Lemma
Let \(\Sch_{Zar}\) be a big Zariski site. Let \(f : T \to S\) be a morphism in \(\Sch_{Zar}\). The functor \(T_{Zar} \to (\Sch/S)_{Zar}\) is cocontinuous and induces a morphism of topoi \[i_f : \Sh(T_{Zar}) \longrightarrow \Sh((\Sch/S)_{Zar})\] For a sheaf \(\mathcal{G}\) on \((\Sch/S)_{Zar}\) we have the formula \((i_f^{-1}\mathcal{G})(U/T) = \mathcal{G}(U/S)\). The functor \(i_f^{-1}\) also has a left adjoint \(i_{f, !}\) which commutes with fibre products and equalizers.
Proof
Denote the functor \(u : T_{Zar} \to (\Sch/S)_{Zar}\). In other words, given and open immersion \(j : U \to T\) corresponding to an object of \(T_{Zar}\) we set \(u(U \to T) = (f \circ j : U \to S)\). This functor commutes with fibre products, see Lemma 020V. Moreover, \(T_{Zar}\) has equalizers (as any two morphisms with the same source and target are the same) and \(u\) commutes with them. It is clearly cocontinuous. It is also continuous as \(u\) transforms coverings to coverings and commutes with fibre products. Hence the lemma follows from Sites, Lemmas 00XR and 00XS.
Lemma
Let \(S\) be a scheme. Let \(\Sch_{Zar}\) be a big Zariski site containing \(S\). The inclusion functor \(S_{Zar} \to (\Sch/S)_{Zar}\) satisfies the hypotheses of Sites, Lemma 00XU and hence induces a morphism of sites \[\pi_S : (\Sch/S)_{Zar} \longrightarrow S_{Zar}\] and a morphism of topoi \[i_S : \Sh(S_{Zar}) \longrightarrow \Sh((\Sch/S)_{Zar})\] such that \(\pi_S \circ i_S = \text{id}\). Moreover, \(i_S = i_{\text{id}_S}\) with \(i_{\text{id}_S}\) as in Lemma 020Y. In particular the functor \(i_S^{-1} = \pi_{S, *}\) is described by the rule \(i_S^{-1}(\mathcal{G})(U/S) = \mathcal{G}(U/S)\).
Proof
In this case the functor \(u : S_{Zar} \to (\Sch/S)_{Zar}\), in addition to the properties seen in the proof of Lemma 020Y above, also is fully faithful and transforms the final object into the final object. The lemma follows.
Definition
In the situation of Lemma 020Z the functor \(i_S^{-1} = \pi_{S, *}\) is often called the restriction to the small Zariski site, and for a sheaf \(\mathcal{F}\) on the big Zariski site we denote \(\mathcal{F}|_{S_{Zar}}\) this restriction.
With this notation in place we have for a sheaf \(\mathcal{F}\) on the big site and a sheaf \(\mathcal{G}\) on the small site that \[\begin{align*} \Mor_{\Sh(S_{Zar})}(\mathcal{F}|_{S_{Zar}}, \mathcal{G}) & = \Mor_{\Sh((\Sch/S)_{Zar})}(\mathcal{F}, i_{S, *}\mathcal{G}) \\ \Mor_{\Sh(S_{Zar})}(\mathcal{G}, \mathcal{F}|_{S_{Zar}}) & = \Mor_{\Sh((\Sch/S)_{Zar})}(\pi_S^{-1}\mathcal{G}, \mathcal{F}) \end{align*}\] Moreover, we have \((i_{S, *}\mathcal{G})|_{S_{Zar}} = \mathcal{G}\) and we have \((\pi_S^{-1}\mathcal{G})|_{S_{Zar}} = \mathcal{G}\).
Lemma
Let \(\Sch_{Zar}\) be a big Zariski site. Let \(f : T \to S\) be a morphism in \(\Sch_{Zar}\). The functor \[u : (\Sch/T)_{Zar} \longrightarrow (\Sch/S)_{Zar}, \quad V/T \longmapsto V/S\] is cocontinuous, and has a continuous right adjoint \[v : (\Sch/S)_{Zar} \longrightarrow (\Sch/T)_{Zar}, \quad (U \to S) \longmapsto (U \times_S T \to T).\] They induce the same morphism of topoi \[f_{big} : \Sh((\Sch/T)_{Zar}) \longrightarrow \Sh((\Sch/S)_{Zar})\] 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.
Proof
The functor \(u\) is cocontinuous, continuous, and commutes with fibre products and equalizers (details omitted; compare with proof of Lemma 020Y). Hence Sites, Lemmas 00XR and 00XS 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 \(\Mor_S(u(U), V) = \Mor_T(U, V \times_S T)\) as desired. Thus we may apply Sites, Lemmas 00XX and 00XY to get the formula for \(f_{big, *}\).
Lemma
Let \(\Sch_{Zar}\) be a big Zariski site. Let \(f : T \to S\) be a morphism in \(\Sch_{Zar}\).
We have \(i_f = f_{big} \circ i_T\) with \(i_f\) as in Lemma 020Y and \(i_T\) as in Lemma 020Z.
The functor \(S_{Zar} \to T_{Zar}\), \((U \to S) \mapsto (U \times_S T \to T)\) is continuous and induces a morphism of sites \[f_{small} : T_{Zar} \longrightarrow S_{Zar}\] The functors \(f_{small}^{-1}\) and \(f_{small, *}\) agree with the usual notions \(f^{-1}\) and \(f_*\) if we identify sheaves on \(T_{Zar}\), resp. \(S_{Zar}\) with sheaves on \(T\), resp. \(S\) via Lemma 020X.
We have a commutative diagram of morphisms of sites \[\xymatrix{ T_{Zar} \ar[d]_{f_{small}} & (\Sch/T)_{Zar} \ar[d]^{f_{big}} \ar[l]^{\pi_T} \\ S_{Zar} & (\Sch/S)_{Zar} \ar[l]_{\pi_S} }\] so that \(f_{small} \circ \pi_T = \pi_S \circ f_{big}\) as morphisms of topoi.
We have \(f_{small} = \pi_S \circ f_{big} \circ i_T = \pi_S \circ i_f\).
Proof
The equality \(i_f = f_{big} \circ i_T\) follows from the equality \(i_f^{-1} = i_T^{-1} \circ f_{big}^{-1}\) which is clear from the descriptions of these functors above. Thus we see (1).
Statement (2): See Sites, Example 00X2.
Part (3) follows because \(\pi_S\) and \(\pi_T\) are given by the inclusion functors and \(f_{small}\) and \(f_{big}\) by the base change functor \(U \mapsto U \times_S T\).
Statement (4) follows from (3) by precomposing with \(i_T\).
In the situation of the lemma, using the terminology of Definition 04BS we have: for \(\mathcal{F}\) a sheaf on the big Zariski site of \(T\) \[(f_{big, *}\mathcal{F})|_{S_{Zar}} = f_{small, *}(\mathcal{F}|_{T_{Zar}}),\] This equality is clear from the commutativity of the diagram of sites of the lemma, since restriction to the small Zariski site of \(T\), resp. \(S\) is given by \(\pi_{T, *}\), resp. \(\pi_{S, *}\). A similar formula involving pullbacks and restrictions is false.
Lemma
Given schemes \(X\), \(Y\), \(Z\) in \((\Sch/S)_{Zar}\) and morphisms \(f : X \to Y\), \(g : Y \to Z\) we have \(g_{big} \circ f_{big} = (g \circ f)_{big}\) and \(g_{small} \circ f_{small} = (g \circ f)_{small}\).
Proof
This follows from the simple description of pushforward and pullback for the functors on the big sites from Lemma 0210. For the functors on the small sites this is Sheaves, Lemma 008E via the identification of Lemma 020X.
Lemma
Let \(\Sch_{Zar}\) be a big Zariski site. Consider a cartesian diagram \[\xymatrix{ T' \ar[r]_{g'} \ar[d]_{f'} & T \ar[d]^f \\ S' \ar[r]^g & S }\] in \(\Sch_{Zar}\). Then \(i_g^{-1} \circ f_{big, *} = f'_{small, *} \circ (i_{g'})^{-1}\) and \(g_{big}^{-1} \circ f_{big, *} = f'_{big, *} \circ (g'_{big})^{-1}\).
Proof
Since the diagram is cartesian, we have for \(U'/S'\) that \(U' \times_{S'} T' = U' \times_S T\). Hence both \(i_g^{-1} \circ f_{big, *}\) and \(f'_{small, *} \circ (i_{g'})^{-1}\) send a sheaf \(\mathcal{F}\) on \((\Sch/T)_{Zar}\) to the sheaf \(U' \mapsto \mathcal{F}(U' \times_{S'} T')\) on \(S'_{Zar}\) (use Lemmas 020Y and 0211). The second equality can be proved in the same manner or can be deduced from the very general Sites, Lemma 03CF.
We can think about a sheaf on the big Zariski site of \(S\) as a collection of “usual” sheaves on all schemes over \(S\).
Lemma
Let \(S\) be a scheme contained in a big Zariski site \(\Sch_{Zar}\). A sheaf \(\mathcal{F}\) on the big Zariski site \((\Sch/S)_{Zar}\) is given by the following data:
for every \(T/S \in \Ob((\Sch/S)_{Zar})\) a sheaf \(\mathcal{F}_T\) on \(T\),
for every \(f : T' \to T\) in \((\Sch/S)_{Zar}\) a map \(c_f : f^{-1}\mathcal{F}_T \to \mathcal{F}_{T'}\).
These data are subject to the following conditions:
given any \(f : T' \to T\) and \(g : T'' \to T'\) in \((\Sch/S)_{Zar}\) the composition \(c_g \circ g^{-1}c_f\) is equal to \(c_{f \circ g}\), and
if \(f : T' \to T\) in \((\Sch/S)_{Zar}\) is an open immersion then \(c_f\) is an isomorphism.
Proof
This lemma follows from a purely sheaf theoretic statement discussed in Sites, Remark 0GWL. We also give a direct proof in this case.
Given a sheaf \(\mathcal{F}\) on \((\Sch/S)_{Zar}\) we set \(\mathcal{F}_T = i_p^{-1}\mathcal{F}\) where \(p : T \to S\) is the structure morphism. Note that \(\mathcal{F}_T(U) = \mathcal{F}(U'/S)\) for any open \(U \subset T\), and \(U' \to T\) an open immersion in \((\Sch/T)_{Zar}\) with image \(U\), see Lemmas 020X and 020Y. Hence given \(f : T' \to T\) over \(S\) and \(U, U' \to T\) we get a canonical map \(\mathcal{F}_T(U) = \mathcal{F}(U'/S) \to \mathcal{F}(U'\times_T T'/S) = \mathcal{F}_{T'}(f^{-1}(U))\) where the middle is the restriction map of \(\mathcal{F}\) with respect to the morphism \(U' \times_T T' \to U'\) over \(S\). The collection of these maps are compatible with restrictions, and hence define an \(f\)-map \(c_f\) from \(\mathcal{F}_T\) to \(\mathcal{F}_{T'}\), see Sheaves, Definition 008J and the discussion surrounding it. It is clear that \(c_{f \circ g}\) is the composition of \(c_f\) and \(c_g\), since composition of restriction maps of \(\mathcal{F}\) gives restriction maps.
Conversely, given a system \((\mathcal{F}_T, c_f)\) as in the lemma we may define a presheaf \(\mathcal{F}\) on \((\Sch/S)_{Zar}\) by simply setting \(\mathcal{F}(T/S) = \mathcal{F}_T(T)\). As restriction mapping, given \(f : T' \to T\) we set for \(s \in \mathcal{F}(T)\) the pullback \(f^*(s)\) equal to \(c_f(s)\) (where we think of \(c_f\) as an \(f\)-map again). The condition on the \(c_f\) guarantees that pullbacks satisfy the required functoriality property. We omit the verification that this is a sheaf. It is clear that the constructions so defined are mutually inverse.
The étale topology
Let \(S\) be a scheme. We would like to define the étale-topology on the category of schemes over \(S\). According to our general principle we first introduce the notion of an étale covering.
Definition
Let \(T\) be a scheme. An étale 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 étale and such that \(T = \bigcup f_i(T_i)\).
Lemma
Any Zariski covering is an étale covering.
Proof
This is clear from the definitions and the fact that an open immersion is an étale morphism, see Morphisms, Lemma 02GP.
Next, we show that this notion satisfies the conditions of Sites, Definition 00VH.
Lemma
Let \(T\) be a scheme.
If \(T' \to T\) is an isomorphism then \(\{T' \to T\}\) is an étale covering of \(T\).
If \(\{T_i \to T\}_{i\in I}\) is an étale covering and for each \(i\) we have an étale covering \(\{T_{ij} \to T_i\}_{j\in J_i}\), then \(\{T_{ij} \to T\}_{i \in I, j\in J_i}\) is an étale covering.
If \(\{T_i \to T\}_{i\in I}\) is an étale covering and \(T' \to T\) is a morphism of schemes then \(\{T' \times_T T_i \to T'\}_{i\in I}\) is an étale covering.
Proof
Omitted.
Lemma
Let \(T\) be an affine scheme. Let \(\{T_i \to T\}_{i \in I}\) be an étale covering of \(T\). Then there exists an étale 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
Omitted.
Thus we define the corresponding standard coverings of affines as follows.
Definition
Let \(T\) be an affine scheme. A standard étale covering of \(T\) is a family \(\{f_j : U_j \to T\}_{j = 1, \ldots, m}\) with each \(U_j\) is affine and étale over \(T\) and \(T = \bigcup f_j(U_j)\).
In the definition above we do not assume the morphisms \(f_j\) are standard étale. The reason is that if we did then the standard étale coverings would not define a site on \(\textit{Aff}/S\), for example because of Algebra, Lemma 00UC part (4). On the other hand, an étale morphism of affines is automatically standard smooth, see Algebra, Lemma 00U9. Hence a standard étale covering is a standard smooth covering and a standard syntomic covering.
Definition
A big étale site is any site \(\Sch_\etale\) as in Sites, Definition 00VH constructed as follows:
Choose any set of schemes \(S_0\), and any set of étale coverings \(\text{Cov}_0\) among these schemes.
As underlying category take any category \(\Sch_\alpha\) constructed as in Sets, Lemma 000J starting with the set \(S_0\).
Choose any set of coverings as in Sets, Lemma 000X starting with the category \(\Sch_\alpha\) and the class of étale coverings, and the set \(\text{Cov}_0\) chosen above.
See the remarks following Definition 020S for motivation and explanation regarding the definition of big sites.
Before we continue with the introduction of the big étale site of a scheme \(S\), let us point out that the topology on a big étale site \(\Sch_\etale\) is in some sense induced from the étale topology on the category of all schemes.
Lemma
Let \(\Sch_\etale\) be a big étale site as in Definition 021A. Let \(T \in \Ob(\Sch_\etale)\). Let \(\{T_i \to T\}_{i \in I}\) be an arbitrary étale covering of \(T\).
There exists a covering \(\{U_j \to T\}_{j \in J}\) of \(T\) in the site \(\Sch_\etale\) which refines \(\{T_i \to T\}_{i \in I}\).
If \(\{T_i \to T\}_{i \in I}\) is a standard étale covering, then it is tautologically equivalent to a covering in \(\Sch_\etale\).
If \(\{T_i \to T\}_{i \in I}\) is a Zariski covering, then it is tautologically equivalent to a covering in \(\Sch_\etale\).
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 0217 the refinement \(\{T_{ij} \to T\}_{i \in I, j \in J_i}\) is an étale 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 000R we see that \(W_i\) is isomorphic to an object of \(\Sch_\etale\). But then \(T_i\) as a finite type scheme over \(W_i\) is isomorphic to an object \(V_i\) of \(\Sch_\etale\) by a second application of Sets, Lemma 000R. 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 \(\Sch_\etale\) by Sets, Lemma 000X. 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 \(\Sch_\etale\) by Sets, Lemma 000R, and another application of Sets, Lemma 000X gives what we want.
Definition
Let \(S\) be a scheme. Let \(\Sch_\etale\) be a big étale site containing \(S\).
The big étale site of \(S\), denoted \((\Sch/S)_\etale\), is the site \(\Sch_\etale/S\) introduced in Sites, Section 00XZ.
The small étale site of \(S\), which we denote \(S_\etale\), is the full subcategory of \((\Sch/S)_\etale\) whose objects are those \(U/S\) such that \(U \to S\) is étale. A covering of \(S_\etale\) is any covering \(\{U_i \to U\}\) of \((\Sch/S)_\etale\) with \(U \in \Ob(S_\etale)\).
The big affine étale site of \(S\), denoted \((\textit{Aff}/S)_\etale\), is the full subcategory of \((\Sch/S)_\etale\) whose objects are those \(U/S\) such that \(U\) is an affine scheme. A covering of \((\textit{Aff}/S)_\etale\) is any covering \(\{U_i \to U\}\) of \((\Sch/S)_\etale\) with \(U \in \Ob((\textit{Aff}/S)_\etale)\) which is a standard étale covering.
The small affine étale site of \(S\), denoted \(S_{affine, \etale}\), is the full subcategory of \(S_\etale\) whose objects are those \(U/S\) such that \(U\) is an affine scheme. A covering of \(S_{affine, \etale}\) is any covering \(\{U_i \to U\}\) of \(S_\etale\) with \(U \in \Ob(S_{affine, \etale})\) which is a standard étale covering.
It is not completely clear that the big affine étale site, the small étale site, and the small affine étale site are sites. We check this now.
Lemma
Let \(S\) be a scheme. Let \(\Sch_\etale\) be a big étale site containing \(S\). The structures \(S_\etale\), \((\textit{Aff}/S)_\etale\), and \(S_{affine, \etale}\) are sites.
Proof
Let us show that \(S_\etale\) is a site. It is a category with a given set of families of morphisms with fixed target. Thus we have to show properties (1), (2) and (3) of Sites, Definition 00VH. Since \((\Sch/S)_\etale\) is a site, it suffices to prove that given any covering \(\{U_i \to U\}\) of \((\Sch/S)_\etale\) with \(U \in \Ob(S_\etale)\) we also have \(U_i \in \Ob(S_\etale)\). This follows from the definitions as the composition of étale morphisms is an étale morphism.
Let us show that \((\textit{Aff}/S)_\etale\) is a site. Reasoning as above, it suffices to show that the collection of standard étale coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 00VH. This is clear since for example, given a standard étale covering \(\{T_i \to T\}_{i\in I}\) and for each \(i\) we have a standard étale 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 étale covering because \(\bigcup_{i\in I} J_i\) is finite and each \(T_{ij}\) is affine.
We omit the proof that \(S_{affine, \etale}\) is a site.
Lemma
Let \(S\) be a scheme. Let \(\Sch_\etale\) be a big étale site containing \(S\). The underlying categories of the sites \(\Sch_\etale\), \((\Sch/S)_\etale\), \(S_\etale\), \((\textit{Aff}/S)_\etale\), and \(S_{affine, \etale}\) have fibre products. In each case the obvious functor into the category \(\Sch\) of all schemes commutes with taking fibre products. The categories \((\Sch/S)_\etale\), and \(S_\etale\) both have a final object, namely \(S/S\).
Proof
For \(\Sch_\etale\) it is true by construction, see Sets, Lemma 000R. Suppose we have \(U \to S\), \(V \to U\), \(W \to U\) morphisms of schemes with \(U, V, W \in \Ob(\Sch_\etale)\). The fibre product \(V \times_U W\) in \(\Sch_\etale\) is a fibre product in \(\Sch\) 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 \((\Sch/S)_\etale\). This proves the result for \((\Sch/S)_\etale\). If \(U \to S\), \(V \to U\) and \(W \to U\) are étale then so is \(V \times_U W \to S\) and hence we get the result for \(S_\etale\). If \(U, V, W\) are affine, so is \(V \times_U W\) and hence the result for \((\textit{Aff}/S)_\etale\) and \(S_{affine, \etale}\).
Next, we check that the big, resp. small affine site defines the same topos as the big, resp. small site.
Lemma
Let \(S\) be a scheme. Let \(\Sch_\etale\) be a big étale site containing \(S\). The functor \((\textit{Aff}/S)_\etale \to (\Sch/S)_\etale\) is special cocontinuous and induces an equivalence of topoi from \(\Sh((\textit{Aff}/S)_\etale)\) to \(\Sh((\Sch/S)_\etale)\).
Proof
The notion of a special cocontinuous functor is introduced in Sites, Definition 03CG. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 03A0. Denote the inclusion functor \(u : (\textit{Aff}/S)_\etale \to (\Sch/S)_\etale\). Being cocontinuous just means that any étale covering of \(T/S\), \(T\) affine, can be refined by a standard étale covering of \(T\). This is the content of Lemma 0218. Hence (1) holds. We see \(u\) is continuous simply because a standard étale covering is a étale 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.
Lemma
Let \(S\) be a scheme. Let \(\Sch_\etale\) be a big étale site containing \(S\). The functor \(S_{affine, \etale} \to S_\etale\) is special cocontinuous and induces an equivalence of topoi from \(\Sh(S_{affine, \etale})\) to \(\Sh(S_\etale)\).
Proof
Omitted. Hint: compare with the proof of Lemma 021E.
Next, we establish some relationships between the topoi associated to these sites.
Lemma
Let \(\Sch_\etale\) be a big étale site. Let \(f : T \to S\) be a morphism in \(\Sch_\etale\). The functor \(T_\etale \to (\Sch/S)_\etale\) is cocontinuous and induces a morphism of topoi \[i_f : \Sh(T_\etale) \longrightarrow \Sh((\Sch/S)_\etale)\] For a sheaf \(\mathcal{G}\) on \((\Sch/S)_\etale\) we have the formula \((i_f^{-1}\mathcal{G})(U/T) = \mathcal{G}(U/S)\). The functor \(i_f^{-1}\) also has a left adjoint \(i_{f, !}\) which commutes with fibre products and equalizers.
Proof
Denote the functor \(u : T_\etale \to (\Sch/S)_\etale\). In other words, given an étale morphism \(j : U \to T\) corresponding to an object of \(T_\etale\) we set \(u(U \to T) = (f \circ j : U \to S)\). This functor commutes with fibre products, see Lemma 021D. Let \(a, b : U \to V\) be two morphisms in \(T_\etale\). In this case the equalizer of \(a\) and \(b\) (in the category of schemes) is \[V \times_{\Delta_{V/T}, V \times_T V, (a, b)} U\] which is a fibre product of schemes étale over \(T\), hence étale over \(T\). Thus \(T_\etale\) has equalizers and \(u\) commutes with them. It is clearly cocontinuous. It is also continuous as \(u\) transforms coverings to coverings and commutes with fibre products. Hence the Lemma follows from Sites, Lemmas 00XR and 00XS.
Lemma
Let \(S\) be a scheme. Let \(\Sch_\etale\) be a big étale site containing \(S\). The inclusion functor \(S_\etale \to (\Sch/S)_\etale\) satisfies the hypotheses of Sites, Lemma 00XU and hence induces a morphism of sites \[\pi_S : (\Sch/S)_\etale \longrightarrow S_\etale\] and a morphism of topoi \[i_S : \Sh(S_\etale) \longrightarrow \Sh((\Sch/S)_\etale)\] such that \(\pi_S \circ i_S = \text{id}\). Moreover, \(i_S = i_{\text{id}_S}\) with \(i_{\text{id}_S}\) as in Lemma 021F. In particular the functor \(i_S^{-1} = \pi_{S, *}\) is described by the rule \(i_S^{-1}(\mathcal{G})(U/S) = \mathcal{G}(U/S)\).
Proof
In this case the functor \(u : S_\etale \to (\Sch/S)_\etale\), in addition to the properties seen in the proof of Lemma 021F above, also is fully faithful and transforms the final object into the final object. The lemma follows from Sites, Lemma 00XU.
Definition
In the situation of Lemma 021G the functor \(i_S^{-1} = \pi_{S, *}\) is often called the restriction to the small étale site, and for a sheaf \(\mathcal{F}\) on the big étale site we denote \(\mathcal{F}|_{S_\etale}\) this restriction.
With this notation in place we have for a sheaf \(\mathcal{F}\) on the big site and a sheaf \(\mathcal{G}\) on the small site that \[\begin{align*} \Mor_{\Sh(S_\etale)}( \mathcal{F}|_{S_\etale}, \mathcal{G}) & = \Mor_{\Sh((\Sch/S)_\etale)}( \mathcal{F}, i_{S, *}\mathcal{G}) \\ \Mor_{\Sh(S_\etale)}( \mathcal{G}, \mathcal{F}|_{S_\etale}) & = \Mor_{\Sh((\Sch/S)_\etale)}( \pi_S^{-1}\mathcal{G}, \mathcal{F}) \end{align*}\] Moreover, we have \((i_{S, *}\mathcal{G})|_{S_\etale} = \mathcal{G}\) and we have \((\pi_S^{-1}\mathcal{G})|_{S_\etale} = \mathcal{G}\).
Lemma
Let \(\Sch_\etale\) be a big étale site. Let \(f : T \to S\) be a morphism in \(\Sch_\etale\). The functor \[u : (\Sch/T)_\etale \longrightarrow (\Sch/S)_\etale, \quad V/T \longmapsto V/S\] is cocontinuous, and has a continuous right adjoint \[v : (\Sch/S)_\etale \longrightarrow (\Sch/T)_\etale, \quad (U \to S) \longmapsto (U \times_S T \to T).\] They induce the same morphism of topoi \[f_{big} : \Sh((\Sch/T)_\etale) \longrightarrow \Sh((\Sch/S)_\etale)\] 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.
Proof
The functor \(u\) is cocontinuous, continuous and commutes with fibre products and equalizers (details omitted; compare with the proof of Lemma 021F). Hence Sites, Lemmas 00XR and 00XS 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 \(\Mor_S(u(U), V) = \Mor_T(U, V \times_S T)\) as desired. Thus we may apply Sites, Lemmas 00XX and 00XY to get the formula for \(f_{big, *}\).
Lemma
Let \(\Sch_\etale\) be a big étale site. Let \(f : T \to S\) be a morphism in \(\Sch_\etale\).
We have \(i_f = f_{big} \circ i_T\) with \(i_f\) as in Lemma 021F and \(i_T\) as in Lemma 021G.
The functor \(S_\etale \to T_\etale\), \((U \to S) \mapsto (U \times_S T \to T)\) is continuous and induces a morphism of sites \[f_{small} : T_\etale \longrightarrow S_\etale\] We have \(f_{small, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times_S T/T)\).
We have a commutative diagram of morphisms of sites \[\xymatrix{ T_\etale \ar[d]_{f_{small}} & (\Sch/T)_\etale \ar[d]^{f_{big}} \ar[l]^{\pi_T}\\ S_\etale & (\Sch/S)_\etale \ar[l]_{\pi_S} }\] so that \(f_{small} \circ \pi_T = \pi_S \circ f_{big}\) as morphisms of topoi.
We have \(f_{small} = \pi_S \circ f_{big} \circ i_T = \pi_S \circ i_f\).
Proof
The equality \(i_f = f_{big} \circ i_T\) follows from the equality \(i_f^{-1} = i_T^{-1} \circ f_{big}^{-1}\) which is clear from the descriptions of these functors above. Thus we see (1).
The functor \(u : S_\etale \to T_\etale\), \(u(U \to S) = (U \times_S T \to T)\) transforms coverings into coverings and commutes with fibre products, see Lemma 0217 (3) and 021D. Moreover, both \(S_\etale\), \(T_\etale\) have final objects, namely \(S/S\) and \(T/T\) and \(u(S/S) = T/T\). Hence by Sites, Proposition 00X6 the functor \(u\) corresponds to a morphism of sites \(T_\etale \to S_\etale\). This in turn gives rise to the morphism of topoi, see Sites, Lemma 00XC. The description of the pushforward is clear from these references.
Part (3) follows because \(\pi_S\) and \(\pi_T\) are given by the inclusion functors and \(f_{small}\) and \(f_{big}\) by the base change functors \(U \mapsto U \times_S T\).
Statement (4) follows from (3) by precomposing with \(i_T\).
In the situation of the lemma, using the terminology of Definition 04BT we have: for \(\mathcal{F}\) a sheaf on the big étale site of \(T\) \[(f_{big, *}\mathcal{F})|_{S_\etale} = f_{small, *}(\mathcal{F}|_{T_\etale}),\] This equality is clear from the commutativity of the diagram of sites of the lemma, since restriction to the small étale site of \(T\), resp. \(S\) is given by \(\pi_{T, *}\), resp. \(\pi_{S, *}\). A similar formula involving pullbacks and restrictions is false.
Lemma
Given schemes \(X\), \(Y\), \(Z\) in \(\Sch_\etale\) and morphisms \(f : X \to Y\), \(g : Y \to Z\) we have \(g_{big} \circ f_{big} = (g \circ f)_{big}\) and \(g_{small} \circ f_{small} = (g \circ f)_{small}\).
Proof
This follows from the simple description of pushforward and pullback for the functors on the big sites from Lemma 021H. For the functors on the small sites this follows from the description of the pushforward functors in Lemma 021I.
Lemma
Let \(\Sch_\etale\) be a big étale site. Consider a cartesian diagram \[\xymatrix{ T' \ar[r]_{g'} \ar[d]_{f'} & T \ar[d]^f \\ S' \ar[r]^g & S }\] in \(\Sch_\etale\). Then \(i_g^{-1} \circ f_{big, *} = f'_{small, *} \circ (i_{g'})^{-1}\) and \(g_{big}^{-1} \circ f_{big, *} = f'_{big, *} \circ (g'_{big})^{-1}\).
Proof
Since the diagram is cartesian, we have for \(U'/S'\) that \(U' \times_{S'} T' = U' \times_S T\). Hence both \(i_g^{-1} \circ f_{big, *}\) and \(f'_{small, *} \circ (i_{g'})^{-1}\) send a sheaf \(\mathcal{F}\) on \((\Sch/T)_\etale\) to the sheaf \(U' \mapsto \mathcal{F}(U' \times_{S'} T')\) on \(S'_\etale\) (use Lemmas 021F and 021H). The second equality can be proved in the same manner or can be deduced from the very general Sites, Lemma 03CF.
We can think about a sheaf on the big étale site of \(S\) as a collection of “usual” sheaves on all schemes over \(S\).
Lemma
Let \(S\) be a scheme contained in a big étale site \(\Sch_\etale\). A sheaf \(\mathcal{F}\) on the big étale site \((\Sch/S)_\etale\) is given by the following data:
for every \(T/S \in \Ob((\Sch/S)_\etale)\) a sheaf \(\mathcal{F}_T\) on \(T_\etale\),
for every \(f : T' \to T\) in \((\Sch/S)_\etale\) a map \(c_f : f_{small}^{-1}\mathcal{F}_T \to \mathcal{F}_{T'}\).
These data are subject to the following conditions:
given any \(f : T' \to T\) and \(g : T'' \to T'\) in \((\Sch/S)_\etale\) the composition \(c_g \circ g_{small}^{-1}c_f\) is equal to \(c_{f \circ g}\), and
if \(f : T' \to T\) in \((\Sch/S)_\etale\) is étale then \(c_f\) is an isomorphism.
Proof
This lemma follows from a purely sheaf theoretic statement discussed in Sites, Remark 0GWL. We also give a direct proof in this case.
Given a sheaf \(\mathcal{F}\) on \((\Sch/S)_\etale\) we set \(\mathcal{F}_T = i_p^{-1}\mathcal{F}\) where \(p : T \to S\) is the structure morphism. Note that \(\mathcal{F}_T(U) = \mathcal{F}(U/S)\) for any \(U \to T\) in \(T_\etale\) see Lemma 021F. Hence given \(f : T' \to T\) over \(S\) and \(U \to T\) we get a canonical map \(\mathcal{F}_T(U) = \mathcal{F}(U/S) \to \mathcal{F}(U \times_T T'/S) = \mathcal{F}_{T'}(U \times_T T')\) where the middle is the restriction map of \(\mathcal{F}\) with respect to the morphism \(U \times_T T' \to U\) over \(S\). The collection of these maps are compatible with restrictions, and hence define a map \(c'_f : \mathcal{F}_T \to f_{small, *}\mathcal{F}_{T'}\) where \(u : T_\etale \to T'_\etale\) is the base change functor associated to \(f\). By adjunction of \(f_{small, *}\) (see Sites, Section 00WU) with \(f_{small}^{-1}\) this is the same as a map \(c_f : f_{small}^{-1}\mathcal{F}_T \to \mathcal{F}_{T'}\). It is clear that \(c'_{f \circ g}\) is the composition of \(c'_f\) and \(f_{small, *}c'_g\), since composition of restriction maps of \(\mathcal{F}\) gives restriction maps, and this gives the desired relationship among \(c_f\), \(c_g\) and \(c_{f \circ g}\).
Conversely, given a system \((\mathcal{F}_T, c_f)\) as in the lemma we may define a presheaf \(\mathcal{F}\) on \((\Sch/S)_\etale\) by simply setting \(\mathcal{F}(T/S) = \mathcal{F}_T(T)\). As restriction mapping, given \(f : T' \to T\) we set for \(s \in \mathcal{F}(T)\) the pullback \(f^*(s)\) equal to \(c_f(s)\) where we think of \(c_f\) as a map \(\mathcal{F}_T \to f_{small, *}\mathcal{F}_{T'}\) again. The condition on the \(c_f\) guarantees that pullbacks satisfy the required functoriality property. We omit the verification that this is a sheaf. It is clear that the constructions so defined are mutually inverse.
The smooth topology
In this section we define the smooth topology. This is a bit pointless as it will turn out later (see More on Morphisms, Section 055S) that this topology defines the same topos as the étale topology. But still it makes sense and it is used occasionally.
Definition
Let \(T\) be a scheme. A smooth 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 smooth and such that \(T = \bigcup f_i(T_i)\).
Lemma
Any étale covering is a smooth covering, and a fortiori, any Zariski covering is a smooth covering.
Proof
This is clear from the definitions, the fact that an étale morphism is smooth see Morphisms, Definition 02GI and Lemma 0216.
Next, we show that this notion satisfies the conditions of Sites, Definition 00VH.
Lemma
Let \(T\) be a scheme.
If \(T' \to T\) is an isomorphism then \(\{T' \to T\}\) is a smooth covering of \(T\).
If \(\{T_i \to T\}_{i\in I}\) is a smooth covering and for each \(i\) we have a smooth covering \(\{T_{ij} \to T_i\}_{j\in J_i}\), then \(\{T_{ij} \to T\}_{i \in I, j\in J_i}\) is a smooth covering.
If \(\{T_i \to T\}_{i\in I}\) is a smooth covering and \(T' \to T\) is a morphism of schemes then \(\{T' \times_T T_i \to T'\}_{i\in I}\) is a smooth covering.
Proof
Omitted.
Lemma
Let \(T\) be an affine scheme. Let \(\{T_i \to T\}_{i \in I}\) be a smooth covering of \(T\). Then there exists a smooth 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 smooth, see Morphisms, Definition 01V5. Moreover, we may choose each \(U_j\) to be open affine in one of the \(T_i\).
Proof
Omitted, but see Algebra, Lemma 00TA.
Thus we define the corresponding standard coverings of affines as follows.
Definition
Let \(T\) be an affine scheme. A standard smooth 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 smooth and \(T = \bigcup f_j(U_j)\).
Definition
A big smooth site is any site \(\Sch_{smooth}\) as in Sites, Definition 00VH constructed as follows:
Choose any set of schemes \(S_0\), and any set of smooth coverings \(\text{Cov}_0\) among these schemes.
As underlying category take any category \(\Sch_\alpha\) constructed as in Sets, Lemma 000J starting with the set \(S_0\).
Choose any set of coverings as in Sets, Lemma 000X starting with the category \(\Sch_\alpha\) and the class of smooth coverings, and the set \(\text{Cov}_0\) chosen above.
See the remarks following Definition 020S for motivation and explanation regarding the definition of big sites.
Before we continue with the introduction of the big smooth site of a scheme \(S\), let us point out that the topology on a big smooth site \(\Sch_{smooth}\) is in some sense induced from the smooth topology on the category of all schemes.
Lemma
Let \(\Sch_{smooth}\) be a big smooth site as in Definition 03WY. Let \(T \in \Ob(\Sch_{smooth})\). Let \(\{T_i \to T\}_{i \in I}\) be an arbitrary smooth covering of \(T\).
There exists a covering \(\{U_j \to T\}_{j \in J}\) of \(T\) in the site \(\Sch_{smooth}\) which refines \(\{T_i \to T\}_{i \in I}\).
If \(\{T_i \to T\}_{i \in I}\) is a standard smooth covering, then it is tautologically equivalent to a covering of \(\Sch_{smooth}\).
If \(\{T_i \to T\}_{i \in I}\) is a Zariski covering, then it is tautologically equivalent to a covering of \(\Sch_{smooth}\).
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 0221 the refinement \(\{T_{ij} \to T\}_{i \in I, j \in J_i}\) is a smooth 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 000R we see that \(W_i\) is isomorphic to an object of \(\Sch_{smooth}\). But then \(T_i\) as a finite type scheme over \(W_i\) is isomorphic to an object \(V_i\) of \(\Sch_{smooth}\) by a second application of Sets, Lemma 000R. 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 \(\Sch_{smooth}\) by Sets, Lemma 000X. 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 \(\Sch_{smooth}\) by Sets, Lemma 000R, and another application of Sets, Lemma 000X gives what we want.
Definition
Let \(S\) be a scheme. Let \(\Sch_{smooth}\) be a big smooth site containing \(S\).
The big smooth site of \(S\), denoted \((\Sch/S)_{smooth}\), is the site \(\Sch_{smooth}/S\) introduced in Sites, Section 00XZ.
The big affine smooth site of \(S\), denoted \((\textit{Aff}/S)_{smooth}\), is the full subcategory of \((\Sch/S)_{smooth}\) whose objects are affine \(U/S\). A covering of \((\textit{Aff}/S)_{smooth}\) is any covering \(\{U_i \to U\}\) of \((\Sch/S)_{smooth}\) which is a standard smooth covering.
Next, we check that the big affine site defines the same topos as the big site.
Lemma
Let \(S\) be a scheme. Let \(\Sch_{smooth}\) be a big smooth site containing \(S\). The functor \((\textit{Aff}/S)_{smooth} \to (\Sch/S)_{smooth}\) is special cocontinuous and induces an equivalence of topoi from \(\Sh((\textit{Aff}/S)_{smooth})\) to \(\Sh((\Sch/S)_{smooth})\).
Proof
The notion of a special cocontinuous functor is introduced in Sites, Definition 03CG. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 03A0. Denote the inclusion functor \(u : (\textit{Aff}/S)_{smooth} \to (\Sch/S)_{smooth}\). Being cocontinuous just means that any smooth covering of \(T/S\), \(T\) affine, can be refined by a standard smooth covering of \(T\). This is the content of Lemma 0222. Hence (1) holds. We see \(u\) is continuous simply because a standard smooth covering is a smooth 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.
Lemma
Let \(\Sch_{smooth}\) be a big smooth site. Let \(f : T \to S\) be a morphism in \(\Sch_{smooth}\). The functor \[u : (\Sch/T)_{smooth} \longrightarrow (\Sch/S)_{smooth}, \quad V/T \longmapsto V/S\] is cocontinuous, and has a continuous right adjoint \[v : (\Sch/S)_{smooth} \longrightarrow (\Sch/T)_{smooth}, \quad (U \to S) \longmapsto (U \times_S T \to T).\] They induce the same morphism of topoi \[f_{big} : \Sh((\Sch/T)_{smooth}) \longrightarrow \Sh((\Sch/S)_{smooth})\] 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.
Proof
The functor \(u\) is cocontinuous, continuous, and commutes with fibre products and equalizers. Hence Sites, Lemmas 00XR and 00XS 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 \(\Mor_S(u(U), V) = \Mor_T(U, V \times_S T)\) as desired. Thus we may apply Sites, Lemmas 00XX and 00XY to get the formula for \(f_{big, *}\).
To be continued...
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.
Definition
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 \(T = \bigcup f_i(T_i)\).
Lemma
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 01VD and Lemma 0220.
Next, we show that this notion satisfies the conditions of Sites, Definition 00VH.
Lemma
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.
Lemma
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 01UC. Moreover, we may choose each \(U_j\) to be open affine in one of the \(T_i\).
Proof
Omitted, but see Algebra, Lemma 00SY.
Thus we define the corresponding standard coverings of affines as follows.
Definition
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
A big syntomic site is any site \(\Sch_{syntomic}\) as in Sites, Definition 00VH 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 \(\Sch_\alpha\) constructed as in Sets, Lemma 000J starting with the set \(S_0\).
Choose any set of coverings as in Sets, Lemma 000X starting with the category \(\Sch_\alpha\) and the class of syntomic coverings, and the set \(\text{Cov}_0\) chosen above.
See the remarks following Definition 020S 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 \(\Sch_{syntomic}\) is in some sense induced from the syntomic topology on the category of all schemes.
Lemma
Let \(\Sch_{syntomic}\) be a big syntomic site as in Definition 03X1. Let \(T \in \Ob(\Sch_{syntomic})\). 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 \(\Sch_{syntomic}\) 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 \(\Sch_{syntomic}\).
If \(\{T_i \to T\}_{i \in I}\) is a Zariski covering, then it is tautologically equivalent to a covering in \(\Sch_{syntomic}\).
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 0227 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 000R we see that \(W_i\) is isomorphic to an object of \(\Sch_{syntomic}\). But then \(T_i\) as a finite type scheme over \(W_i\) is isomorphic to an object \(V_i\) of \(\Sch_{syntomic}\) by a second application of Sets, Lemma 000R. 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 \(\Sch_{syntomic}\) by Sets, Lemma 000X. 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 \(\Sch_{syntomic}\) by Sets, Lemma 000R, and another application of Sets, Lemma 000X gives what we want.
Definition
Let \(S\) be a scheme. Let \(\Sch_{syntomic}\) be a big syntomic site containing \(S\).
The big syntomic site of \(S\), denoted \((\Sch/S)_{syntomic}\), is the site \(\Sch_{syntomic}/S\) introduced in Sites, Section 00XZ.
The big affine syntomic site of \(S\), denoted \((\textit{Aff}/S)_{syntomic}\), is the full subcategory of \((\Sch/S)_{syntomic}\) whose objects are affine \(U/S\). A covering of \((\textit{Aff}/S)_{syntomic}\) is any covering \(\{U_i \to U\}\) of \((\Sch/S)_{syntomic}\) which is a standard syntomic covering.
Next, we check that the big affine site defines the same topos as the big site.
Lemma
Let \(S\) be a scheme. Let \(\Sch_{syntomic}\) be a big syntomic site containing \(S\). The functor \((\textit{Aff}/S)_{syntomic} \to (\Sch/S)_{syntomic}\) is special cocontinuous and induces an equivalence of topoi from \(\Sh((\textit{Aff}/S)_{syntomic})\) to \(\Sh((\Sch/S)_{syntomic})\).
Proof
The notion of a special cocontinuous functor is introduced in Sites, Definition 03CG. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 03A0. Denote the inclusion functor \(u : (\textit{Aff}/S)_{syntomic} \to (\Sch/S)_{syntomic}\). 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 0228. 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.
Lemma
Let \(\Sch_{syntomic}\) be a big syntomic site. Let \(f : T \to S\) be a morphism in \(\Sch_{syntomic}\). The functor \[u : (\Sch/T)_{syntomic} \longrightarrow (\Sch/S)_{syntomic}, \quad V/T \longmapsto V/S\] is cocontinuous, and has a continuous right adjoint \[v : (\Sch/S)_{syntomic} \longrightarrow (\Sch/T)_{syntomic}, \quad (U \to S) \longmapsto (U \times_S T \to T).\] They induce the same morphism of topoi \[f_{big} : \Sh((\Sch/T)_{syntomic}) \longrightarrow \Sh((\Sch/S)_{syntomic})\] 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.
Proof
The functor \(u\) is cocontinuous, continuous, and commutes with fibre products and equalizers. Hence Sites, Lemmas 00XR and 00XS 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 \(\Mor_S(u(U), V) = \Mor_T(U, V \times_S T)\) as desired. Thus we may apply Sites, Lemmas 00XX and 00XY to get the formula for \(f_{big, *}\).
To be continued...
The fppf topology
Let \(S\) be a scheme. We would like to define the fppf-topology3 on the category of schemes over \(S\). According to our general principle we first introduce the notion of an fppf-covering.
Definition
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 \(T = \bigcup f_i(T_i)\).
Lemma
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 01UK and 01UL, and Lemma 0226.
Next, we show that this notion satisfies the conditions of Sites, Definition 00VH.
Lemma
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 01U7) and the composition of morphisms which are locally of finite presentation is locally of finite presentation (see Morphisms, Lemma 01TR). The third follows as the base change of a flat morphism is flat (see Morphisms, Lemma 01U9) and the base change of a morphism which is locally of finite presentation is locally of finite presentation (see Morphisms, Lemma 01TS). Moreover, the base change of a surjective family of morphisms is surjective (proof omitted).
Lemma
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 01UA.
Thus we define the corresponding standard coverings of affines as follows.
Definition
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
A big fppf site is any site \(\Sch_{fppf}\) as in Sites, Definition 00VH 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 \(\Sch_\alpha\) constructed as in Sets, Lemma 000J starting with the set \(S_0\).
Choose any set of coverings as in Sets, Lemma 000X starting with the category \(\Sch_\alpha\) and the class of fppf coverings, and the set \(\text{Cov}_0\) chosen above.
See the remarks following Definition 020S 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 \(\Sch_{fppf}\) is in some sense induced from the fppf topology on the category of all schemes.
Lemma
Let \(\Sch_{fppf}\) be a big fppf site as in Definition 021R. Let \(T \in \Ob(\Sch_{fppf})\). 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 \(\Sch_{fppf}\) 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 \(\Sch_{fppf}\).
If \(\{T_i \to T\}_{i \in I}\) is a Zariski covering, then it is tautologically equivalent to a covering of \(\Sch_{fppf}\).
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 021O 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 000R we see that \(W_i\) is isomorphic to an object of \(\Sch_{fppf}\). But then \(T_i\) as a finite type scheme over \(W_i\) is isomorphic to an object \(V_i\) of \(\Sch_{fppf}\) by a second application of Sets, Lemma 000R. 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 \(\Sch_{fppf}\) by Sets, Lemma 000X. 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 \(\Sch_{fppf}\) by Sets, Lemma 000R, and another application of Sets, Lemma 000X gives what we want.
Definition
Let \(S\) be a scheme. Let \(\Sch_{fppf}\) be a big fppf site containing \(S\).
The big fppf site of \(S\), denoted \((\Sch/S)_{fppf}\), is the site \(\Sch_{fppf}/S\) introduced in Sites, Section 00XZ.
The big affine fppf site of \(S\), denoted \((\textit{Aff}/S)_{fppf}\), is the full subcategory of \((\Sch/S)_{fppf}\) whose objects are affine \(U/S\). A covering of \((\textit{Aff}/S)_{fppf}\) is any covering \(\{U_i \to U\}\) of \((\Sch/S)_{fppf}\) 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
Let \(S\) be a scheme. Let \(\Sch_{fppf}\) 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 021C it suffices to show that the collection of standard fppf coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 00VH. 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.
Lemma
Let \(S\) be a scheme. Let \(\Sch_{fppf}\) be a big fppf site containing \(S\). The underlying categories of the sites \(\Sch_{fppf}\), \((\Sch/S)_{fppf}\), and \((\textit{Aff}/S)_{fppf}\) have fibre products. In each case the obvious functor into the category \(\Sch\) of all schemes commutes with taking fibre products. The category \((\Sch/S)_{fppf}\) has a final object, namely \(S/S\).
Proof
For \(\Sch_{fppf}\) it is true by construction, see Sets, Lemma 000R. Suppose we have \(U \to S\), \(V \to U\), \(W \to U\) morphisms of schemes with \(U, V, W \in \Ob(\Sch_{fppf})\). The fibre product \(V \times_U W\) in \(\Sch_{fppf}\) is a fibre product in \(\Sch\) 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 \((\Sch/S)_{fppf}\). This proves the result for \((\Sch/S)_{fppf}\). If \(U, V, W\) are affine, so is \(V \times_U W\) and hence the result for \((\textit{Aff}/S)_{fppf}\).
Next, we check that the big affine site defines the same topos as the big site.
Lemma
Let \(S\) be a scheme. Let \(\Sch_{fppf}\) be a big fppf site containing \(S\). The functor \((\textit{Aff}/S)_{fppf} \to (\Sch/S)_{fppf}\) is cocontinuous and induces an equivalence of topoi from \(\Sh((\textit{Aff}/S)_{fppf})\) to \(\Sh((\Sch/S)_{fppf})\).
Proof
The notion of a special cocontinuous functor is introduced in Sites, Definition 03CG. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 03A0. Denote the inclusion functor \(u : (\textit{Aff}/S)_{fppf} \to (\Sch/S)_{fppf}\). 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 021P. 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.
Next, we establish some relationships between the topoi associated to these sites.
Lemma
Let \(\Sch_{fppf}\) be a big fppf site. Let \(f : T \to S\) be a morphism in \(\Sch_{fppf}\). The functor \[u : (\Sch/T)_{fppf} \longrightarrow (\Sch/S)_{fppf}, \quad V/T \longmapsto V/S\] is cocontinuous, and has a continuous right adjoint \[v : (\Sch/S)_{fppf} \longrightarrow (\Sch/T)_{fppf}, \quad (U \to S) \longmapsto (U \times_S T \to T).\] They induce the same morphism of topoi \[f_{big} : \Sh((\Sch/T)_{fppf}) \longrightarrow \Sh((\Sch/S)_{fppf})\] 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.
Proof
The functor \(u\) is cocontinuous, continuous, and commutes with fibre products and equalizers. Hence Sites, Lemmas 00XR and 00XS 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 \(\Mor_S(u(U), V) = \Mor_T(U, V \times_S T)\) as desired. Thus we may apply Sites, Lemmas 00XX and 00XY to get the formula for \(f_{big, *}\).
Lemma
Given schemes \(X\), \(Y\), \(Z\) in \((\Sch/S)_{fppf}\) 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 021W.
The ph topology
In this section we define the ph topology. This is the topology generated by Zariski coverings and proper surjective morphisms, see Lemma 0DBQ.
We borrow our notation/terminology from the paper [ph] by Goodwillie and Lichtenbaum. These authors show that if we restrict to the subcategory of Noetherian schemes, then the ph topology is the same as the “h topology” as originally defined by Voevodsky: this is the topology generated by Zariski open coverings and finite type morphisms which are universally submersive. They also show that the two topologies do not agree on non-Noetherian schemes, see [ph, Example 4.5]. We return to (our version of) the h topology in More on Flatness, Section 0ETQ.
Before we can define the coverings in our topology we need to do a bit of work.
Definition
Let \(T\) be an affine scheme. A standard ph covering is a family \(\{f_j : U_j \to T\}_{j = 1, \ldots, m}\) constructed from a proper surjective morphism \(f : U \to T\) and an affine open covering \(U = \bigcup_{j = 1, \ldots, m} U_j\) by setting \(f_j = f|_{U_j}\).
It follows immediately from Chow’s lemma that we can refine a standard ph covering by a standard ph covering corresponding to a surjective projective morphism.
Lemma
Let \(\{f_j : U_j \to T\}_{j = 1, \ldots, m}\) be a standard ph covering. Let \(T' \to T\) be a morphism of affine schemes. Then \(\{U_j \times_T T' \to T'\}_{j = 1, \ldots, m}\) is a standard ph covering.
Proof
Let \(f : U \to T\) be proper surjective and let an affine open covering \(U = \bigcup_{j = 1, \ldots, m} U_j\) be given as in Definition 0DBD. Then \(U \times_T T' \to T'\) is proper surjective (Morphisms, Lemmas 01S1 and 01W4). Also, \(U \times_T T' = \bigcup_{j = 1, \ldots, m} U_j \times_T T'\) is an affine open covering. This concludes the proof.
Lemma
Let \(T\) be an affine scheme. Each of the following types of families of maps with target \(T\) has a refinement by a standard ph covering:
any Zariski open covering of \(T\),
\(\{W_{ji} \to T\}_{j = 1, \ldots, m, i = 1, \ldots n_j}\) where \(\{W_{ji} \to U_j\}_{i = 1, \ldots, n_j}\) and \(\{U_j \to T\}_{j = 1, \ldots, m}\) are standard ph coverings.
Proof
Part (1) follows from the fact that any Zariski open covering of \(T\) can be refined by a finite affine open covering.
Proof of (2). Choose \(U \to T\) proper surjective and \(U = \bigcup_{j = 1, \ldots, m} U_j\) as in Definition 0DBD. Choose \(W_j \to U_j\) proper surjective and \(W_j = \bigcup W_{ji}\) as in Definition 0DBD. By Chow’s lemma (Limits, Lemma 0202) we can find \(W'_j \to W_j\) proper surjective and closed immersions \(W'_j \to \mathbf{P}^{e_j}_{U_j}\). Thus, after replacing \(W_j\) by \(W'_j\) and \(W_j = \bigcup W_{ji}\) by a suitable affine open covering of \(W'_j\), we may assume there is a closed immersion \(W_j \subset \mathbf{P}^{e_j}_{U_j}\) for all \(j = 1, \ldots, m\).
Let \(\overline{W}_j \subset \mathbf{P}^{e_j}_U\) be the scheme theoretic closure of \(W_j\). Then \(W_j \subset \overline{W}_j\) is an open subscheme; in fact \(W_j\) is the inverse image of \(U_j \subset U\) under the morphism \(\overline{W}_j \to U\). (To see this use that \(W_j \to \mathbf{P}^{e_j}_U\) is quasi-compact and hence formation of the scheme theoretic image commutes with restriction to opens, see Morphisms, Section 01R5.) Let \(Z_j = U \setminus U_j\) with reduced induced closed subscheme structure. Then \[V_j = \overline{W}_j \amalg Z_j \to U\] is proper surjective and the open subscheme \(W_j \subset V_j\) is the inverse image of \(U_j\). Hence for \(v \in V_j\), \(v \not \in W_j\) we can pick an affine open neighbourhood \(v \in V_{j, v} \subset V_j\) which maps into \(U_{j'}\) for some \(1 \leq j' \leq m\).
To finish the proof we consider the proper surjective morphism \[V = V_1 \times_U V_2 \times_U \ldots \times_U V_m \longrightarrow U \longrightarrow T\] and the covering of \(V\) by the affine opens \[V_{1, v_1} \times_U \ldots \times_U V_{j - 1, v_{j - 1}} \times_U W_{j i} \times_U V_{j + 1, v_{j + 1}} \times_U \ldots \times_U V_{m, v_m}\] These do indeed form a covering, because each point of \(U\) is in some \(U_j\) and the inverse image of \(U_j\) in \(V\) is equal to \(V_1 \times \ldots \times V_{j - 1} \times W_j \times V_{j + 1} \times \ldots \times V_m\). Observe that the morphism from the affine open displayed above to \(T\) factors through \(W_{ji}\) thus we obtain a refinement. Finally, we only need a finite number of these affine opens as \(V\) is quasi-compact (as a scheme proper over the affine scheme \(T\)).
Definition
Let \(T\) be a scheme. A ph covering of \(T\) is a family of morphisms \(\{f_i : T_i \to T\}_{i \in I}\) of schemes such that \(f_i\) is locally of finite type and such that for every affine open \(U \subset T\) there exists a standard ph covering \(\{U_j \to U\}_{j = 1, \ldots, m}\) refining the family \(\{T_i \times_T U \to U\}_{i \in I}\).
A standard ph covering is a ph covering by Lemma 0DBE.
Lemma
A Zariski covering is a ph covering4.
Proof
This is true because a Zariski covering of an affine scheme can be refined by a standard ph covering by Lemma 0DBF.
Lemma
Let \(f : Y \to X\) be a surjective proper morphism of schemes. Then \(\{Y \to X\}\) is a ph covering.
Proof
Omitted.
Lemma
Let \(T\) be a scheme. Let \(\{f_i : T_i \to T\}_{i \in I}\) be a family of morphisms such that \(f_i\) is locally of finite type for all \(i\). The following are equivalent
\(\{T_i \to T\}_{i \in I}\) is a ph covering,
there is a ph covering which refines \(\{T_i \to T\}_{i \in I}\), and
\(\{\coprod_{i \in I} T_i \to T\}\) is a ph covering.
Proof
The equivalence of (1) and (2) follows immediately from Definition 0DBG and the fact that a refinement of a refinement is a refinement. Because of the equivalence of (1) and (2) and since \(\{T_i \to T\}_{i \in I}\) refines \(\{\coprod_{i \in I} T_i \to T\}\) we see that (1) implies (3). Finally, assume (3) holds. Let \(U \subset T\) be an affine open and let \(\{U_j \to U\}_{j = 1, \ldots, m}\) be a standard ph covering which refines \(\{U \times_T \coprod_{i \in I} T_i \to U\}\). This means that for each \(j\) we have a morphism \[h_j : U_j \longrightarrow U \times_T \coprod\nolimits_{i \in I} T_i = \coprod\nolimits_{i \in I} U \times_T T_i\] over \(U\). Since \(U_j\) is quasi-compact, we get disjoint union decompositions \(U_j = \coprod_{i \in I} U_{j, i}\) by open and closed subschemes almost all of which are empty such that \(h_j|_{U_{j, i}}\) maps \(U_{j, i}\) into \(U \times_T T_i\). It follows that \[\{U_{j, i} \to U\}_{j = 1, \ldots, m,\ i \in I,\ U_{j, i} \not = \emptyset}\] is a standard ph covering (small detail omitted) refining \(\{U \times_T T_i \to U\}_{i \in I}\). Thus (1) holds.
Next, we show that this notion satisfies the conditions of Sites, Definition 00VH.
Lemma
Let \(T\) be a scheme.
If \(T' \to T\) is an isomorphism then \(\{T' \to T\}\) is a ph covering of \(T\).
If \(\{T_i \to T\}_{i\in I}\) is a ph covering and for each \(i\) we have a ph covering \(\{T_{ij} \to T_i\}_{j\in J_i}\), then \(\{T_{ij} \to T\}_{i \in I, j\in J_i}\) is a ph covering.
If \(\{T_i \to T\}_{i\in I}\) is a ph covering and \(T' \to T\) is a morphism of schemes then \(\{T' \times_T T_i \to T'\}_{i\in I}\) is a ph covering.
Proof
Assertion (1) is clear.
Proof of (3). The base change \(T_i \times_T T' \to T'\) is locally of finite type by Morphisms, Lemma 01T4. hence we only need to check the condition on affine opens. Let \(U' \subset T'\) be an affine open subscheme. Since \(U'\) is quasi-compact we can find a finite affine open covering \(U' = U'_1 \cup \ldots \cup U'_m\) such that \(U'_j \to T\) maps into an affine open \(U_j \subset T\). Choose a standard ph covering \(\{U_{jl} \to U_j\}_{l = 1, \ldots, n_j}\) refining \(\{T_i \times_T U_j \to U_j\}\). By Lemma 0DBE the base change \(\{U_{jl} \times_{U_j} U'_j \to U'_j\}\) is a standard ph covering. Note that \(\{U'_j \to U'\}\) is a standard ph covering as well. By Lemma 0DBF the family \(\{U_{jl} \times_{U_j} U'_j \to U'\}\) can be refined by a standard ph covering. Since \(\{U_{jl} \times_{U_j} U'_j \to U'\}\) refines \(\{T_i \times_T U' \to U'\}\) we conclude.
Proof of (2). Composition preserves being locally of finite type, see Morphisms, Lemma 01T3. Hence we only need to check the condition on affine opens. Let \(U \subset T\) be affine open. First we pick a standard ph covering \(\{U_k \to U\}_{k = 1, \ldots, m}\) refining \(\{T_i \times_T U \to U\}\). Say the refinement is given by morphisms \(U_k \to T_{i_k}\) over \(T\). Then \[\{T_{i_kj} \times_{T_{i_k}} U_k \to U_k\}_{j \in J_{i_k}}\] is a ph covering by part (3). As \(U_k\) is affine, we can find a standard ph covering \(\{U_{ka} \to U_k\}_{a = 1, \ldots, b_k}\) refining this family. Then we apply Lemma 0DBF to see that \(\{U_{ka} \to U\}\) can be refined by a standard ph covering. Since \(\{U_{ka} \to U\}\) refines \(\{T_{ij} \times_T U \to U\}\) this finishes the proof.
Definition
A big ph site is any site \(\Sch_{ph}\) as in Sites, Definition 00VH constructed as follows:
Choose any set of schemes \(S_0\), and any set of ph coverings \(\text{Cov}_0\) among these schemes.
As underlying category take any category \(\Sch_\alpha\) constructed as in Sets, Lemma 000J starting with the set \(S_0\).
Choose any set of coverings as in Sets, Lemma 000X starting with the category \(\Sch_\alpha\) and the class of ph coverings, and the set \(\text{Cov}_0\) chosen above.
See the remarks following Definition 020S for motivation and explanation regarding the definition of big sites.
Before we continue with the introduction of the big ph site of a scheme \(S\), let us point out that the topology on a big ph site \(\Sch_{ph}\) is in some sense induced from the ph topology on the category of all schemes.
Lemma
Let \(\Sch_{ph}\) be a big ph site as in Definition 0DBJ. Let \(T \in \Ob(\Sch_{ph})\). Let \(\{T_i \to T\}_{i \in I}\) be an arbitrary ph covering of \(T\).
There exists a covering \(\{U_j \to T\}_{j \in J}\) of \(T\) in the site \(\Sch_{ph}\) which refines \(\{T_i \to T\}_{i \in I}\).
If \(\{T_i \to T\}_{i \in I}\) is a standard ph covering, then it is tautologically equivalent to a covering of \(\Sch_{ph}\).
If \(\{T_i \to T\}_{i \in I}\) is a Zariski covering, then it is tautologically equivalent to a covering of \(\Sch_{ph}\).
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 Lemmas 0DBH and 0DBI the refinement \(\{T_{ij} \to T\}_{i \in I, j \in J_i}\) is a ph 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 000R we see that \(W_i\) is isomorphic to an object of \(\Sch_{ph}\). But then \(T_i\) as a finite type scheme over \(W_i\) is isomorphic to an object \(V_i\) of \(\Sch_{ph}\) by a second application of Sets, Lemma 000R. 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 \(\Sch_{ph}\) by Sets, Lemma 000X. 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 \(\Sch_{ph}\) by Sets, Lemma 000R, and another application of Sets, Lemma 000X gives what we want.
Definition
Let \(S\) be a scheme. Let \(\Sch_{ph}\) be a big ph site containing \(S\).
The big ph site of \(S\), denoted \((\Sch/S)_{ph}\), is the site \(\Sch_{ph}/S\) introduced in Sites, Section 00XZ.
The big affine ph site of \(S\), denoted \((\textit{Aff}/S)_{ph}\), is the full subcategory of \((\Sch/S)_{ph}\) whose objects are affine \(U/S\). A covering of \((\textit{Aff}/S)_{ph}\) is any finite covering \(\{U_i \to U\}\) of \((\Sch/S)_{ph}\) with \(U_i\) and \(U\) affine.
Observe that the coverings in \((\textit{Aff}/S)_{ph}\) are not given by standard ph coverings. The reason is simply that this would fail the second axiom of Sites, Definition 00VH. Rather, the coverings in \((\textit{Aff}/S)_{ph}\) are those finite families \(\{U_i \to U\}\) of finite type morphisms between affine objects of \((\Sch/S)_{ph}\) which can be refined by a standard ph covering. We explicitly state and prove that the big affine ph site is a site.
Lemma
Let \(S\) be a scheme. Let \(\Sch_{ph}\) be a big ph site containing \(S\). Then \((\textit{Aff}/S)_{ph}\) is a site.
Proof
Reasoning as in the proof of Lemma 021C it suffices to show that the collection of finite ph coverings \(\{U_i \to U\}\) with \(U\), \(U_i\) affine satisfies properties (1), (2) and (3) of Sites, Definition 00VH. This is clear since for example, given a finite ph covering \(\{T_i \to T\}_{i\in I}\) with \(T_i, T\) affine, and for each \(i\) a finite ph covering \(\{T_{ij} \to T_i\}_{j\in J_i}\) with \(T_{ij}\) affine , then \(\{T_{ij} \to T\}_{i \in I, j\in J_i}\) is a ph covering (Lemma 0DBI), \(\bigcup_{i\in I} J_i\) is finite and each \(T_{ij}\) is affine.
Lemma
Let \(S\) be a scheme. Let \(\Sch_{ph}\) be a big ph site containing \(S\). The underlying categories of the sites \(\Sch_{ph}\), \((\Sch/S)_{ph}\), and \((\textit{Aff}/S)_{ph}\) have fibre products. In each case the obvious functor into the category \(\Sch\) of all schemes commutes with taking fibre products. The category \((\Sch/S)_{ph}\) has a final object, namely \(S/S\).
Proof
For \(\Sch_{ph}\) it is true by construction, see Sets, Lemma 000R. Suppose we have \(U \to S\), \(V \to U\), \(W \to U\) morphisms of schemes with \(U, V, W \in \Ob(\Sch_{ph})\). The fibre product \(V \times_U W\) in \(\Sch_{ph}\) is a fibre product in \(\Sch\) 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 \((\Sch/S)_{ph}\). This proves the result for \((\Sch/S)_{ph}\). If \(U, V, W\) are affine, so is \(V \times_U W\) and hence the result for \((\textit{Aff}/S)_{ph}\).
Next, we check that the big affine site defines the same topos as the big site.
Lemma
Let \(S\) be a scheme. Let \(\Sch_{ph}\) be a big ph site containing \(S\). The functor \((\textit{Aff}/S)_{ph} \to (\Sch/S)_{ph}\) is cocontinuous and induces an equivalence of topoi from \(\Sh((\textit{Aff}/S)_{ph})\) to \(\Sh((\Sch/S)_{ph})\).
Proof
The notion of a special cocontinuous functor is introduced in Sites, Definition 03CG. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 03A0. Denote the inclusion functor \(u : (\textit{Aff}/S)_{ph} \to (\Sch/S)_{ph}\). Being cocontinuous follows because any ph covering of \(T/S\), \(T\) affine, can be refined by a standard ph covering of \(T\) by definition. Hence (1) holds. We see \(u\) is continuous simply because a finite ph covering of an affine by affines is a ph 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 (which is a ph covering).
Lemma
Let \(\mathcal{F}\) be a presheaf on \((\Sch/S)_{ph}\). Then \(\mathcal{F}\) is a sheaf if and only if
\(\mathcal{F}\) satisfies the sheaf condition for Zariski coverings, and
if \(f : V \to U\) is proper surjective, then \(\mathcal{F}(U)\) maps bijectively to the equalizer of the two maps \(\mathcal{F}(V) \to \mathcal{F}(V \times_U V)\).
Moreover, in the presence of (1) property (2) is equivalent to property
the sheaf property for \(\{V \to U\}\) as in (2) with \(U\) affine.
Proof
We will show that if (1) and (2) hold, then \(\mathcal{F}\) is sheaf. Let \(\{T_i \to T\}\) be a ph covering, i.e., a covering in \((\Sch/S)_{ph}\). We will verify the sheaf condition for this covering. Let \(s_i \in \mathcal{F}(T_i)\) be sections which restrict to the same section over \(T_i \times_T T_{i'}\). We will show that there exists a unique section \(s \in \mathcal{F}(T)\) restricting to \(s_i\) over \(T_i\). Let \(T = \bigcup U_j\) be an affine open covering. By property (1) it suffices to produce sections \(s_j \in \mathcal{F}(U_j)\) which agree on \(U_j \cap U_{j'}\) in order to produce \(s\). Consider the ph coverings \(\{T_i \times_T U_j \to U_j\}\). Then \(s_{ji} = s_i|_{T_i \times_T U_j}\) are sections agreeing over \((T_i \times_T U_j) \times_{U_j} (T_{i'} \times_T U_j)\). Choose a proper surjective morphism \(V_j \to U_j\) and a finite affine open covering \(V_j = \bigcup V_{jk}\) such that the standard ph covering \(\{V_{jk} \to U_j\}\) refines \(\{T_i \times_T U_j \to U_j\}\). If \(s_{jk} \in \mathcal{F}(V_{jk})\) denotes the pullback of \(s_{ji}\) to \(V_{jk}\) by the implied morphisms, then we find that \(s_{jk}\) glue to a section \(s'_j \in \mathcal{F}(V_j)\). Using the agreement on overlaps once more, we find that \(s'_j\) is in the equalizer of the two maps \(\mathcal{F}(V_j) \to \mathcal{F}(V_j \times_{U_j} V_j)\). Hence by (2) we find that \(s'_j\) comes from a unique section \(s_j \in \mathcal{F}(U_j)\). We omit the verification that these sections \(s_j\) have all the desired properties.
Proof of the equivalence of (2) and (2’) in the presence of (1). Suppose \(V \to U\) is a morphism of \((\Sch/S)_{ph}\) which is proper and surjective. Choose an affine open covering \(U = \bigcup U_i\) and set \(V_i = V \times_U U_i\). Then we see that \(\mathcal{F}(U) \to \mathcal{F}(V)\) is injective because we know \(\mathcal{F}(U_i) \to \mathcal{F}(V_i)\) is injective by (2’) and we know \(\mathcal{F}(U) \to \prod \mathcal{F}(U_i)\) is injective by (1). Finally, suppose that we are given an \(t \in \mathcal{F}(V)\) in the equalizer of the two maps \(\mathcal{F}(V) \to \mathcal{F}(V \times_U V)\). Then \(t|_{V_i}\) is in the equalizer of the two maps \(\mathcal{F}(V_i) \to \mathcal{F}(V_i \times_{U_i} V_i)\) for all \(i\). Hence we obtain a unique section \(s_i \in \mathcal{F}(U_i)\) mapping to \(t|_{V_i}\) for all \(i\) by (2’). We omit the verification that \(s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}\) for all \(i, j\); this uses the uniqueness property just shown. By the sheaf property for the covering \(U = \bigcup U_i\) we obtain a section \(s \in \mathcal{F}(U)\). We omit the proof that \(s\) maps to \(t\) in \(\mathcal{F}(V)\).
Next, we establish some relationships between the topoi associated to these sites.
Lemma
Let \(\Sch_{ph}\) be a big ph site. Let \(f : T \to S\) be a morphism in \(\Sch_{ph}\). The functor \[u : (\Sch/T)_{ph} \longrightarrow (\Sch/S)_{ph}, \quad V/T \longmapsto V/S\] is cocontinuous, and has a continuous right adjoint \[v : (\Sch/S)_{ph} \longrightarrow (\Sch/T)_{ph}, \quad (U \to S) \longmapsto (U \times_S T \to T).\] They induce the same morphism of topoi \[f_{big} : \Sh((\Sch/T)_{ph}) \longrightarrow \Sh((\Sch/S)_{ph})\] 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.
Proof
The functor \(u\) is cocontinuous, continuous, and commutes with fibre products and equalizers. Hence Sites, Lemmas 00XR and 00XS 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 \(\Mor_S(u(U), V) = \Mor_T(U, V \times_S T)\) as desired. Thus we may apply Sites, Lemmas 00XX and 00XY to get the formula for \(f_{big, *}\).
Lemma
Given schemes \(X\), \(Y\), \(Z\) in \((\Sch/S)_{ph}\) 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 0DBR.
The fpqc topology
Definition
Let \(T\) be a scheme. An fpqc 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 and such that for every affine open \(U \subset T\) there exists \(n \geq 0\), a map \(a : \{1, \ldots, n\} \to I\) and affine opens \(V_j \subset T_{a(j)}\), \(j = 1, \ldots, n\) with \(\bigcup_{j = 1}^n f_{a(j)}(V_j) = U\).
To be sure this condition implies that \(T = \bigcup f_i(T_i)\). It is slightly harder to recognize an fpqc covering, hence we provide some lemmas to do so.
Lemma
Let \(T\) be a scheme. Let \(\{f_i : T_i \to T\}_{i \in I}\) be a family of morphisms of schemes with target \(T\). The following are equivalent
\(\{f_i : T_i \to T\}_{i \in I}\) is an fpqc covering,
each \(f_i\) is flat and for every affine open \(U \subset T\) there exist quasi-compact opens \(U_i \subset T_i\) which are almost all empty, such that \(U = \bigcup f_i(U_i)\),
each \(f_i\) is flat and there exists an affine open covering \(T = \bigcup_{\alpha \in A} U_\alpha\) and for each \(\alpha \in A\) there exist \(i_{\alpha, 1}, \ldots, i_{\alpha, n(\alpha)} \in I\) and quasi-compact opens \(U_{\alpha, j} \subset T_{i_{\alpha, j}}\) such that \(U_\alpha = \bigcup_{j = 1, \ldots, n(\alpha)} f_{i_{\alpha, j}}(U_{\alpha, j})\).
If \(T\) is quasi-separated, these are also equivalent to
each \(f_i\) is flat, and for every \(t \in T\) there exist \(i_1, \ldots, i_n \in I\) and quasi-compact opens \(U_j \subset T_{i_j}\) such that \(\bigcup_{j = 1, \ldots, n} f_{i_j}(U_j)\) is a (not necessarily open) neighbourhood of \(t\) in \(T\).
Proof
We omit the proof of the equivalence of (1), (2), and (3). From now on assume \(T\) is quasi-separated. We prove (4) implies (2). Let \(U \subset T\) be an affine open. To prove (2) it suffices to show that for every \(t \in U\) there exist finitely many quasi-compact opens \(U_j \subset T_{i_j}\) such that \(f_{i_j}(U_j) \subset U\) and such that \(\bigcup f_{i_j}(U_j)\) is a neighbourhood of \(t\) in \(U\). By assumption there do exist finitely many quasi-compact opens \(U'_j \subset T_{i_j}\) such that such that \(\bigcup f_{i_j}(U'_j)\) is a neighbourhood of \(t\) in \(T\). Since \(T\) is quasi-separated we see that \(U_j = U'_j \cap f_{i_j}^{-1}(U) = (f_{i_j}|_{U'_j})^{-1}(U)\) is quasi-compact by Schemes, Lemma 03GI. Thus (2) holds. Since it is clear that (2) implies (4) the proof is finished.
Example
Let \(X = X_1 \cup X_2\) where \(X_1 = X_2 = \Spec(k[t_1,t_2,\dots])\) be the non-quasi-separated scheme from Schemes, Example 01KL, and \(Y = X_1 \amalg \Spec(\mathcal{O}_{X_2, 0})\). Then the obvious morphism \(f : Y \to X\) is flat and surjective, and (since \(Y\) is quasi-compact) trivially satisfies (4) of Lemma 03L7. But \(\{f : Y \to X\}\) isn’t an fpqc covering. Namely, we claim there is no quasi-compact open \(W\) of \(Y\) whose image by \(f\) is \(X_2\). Namely, \(f^{-1}(X_2) = X_1 \setminus \{0\} \amalg \Spec(\mathcal{O}_{X_2, 0})\). Hence if \(W\) exists, then \(W = U \amalg \Spec(\mathcal{O}_{X_2, 0})\) for some quasi-compact open \(U\) of \(X_1 \setminus \{0\} = X_2 \setminus \{0\}\). But then we may write \(U = D(f_1) \cup \ldots \cup D(f_n)\) for some \(f_i \in k[t_1, t_2, \ldots]\) vanishing at \(0\). Suppose that \(f_1, \ldots, f_n\) use only the variables \(t_1, \ldots, t_m\). Then \(p = (0, \ldots, 0, 1, 0, \ldots)\) with \(1\) in the \((m + 1)\)st spot is a closed point of \(X_2\) not in the image of \(W\).
Lemma
Let \(T\) be a scheme. Let \(\{f_i : T_i \to T\}_{i \in I}\) be a family of morphisms of schemes with target \(T\). The following are equivalent
\(\{f_i : T_i \to T\}_{i \in I}\) is an fpqc covering, and
setting \(T' = \coprod_{i \in I} T_i\), and \(f = \coprod_{i \in I} f_i\) the family \(\{f : T' \to T\}\) is an fpqc covering.
Proof
Suppose that \(U \subset T\) is an affine open. If (1) holds, then we find \(i_1, \ldots, i_n \in I\) and affine opens \(U_j \subset T_{i_j}\) such that \(U = \bigcup_{j = 1, \ldots, n} f_{i_j}(U_j)\). Then \(U_1 \amalg \ldots \amalg U_n \subset T'\) is a quasi-compact open surjecting onto \(U\). Thus \(\{f : T' \to T\}\) is an fpqc covering by Lemma 03L7. Conversely, if (2) holds then there exists a quasi-compact open \(U' \subset T'\) with \(U = f(U')\). Then \(U_j = U' \cap T_j\) is quasi-compact open in \(T_j\) and empty for almost all \(j\). By Lemma 03L7 we see that (1) holds.
Lemma
Let \(T\) be a scheme. Let \(\{f_i : T_i \to T\}_{i \in I}\) be a family of morphisms of schemes with target \(T\). Assume that
each \(f_i\) is flat, and
the family \(\{f_i : T_i \to T\}_{i \in I}\) can be refined by an fpqc covering of \(T\).
Then \(\{f_i : T_i \to T\}_{i \in I}\) is an fpqc covering of \(T\).
Proof
Let \(\{g_j : X_j \to T\}_{j \in J}\) be an fpqc covering refining \(\{f_i : T_i \to T\}\). Suppose that \(U \subset T\) is affine open. Choose \(j_1, \ldots, j_m \in J\) and \(V_k \subset X_{j_k}\) affine open such that \(U = \bigcup g_{j_k}(V_k)\). For each \(j\) pick \(i_j \in I\) and a morphism \(h_j : X_j \to T_{i_j}\) such that \(g_j = f_{i_j} \circ h_j\). Since \(h_{j_k}(V_k)\) is quasi-compact we can find a quasi-compact open \(h_{j_k}(V_k) \subset U_k \subset f_{i_{j_k}}^{-1}(U)\). Then \(U = \bigcup f_{i_{j_k}}(U_k)\). We conclude that \(\{f_i : T_i \to T\}_{i \in I}\) is an fpqc covering by Lemma 03L7.
Lemma
Let \(T\) be a scheme. Let \(\{f_i : T_i \to T\}_{i \in I}\) be a family of morphisms of schemes with target \(T\). Assume that
each \(f_i\) is flat, and
there exists an fpqc covering \(\{g_j : S_j \to T\}_{j \in J}\) such that each \(\{S_j \times_T T_i \to S_j\}_{i \in I}\) is an fpqc covering.
Then \(\{f_i : T_i \to T\}_{i \in I}\) is an fpqc covering of \(T\).
Proof
We will use Lemma 03L7 without further mention. Let \(U \subset T\) be an affine open. By (2) we can find quasi-compact opens \(V_j \subset S_j\) for \(j \in J\), almost all empty, such that \(U = \bigcup g_j(V_j)\). Then for each \(j\) we can choose quasi-compact opens \(W_{ij} \subset S_j \times_T T_i\) for \(i \in I\), almost all empty, with \(V_j = \bigcup_i \text{pr}_1(W_{ij})\). Thus \(\{S_j \times_T T_i \to T\}\) is an fpqc covering. Since this covering refines \(\{f_i : T_i \to T\}\) we conclude by Lemma 03L8.
Lemma
Any fppf covering is an fpqc covering, and a fortiori, any syntomic, smooth, étale or Zariski covering is an fpqc covering.
Proof
We will show that an fppf covering is an fpqc covering, and then the rest follows from Lemma 021N. Let \(\{f_i : U_i \to U\}_{i \in I}\) be an fppf covering. By definition this means that the \(f_i\) are flat which checks the first condition of Definition 022B. To check the second, let \(V \subset U\) be an affine open subset. Write \(f_i^{-1}(V) = \bigcup_{j \in J_i} V_{ij}\) for some affine opens \(V_{ij} \subset U_i\). Since each \(f_i\) is open (Morphisms, Lemma 01UA), we see that \(V = \bigcup_{i\in I} \bigcup_{j \in J_i} f_i(V_{ij})\) is an open covering of \(V\). Since \(V\) is quasi-compact, this covering has a finite refinement. This finishes the proof.
The fpqc5 topology cannot be treated in the same way as the fppf topology6. Namely, suppose that \(R\) is a nonzero ring. We will see in Lemma 0BBK that there does not exist a set \(A\) of fpqc-coverings of \(\Spec(R)\) such that every fpqc-covering can be refined by an element of \(A\). If \(R = k\) is a field, then the reason for this unboundedness is that there does not exist a field extension of \(k\) such that every field extension of \(k\) is contained in it.
If you ignore set theoretic difficulties, then you run into presheaves which do not have a sheafification, see [Waterhouse-fpqc-sheafification, Theorem 5.5]. A mildly interesting option is to consider only those faithfully flat ring extensions \(R \to R'\) where the cardinality of \(R'\) is suitably bounded. (And if you consider all schemes in a fixed universe as in SGA4 then you are bounding the cardinality by a strongly inaccessible cardinal.) However, it is not so clear what happens if you change the cardinal to a bigger one.
For these reasons we do not introduce fpqc sites and we will not consider cohomology with respect to the fpqc-topology.
On the other hand, given a contravariant functor \(F : \Sch^{opp} \to \textit{Sets}\) it does make sense to ask whether \(F\) satisfies the sheaf property for the fpqc topology, see below. Moreover, we can wonder about descent of object in the fpqc topology, etc. Simply put, for certain results the correct generality is to work with fpqc coverings.
Lemma
Let \(T\) be a scheme.
If \(T' \to T\) is an isomorphism then \(\{T' \to T\}\) is an fpqc covering of \(T\).
If \(\{T_i \to T\}_{i\in I}\) is an fpqc covering and for each \(i\) we have an fpqc covering \(\{T_{ij} \to T_i\}_{j\in J_i}\), then \(\{T_{ij} \to T\}_{i \in I, j\in J_i}\) is an fpqc covering.
If \(\{T_i \to T\}_{i\in I}\) is an fpqc covering and \(T' \to T\) is a morphism of schemes then \(\{T' \times_T T_i \to T'\}_{i\in I}\) is an fpqc covering.
Proof
Part (1) is immediate. Recall that the composition of flat morphisms is flat and that the base change of a flat morphism is flat (Morphisms, Lemmas 01U9 and 01U7). Thus we can apply Lemma 03L7 in each case to check that our families of morphisms are fpqc coverings.
Proof of (2). Assume \(\{T_i \to T\}_{i\in I}\) is an fpqc covering and for each \(i\) we have an fpqc covering \(\{f_{ij} : T_{ij} \to T_i\}_{j\in J_i}\). Let \(U \subset T\) be an affine open. We can find quasi-compact opens \(U_i \subset T_i\) for \(i \in I\), almost all empty, such that \(U = \bigcup f_i(U_i)\). Then for each \(i\) we can choose quasi-compact opens \(U_{ij} \subset T_{ij}\) for \(j \in J_i\), almost all empty, with \(U_i = \bigcup_j f_{ij}(U_{ij})\). Thus \(\{T_{ij} \to T\}\) is an fpqc covering.
Proof of (3). Assume \(\{T_i \to T\}_{i\in I}\) is an fpqc covering and \(T' \to T\) is a morphism of schemes. Let \(U' \subset T'\) be an affine open which maps into the affine open \(U \subset T\). Choose quasi-compact opens \(U_i \subset T_i\), almost all empty, such that \(U = \bigcup f_i(U_i)\). Then \(U' \times_U U_i\) is a quasi-compact open of \(T' \times_T T_i\) and \(U' = \bigcup \text{pr}_1(U' \times_U U_i)\). Since \(T'\) can be covered by such affine opens \(U' \subset T'\) we see that \(\{T' \times_T T_i \to T'\}_{i\in I}\) is an fpqc covering by Lemma 03L7.
Lemma
Let \(T\) be an affine scheme. Let \(\{T_i \to T\}_{i \in I}\) be an fpqc covering of \(T\). Then there exists an fpqc covering \(\{U_j \to T\}_{j = 1, \ldots, n}\) 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 definition.
Definition
Let \(T\) be an affine scheme. A standard fpqc covering of \(T\) is a family \(\{f_j : U_j \to T\}_{j = 1, \ldots, n}\) with each \(U_j\) is affine, flat over \(T\) and \(T = \bigcup f_j(U_j)\).
Since we do not introduce the affine site we have to show directly that the collection of all standard fpqc coverings satisfies the axioms.
Lemma
Let \(T\) be an affine scheme.
If \(T' \to T\) is an isomorphism then \(\{T' \to T\}\) is a standard fpqc covering of \(T\).
If \(\{T_i \to T\}_{i\in I}\) is a standard fpqc covering and for each \(i\) we have a standard fpqc 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 fpqc covering.
If \(\{T_i \to T\}_{i\in I}\) is a standard fpqc covering and \(T' \to T\) is a morphism of affine schemes then \(\{T' \times_T T_i \to T'\}_{i\in I}\) is a standard fpqc covering.
Proof
This follows formally from the fact that compositions and base changes of flat morphisms are flat (Morphisms, Lemmas 01U9 and 01U7) and that fibre products of affine schemes are affine (Schemes, Lemma 01JQ).
Lemma
Let \(T\) be a scheme. Let \(\{f_i : T_i \to T\}_{i \in I}\) be a family of morphisms of schemes with target \(T\). Assume that
each \(f_i\) is flat, and
every affine scheme \(Z\) and morphism \(h : Z \to T\) there exists a standard fpqc covering \(\{Z_j \to Z\}_{j = 1, \ldots, n}\) which refines the family \(\{T_i \times_T Z \to Z\}_{i \in I}\).
Then \(\{f_i : T_i \to T\}_{i \in I}\) is an fpqc covering of \(T\).
Proof
Let \(T = \bigcup U_\alpha\) be an affine open covering. For each \(\alpha\) the pullback family \(\{T_i \times_T U_\alpha \to U_\alpha\}\) can be refined by a standard fpqc covering, hence is an fpqc covering by Lemma 03L8. As \(\{U_\alpha \to T\}\) is an fpqc covering we conclude that \(\{T_i \to T\}\) is an fpqc covering by Lemma 03L9.
Definition
Let \(F\) be a contravariant functor on the category of schemes with values in sets.
Let \(\{U_i \to T\}_{i \in I}\) be a family of morphisms of schemes with fixed target. We say that \(F\) satisfies the sheaf property for the given family if for any collection of elements \(\xi_i \in F(U_i)\) such that \(\xi_i|_{U_i \times_T U_j} = \xi_j|_{U_i \times_T U_j}\) there exists a unique element \(\xi \in F(T)\) such that \(\xi_i = \xi|_{U_i}\) in \(F(U_i)\).
We say that \(F\) satisfies the sheaf property for the fpqc topology if it satisfies the sheaf property for any fpqc covering.
We try to avoid using the terminology “\(F\) is a sheaf” in this situation since we are not defining a category of fpqc sheaves as we explained above.
Lemma
Let \(F\) be a contravariant functor on the category of schemes with values in sets. Then \(F\) satisfies the sheaf property for the fpqc topology if and only if it satisfies
the sheaf property for every Zariski covering, and
the sheaf property for any standard fpqc covering.
Moreover, in the presence of (1) property (2) is equivalent to property
the sheaf property for \(\{V \to U\}\) with \(V\), \(U\) affine and \(V \to U\) faithfully flat.
Proof
Assume (1) and (2) hold. Let \(\{f_i : T_i \to T\}_{i \in I}\) be an fpqc covering. Let \(s_i \in F(T_i)\) be a family of elements such that \(s_i\) and \(s_j\) map to the same element of \(F(T_i \times_T T_j)\). Let \(W \subset T\) be the maximal open subset such that there exists a unique \(s \in F(W)\) with \(s|_{f_i^{-1}(W)} = s_i|_{f_i^{-1}(W)}\) for all \(i\). Such a maximal open exists because \(F\) satisfies the sheaf property for Zariski coverings; in fact \(W\) is the union of all opens with this property. Let \(t \in T\). We will show \(t \in W\). To do this we pick an affine open \(t \in U \subset T\) and we will show there is a unique \(s \in F(U)\) with \(s|_{f_i^{-1}(U)} = s_i|_{f_i^{-1}(U)}\) for all \(i\).
By Lemma 022E we can find a standard fpqc covering \(\{U_j \to U\}_{j = 1, \ldots, n}\) refining \(\{U \times_T T_i \to U\}\), say by morphisms \(h_j : U_j \to T_{i_j}\). By (2) we obtain a unique element \(s \in F(U)\) such that \(s|_{U_j} = F(h_j)(s_{i_j})\). Note that for any scheme \(V \to U\) over \(U\) there is a unique section \(s_V \in F(V)\) which restricts to \(F(h_j \circ \text{pr}_2)(s_{i_j})\) on \(V \times_U U_j\) for \(j = 1, \ldots, n\). Namely, this is true if \(V\) is affine by (2) as \(\{V \times_U U_j \to V\}\) is a standard fpqc covering and in general this follows from (1) and the affine case by choosing an affine open covering of \(V\). In particular, \(s_V = s|_V\). Now, taking \(V = U \times_T T_i\) and using that \(s_{i_j}|_{T_{i_j} \times_T T_i} = s_i|_{T_{i_j} \times_T T_i}\) we conclude that \(s|_{U \times_T T_i} = s_V = s_i|_{U \times_T T_i}\) which is what we had to show.
Proof of the equivalence of (2) and (2’) in the presence of (1). Suppose \(\{T_i \to T\}\) is a standard fpqc covering, then \(\coprod T_i \to T\) is a faithfully flat morphism of affine schemes. In the presence of (1) we have \(F(\coprod T_i) = \prod F(T_i)\) and similarly \(F((\coprod T_i) \times_T (\coprod T_i)) = \prod F(T_i \times_T T_{i'})\). Thus the sheaf condition for \(\{T_i \to T\}\) and \(\{\coprod T_i \to T\}\) is the same.
The following lemma is here just to point out set theoretical difficulties do indeed arise and should be ignored by most readers.
Lemma
Let \(R\) be a nonzero ring. There does not exist a set \(A\) of fpqc-coverings of \(\Spec(R)\) such that every fpqc-covering can be refined by an element of \(A\).
Proof
Let us first explain this when \(R = k\) is a field. For any set \(I\) consider the purely transcendental field extension \(k_I = k(\{t_i\}_{i \in I})/k\). Since \(k \to k_I\) is faithfully flat we see that \(\{\Spec(k_I) \to \Spec(k)\}\) is an fpqc covering. Let \(A\) be a set and for each \(\alpha \in A\) let \(\mathcal{U}_\alpha = \{S_{\alpha, j} \to \Spec(k)\}_{j \in J_\alpha}\) be an fpqc covering. If \(\mathcal{U}_\alpha\) refines \(\{\Spec(k_I) \to \Spec(k)\}\) then the morphisms \(S_{\alpha, j} \to \Spec(k)\) factor through \(\Spec(k_I)\). Since \(\mathcal{U}_\alpha\) is a covering, at least some \(S_{\alpha, j}\) is nonempty. Pick a point \(s \in S_{\alpha, j}\). Since we have the factorization \(S_{\alpha, j} \to \Spec(k_I) \to \Spec(k)\) we obtain a homomorphism of fields \(k_I \to \kappa(s)\). In particular, we see that the cardinality of \(\kappa(s)\) is at least the cardinality of \(I\). Thus if we take \(I\) to be a set of cardinality bigger than the cardinalities of the residue fields of all the schemes \(S_{\alpha, j}\), then such a factorization does not exist and the lemma holds for \(R = k\).
General case. Since \(R\) is nonzero it has a maximal prime ideal \(\mathfrak m\) with residue field \(\kappa\). Let \(I\) be a set and consider \(R_I = S_I^{-1} R[\{t_i\}_{i \in I}]\) where \(S_I \subset R[\{t_i\}_{i \in I}]\) is the multiplicative subset of \(f \in R[\{t_i\}_{i \in I}]\) such that \(f\) maps to a nonzero element of \(R/\mathfrak p[\{t_i\}_{i \in I}]\) for all primes \(\mathfrak p\) of \(R\). Then \(R_I\) is a faithfully flat \(R\)-algebra and \(\{\Spec(R_I) \to \Spec(R)\}\) is an fpqc covering. We leave it as an exercise to the reader to show that \(R_I \otimes_R \kappa \cong \kappa(\{t_i\}_{i \in I}) = \kappa_I\) with notation as above (hint: use that \(R \to \kappa\) is surjective and that any \(f \in R[\{t_i\}_{i \in I}]\) one of whose monomials occurs with coefficient \(1\) is an element of \(S_I\)). Let \(A\) be a set and for each \(\alpha \in A\) let \(\mathcal{U}_\alpha = \{S_{\alpha, j} \to \Spec(R)\}_{j \in J_\alpha}\) be an fpqc covering. If \(\mathcal{U}_\alpha\) refines \(\{\Spec(R_I) \to \Spec(R)\}\), then by base change we conclude that \(\{S_{\alpha, j} \times_{\Spec(R)} \Spec(\kappa) \to \Spec(\kappa)\}\) refines \(\{\Spec(\kappa_I) \to \Spec(\kappa)\}\). Hence by the result of the previous paragraph, there exists an \(I\) such that this is not the case and the lemma is proved.
The V topology
The V topology is stronger than all other topologies in this chapter. Roughly speaking it is generated by Zariski coverings and by quasi-compact morphisms satisfying a lifting property for specializations (Lemma 0ETN). However, the procedure we will use to define V coverings is a bit different. We will first define standard V coverings of affines and then use these to define V coverings in general. Typographical point: in the literature sometimes “\(v\)-covering” is used instead of “V covering”.
Definition
Let \(T\) be an affine scheme. A standard V covering is a finite family \(\{T_j \to T\}_{j = 1, \ldots, m}\) with \(T_j\) affine such that for every morphism \(g : \Spec(V) \to T\) where \(V\) is a valuation ring, there is an extension \(V \subset W\) of valuation rings (More on Algebra, Definition 0ASG), an index \(1 \leq j \leq m\), and a commutative diagram \[\xymatrix{ \Spec(W) \ar[r] \ar[d] & T_j \ar[d] \\ \Spec(V) \ar[r]^g & T }\]
We first prove a few basic lemmas about this notion.
Lemma
A standard fpqc covering is a standard V covering.
Proof
Let \(\{X_i \to X\}_{i = 1, \ldots, n}\) be a standard fpqc covering (Definition 022F). Let \(g : \Spec(V) \to X\) be a morphism where \(V\) is a valuation ring. Let \(x \in X\) be the image of the closed point of \(\Spec(V)\). Choose an \(i\) and a point \(x_i \in X_i\) mapping to \(x\). Then \(\Spec(V) \times_X X_i\) has a point \(x'_i\) mapping to the closed point of \(\Spec(V)\). Since \(\Spec(V) \times_X X_i \to \Spec(V)\) is flat we can find a specialization \(x''_i \leadsto x'_i\) of points of \(\Spec(V) \times_X X_i\) with \(x''_i\) mapping to the generic point of \(\Spec(V)\), see Morphisms, Lemma 03HV. By Schemes, Lemma 01J8 we can choose a valuation ring \(W\) and a morphism \(h : \Spec(W) \to \Spec(V) \times_X X_i\) such that \(h\) maps the generic point of \(\Spec(W)\) to \(x''_i\) and the closed point of \(\Spec(W)\) to \(x'_i\). We obtain a commutative diagram \[\xymatrix{ \Spec(W) \ar[r] \ar[d] & X_i \ar[d] \\ \Spec(V) \ar[r] & X }\] where \(V \to W\) is an extension of valuation rings. This proves the lemma.
Lemma
A standard ph covering is a standard V covering.
Proof
Let \(T\) be an affine scheme. Let \(f : U \to T\) be a proper surjective morphism. Let \(U = \bigcup_{j = 1, \ldots, m} U_j\) be a finite affine open covering. We have to show that \(\{U_j \to T\}\) is a standard V covering, see Definition 0DBD. Let \(g : \Spec(V) \to T\) be a morphism where \(V\) is a valuation ring with fraction field \(K\). Since \(U \to T\) is surjective, we may choose a field extension \(L/K\) and a commutative diagram \[\xymatrix{ \Spec(L) \ar[rr] \ar[d] & & U \ar[d] \\ \Spec(K) \ar[r] & \Spec(V) \ar[r]^g & T }\] By Algebra, Lemma 00IA we can choose a valuation ring \(W \subset L\) dominating \(V\). By the valuative criterion of properness (Morphisms, Lemma 0BX5) we can then find the morphism \(h\) in the commutative diagram \[\xymatrix{ \Spec(L) \ar[r] \ar[d] & \Spec(W) \ar[r]_h \ar[d] & U \ar[d] \\ \Spec(K) \ar[r] & \Spec(V) \ar[r]^g & T }\] Since \(\Spec(W)\) has a unique closed point, we see that \(\Im(h)\) is contained in \(U_j\) for some \(j\). Thus \(h : \Spec(W) \to U_j\) is the desired lift and we conclude \(\{U_j \to T\}\) is a standard V covering.
Lemma
Let \(\{T_j \to T\}_{j = 1, \ldots, m}\) be a standard V covering. Let \(T' \to T\) be a morphism of affine schemes. Then \(\{T_j \times_T T' \to T'\}_{j = 1, \ldots, m}\) is a standard V covering.
Proof
Let \(\Spec(V) \to T'\) be a morphism where \(V\) is a valuation ring. By assumption we can find an extension of valuation rings \(V \subset W\), an \(i\), and a commutative diagram \[\xymatrix{ \Spec(W) \ar[r] \ar[d] & T_i \ar[d] \\ \Spec(V) \ar[r] & T }\] By the universal property of fibre products we obtain a morphism \(\Spec(W) \to T' \times_T T_i\) as desired.
Lemma
Let \(T\) be an affine scheme. Let \(\{T_j \to T\}_{j = 1, \ldots, m}\) be a standard V covering. Let \(\{T_{ji} \to T_j\}_{i = 1, \ldots n_j}\) be a standard V covering. Then \(\{T_{ji} \to T\}_{i, j}\) is a standard V covering.
Proof
This follows formally from the observation that if \(V \subset W\) and \(W \subset \Omega\) are extensions of valuation rings, then \(V \subset \Omega\) is an extension of valuation rings.
Lemma
Let \(T\) be an affine scheme. Let \(\{T_j \to T\}_{j = 1, \ldots, m}\) be a family of morphisms with \(T_j\) affine for all \(j\). The following are equivalent
\(\{T_j \to T\}_{j = 1, \ldots, m}\) is a standard V covering,
there is a standard V covering which refines \(\{T_j \to T\}_{j = 1, \ldots, m}\), and
\(\{\coprod_{j = 1, \ldots, m} T_j \to T\}\) is a standard V covering.
Proof
Omitted. Hints: This follows almost immediately from the definition. The only slightly interesting point is that a morphism from the spectrum of a local ring into \(\coprod_{j = 1, \ldots, m} T_j\) must factor through some \(T_j\).
Definition
Let \(T\) be a scheme. A V covering of \(T\) is a family of morphisms \(\{T_i \to T\}_{i \in I}\) of schemes such that for every affine open \(U \subset T\) there exists a standard V covering \(\{U_j \to U\}_{j = 1, \ldots, m}\) refining the family \(\{T_i \times_T U \to U\}_{i \in I}\).
The V topology has the same set theoretical problems as the fpqc topology. Thus we refrain from defining V sites and we will not consider cohomology with respect to the V topology. On the other hand, given a \(F : \Sch^{opp} \to \textit{Sets}\) it does make sense to ask whether \(F\) satisfies the sheaf property for the V topology, see below. Moreover, we can wonder about descent of object in the V topology, etc.
Lemma
Let \(T\) be a scheme. Let \(\{f_i : T_i \to T\}_{i \in I}\) be a family of morphisms. The following are equivalent
\(\{T_i \to T\}_{i \in I}\) is a V covering,
there is a V covering which refines \(\{T_i \to T\}_{i \in I}\), and
\(\{\coprod_{i \in I} T_i \to T\}\) is a V covering.
Proof
Omitted. Hint: compare with the proof of Lemma 0ET9.
Lemma
Let \(T\) be a scheme.
If \(T' \to T\) is an isomorphism then \(\{T' \to T\}\) is a V covering of \(T\).
If \(\{T_i \to T\}_{i\in I}\) is a V covering and for each \(i\) we have a V covering \(\{T_{ij} \to T_i\}_{j\in J_i}\), then \(\{T_{ij} \to T\}_{i \in I, j\in J_i}\) is a V covering.
If \(\{T_i \to T\}_{i\in I}\) is a V covering and \(T' \to T\) is a morphism of schemes then \(\{T' \times_T T_i \to T'\}_{i\in I}\) is a V covering.
Proof
Assertion (1) is clear.
Proof of (3). Let \(U' \subset T'\) be an affine open subscheme. Since \(U'\) is quasi-compact we can find a finite affine open covering \(U' = U'_1 \cup \ldots \cup U'_m\) such that \(U'_j \to T\) maps into an affine open \(U_j \subset T\). Choose a standard V covering \(\{U_{jl} \to U_j\}_{l = 1, \ldots, n_j}\) refining \(\{T_i \times_T U_j \to U_j\}\). By Lemma 0ETE the base change \(\{U_{jl} \times_{U_j} U'_j \to U'_j\}\) is a standard V covering. Note that \(\{U'_j \to U'\}\) is a standard V covering (for example by Lemma 0ETC). By Lemma 0ETF the family \(\{U_{jl} \times_{U_j} U'_j \to U'\}\) is a standard V covering. Since \(\{U_{jl} \times_{U_j} U'_j \to U'\}\) refines \(\{T_i \times_T U' \to U'\}\) we conclude.
Proof of (2). Let \(U \subset T\) be affine open. First we pick a standard V covering \(\{U_k \to U\}_{k = 1, \ldots, m}\) refining \(\{T_i \times_T U \to U\}\). Say the refinement is given by morphisms \(U_k \to T_{i_k}\) over \(T\). Then \[\{T_{i_kj} \times_{T_{i_k}} U_k \to U_k\}_{j \in J_{i_k}}\] is a V covering by part (3). As \(U_k\) is affine, we can find a standard V covering \(\{U_{ka} \to U_k\}_{a = 1, \ldots, b_k}\) refining this family. Then we apply Lemma 0ETF to see that \(\{U_{ka} \to U\}\) is a standard V covering which refines \(\{T_{ij} \times_T U \to U\}\). This finishes the proof.
Lemma
Any fpqc covering is a V covering. A fortiori, any fppf, syntomic, smooth, étale or Zariski covering is a V covering. Also, a ph covering is a V covering.
Proof
An fpqc covering can affine locally be refined by a standard fpqc covering, see Lemmas 022E. A standard fpqc covering is a standard V covering, see Lemma 0ETC. Hence the first statement follows from our definition of V covers in terms of standard V coverings. The conclusion for fppf, syntomic, smooth, étale or Zariski coverings follows as these are fpqc coverings, see Lemma 022C.
The statement on ph coverings follows from Lemma 0ETD in the same manner.
Definition
Let \(F\) be a contravariant functor on the category of schemes with values in sets. We say that \(F\) satisfies the sheaf property for the V topology if it satisfies the sheaf property for any V covering (see Definition 022G).
We try to avoid using the terminology “\(F\) is a sheaf” in this situation since we are not defining a category of V sheaves as we explained above.
Lemma
Let \(F\) be a contravariant functor on the category of schemes with values in sets. Then \(F\) satisfies the sheaf property for the V topology if and only if it satisfies
the sheaf property for every Zariski covering, and
the sheaf property for any standard V covering.
Moreover, in the presence of (1) property (2) is equivalent to property
the sheaf property for a standard V covering of the form \(\{V \to U\}\), i.e., consisting of a single arrow.
Proof
Assume (1) and (2) hold. Let \(\{f_i : T_i \to T\}_{i \in I}\) be a V covering. Let \(s_i \in F(T_i)\) be a family of elements such that \(s_i\) and \(s_j\) map to the same element of \(F(T_i \times_T T_j)\). Let \(W \subset T\) be the maximal open subset such that there exists a unique \(s \in F(W)\) with \(s|_{f_i^{-1}(W)} = s_i|_{f_i^{-1}(W)}\) for all \(i\). Such a maximal open exists because \(F\) satisfies the sheaf property for Zariski coverings; in fact \(W\) is the union of all opens with this property. Let \(t \in T\). We will show \(t \in W\). To do this we pick an affine open \(t \in U \subset T\) and we will show there is a unique \(s \in F(U)\) with \(s|_{f_i^{-1}(U)} = s_i|_{f_i^{-1}(U)}\) for all \(i\).
We can find a standard V covering \(\{U_j \to U\}_{j = 1, \ldots, n}\) refining \(\{U \times_T T_i \to U\}\), say by morphisms \(h_j : U_j \to T_{i_j}\). By (2) we obtain a unique element \(s \in F(U)\) such that \(s|_{U_j} = F(h_j)(s_{i_j})\). Note that for any scheme \(V \to U\) over \(U\) there is a unique section \(s_V \in F(V)\) which restricts to \(F(h_j \circ \text{pr}_2)(s_{i_j})\) on \(V \times_U U_j\) for \(j = 1, \ldots, n\). Namely, this is true if \(V\) is affine by (2) as \(\{V \times_U U_j \to V\}\) is a standard V covering (Lemma 0ETE) and in general this follows from (1) and the affine case by choosing an affine open covering of \(V\). In particular, \(s_V = s|_V\). Now, taking \(V = U \times_T T_i\) and using that \(s_{i_j}|_{T_{i_j} \times_T T_i} = s_i|_{T_{i_j} \times_T T_i}\) we conclude that \(s|_{U \times_T T_i} = s_V = s_i|_{U \times_T T_i}\) which is what we had to show.
Proof of the equivalence of (2) and (2’) in the presence of (1). Suppose \(\{T_i \to T\}_{i = 1, \ldots, n}\) is a standard V covering, then \(\coprod_{i = 1, \ldots, n} T_i \to T\) is a morphism of affine schemes which is clearly also a standard V covering. In the presence of (1) we have \(F(\coprod T_i) = \prod F(T_i)\) and similarly \(F((\coprod T_i) \times_T (\coprod T_i)) = \prod F(T_i \times_T T_{i'})\). Thus the sheaf condition for \(\{T_i \to T\}\) and \(\{\coprod T_i \to T\}\) is the same.
The following lemma shows that being a V covering is related to the possibility of lifting specializations.
Lemma
Let \(X \to Y\) be a quasi-compact morphism of schemes. The following are equivalent
\(\{X \to Y\}\) is a V covering,
for any valuation ring \(V\) and morphism \(g : \Spec(V) \to Y\) there exists an extension of valuation rings \(V \subset W\) and a commutative diagram \[\xymatrix{ \Spec(W) \ar[r] \ar[d] & X \ar[d] \\ \Spec(V) \ar[r] & Y }\]
for any morphism \(Z \to Y\) and specialization \(z' \leadsto z\) of points in \(Z\), there is a specialization \(w' \leadsto w\) of points in \(Z \times_Y X\) mapping to \(z' \leadsto z\).
Proof
Assume (1) and let \(g : \Spec(V) \to Y\) be as in (2). Since \(V\) is a local ring there is an affine open \(U \subset Y\) such that \(g\) factors through \(U\). By Definition 0ETH we can find a standard V covering \(\{U_j \to U\}\) refining \(\{X \times_Y U \to U\}\). By Definition 0ETB we can find a \(j\), an extension of valuation rings \(V \subset W\) and a commutative diagram \[\xymatrix{ \Spec(W) \ar[r] \ar[d] & U_j \ar[d] \ar@{..>}[r] & X \ar[ld] \\ \Spec(V) \ar[r] & Y }\] We have the dotted arrow making the diagram commute by the refinement property of the covering and we see that (2) holds.
Assume (2) and let \(Z \to Y\) and \(z' \leadsto z\) be as in (3). By Schemes, Lemma 01J8 we can find a valuation ring \(V\) and a morphism \(\Spec(V) \to Z\) such that the closed point of \(\Spec(V)\) maps to \(z\) and the generic point of \(\Spec(V)\) maps to \(z'\). By (2) we can find an extension of valuation rings \(V \subset W\) and a commutative diagram \[\xymatrix{ \Spec(W) \ar[rr] \ar[d] & & X \ar[d] \\ \Spec(V) \ar[r] & Z \ar[r] & Y }\] The generic and closed points of \(\Spec(W)\) map to points \(w' \leadsto w\) in \(Z \times_Y X\) via the induced morphism \(\Spec(W) \to Z \times_Y X\). This shows that (3) holds.
Assume (3) holds and let \(U \subset Y\) be an affine open. Choose a finite affine open covering \(U \times_Y X = \bigcup_{j = 1, \ldots, m} U_j\). This is possible as \(X \to Y\) is quasi-compact. We claim that \(\{U_j \to U\}\) is a standard V covering. The claim implies (1) is true and finishes the proof of the lemma. In order to prove the claim, let \(V\) be a valuation ring and let \(g : \Spec(V) \to U\) be a morphism. By (3) we find a specialization \(w' \leadsto w\) of points of \[T = \Spec(V) \times_Y X = \Spec(V) \times_U (U \times_Y X)\] such that \(w'\) maps to the generic point of \(\Spec(V)\) and \(w\) maps to the closed point of \(\Spec(V)\). By Schemes, Lemma 01J8 we can find a valuation ring \(W\) and a morphism \(\Spec(W) \to T\) such that the generic point of \(\Spec(W)\) maps to \(w'\) and the closed point of \(\Spec(W)\) maps to \(w\). The composition \(\Spec(W) \to T \to \Spec(V)\) corresponds to an inclusion \(V \subset W\) which presents \(W\) as an extension of the valuation ring \(V\). Since \(T = \bigcup \Spec(V) \times_U U_j\) is an open covering, we see that \(\Spec(W) \to T\) factors through \(\Spec(V) \times_U U_j\) for some \(j\). Thus we obtain a commutative diagram \[\xymatrix{ \Spec(W) \ar[d] \ar[r] & U_j \ar[d] \\ \Spec(V) \ar[r] & U }\] and the proof of the claim is complete.
A V covering gives a universally submersive family of maps. The converse of this lemma is false, see Examples, Section 0EU8.
Lemma
Let \(\{f_i : X_i \to X\}_{i \in I}\) be a V covering. Then \[\coprod\nolimits_{i \in I} f_i : \coprod\nolimits_{i \in I} X_i \longrightarrow X\] is a universally submersive morphism of schemes (Morphisms, Definition 040H).
Proof
We will use without further mention that the base change of a V covering is a V covering (Lemma 0ETJ). In particular it suffices to show that the morphism is submersive. Being submersive is clearly Zariski local on the base. Thus we may assume \(X\) is affine. Then \(\{X_i \to X\}\) can be refined by a standard V covering \(\{Y_j \to X\}\). If we can show that \(\coprod Y_j \to X\) is submersive, then since there is a factorization \(\coprod Y_j \to \coprod X_i \to X\) we conclude that \(\coprod X_i \to X\) is submersive. Set \(Y = \coprod Y_j\) and consider the morphism of affines \(f : Y \to X\). By Lemma 0ETN we know that we can lift any specialization \(x' \leadsto x\) in \(X\) to some specialization \(y' \leadsto y\) in \(Y\). Thus if \(T \subset X\) is a subset such that \(f^{-1}(T)\) is closed in \(Y\), then \(T \subset X\) is closed under specialization. Since \(f^{-1}(T) \subset Y\) with the reduced induced closed subscheme structure is an affine scheme, we conclude that \(T \subset X\) is closed by Algebra, Lemma 00HY. Hence \(f\) is submersive.
Change of topologies
Let \(f : X \to Y\) be a morphism of schemes over a base scheme \(S\). In this case we have the following morphisms of sites7 (with suitable choices of sites as in Remark 03FF below):
\((\Sch/X)_{fppf} \longrightarrow (\Sch/Y)_{fppf}\),
\((\Sch/X)_{fppf} \longrightarrow (\Sch/Y)_{syntomic}\),
\((\Sch/X)_{fppf} \longrightarrow (\Sch/Y)_{smooth}\),
\((\Sch/X)_{fppf} \longrightarrow (\Sch/Y)_\etale\),
\((\Sch/X)_{fppf} \longrightarrow (\Sch/Y)_{Zar}\),
\((\Sch/X)_{syntomic} \longrightarrow (\Sch/Y)_{syntomic}\),
\((\Sch/X)_{syntomic} \longrightarrow (\Sch/Y)_{smooth}\),
\((\Sch/X)_{syntomic} \longrightarrow (\Sch/Y)_\etale\),
\((\Sch/X)_{syntomic} \longrightarrow (\Sch/Y)_{Zar}\),
\((\Sch/X)_{smooth} \longrightarrow (\Sch/Y)_{smooth}\),
\((\Sch/X)_{smooth} \longrightarrow (\Sch/Y)_\etale\),
\((\Sch/X)_{smooth} \longrightarrow (\Sch/Y)_{Zar}\),
\((\Sch/X)_\etale \longrightarrow (\Sch/Y)_\etale\),
\((\Sch/X)_\etale \longrightarrow (\Sch/Y)_{Zar}\),
\((\Sch/X)_{Zar} \longrightarrow (\Sch/Y)_{Zar}\),
\((\Sch/X)_{fppf} \longrightarrow Y_\etale\),
\((\Sch/X)_{syntomic} \longrightarrow Y_\etale\),
\((\Sch/X)_{smooth} \longrightarrow Y_\etale\),
\((\Sch/X)_\etale \longrightarrow Y_\etale\),
\((\Sch/X)_{fppf} \longrightarrow Y_{Zar}\),
\((\Sch/X)_{syntomic} \longrightarrow Y_{Zar}\),
\((\Sch/X)_{smooth} \longrightarrow Y_{Zar}\),
\((\Sch/X)_\etale \longrightarrow Y_{Zar}\),
\((\Sch/X)_{Zar} \longrightarrow Y_{Zar}\),
\(X_\etale \longrightarrow Y_\etale\),
\(X_\etale \longrightarrow Y_{Zar}\),
\(X_{Zar} \longrightarrow Y_{Zar}\),
In each case the underlying continuous functor \(\Sch/Y \to \Sch/X\), or \(Y_\tau \to \Sch/X\) is the functor \(Y'/Y \mapsto X \times_Y Y'/X\). Namely, in the sections above we have seen the morphisms \(f_{big} : (\Sch/X)_\tau \to (\Sch/Y)_\tau\) and \(f_{small} : X_\tau \to Y_\tau\) for \(\tau\) as above. We also have seen the morphisms of sites \(\pi_Y : (\Sch/Y)_\tau \to Y_\tau\) for \(\tau \in \{\etale, Zariski\}\). On the other hand, it is clear that the identity functor \((\Sch/X)_\tau \to (\Sch/X)_{\tau'}\) defines a morphism of sites when \(\tau\) is a stronger topology than \(\tau'\). Hence composing these gives the list of possible morphisms above.
Because of the simple description of the underlying functor it is clear that given morphisms of schemes \(X \to Y \to Z\) the composition of two of the morphisms of sites above, e.g., \[(\Sch/X)_{\tau_0} \longrightarrow (\Sch/Y)_{\tau_1} \longrightarrow (\Sch/Z)_{\tau_2}\] is the corresponding morphism of sites associated to the morphism of schemes \(X \to Z\).
Remark
Take any category \(\Sch_\alpha\) constructed as in Sets, Lemma 000J starting with the set of schemes \(\{X, Y, S\}\). Choose any set of coverings \(\text{Cov}_{fppf}\) on \(\Sch_\alpha\) as in Sets, Lemma 000X starting with the category \(\Sch_\alpha\) and the class of fppf coverings. Let \(\Sch_{fppf}\) denote the big fppf site so obtained. Next, for \(\tau \in \{Zariski, \etale, smooth, syntomic\}\) let \(\Sch_\tau\) have the same underlying category as \(\Sch_{fppf}\) with coverings \(\text{Cov}_\tau \subset \text{Cov}_{fppf}\) simply the subset of \(\tau\)-coverings. It is straightforward to check that this gives rise to a big site \(\Sch_\tau\).
Change of big sites
In this section we explain what happens on changing the big Zariski/fppf/étale sites.
Let \(\tau, \tau' \in \{Zariski, \etale, smooth, syntomic, fppf\}\). Given two big sites \(\Sch_\tau\) and \(\Sch'_{\tau'}\) we say that \(\Sch_\tau\) is contained in \(\Sch'_{\tau'}\) if \(\Ob(\Sch_\tau) \subset \Ob(\Sch'_{\tau'})\) and \(\text{Cov}(\Sch_\tau) \subset \text{Cov}(\Sch'_{\tau'})\). In this case \(\tau'\) is stronger than \(\tau\), for example, no fppf site can be contained in an étale site.
Lemma
Any set of big Zariski sites is contained in a common big Zariski site. The same is true, mutatis mutandis, for big fppf and big étale sites.
Proof
This is true because the union of a set of sets is a set, and the constructions in Sets, Lemmas 000J and 000X allow one to start with any initially given set of schemes and coverings.
Lemma
Let \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\). Suppose given big sites \(\Sch_\tau\) and \(\Sch'_\tau\). Assume that \(\Sch_\tau\) is contained in \(\Sch'_\tau\). The inclusion functor \(\Sch_\tau \to \Sch'_\tau\) satisfies the assumptions of Sites, Lemma 00XU. There are morphisms of topoi \[\begin{eqnarray*} g : \Sh(\Sch_\tau) & \longrightarrow & \Sh(\Sch'_\tau) \\ f : \Sh(\Sch'_\tau) & \longrightarrow & \Sh(\Sch_\tau) \end{eqnarray*}\] such that \(f \circ g \cong \text{id}\). For any object \(S\) of \(\Sch_\tau\) the inclusion functor \((\Sch/S)_\tau \to (\Sch'/S)_\tau\) satisfies the assumptions of Sites, Lemma 00XU also. Hence similarly we obtain morphisms \[\begin{eqnarray*} g : \Sh((\Sch/S)_\tau) & \longrightarrow & \Sh((\Sch'/S)_\tau) \\ f : \Sh((\Sch'/S)_\tau) & \longrightarrow & \Sh((\Sch/S)_\tau) \end{eqnarray*}\] with \(f \circ g \cong \text{id}\).
Proof
Assumptions (b), (c), and (e) of Sites, Lemma 00XU are immediate for the functors \(\Sch_\tau \to \Sch'_\tau\) and \((\Sch/S)_\tau \to (\Sch'/S)_\tau\). Property (a) holds by Lemma 03WV, 03WW, 03WZ, 03X2, or 03WX. Property (d) holds because fibre products in the categories \(\Sch_\tau\), \(\Sch'_\tau\) exist and are compatible with fibre products in the category of schemes.
Discussion: The functor \(g^{-1} = f_*\) is simply the restriction functor which associates to a sheaf \(\mathcal{G}\) on \(\Sch'_\tau\) the restriction \(\mathcal{G}|_{\Sch_\tau}\). Hence this lemma simply says that given any sheaf of sets \(\mathcal{F}\) on \(\Sch_\tau\) there exists a canonical sheaf \(\mathcal{F}'\) on \(\Sch'_\tau\) such that \(\mathcal{F}'|_{\Sch_\tau} = \mathcal{F}\). In fact the sheaf \(\mathcal{F}'\) has the following description: it is the sheafification of the presheaf \[\Sch'_\tau \longrightarrow \textit{Sets}, \quad V \longmapsto \colim_{V \to U} \mathcal{F}(U)\] where \(U\) is an object of \(\Sch_\tau\). This is true because \(\mathcal{F}' = f^{-1}\mathcal{F} = (u_p\mathcal{F})^\#\) according to Sites, Lemmas 00XR and 00XU.
Extending functors
Let us start with a simple example which explains what we are doing. Let \(R\) be a ring. Suppose \(F\) is a functor defined on the category \(\mathcal{C}\) of \(R\)-algebras of the form \[A = R[x_1, \ldots, x_n]/(f_1, \ldots, f_m)\] for \(n, m \geq 0\) integers and \(f_1, \ldots, f_m \in R[x_1, \ldots, x_n]\) elements. Then for any \(R\)-algebra \(B\) we can define \[F'(B) = \colim_{A \to B,\ A \in \mathcal{C}} F(A)\] It turns out \(F'\) is the unique functor on the category of all \(R\)-algebras which extends \(F\) and commutes with filtered colimits. The same procedure works in the category of schemes if we impose that our functor is a Zariski sheaf.
Lemma
Let \(S\) be a scheme. Let \(\mathcal{C}\) be a full subcategory of the category \(\Sch/S\) of all schemes over \(S\). Assume
if \(X \to S\) is an object of \(\mathcal{C}\) and \(U \subset X\) is an affine open, then \(U \to S\) is isomorphic to an object of \(\mathcal{C}\),
if \(V\) is an affine scheme lying over an affine open \(U \subset S\) such that \(V \to U\) is of finite presentation, then \(V \to S\) is isomorphic to an object of \(\mathcal{C}\).
Let \(F : \mathcal{C}^{opp} \to \textit{Sets}\) be a functor. Assume
for any Zariski covering \(\{f_i : X_i \to X\}_{i \in I}\) with \(X, X_i\) objects of \(\mathcal{C}\) we have the sheaf condition for \(F\) and this family8,
if \(X = \lim X_i\) is a directed limit of affine schemes over \(S\) with \(X, X_i\) objects of \(\mathcal{C}\), then \(F(X) = \colim F(X_i)\).
Then there is a unique way to extend \(F\) to a functor \(F' : (\Sch/S)^{opp} \to \textit{Sets}\) satisfying the analogues of (a) and (b), i.e., \(F'\) satisfies the sheaf condition for any Zariski covering and \(F'(X) = \colim F'(X_i)\) whenever \(X = \lim X_i\) is a directed limit of affine schemes over \(S\).
Proof
The idea will be to first extend \(F\) to a sufficiently large collection of affine schemes over \(S\) and then use the Zariski sheaf property to extend to all schemes.
Suppose that \(V\) is an affine scheme over \(S\) whose structure morphism \(V \to S\) factors through some affine open \(U \subset S\). In this case we can write \[V = \lim V_i\] as a cofiltered limit with \(V_i \to U\) of finite presentation and \(V_i\) affine. See Algebra, Lemma 00QN. By conditions (1) and (2) we may replace our \(V_i\) by objects of \(\mathcal{C}\). Observe that \(V_i \to S\) is locally of finite presentation (if \(S\) is quasi-separated, then these morphisms are actually of finite presentation). Then we set \[F'(V) = \colim F(V_i)\] Actually, we can give a more canonical expression, namely \[F'(V) = \colim_{V \to V'} F(V')\] where the colimit is over the category of morphisms \(V \to V'\) over \(S\) where \(V'\) is an object of \(\mathcal{C}\) whose structure morphism \(V' \to S\) is locally of finite presentation. The reason this is the same as the first formula is that by Limits, Proposition 01ZC our inverse system \(V_i\) is cofinal in this category! Finally, note that if \(V\) were an object of \(\mathcal{C}\), then \(F'(V) = F(V)\) by assumption (b).
The second formula turns \(F'\) into a contravariant functor on the category formed by affine schemes \(V\) over \(S\) whose structure morphism factors through an affine open of \(S\). Let \(V\) be such an affine scheme over \(S\) and suppose that \(V = \bigcup_{k = 1, \ldots, n} V_k\) is a finite open covering by affines. Then it makes sense to ask if the sheaf condition holds for \(F'\) and this open covering. This is true and easy to show: write \(V = \lim V_i\) as in the previous paragraph. By Limits, Lemma 01Z4 for all sufficiently large \(i\) we can find affine opens \(V_{i, k} \subset V_i\) compatible with transition maps pulling back to \(V_k\) in \(V\). Thus \[F'(V_k) = \colim F(V_{i, k}) \quad\text{and}\quad F'(V_k \cap V_l) = \colim F(V_{i, k} \cap V_{i, l})\] Strictly speaking in these formulas we need to replace \(V_{i, k}\) and \(V_{i, k} \cap V_{i, l}\) by isomorphic affine objects of \(\mathcal{C}\) before applying the functor \(F\). Since \(I\) is directed the colimits pass through equalizers. Hence the sheaf condition (a) for \(F\) and the Zariski coverings \(\{V_{i, k} \to V_i\}\) implies the sheaf condition for \(F'\) and this covering.
Let \(X\) be a general scheme over \(S\). Let \(\mathcal{B}_X\) denote the collection of affine opens of \(X\) whose structure morphism to \(S\) maps into an affine open of \(S\). It is clear that \(\mathcal{B}_X\) is a basis for the topology of \(X\). By the result of the previous paragraph and Sheaves, Lemma 009L we see that \(F'\) is a sheaf on \(\mathcal{B}_X\). Hence \(F'\) restricted to \(\mathcal{B}_X\) extends uniquely to a sheaf \(F'_X\) on \(X\), see Sheaves, Lemma 009N. If \(X\) is an object of \(\mathcal{C}\) then we have a canonical identification \(F'_X(X) = F(X)\) by the agreement of \(F'\) and \(F\) on the objects for which they are both defined and the fact that \(F\) satisfies the sheaf condition for Zariski coverings.
Let \(f : X \to Y\) be a morphism of schemes over \(S\). We get a unique \(f\)-map from \(F'_Y\) to \(F'_X\) compatible with the maps \(F'(V) \to F'(U)\) for all \(U \in \mathcal{B}_X\) and \(V \in \mathcal{B}_Y\) with \(f(U) \subset V\), see Sheaves, Lemma 009X. We omit the verification that these maps compose correctly given morphisms \(X \to Y \to Z\) of schemes over \(S\). We also omit the verification that if \(f\) is a morphism of \(\mathcal{C}\), then the induced map \(F'_Y(Y) \to F'_X(X)\) is the same as the map \(F(Y) \to F(X)\) via the identifications \(F'_X(X) = F(X)\) and \(F'_Y(Y) = F(Y)\) above. In this way we see that the desired extension of \(F\) is the functor which sends \(X/S\) to \(F'_X(X)\).
Property (a) for the functor \(X \mapsto F'_X(X)\) is almost immediate from the construction; we omit the details. Suppose that \(X = \lim_{i \in I} X_i\) is a directed limit of affine schemes over \(S\). We have to show that \[F'_X(X) = \colim_{i \in I} F'_{X_i}(X_i)\] First assume that there is some \(i \in I\) such that \(X_i \to S\) factors through an affine open \(U \subset S\). Then \(F'\) is defined on \(X\) and on \(X_{i'}\) for \(i' \geq i\) and we see that \(F'_{X_{i'}}(X_{i'}) = F'(X_{i'})\) for \(i' \geq i\) and \(F'_X(X) = F'(X)\). In this case every arrow \(X \to V\) with \(V\) locally of finite presentation over \(S\) factors as \(X \to X_{i'} \to V\) for some \(i' \geq i\), see Limits, Proposition 01ZC. Thus we have \[\begin{align*} F'_X(X) & = F'(X) \\ & = \colim_{X \to V} F(V) \\ & = \colim_{i' \geq i} \colim_{X_{i'} \to V} F(V) \\ & = \colim_{i' \geq i} F'(X_{i'}) \\ & = \colim_{i' \geq i} F'_{X_{i'}}(X_{i'}) \\ & = \colim_{i' \in I} F'_{X_{i'}}(X_{i'}) \end{align*}\] as desired. Finally, in general we pick any \(i \in I\) and we choose a finite affine open covering \(X_i = X_{i, 1} \cup \ldots \cup X_{i, n}\) such that \(X_{i, k} \to S\) factors through an affine open of \(S\). Let \(X_k \subset X\) and \(X_{i', k}\) for \(i' \geq i\) be the inverse images of \(X_{i, k}\). By the previous case we see that \[F'_{X_k}(X_k) = \colim_{i' \geq i} F'_{X_{i', k}}(X_{i', k})\] and \[F'_{X_k \cap X_l}(X_k \cap X_l) = \colim_{i' \geq i} F'_{X_{i', k} \cap X_{i', l}}(X_{i', k} \cap X_{i', l})\] By the sheaf property and exactness of filtered colimits we find that \(F'_X(X) = \colim_{i \in I} F'_{X_i}(X_i)\) also in this case. This finishes the proof of property (b) and hence finishes the proof of the lemma.
Lemma
Let \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\). Let \(T\) be an affine scheme which is written as a limit \(T = \lim_{i \in I} T_i\) of a directed inverse system of affine schemes.
Let \(\mathcal{V} = \{V_j \to T\}_{j = 1, \ldots, m}\) be a standard \(\tau\)-covering of \(T\), see Definitions 020R, 0219, 0223, 0229, and 021Q. Then there exists an index \(i\) and a standard \(\tau\)-covering \(\mathcal{V}_i = \{V_{i, j} \to T_i\}_{j = 1, \ldots, m}\) whose base change \(T \times_{T_i} \mathcal{V}_i\) to \(T\) is isomorphic to \(\mathcal{V}\).
Let \(\mathcal{V}_i\), \(\mathcal{V}'_i\) be a pair of standard \(\tau\)-coverings of \(T_i\). If \(f : T \times_{T_i} \mathcal{V}_i \to T \times_{T_i} \mathcal{V}'_i\) is a morphism of coverings of \(T\), then there exists an index \(i' \geq i\) and a morphism \(f_{i'} : T_{i'} \times_{T_i} \mathcal{V}_i \to T_{i'} \times_{T_i} \mathcal{V}'_i\) whose base change to \(T\) is \(f\).
If \(f, g : \mathcal{V}_i \to \mathcal{V}'_i\) are morphisms of standard \(\tau\)-coverings of \(T_i\) whose base changes \(f_T, g_T\) to \(T\) are equal then there exists an index \(i' \geq i\) such that \(f_{T_{i'}} = g_{T_{i'}}\).
In other words, the category of standard \(\tau\)-coverings of \(T\) is the colimit over \(I\) of the categories of standard \(\tau\)-coverings of \(T_i\).
Proof
Let us prove this for \(\tau = fppf\). By Limits, Lemma 01ZM the category of schemes of finite presentation over \(T\) is the colimit over \(I\) of the categories of finite presentation over \(T_i\). By Limits, Lemmas 01ZN and 04AI the same is true for category of schemes which are affine, flat and of finite presentation over \(T\). To finish the proof of the lemma it suffices to show that if \(\{V_{j, i} \to T_i\}_{j = 1, \ldots, m}\) is a finite family of flat finitely presented morphisms with \(V_{j, i}\) affine, and the base change \(\coprod_j T \times_{T_i} V_{j, i} \to T\) is surjective, then for some \(i' \geq i\) the morphism \(\coprod T_{i'} \times_{T_i} V_{j, i} \to T_{i'}\) is surjective. Denote \(W_{i'} \subset T_{i'}\), resp. \(W \subset T\) the image. Of course \(W = T\) by assumption. Since the morphisms are flat and of finite presentation we see that \(W_i\) is a quasi-compact open of \(T_i\), see Morphisms, Lemma 01UA. Moreover, \(W = T \times_{T_i} W_i\) (formation of image commutes with base change). Hence by Limits, Lemma 01Z4 we conclude that \(W_{i'} = T_{i'}\) for some large enough \(i'\) and we win.
For \(\tau \in \{Zariski, \etale, smooth, syntomic\}\) a standard \(\tau\)-covering is a standard fppf covering. Hence the fully faithfulness of the functor holds. The only issue is to show that given a standard fppf covering \(\mathcal{V}_i\) for some \(i\) such that \(\mathcal{V}_i \times_{T_i} T\) is a standard \(\tau\)-covering, then \(\mathcal{V}_i \times_{T_i} T_{i'}\) is a standard \(\tau\)-covering for all \(i' \gg i\). This follows immediately from Limits, Lemmas 0EUU, 07RP, 0C0C, and 0C3L.
Lemma
Let \(S\), \(\mathcal{C}\), \(F\) satisfy conditions (1), (2), (a), and (b) of Lemma 0EUW and denote \(F' : (\Sch/S)^{opp} \to \textit{Sets}\) the unique extension constructed in the lemma. Let \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\). Assume
for any standard \(\tau\)-covering \(\{V_i \to V\}_{i = 1, \ldots, n}\) of affines in \(\Sch/S\) such that \(V \to S\) factors through an affine open \(U \subset S\) and \(V \to U\) is of finite presentation, the sheaf condition hold for \(F\) and \(\{V_i \to V\}_{i = 1, \ldots, n}\)9.
Then \(F'\) satisfies the sheaf condition for all \(\tau\)-coverings.
Proof
Let \(X\) be a scheme over \(S\) and let \(\{X_i \to X\}_{i \in I}\) be a \(\tau\)-covering. Let \(s_i \in F'(X_i)\) be elements such that \(s_i\) and \(s_j\) map to the same element of \(F'(X_i \times_X X_j)\) for all \(i, j \in I\). We have to show that there is a unique element \(s \in F'(X)\) restricting to \(s_i \in F'(X_i)\) for all \(i \in I\).
Special case: \(X\) is an affine such that the structure morphism maps into an affine open \(U\) of \(S\) and the covering \(\{X_i \to X\}_{i \in I}\) is a standard \(\tau\)-covering. In this case we can write \[X = \lim V_k\] as a cofiltered limit with \(V_k \to U\) of finite presentation and \(V_k\) affine. See Algebra, Lemma 00QN. By Lemma 049N there exists a \(k\) and a standard \(\tau\)-covering \(\{V_{k, i} \to V_k\}_{i \in I}\) whose base change to \(X\) is the given covering. For \(k' \geq k\) denote \(\{V_{k', i} \to V_{k'}\}_{i \in I}\) the base change to \(V_{k'}\) of our covering. Then we see that \[\begin{align*} F'(X) & = \colim_{k' \geq k} F(V_{k'}) \\ & = \colim_{k' \geq k} \text{Equalizer}( \xymatrix{ \prod F(V_{k', i}) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod F(V_{k', i} \times_{V_{k'}} V_{k', j}) }) \\ & = \text{Equalizer}( \xymatrix{ \colim_{k' \geq k} \prod F(V_{k', i}) \ar@<1ex>[r] \ar@<-1ex>[r] & \colim_{k' \geq k} \prod F(V_{k', i} \times_{V_{k'}} V_{k', j}) }) \\ & = \text{Equalizer}( \xymatrix{ \prod F'(X_i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod F'(X_i \times_X X_j) }) \end{align*}\] The first equality holds by construction of \(F'\). The second holds by assumption (c). The third holds because filtered colimits are exact. The fourth again holds by construction of \(F'\). In this way we find that the sheaf property holds for \(F'\) with respect to \(\{X_i \to X\}_{i \in I}\).
General case. Choose an affine open covering \(X = \bigcup U_k\) such that each \(U_k\) maps into an affine open of \(S\). For every \(k\) we may choose a standard \(\tau\)-covering \(\{V_{k, j} \to U_k\}_{j = 1, \ldots, m_k}\) which refines \(\{X_i \times_X U_k \to U_k\}_{i \in I}\). For each \(j \in \{1, \ldots, m_k\}\) choose an index \(i_{k, j} \in I\) and a morphism \(g_{k, j} : V_{k, j} \to X_{i_{k, j}}\) over \(X\). Let \(s_{k, j}\) be the element of \(F'(V_{k, j})\) we get by restricting \(s_{i_{k, j}}\) via \(g_{k, j}\). Observe that \(s_{k, j}\) and \(s_{k', j'}\) restrict to the same element of \(F'(V_{k, j} \times_X V_{k', j'})\) for all \(k\) and \(k'\) and all \(j \in \{1, \ldots, m_k\}\) and \(j' \in \{1, \ldots, m_{k'}\}\); verification omitted. In particular, by the result of the previous paragraph there is a unique element \(s_k \in F'(U_k)\) restricting to \(s_{k, j}\) for all \(j\). With this notation we are ready to finish the proof.
Proof of uniqueness of \(s\): this is true because \(F'\) satisfies the sheaf property for Zariski coverings and \(s|_{U_k}\) must be equal to \(s_k\) because both restrict to \(s_{k, j}\) for all \(j\). This uniqueness then shows that \(s_k\) and \(s_{k'}\) must restrict to the same section of \(F'\) over (the non-affine scheme) \(U_k \cap U_{k'}\) because these sections restrict to the same section over the \(\tau\)-covering \(\{V_{k, j} \times_X V_{k', j'} \to U_k \cap U_{k'}\}\). Thus by the sheaf property for Zariski coverings, there is a unique section \(s\) of \(F'\) over \(X\) whose restriction to \(U_k\) is \(s_k\). We omit the verification (similar to the above) that \(s\) restricts to \(s_i\) over \(X_i\).
Lemma
Let \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\). Let \(S\) be a scheme contained in a big site \(\Sch_\tau\). Let \(F : (\Sch/S)_\tau^{opp} \to \textit{Sets}\) be a \(\tau\)-sheaf satisfying property (b) of Lemma 0EUW with \(\mathcal{C} = (\Sch/S)_\tau\). Then the extension \(F'\) of \(F\) to the category of all schemes over \(S\) satisfies the sheaf condition for all \(\tau\)-coverings.
Proof
This follows from Lemma 0GDW applied with \(\mathcal{C} = (\Sch/S)_\tau\). Conditions (1), (2), (a), and (b) of Lemma 0EUW hold; we omit the details. Thus we get our unique extension \(F'\) to the category of all schemes over \(S\). Finally, observe that any standard \(\tau\)-covering is tautologically equivalent to a covering in \((\Sch/S)_\tau\), see Sets, Lemma 000R as well as Lemmas 03WV, 03WW, 03WZ, 03X2, and 03WX. By Sites, Lemma 00VW the sheaf property passes through tautological equivalence of coverings. Hence the fact that \(F\) is a \(\tau\)-sheaf implies that property (c) of Lemma 0GDW holds and we conclude.
The words big and small here do not relate to bigness/smallness of the corresponding categories.↩︎
In the case of the ph topology we deviate very slightly from this approach, see Definition 0DBL and the surrounding discussion.↩︎
The letters fppf stand for “fidèlement plat de présentation finie”.↩︎
We will see in More on Morphisms, Lemma 0DBT that fppf coverings (and hence syntomic, smooth, or étale coverings) are ph coverings as well.↩︎
The letters fpqc stand for “fidèlement plat quasi-compacte”.↩︎
A more precise statement would be that the analogue of Lemma 03WX for the fpqc topology does not hold.↩︎
We have not included the comparison between the ph topology and the others; for this see More on Morphisms, Remark 0DBU.↩︎
As we do not know that \(X_i \times_X X_j\) is in \(\mathcal{C}\) this has to be interpreted as follows: by property (1) there exist Zariski coverings \(\{U_{ijk} \to X_i \times_X X_j\}_{k \in K_{ij}}\) with \(U_{ijk}\) an object of \(\mathcal{C}\). Then the sheaf condition says that \(F(X)\) is the equalizer of the two maps from \(\prod F(X_i)\) to \(\prod F(U_{ijk})\).↩︎
This makes sense as \(V\), \(V_i\), and \(V_i \times_V V_j\) are isomorphic to objects of \(\mathcal{C}\) by (2).↩︎