Definition 72.8.1. 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 $\{ X_ i \times _ X U \to U\} _{i \in I}$.

## 72.8 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 72.8.7.

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 72.8.2. 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 72.7.2. 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 72.7.4). 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 72.8.3. 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 68.18.1) 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 72.8.4. 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 66.23.3 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 72.8.5. 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 72.2 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 72.8.1.

Having defined this, we can localize to get the ph site of an algebraic space.

Definition 72.8.6. Let $S$ be a scheme. Let $(\textit{Spaces}/S)_{ph}$ be as in Definition 72.8.5. 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 72.8.7. 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 72.8.8. 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 72.8.9. 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 72.8.8.
$\square$

Lemma 72.8.10. 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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