## 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.

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

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 $S$.

1. If $X' \to X$ is an isomorphism then $\{ X' \to X\}$ is a ph covering of $X$.

2. 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.

3. 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:

1. Choose a big ph site $(\mathit{Sch}/S)_{ph}$ as in Topologies, Section 34.8.

2. 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).

3. 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

1. $\mathcal{F}$ satisfies the sheaf condition for étale coverings, and

2. 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

$u : (\textit{Spaces}/Y)_{ph} \longrightarrow (\textit{Spaces}/X)_{ph}, \quad V/Y \longmapsto V/X$

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

$f_{big} : \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/Y)_{ph}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{ph})$

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 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

$P(U_{i_0} \times _ U \ldots \times _ U U_{i_ p}) \text{ for all } p \geq 0,\ i_0, \ldots , i_ p \in I \Rightarrow P(U)$

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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).