Introduction
In this chapter we introduce some topologies on the category of algebraic spaces. Compare with the material in [SGA1], [Ner], [LM-B] and [Kn]. 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. This discussion will make little or no sense unless the reader has read Topologies, Section 020M.
Let \(S\) be a base scheme. Take any category \(\Sch_\alpha\) constructed as in Sets, Lemma 000J starting with \(S\) and any set of schemes over \(S\) you want to be included. 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, and let \((\Sch/S)_{fppf}\) denote the corresponding big fppf site of \(S\). (The above is entirely as prescribed in Topologies, Section 021L.)
Given choices as above the category of algebraic spaces over \(S\) has a set of isomorphism classes. One way to see this is to use the fact that any algebraic space over \(S\) is of the form \(U/R\) for some étale equivalence relation \(j : R \to U \times_S U\) with \(U, R \in \Ob((\Sch/S)_{fppf})\), see Spaces, Lemma 0262. Hence we can find a full subcategory \(\textit{Spaces}/S\) of the category of algebraic spaces over \(S\) which has a set of objects such that each algebraic space is isomorphic to an object of \(\textit{Spaces}/S\). We fix a choice of such a category.
In the sections below, given a topology \(\tau\), the big site \((\textit{Spaces}/S)_\tau\) (resp. the big site \((\textit{Spaces}/X)_\tau\) of an algebraic space \(X\) over \(S\)) has as underlying category the category \(\textit{Spaces}/S\) (resp. the subcategory \(\textit{Spaces}/X\) of \(\textit{Spaces}/S\), see Categories, Example 001G). The procedure for turning this into a site is as usual by defining a class of \(\tau\)-coverings and using Sets, Lemma 000X to choose a sufficiently large set of coverings which defines the topology.
We point out that the small étale site \(X_\etale\) of an algebraic space \(X\) has already been defined in Properties of Spaces, Definition 03ED. Its objects are schemes étale over \(X\), of which there are plenty by definition of an algebraic spaces. However, a more natural site, from the perspective of this chapter (compare Topologies, Definition 021B) is the site \(X_{spaces, \etale}\) of Properties of Spaces, Definition 03G0. These two sites define the same topos, see Properties of Spaces, Lemma 03G1. We will not redefine these in this chapter; instead we will simply use them.
Zariski topology
In Spaces, Section 02YT we introduced the notion of a Zariski covering of an algebraic space by open subspaces. Here is the corresponding notion with open subspaces replaced by open immersions.
Definition
Let \(S\) be a scheme, and let \(X\) be an algebraic space over \(S\). A Zariski covering of \(X\) is a family of morphisms \(\{f_i : X_i \to X\}_{i \in I}\) of algebraic spaces over \(S\) such that each \(f_i\) is an open immersion and such that \[|X| = \bigcup\nolimits_{i \in I} |f_i|(|X_i|),\] i.e., the morphisms are jointly surjective.
Although Zariski coverings are occasionally useful the corresponding topology on the category of algebraic spaces is really too coarse, and not particularly useful. Still, it does define a site.
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\).
If \(X' \to X\) is an isomorphism then \(\{X' \to X\}\) is a Zariski covering of \(X\).
If \(\{X_i \to X\}_{i\in I}\) is a Zariski covering and for each \(i\) we have a Zariski covering \(\{X_{ij} \to X_i\}_{j\in J_i}\), then \(\{X_{ij} \to X\}_{i \in I, j\in J_i}\) is a Zariski covering.
If \(\{X_i \to X\}_{i\in I}\) is a Zariski covering and \(X' \to X\) is a morphism of algebraic spaces then \(\{X' \times_X X_i \to X'\}_{i\in I}\) is a Zariski covering.
Proof
Omitted.
Étale topology
In this section we discuss the notion of a étale covering of algebraic spaces, and we define the big étale site of an algebraic space. Please compare with Topologies, Section 0214.
Definition
Let \(S\) be a scheme, and let \(X\) be an algebraic space over \(S\). An étale covering of \(X\) is a family of morphisms \(\{f_i : X_i \to X\}_{i \in I}\) of algebraic spaces over \(S\) such that each \(f_i\) is étale and such that \[|X| = \bigcup\nolimits_{i \in I} |f_i|(|X_i|),\] i.e., the morphisms are jointly surjective.
This is exactly the same as Topologies, Definition 0215. In particular, if \(X\) and all the \(X_i\) are schemes, then we recover the usual notion of a étale covering of schemes.
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 (this follows from Morphisms, Lemma 02GP via Spaces, Lemma 02YO as immersions are representable).
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\).
If \(X' \to X\) is an isomorphism then \(\{X' \to X\}\) is a étale covering of \(X\).
If \(\{X_i \to X\}_{i\in I}\) is a étale covering and for each \(i\) we have a étale covering \(\{X_{ij} \to X_i\}_{j\in J_i}\), then \(\{X_{ij} \to X\}_{i \in I, j\in J_i}\) is a étale covering.
If \(\{X_i \to X\}_{i\in I}\) is a étale covering and \(X' \to X\) is a morphism of algebraic spaces then \(\{X' \times_X X_i \to X'\}_{i\in I}\) is a étale covering.
Proof
Omitted.
The following lemma tells us that the sites \((\textit{Spaces}/X)_\etale\) and \((\textit{Spaces}/X)_{smooth}\) have the same categories of sheaves.
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\{X_i \to X\}_{i \in I}\) be a smooth covering of \(X\). Then there exists an étale covering \(\{U_j \to X\}_{j \in J}\) of \(X\) which refines \(\{X_i \to X\}_{i \in I}\).
Proof
First choose a scheme \(U\) and a surjective étale morphism \(U \to X\). For each \(i\) choose a scheme \(W_i\) and a surjective étale morphism \(W_i \to X_i\). Then \(\{W_i \to X\}_{i \in I}\) is a smooth covering which refines \(\{X_i \to X\}_{i \in I}\). Hence \(\{W_i \times_X U \to U\}_{i \in I}\) is a smooth covering of schemes. By More on Morphisms, Lemma 055V we can choose an étale covering \(\{U_j \to U\}\) which refines \(\{W_i \times_X U \to U\}\). Then \(\{U_j \to X\}_{j \in J}\) is an étale covering refining \(\{X_i \to X\}_{i \in I}\).
Definition
Let \(S\) be a scheme. A big étale site \((\textit{Spaces}/S)_\etale\) is any site constructed as follows:
Choose a big étale site \((\Sch/S)_\etale\) as in Topologies, Section 0214.
As underlying category take the category \(\textit{Spaces}/S\) of algebraic spaces over \(S\) (see discussion in Section 03Y6 why this is a set).
Choose any set of coverings as in Sets, Lemma 000X starting with the category \(\textit{Spaces}/S\) and the class of étale coverings of Definition 041E.
Having defined this, we can localize to get the étale site of an algebraic space.
Definition
Let \(S\) be a scheme. Let \((\textit{Spaces}/S)_\etale\) be as in Definition 0DBX. Let \(X\) be an algebraic space over \(S\), i.e., an object of \((\textit{Spaces}/S)_\etale\). Then the big étale site \((\textit{Spaces}/X)_\etale\) of \(X\) is the localization of the site \((\textit{Spaces}/S)_\etale\) at \(X\) introduced in Sites, Section 00XZ.
Recall that given an algebraic space \(X\) over \(S\) as in the definition, we already have defined the small étale sites \(X_{spaces, \etale}\) and \(X_\etale\), see Properties of Spaces, Section 03EB. We will silently identify the corresponding topoi using the inclusion functor \(X_\etale \subset X_{spaces, \etale}\) (Properties of Spaces, Lemma 03G1) and we will call it the small étale topos of \(X\). Next, we establish some relationships between the topoi associated to these sites.
Lemma
Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism of \((\textit{Spaces}/S)_\etale\). The inclusion functor \(Y_{spaces, \etale} \to (\textit{Spaces}/X)_\etale\) is cocontinuous and induces a morphism of topoi \[i_f : \Sh(Y_\etale) \longrightarrow \Sh((\textit{Spaces}/X)_\etale)\] For a sheaf \(\mathcal{G}\) on \((\textit{Spaces}/X)_\etale\) we have the formula \((i_f^{-1}\mathcal{G})(U/Y) = \mathcal{G}(U/X)\). The functor \(i_f^{-1}\) also has a left adjoint \(i_{f, !}\) which commutes with fibre products and equalizers.
Proof
Denote the functor \(u : Y_{spaces, \etale} \to (\textit{Spaces}/X)_\etale\). In other words, given an étale morphism \(j : U \to Y\) corresponding to an object of \(Y_{spaces, \etale}\) we set \(u(U \to T) = (f \circ j : U \to S)\). The category \(Y_{spaces, \etale}\) has fibre products and equalizers and \(u\) commutes with them. It is immediate that \(u\) cocontinuous. The functor \(u\) 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 \(X\) be an object of \((\textit{Spaces}/S)_\etale\). The inclusion functor \(X_{spaces, \etale} \to (\textit{Spaces}/X)_\etale\) satisfies the hypotheses of Sites, Lemma 00XU and hence induces a morphism of sites \[\pi_X : (\textit{Spaces}/X)_\etale \longrightarrow X_{spaces, \etale}\] and a morphism of topoi \[i_X : \Sh(X_\etale) \longrightarrow \Sh((\textit{Spaces}/X)_\etale)\] such that \(\pi_X \circ i_X = \text{id}\). Moreover, \(i_X = i_{\text{id}_X}\) with \(i_{\text{id}_X}\) as in Lemma 0DF2. In particular the functor \(i_X^{-1} = \pi_{X, *}\) is described by the rule \(i_X^{-1}(\mathcal{G})(U/X) = \mathcal{G}(U/X)\).
Proof
In this case the functor \(u : X_{spaces, \etale} \to (\textit{Spaces}/X)_\etale\), in addition to the properties seen in the proof of Lemma 0DF2 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 0DF3 the functor \(i_X^{-1} = \pi_{X, *}\) is often called the restriction to the small étale site, and for a sheaf \(\mathcal{F}\) on the big étale site we often denote \(\mathcal{F}|_{X_\etale}\) 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 \[\begin{align*} \Mor_{\Sh(X_\etale)}( \mathcal{F}|_{X_\etale}, \mathcal{G}) & = \Mor_{\Sh((\textit{Spaces}/X)_\etale)}( \mathcal{F}, i_{X, *}\mathcal{G}) \\ \Mor_{\Sh(X_\etale)}( \mathcal{G}, \mathcal{F}|_{X_\etale}) & = \Mor_{\Sh((\textit{Spaces}/X)_\etale)}( \pi_X^{-1}\mathcal{G}, \mathcal{F}) \end{align*}\] Moreover, we have \((i_{X, *}\mathcal{G})|_{X_\etale} = \mathcal{G}\) and we have \((\pi_X^{-1}\mathcal{G})|_{X_\etale} = \mathcal{G}\).
Lemma
Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism in \((\textit{Spaces}/S)_\etale\). The functor \[u : (\textit{Spaces}/Y)_\etale \longrightarrow (\textit{Spaces}/X)_\etale, \quad V/Y \longmapsto V/X\] is cocontinuous, and has a continuous right adjoint \[v : (\textit{Spaces}/X)_\etale \longrightarrow (\textit{Spaces}/Y)_\etale, \quad (U \to X) \longmapsto (U \times_X Y \to Y).\] They induce the same morphism of topoi \[f_{big} : \Sh((\textit{Spaces}/Y)_\etale) \longrightarrow \Sh((\textit{Spaces}/X)_\etale)\] We have \(f_{big}^{-1}(\mathcal{G})(U/Y) = \mathcal{G}(U/X)\). We have \(f_{big, *}(\mathcal{F})(U/X) = \mathcal{F}(U \times_X Y/Y)\). Also, \(f_{big}^{-1}\) has a left adjoint \(f_{big!}\) which commutes with fibre products and equalizers.
Proof
The functor \(u\) is cocontinuous, continuous and commutes with fibre products and equalizers (details omitted; compare with the proof of Lemma 0DF2). 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 \(U/Y\) and \(V/X\) we have \(\Mor_X(u(U), V) = \Mor_Y(U, V \times_X Y)\) as desired. Thus we may apply Sites, Lemmas 00XX and 00XY to get the formula for \(f_{big, *}\).
Lemma
Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism in \((\textit{Spaces}/S)_\etale\).
We have \(i_f = f_{big} \circ i_T\) with \(i_f\) as in Lemma 0DF2 and \(i_T\) as in Lemma 0DF3.
The functor \(X_{spaces, \etale} \to T_{spaces, \etale}\), \((U \to X) \mapsto (U \times_X Y \to Y)\) is continuous and induces a morphism of sites \[f_{spaces, \etale} : Y_{spaces, \etale} \longrightarrow X_{spaces, \etale}\] The corresponding morphism of small étale topoi is denoted \[f_{small} : \Sh(Y_\etale) \to \Sh(X_\etale)\] We have \(f_{small, *}(\mathcal{F})(U/X) = \mathcal{F}(U \times_X Y/Y)\).
We have a commutative diagram of morphisms of sites \[\xymatrix{ Y_{spaces, \etale} \ar[d]_{f_{spaces, \etale}} & (\textit{Spaces}/Y)_\etale \ar[d]^{f_{big}} \ar[l]^-{\pi_Y}\\ X_{spaces, \etale} & (\textit{Spaces}/X)_\etale \ar[l]_-{\pi_X} }\] so that \(f_{small} \circ \pi_Y = \pi_X \circ f_{big}\) as morphisms of topoi.
We have \(f_{small} = \pi_X \circ f_{big} \circ i_Y = \pi_X \circ i_f\).
Proof
The equality \(i_f = f_{big} \circ i_Y\) 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 \(u : X_{spaces, \etale} \to Y_{spaces, \etale}\), \(u(U \to X) = (U \times_X Y \to Y)\) was shown to give rise to a morphism of sites and correspong morphism of small étale topoi in Properties of Spaces, Lemma 03G2. The description of the pushforward is clear.
Part (3) follows because \(\pi_X\) and \(\pi_Y\) are given by the inclusion functors and \(f_{spaces, \etale}\) and \(f_{big}\) by the base change functors \(U \mapsto U \times_X Y\).
Statement (4) follows from (3) by precomposing with \(i_Y\).
In the situation of the lemma, using the terminology of Definition 0DF4 we have: for \(\mathcal{F}\) a sheaf on the big étale site of \(Y\) \[(f_{big, *}\mathcal{F})|_{X_\etale} = f_{small, *}(\mathcal{F}|_{Y_\etale}),\] This equality is clear from the commutativity of the diagram of sites of the lemma, since restriction to the small étale site of \(Y\), resp. \(X\) is given by \(\pi_{Y, *}\), resp. \(\pi_{X, *}\). A similar formula involving pullbacks and restrictions is false.
Lemma
Let \(S\) be a scheme. Given morphisms \(f : X \to Y\), \(g : Y \to Z\) in \((\textit{Spaces}/S)_\etale\) 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 0DF5. For the functors on the small sites this follows from the description of the pushforward functors in Lemma 0DF6.
Lemma
Let \(S\) be a scheme. Consider a cartesian diagram \[\xymatrix{ Y' \ar[r]_{g'} \ar[d]_{f'} & Y \ar[d]^f \\ X' \ar[r]^g & X }\] in \((\textit{Spaces}/S)_\etale\). 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 \(U'/X'\) that \(U' \times_{X'} Y' = U' \times_X Y\). Hence both \(i_g^{-1} \circ f_{big, *}\) and \(f'_{small, *} \circ (i_{g'})^{-1}\) send a sheaf \(\mathcal{F}\) on \((\textit{Spaces}/Y)_\etale\) to the sheaf \(U' \mapsto \mathcal{F}(U' \times_{X'} Y')\) on \(X'_\etale\) (use Lemmas 0DF2 and 0DF5). The second equality can be proved in the same manner or can be deduced from the very general Sites, Lemma 03CF.
Remark
The sites \((\textit{Spaces}/X)_\etale\) and \(X_{spaces, \etale}\) come with structure sheaves. For the small étale site we have seen this in Properties of Spaces, Section 04KD. The structure sheaf \(\mathcal{O}\) on the big étale site \((\textit{Spaces}/X)_\etale\) is defined by assigning to an object \(U\) the global sections of the structure sheaf of \(U\). This makes sense because after all \(U\) is an algebraic space itself hence has a structure sheaf. Since \(\mathcal{O}_U\) is a sheaf on the étale site of \(U\), the presheaf \(\mathcal{O}\) so defined satisfies the sheaf condition for coverings of \(U\), i.e., \(\mathcal{O}\) is a sheaf. We can upgrade the morphisms \(i_f\), \(\pi_X\), \(i_X\), \(f_{small}\), and \(f_{big}\) defined above to morphisms of ringed sites, respectively topoi. Let us deal with these one by one.
In Lemma 0DF2 denote \(\mathcal{O}\) the structure sheaf on \((\textit{Spaces}/X)_\etale\). We have \((i_f^{-1}\mathcal{O})(U/Y) = \mathcal{O}_U(U) = \mathcal{O}_Y(U)\) by construction. Hence an isomorphism \(i_f^\sharp : i_f^{-1}\mathcal{O} \to \mathcal{O}_Y\).
In Lemma 0DF3 it was noted that \(i_X\) is a special case of \(i_f\) with \(f = \text{id}_X\) hence we are back in case (1).
In Lemma 0DF3 the morphism \(\pi_X\) satisfies \((\pi_{X, *}\mathcal{O})(U) = \mathcal{O}(U) = \mathcal{O}_X(U)\). Hence we can use this to define \(\pi_X^\sharp : \mathcal{O}_X \to \pi_{X, *}\mathcal{O}\).
In Lemma 0DF6 the extension of \(f_{small}\) to a morphism of ringed topoi was discussed in Properties of Spaces, Lemma 03G8.
In Lemma 0DF6 the functor \(f_{big}^{-1}\) is simply the restriction via the inclusion functor \((\textit{Spaces}/Y)_\etale \to (\textit{Spaces}/X)_\etale\). Let \(\mathcal{O}_1\) be the structure sheaf on \((\textit{Spaces}/X)_\etale\) and let \(\mathcal{O}_2\) be the structure sheaf on \((\textit{Spaces}/Y)_\etale\). We obtain a canonical isomorphism \(f_{big}^\sharp : f_{big}^{-1}\mathcal{O}_1 \to \mathcal{O}_2\).
Moreover, with these definitions compositions work out correctly too. We omit giving a detailed statement and proof.
Smooth topology
In this section we discuss the notion of a smooth covering of algebraic spaces, and we define the big smooth site of an algebraic space. Please compare with Topologies, Section 021Y.
Definition
Let \(S\) be a scheme, and let \(X\) be an algebraic space over \(S\). A smooth covering of \(X\) is a family of morphisms \(\{f_i : X_i \to X\}_{i \in I}\) of algebraic spaces over \(S\) such that each \(f_i\) is smooth and such that \[|X| = \bigcup\nolimits_{i \in I} |f_i|(|X_i|),\] i.e., the morphisms are jointly surjective.
This is exactly the same as Topologies, Definition 021Z. In particular, if \(X\) and all the \(X_i\) are schemes, then we recover the usual notion of a smooth covering of schemes.
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 (Morphisms of Spaces, Lemma 04XX), and Lemma 0DF1.
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\).
If \(X' \to X\) is an isomorphism then \(\{X' \to X\}\) is a smooth covering of \(X\).
If \(\{X_i \to X\}_{i\in I}\) is a smooth covering and for each \(i\) we have a smooth covering \(\{X_{ij} \to X_i\}_{j\in J_i}\), then \(\{X_{ij} \to X\}_{i \in I, j\in J_i}\) is a smooth covering.
If \(\{X_i \to X\}_{i\in I}\) is a smooth covering and \(X' \to X\) is a morphism of algebraic spaces then \(\{X' \times_X X_i \to X'\}_{i\in I}\) is a smooth covering.
Proof
Omitted.
To be continued...
Syntomic topology
In this section we discuss the notion of a syntomic covering of algebraic spaces, and we define the big syntomic site of an algebraic space. Please compare with Topologies, Section 0224.
Definition
Let \(S\) be a scheme, and let \(X\) be an algebraic space over \(S\). A syntomic covering of \(X\) is a family of morphisms \(\{f_i : X_i \to X\}_{i \in I}\) of algebraic spaces over \(S\) such that each \(f_i\) is syntomic and such that \[|X| = \bigcup\nolimits_{i \in I} |f_i|(|X_i|),\] i.e., the morphisms are jointly surjective.
This is exactly the same as Topologies, Definition 0225. In particular, if \(X\) and all the \(X_i\) are schemes, then we recover the usual notion of a syntomic covering of schemes.
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 (Morphisms of Spaces, Lemma 06CP), and Lemma 0DFA.
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\).
If \(X' \to X\) is an isomorphism then \(\{X' \to X\}\) is a syntomic covering of \(X\).
If \(\{X_i \to X\}_{i\in I}\) is a syntomic covering and for each \(i\) we have a syntomic covering \(\{X_{ij} \to X_i\}_{j\in J_i}\), then \(\{X_{ij} \to X\}_{i \in I, j\in J_i}\) is a syntomic covering.
If \(\{X_i \to X\}_{i\in I}\) is a syntomic covering and \(X' \to X\) is a morphism of algebraic spaces then \(\{X' \times_X X_i \to X'\}_{i\in I}\) is a syntomic covering.
Proof
Omitted.
To be continued...
Fppf topology
In this section we discuss the notion of an fppf covering of algebraic spaces, and we define the big fppf site of an algebraic space. Please compare with Topologies, Section 021L.
Definition
Let \(S\) be a scheme, and let \(X\) be an algebraic space over \(S\). An fppf covering of \(X\) is a family of morphisms \(\{f_i : X_i \to X\}_{i \in I}\) of algebraic spaces over \(S\) such that each \(f_i\) is flat and locally of finite presentation and such that \[|X| = \bigcup\nolimits_{i \in I} |f_i|(|X_i|),\] i.e., the morphisms are jointly surjective.
This is exactly the same as Topologies, Definition 021M. In particular, if \(X\) and all the \(X_i\) are schemes, then we recover the usual notion of an fppf covering of schemes.
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 (Morphisms of Spaces, Lemmas 0DEY and 0DEZ) and Lemma 0DFB.
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\).
If \(X' \to X\) is an isomorphism then \(\{X' \to X\}\) is an fppf covering of \(X\).
If \(\{X_i \to X\}_{i\in I}\) is an fppf covering and for each \(i\) we have an fppf covering \(\{X_{ij} \to X_i\}_{j\in J_i}\), then \(\{X_{ij} \to X\}_{i \in I, j\in J_i}\) is an fppf covering.
If \(\{X_i \to X\}_{i\in I}\) is an fppf covering and \(X' \to X\) is a morphism of algebraic spaces then \(\{X' \times_X X_i \to X'\}_{i\in I}\) is an fppf covering.
Proof
Omitted.
Lemma
Let \(S\) be a scheme, and let \(X\) be an algebraic space over \(S\). Suppose that \(\mathcal{U} = \{f_i : X_i \to X\}_{i \in I}\) is an fppf covering of \(X\). Then there exists a refinement \(\mathcal{V} = \{g_i : T_i \to X\}\) of \(\mathcal{U}\) which is an fppf covering such that each \(T_i\) is a scheme.
Proof
Omitted. Hint: For each \(i\) choose a scheme \(T_i\) and a surjective étale morphism \(T_i \to X_i\). Then check that \(\{T_i \to X\}\) is an fppf covering.
Lemma
Let \(S\) be a scheme. Let \(\{f_i : X_i \to X\}_{i \in I}\) be an fppf covering of algebraic spaces over \(S\). Then the map of sheaves \[\coprod X_i \longrightarrow X\] is surjective.
Proof
This follows from Spaces, Lemma 05VM. See also Spaces, Remark 02YN in case you are confused about the meaning of this lemma.
Definition
Let \(S\) be a scheme. A big fppf site \((\textit{Spaces}/S)_{fppf}\) is any site constructed as follows:
Choose a big fppf site \((\Sch/S)_{fppf}\) as in Topologies, Section 021L.
As underlying category take the category \(\textit{Spaces}/S\) of algebraic spaces over \(S\) (see discussion in Section 03Y6 why this is a set).
Choose any set of coverings as in Sets, Lemma 000X starting with the category \(\textit{Spaces}/S\) and the class of fppf coverings of Definition 03Y8.
Having defined this, we can localize to get the fppf site of an algebraic space.
Definition
Let \(S\) be a scheme. Let \((\textit{Spaces}/S)_{fppf}\) be as in Definition 0DBV. Let \(X\) be an algebraic space over \(S\), i.e., an object of \((\textit{Spaces}/S)_{fppf}\). Then the big fppf site \((\textit{Spaces}/X)_{fppf}\) of \(X\) is the localization of the site \((\textit{Spaces}/S)_{fppf}\) at \(X\) introduced in Sites, Section 00XZ.
Next, we establish some relationships between the topoi associated to these sites.
Lemma
Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism of algebraic spaces over \(S\). The functor \[u : (\textit{Spaces}/Y)_{fppf} \longrightarrow (\textit{Spaces}/X)_{fppf}, \quad V/Y \longmapsto V/X\] is cocontinuous, and has a continuous right adjoint \[v : (\textit{Spaces}/X)_{fppf} \longrightarrow (\textit{Spaces}/Y)_{fppf}, \quad (U \to Y) \longmapsto (U \times_X Y \to Y).\] They induce the same morphism of topoi \[f_{big} : \Sh((\textit{Spaces}/Y)_{fppf}) \longrightarrow \Sh((\textit{Spaces}/X)_{fppf})\] We have \(f_{big}^{-1}(\mathcal{G})(U/Y) = \mathcal{G}(U/X)\). We have \(f_{big, *}(\mathcal{F})(U/X) = \mathcal{F}(U \times_X Y/Y)\). Also, \(f_{big}^{-1}\) has a left adjoint \(f_{big!}\) which commutes with fibre products and equalizers.
Proof
The functor \(u\) is cocontinuous, continuous, and commutes with fibre products and equalizers. Hence Sites, Lemmas 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 \(V/X\) we have \(\Mor_X(u(U), V) = \Mor_Y(U, V \times_X Y)\) as desired. Thus we may apply Sites, Lemmas 00XX and 00XY to get the formula for \(f_{big, *}\).
Lemma
Let \(S\) be a scheme. Given morphisms \(f : X \to Y\), \(g : Y \to Z\) of algebraic spaces over \(S\) we have \(g_{big} \circ f_{big} = (g \circ f)_{big}\).
Proof
This follows from the simple description of pushforward and pullback for the functors on the big sites from Lemma 0DFD.
The ph topology
In this section we define the ph topology. This is the topology generated by étale coverings and proper surjective morphisms, see Lemma 0DFM.
Definition
Let \(S\) be a scheme and let \(X\) be an algebraic space over \(S\). A ph covering of \(X\) is a family of morphisms \(\{X_i \to X\}_{i \in I}\) of algebraic spaces over \(S\) such that \(f_i\) is locally of finite type and such that for every \(U \to X\) with \(U\) affine there exists a standard ph covering \(\{U_j \to U\}_{j = 1, \ldots, m}\) refining the family \(\{X_i \times_X U \to U\}_{i \in I}\).
In other words, there exists indices \(i_1, \ldots, i_m \in I\) and morphisms \(h_j : U_j \to X_{i_j}\) such that \(f_{i_j} \circ h_j = h \circ g_j\). Note that if \(X\) and all \(X_i\) are representable, this is the same as a ph covering of schemes by Topologies, Definition 0DBG.
Lemma
Any fppf covering is a ph covering, and a fortiori, any syntomic, smooth, étale or Zariski covering is a ph covering.
Proof
We will show that an fppf covering is a ph covering, and then the rest follows from Lemma 0DFC. Let \(\{X_i \to X\}_{i \in I}\) be an fppf covering of algebraic spaces over a base scheme \(S\). Let \(U\) be an affine scheme and let \(U \to X\) be a morphism. We can refine the fppf covering \(\{X_i \times_U U \to U\}_{i \in I}\) by an fppf covering \(\{T_i \to U\}_{i \in I}\) where \(T_i\) is a scheme (Lemma 042T). Then we can find a standard ph covering \(\{U_j \to U\}_{j = 1, \ldots, m}\) refining \(\{T_i \to U\}_{i \in I}\) by More on Morphisms, Lemma 0DBT (and the definition of ph coverings for schemes). Thus \(\{X_i \to X\}_{i \in I}\) is a ph covering by definition.
Lemma
Let \(S\) be a scheme. Let \(f : Y \to X\) be a surjective proper morphism of algebraic spaces over \(S\). Then \(\{Y \to X\}\) is a ph covering.
Proof
Let \(U \to X\) be a morphism with \(U\) affine. By Chow’s lemma (in the weak form given as Cohomology of Spaces, Lemma 089J) we see that there is a surjective proper morphism of schemes \(V \to U\) which factors through \(Y \times_X U \to U\). Taking any finite affine open cover of \(V\) we obtain a standard ph covering of \(U\) refining \(\{X \times_Y U \to U\}\) as desired.
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\).
If \(X' \to X\) is an isomorphism then \(\{X' \to X\}\) is a ph covering of \(X\).
If \(\{X_i \to X\}_{i\in I}\) is a ph covering and for each \(i\) we have a ph covering \(\{X_{ij} \to X_i\}_{j\in J_i}\), then \(\{X_{ij} \to X\}_{i \in I, j\in J_i}\) is a ph covering.
If \(\{X_i \to X\}_{i\in I}\) is a ph covering and \(X' \to X\) is a morphism of algebraic spaces then \(\{X' \times_X X_i \to X'\}_{i\in I}\) is a ph covering.
Proof
Part (1) is clear. Consider \(g : X' \to X\) and \(\{X_i \to X\}_{i\in I}\) a ph covering as in (3). By Morphisms of Spaces, Lemma 03XH the morphisms \(X' \times_X X_i \to X'\) are locally of finite type. If \(h' : Z \to X'\) is a morphism from an affine scheme towards \(X'\), then set \(h = g \circ h' : Z \to X\). The assumption on \(\{X_i \to X\}_{i\in I}\) means there exists a standard ph covering \(\{Z_j \to Z\}_{j = 1, \ldots, n}\) and morphisms \(Z_j \to X_{i(j)}\) covering \(h\) for certain \(i(j) \in I\). By the universal property of the fibre product we obtain morphisms \(Z_j \to X' \times_X X_{i(j)}\) over \(h'\) also. Hence \(\{X' \times_X X_i \to X'\}_{i\in I}\) is a ph covering. This proves (3).
Let \(\{X_i \to X\}_{i\in I}\) and \(\{X_{ij} \to X_i\}_{j\in J_i}\) be as in (2). Let \(h : Z \to X\) be a morphism from an affine scheme towards \(X\). By assumption there exists a standard ph covering \(\{Z_j \to Z\}_{j = 1, \ldots, n}\) and morphisms \(h_j : Z_j \to X_{i(j)}\) covering \(h\) for some indices \(i(j) \in I\). By assumption there exist standard ph coverings \(\{Z_{j, l} \to Z_j\}_{l = 1, \ldots, n(j)}\) and morphisms \(Z_{j, l} \to X_{i(j)j(l)}\) covering \(h_j\) for some indices \(j(l) \in J_{i(j)}\). By Topologies, Lemma 0DBF the family \(\{Z_{j, l} \to Z\}\) can be refined by a standard ph covering. Hence we conclude that \(\{X_{ij} \to X\}_{i \in I, j\in J_i}\) is a ph covering.
Definition
Let \(S\) be a scheme. A big ph site \((\textit{Spaces}/S)_{ph}\) is any site constructed as follows:
Choose a big ph site \((\Sch/S)_{ph}\) as in Topologies, Section 0DBC.
As underlying category take the category \(\textit{Spaces}/S\) of algebraic spaces over \(S\) (see discussion in Section 03Y6 why this is a set).
Choose any set of coverings as in Sets, Lemma 000X starting with the category \(\textit{Spaces}/S\) and the class of ph coverings of Definition 0DFG.
Having defined this, we can localize to get the ph site of an algebraic space.
Definition
Let \(S\) be a scheme. Let \((\textit{Spaces}/S)_{ph}\) be as in Definition 0DFK. Let \(X\) be an algebraic space over \(S\), i.e., an object of \((\textit{Spaces}/S)_{ph}\). Then the big ph site \((\textit{Spaces}/X)_{ph}\) of \(X\) is the localization of the site \((\textit{Spaces}/S)_{ph}\) at \(X\) introduced in Sites, Section 00XZ.
Here is the promised characterization of ph sheaves.
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a presheaf on \((\textit{Spaces}/X)_{ph}\). Then \(\mathcal{F}\) is a sheaf if and only if
\(\mathcal{F}\) satisfies the sheaf condition for étale coverings, and
if \(f : V \to U\) is a proper surjective morphism of \((\textit{Spaces}/X)_{ph}\), then \(\mathcal{F}(U)\) maps bijectively to the equalizer of the two maps \(\mathcal{F}(V) \to \mathcal{F}(V \times_U V)\).
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 \((\textit{Spaces}/X)_{ph}\). 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 \(s \in \mathcal{F}\) restricting to \(s_i\) over \(T_i\). Let \(\{U_j \to T\}\) be an étale covering with \(U_j\) affine. 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.
Next, we establish some relationships between the topoi associated to these sites.
Lemma
Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism of algebraic spaces over \(S\). The functor \[u : (\textit{Spaces}/Y)_{ph} \longrightarrow (\textit{Spaces}/X)_{ph}, \quad V/Y \longmapsto V/X\] is cocontinuous, and has a continuous right adjoint \[v : (\textit{Spaces}/X)_{ph} \longrightarrow (\textit{Spaces}/Y)_{ph}, \quad (U \to Y) \longmapsto (U \times_X Y \to Y).\] They induce the same morphism of topoi \[f_{big} : \Sh((\textit{Spaces}/Y)_{ph}) \longrightarrow \Sh((\textit{Spaces}/X)_{ph})\] We have \(f_{big}^{-1}(\mathcal{G})(U/Y) = \mathcal{G}(U/X)\). We have \(f_{big, *}(\mathcal{F})(U/X) = \mathcal{F}(U \times_X Y/Y)\). Also, \(f_{big}^{-1}\) has a left adjoint \(f_{big!}\) which commutes with fibre products and equalizers.
Proof
The functor \(u\) is cocontinuous, continuous, and commutes with fibre products and equalizers. Hence Sites, Lemmas 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 \(V/X\) we have \(\Mor_X(u(U), V) = \Mor_Y(U, V \times_X Y)\) as desired. Thus we may apply Sites, Lemmas 00XX and 00XY to get the formula for \(f_{big, *}\).
Lemma
Let \(S\) be a scheme. Given morphisms \(f : X \to Y\), \(g : Y \to Z\) of algebraic spaces over \(S\) we have \(g_{big} \circ f_{big} = (g \circ f)_{big}\).
Proof
This follows from the simple description of pushforward and pullback for the functors on the big sites from Lemma 0DFN.
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(P\) be a property of objects in \((\textit{Spaces}/X)_{fppf}\) such that whenever \(\{U_i \to U\}\) is a covering in \((\textit{Spaces}/X)_{fppf}\), then \[P(U_{i_0} \times_U \ldots \times_U U_{i_p}) \text{ for all } p \geq 0,\ i_0, \ldots, i_p \in I \Rightarrow P(U)\] If \(P(U)\) for all \(U\) affine and flat, locally of finite presentation over \(X\), then \(P(X)\).
Proof
Let \(U\) be a separated algebraic space locally of finite presentation over \(X\). Then we can choose an étale covering \(\{U_i \to U\}_{i \in I}\) with \(V_i\) affine. Since \(U\) is separated, we conclude that \(U_{i_0} \times_U \ldots \times_U U_{i_p}\) is always affine. Whence \(P(U_{i_0} \times_U \ldots \times_U U_{i_p})\) always. Hence \(P(U)\) holds. Choose a scheme \(U\) which is a disjoint union of affines and a surjective étale morphism \(U \to X\). Then \(U \times_X \ldots \times_X U\) (with \(p + 1\) factors) is a separated algebraic space étale over \(X\). Hence \(P(U \times_X \ldots \times_X U)\) by the above. We conclude that \(P(X)\) is true.
Fpqc topology
We briefly discuss the notion of an fpqc covering of algebraic spaces. Please compare with Topologies, Section 022A. We will show in Descent on Spaces, Proposition 04W8 that quasi-coherent sheaves descent along these.
Definition
Let \(S\) be a scheme, and let \(X\) be an algebraic space over \(S\). An fpqc covering of \(X\) is a family of morphisms \(\{f_i : X_i \to X\}_{i \in I}\) of algebraic spaces such that each \(f_i\) is flat and such that for every affine scheme \(Z\) and morphism \(h : Z \to X\) there exists a standard fpqc covering \(\{g_j : Z_j \to Z\}_{j = 1, \ldots, m}\) which refines the family \(\{X_i \times_X Z \to Z\}_{i \in I}\).
In other words, there exists indices \(i_1, \ldots, i_m \in I\) and morphisms \(h_j : Z_j \to X_{i_j}\) such that \(f_{i_j} \circ h_j = h \circ g_j\). Note that if \(X\) and all \(X_i\) are representable, this is the same as a fpqc covering of schemes by Topologies, Lemma 03LB.
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 0DFC. Let \(\{f_i : U_i \to U\}_{i \in I}\) be an fppf covering of algebraic spaces over \(S\). By definition this means that the \(f_i\) are flat which checks the first condition of Definition 03MQ. To check the second, let \(V \to U\) be a morphism with \(V\) affine. We may choose an étale covering \(\{V_{ij} \to V \times_U U_i\}\) with \(V_{ij}\) affine. Then the compositions \(f_{ij} : V_{ij} \to V \times_U U_i \to V\) are flat and locally of finite presentation as compositions of such (Morphisms of Spaces, Lemmas 03XQ, 03MN, 0467, and 0468). Hence these morphisms are open (Morphisms of Spaces, Lemma 042S) and we see that \(|V| = \bigcup_{i \in I} \bigcup_{j \in J_i} f_{ij}(|V_{ij}|)\) is an open covering of \(|V|\). Since \(|V|\) is quasi-compact, this covering has a finite refinement. Say \(V_{i_1j_1}, \ldots, V_{i_Nj_N}\) do the job. Then \(\{V_{i_kj_k} \to V\}_{k = 1, \ldots, N}\) is a standard fpqc covering of \(V\) refinining the family \(\{U_i \times_U V \to V\}\). This finishes the proof.
Lemma
Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\).
If \(X' \to X\) is an isomorphism then \(\{X' \to X\}\) is an fpqc covering of \(X\).
If \(\{X_i \to X\}_{i\in I}\) is an fpqc covering and for each \(i\) we have an fpqc covering \(\{X_{ij} \to X_i\}_{j\in J_i}\), then \(\{X_{ij} \to X\}_{i \in I, j\in J_i}\) is an fpqc covering.
If \(\{X_i \to X\}_{i\in I}\) is an fpqc covering and \(X' \to X\) is a morphism of algebraic spaces then \(\{X' \times_X X_i \to X'\}_{i\in I}\) is an fpqc covering.
Proof
Part (1) is clear. Consider \(g : X' \to X\) and \(\{X_i \to X\}_{i\in I}\) an fpqc covering as in (3). By Morphisms of Spaces, Lemma 03MO the morphisms \(X' \times_X X_i \to X'\) are flat. If \(h' : Z \to X'\) is a morphism from an affine scheme towards \(X'\), then set \(h = g \circ h' : Z \to X\). The assumption on \(\{X_i \to X\}_{i\in I}\) means there exists a standard fpqc covering \(\{Z_j \to Z\}_{j = 1, \ldots, n}\) and morphisms \(Z_j \to X_{i(j)}\) covering \(h\) for certain \(i(j) \in I\). By the universal property of the fibre product we obtain morphisms \(Z_j \to X' \times_X X_{i(j)}\) over \(h'\) also. Hence \(\{X' \times_X X_i \to X'\}_{i\in I}\) is an fpqc covering. This proves (3).
Let \(\{X_i \to X\}_{i\in I}\) and \(\{X_{ij} \to X_i\}_{j\in J_i}\) be as in (2). Let \(h : Z \to X\) be a morphism from an affine scheme towards \(X\). By assumption there exists a standard fpqc covering \(\{Z_j \to Z\}_{j = 1, \ldots, n}\) and morphisms \(h_j : Z_j \to X_{i(j)}\) covering \(h\) for some indices \(i(j) \in I\). By assumption there exist standard fpqc coverings \(\{Z_{j, l} \to Z_j\}_{l = 1, \ldots, n(j)}\) and morphisms \(Z_{j, l} \to X_{i(j)j(l)}\) covering \(h_j\) for some indices \(j(l) \in J_{i(j)}\). By Topologies, Lemma 03LA the family \(\{Z_{j, l} \to Z\}\) is a standard fpqc covering. Hence we conclude that \(\{X_{ij} \to X\}_{i \in I, j\in J_i}\) is an fpqc covering.
Lemma
Let \(S\) be a scheme, and let \(X\) be an algebraic space over \(S\). Suppose that \(\{f_i : X_i \to X\}_{i \in I}\) is a family of morphisms of algebraic spaces with target \(X\). Let \(U \to X\) be a surjective étale morphism from a scheme towards \(X\). Then \(\{f_i : X_i \to X\}_{i \in I}\) is an fpqc covering of \(X\) if and only if \(\{U \times_X X_i \to U\}_{i \in I}\) is an fpqc covering of \(U\).
Proof
If \(\{X_i \to X\}_{i \in I}\) is an fpqc covering, then so is \(\{U \times_X X_i \to U\}_{i \in I}\) by Lemma 03MR. Assume that \(\{U \times_X X_i \to U\}_{i \in I}\) is an fpqc covering. Let \(h : Z \to X\) be a morphism from an affine scheme towards \(X\). Then we see that \(U \times_X Z \to Z\) is a surjective étale morphism of schemes, in particular open. Hence we can find finitely many affine opens \(W_1, \ldots, W_t\) of \(U \times_X Z\) whose images cover \(Z\). For each \(j\) we may apply the condition that \(\{U \times_X X_i \to U\}_{i \in I}\) is an fpqc covering to the morphism \(W_j \to U\), and obtain a standard fpqc covering \(\{W_{jl} \to W_j\}\) which refines \(\{W_j \times_X X_i \to W_j\}_{i \in I}\). Hence \(\{W_{jl} \to Z\}\) is a standard fpqc covering of \(Z\) (see Topologies, Lemma 03LA) which refines \(\{Z \times_X X_i \to X\}\) and we win.
Lemma
Let \(S\) be a scheme, and let \(X\) be an algebraic space over \(S\). Suppose that \(\mathcal{U} = \{f_i : X_i \to X\}_{i \in I}\) is an fpqc covering of \(X\). Then there exists a refinement \(\mathcal{V} = \{g_i : T_i \to X\}\) of \(\mathcal{U}\) which is an fpqc covering such that each \(T_i\) is a scheme.
Proof
Omitted. Hint: For each \(i\) choose a scheme \(T_i\) and a surjective étale morphism \(T_i \to X_i\). Then check that \(\{T_i \to X\}\) is an fpqc covering.
To be continued...