Definition 73.8.1. 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}.
73.8 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 73.8.7.
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 34.8.4.
Lemma 73.8.2. 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 73.7.2. 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 73.7.4). 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 37.48.7 (and the definition of ph coverings for schemes). Thus \{ X_ i \to X\} _{i \in I} is a ph covering by definition. \square
Lemma 73.8.3. 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 69.18.1) 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. \square
Lemma 73.8.4. 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 67.23.3 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 34.8.3 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. \square
Definition 73.8.5. Let S be a scheme. A big ph site (\textit{Spaces}/S)_{ph} is any site constructed as follows:
Choose a big ph site (\mathit{Sch}/S)_{ph} as in Topologies, Section 34.8.
As underlying category take the category \textit{Spaces}/S of algebraic spaces over S (see discussion in Section 73.2 why this is a set).
Choose any set of coverings as in Sets, Lemma 3.11.1 starting with the category \textit{Spaces}/S and the class of ph coverings of Definition 73.8.1.
Having defined this, we can localize to get the ph site of an algebraic space.
Definition 73.8.6. Let S be a scheme. Let (\textit{Spaces}/S)_{ph} be as in Definition 73.8.5. 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 7.25.
Here is the promised characterization of ph sheaves.
Lemma 73.8.7. 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. \square
Next, we establish some relationships between the topoi associated to these sites.
Lemma 73.8.8. Let S be a scheme. Let f : Y \to X be a morphism of algebraic spaces over S. The functor
is cocontinuous, and has a continuous right adjoint
They induce the same morphism of topoi
We have f_{big}^{-1}(\mathcal{G})(U/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 7.21.5 and 7.21.6 apply and we deduce the formula for f_{big}^{-1} and the existence of f_{big!}. Moreover, the functor v is a right adjoint because given U/T and V/X we have \mathop{\mathrm{Mor}}\nolimits _ X(u(U), V) = \mathop{\mathrm{Mor}}\nolimits _ Y(U, V \times _ X Y) as desired. Thus we may apply Sites, Lemmas 7.22.1 and 7.22.2 to get the formula for f_{big, *}. \square
Lemma 73.8.9. 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 73.8.8. \square
Lemma 73.8.10. 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
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. \square
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