Definition 34.8.1. 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 $f_ j = f|_{U_ j}$.

## 34.8 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 34.8.15.

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 [Example 4.5, ph]. We return to (our version of) the h topology in More on Flatness, Section 38.34.

Before we can define the coverings in our topology we need to do a bit of work.

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 34.8.2. 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 34.8.1. Then $U \times _ T T' \to T'$ is proper surjective (Morphisms, Lemmas 29.9.4 and 29.41.5). Also, $U \times _ T T' = \bigcup _{j = 1, \ldots , m} U_ j \times _ T T'$ is an affine open covering. This concludes the proof.
$\square$

Lemma 34.8.3. 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 (3). Choose $U \to T$ proper surjective and $U = \bigcup _{j = 1, \ldots , m} U_ j$ as in Definition 34.8.1. Choose $W_ j \to U_ j$ proper surjective and $W_ j = \bigcup W_{ji}$ as in Definition 34.8.1. By Chow's lemma (Limits, Lemma 32.12.1) 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 29.6.) Let $Z_ j = U \setminus U_ j$ with reduced induced closed subscheme structure. Then

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

and the covering of $V$ by the affine opens

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 $T$). $\square$

Definition 34.8.4. 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 34.8.2.

Lemma 34.8.5. A Zariski covering is a ph covering^{1}.

**Proof.**
This is true because a Zariski covering of an affine scheme can be refined by a standard ph covering by Lemma 34.8.3.
$\square$

Lemma 34.8.6. Let $f : Y \to X$ be a surjective proper morphism of schemes. Then $\{ Y \to X\} $ is a ph covering.

**Proof.**
Omitted.
$\square$

Lemma 34.8.7. 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 34.8.4 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

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

is a standard ph covering (small detail omitted) refining $\{ U \times _ T T_ i \to U\} _{i \in I}$. Thus (1) holds. $\square$

Next, we show that this notion satisfies the conditions of Sites, Definition 7.6.2.

Lemma 34.8.8. 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 29.15.4. 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 $U' = U'_1 \cup \ldots \cup U'$ 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 34.8.2 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 34.8.3 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 29.15.3. 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

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 34.8.3 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. $\square$

Definition 34.8.9. A *big ph site* is any site $\mathit{Sch}_{ph}$ as in Sites, Definition 7.6.2 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 $\mathit{Sch}_\alpha $ constructed as in Sets, Lemma 3.9.2 starting with the set $S_0$.

Choose any set of coverings as in Sets, Lemma 3.11.1 starting with the category $\mathit{Sch}_\alpha $ and the class of ph coverings, and the set $\text{Cov}_0$ chosen above.

See the remarks following Definition 34.3.5 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 $\mathit{Sch}_{ph}$ is in some sense induced from the ph topology on the category of all schemes.

Lemma 34.8.10. Let $\mathit{Sch}_{ph}$ be a big ph site as in Definition 34.8.9. Let $T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{ph})$. 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 $\mathit{Sch}_{ph}$ 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 $\mathit{Sch}_{ph}$.

If $\{ T_ i \to T\} _{i \in I}$ is a Zariski covering, then it is tautologically equivalent to a covering of $\mathit{Sch}_{ph}$.

**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 34.8.5 and 34.8.8 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 3.9.9 we see that $W_ i$ is isomorphic to an object of $\mathit{Sch}_{ph}$. But then $T_ i$ as a finite type scheme over $W_ i$ is isomorphic to an object $V_ i$ of $\mathit{Sch}_{ph}$ by a second application of Sets, Lemma 3.9.9. 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 $\mathit{Sch}_{ph}$ by Sets, Lemma 3.9.9. 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 $\mathit{Sch}_{ph}$ by Sets, Lemma 3.9.9, and another application of Sets, Lemma 3.11.1 gives what we want.
$\square$

Definition 34.8.11. Let $S$ be a scheme. Let $\mathit{Sch}_{ph}$ be a big ph site containing $S$.

The

*big ph site of $S$*, denoted $(\mathit{Sch}/S)_{ph}$, is the site $\mathit{Sch}_{ph}/S$ introduced in Sites, Section 7.25.The

*big affine ph site of $S$*, denoted $(\textit{Aff}/S)_{ph}$, is the full subcategory of $(\mathit{Sch}/S)_{ph}$ whose objects are affine $U/S$. A covering of $(\textit{Aff}/S)_{ph}$ is any finite covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{ph}$ 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 7.6.2. Rather, the coverings in $(\textit{Aff}/S)_{ph}$ are those finite families $\{ U_ i \to U\} $ of finite type morphisms between affine objects of $(\mathit{Sch}/S)_{ph}$ which can be refined by a standard ph covering. We explicitly state and prove that the big affine ph site is a site.

Lemma 34.8.12. Let $S$ be a scheme. Let $\mathit{Sch}_{ph}$ be a big ph site containing $S$. Then $(\textit{Aff}/S)_{ph}$ is a site.

**Proof.**
Reasoning as in the proof of Lemma 34.4.9 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 7.6.2. 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 34.8.8), $\bigcup _{i\in I} J_ i$ is finite and each $T_{ij}$ is affine.
$\square$

Lemma 34.8.13. Let $S$ be a scheme. Let $\mathit{Sch}_{ph}$ be a big ph site containing $S$. The underlying categories of the sites $\mathit{Sch}_{ph}$, $(\mathit{Sch}/S)_{ph}$, and $(\textit{Aff}/S)_{ph}$ have fibre products. In each case the obvious functor into the category $\mathit{Sch}$ of all schemes commutes with taking fibre products. The category $(\mathit{Sch}/S)_{ph}$ has a final object, namely $S/S$.

**Proof.**
For $\mathit{Sch}_{ph}$ it is true by construction, see Sets, Lemma 3.9.9. Suppose we have $U \to S$, $V \to U$, $W \to U$ morphisms of schemes with $U, V, W \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{ph})$. The fibre product $V \times _ U W$ in $\mathit{Sch}_{ph}$ is a fibre product in $\mathit{Sch}$ 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 $(\mathit{Sch}/S)_{ph}$. This proves the result for $(\mathit{Sch}/S)_{ph}$. If $U, V, W$ are affine, so is $V \times _ U W$ and hence the result for $(\textit{Aff}/S)_{ph}$.
$\square$

Next, we check that the big affine site defines the same topos as the big site.

Lemma 34.8.14. Let $S$ be a scheme. Let $\mathit{Sch}_{ph}$ be a big ph site containing $S$. The functor $(\textit{Aff}/S)_{ph} \to (\mathit{Sch}/S)_{ph}$ is cocontinuous and induces an equivalence of topoi from $\mathop{\mathit{Sh}}\nolimits ((\textit{Aff}/S)_{ph})$ to $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{ph})$.

**Proof.**
The notion of a special cocontinuous functor is introduced in Sites, Definition 7.29.2. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 7.29.1. Denote the inclusion functor $u : (\textit{Aff}/S)_{ph} \to (\mathit{Sch}/S)_{ph}$. 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).
$\square$

Lemma 34.8.15. Let $\mathcal{F}$ be a presheaf on $(\mathit{Sch}/S)_{ph}$. 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 $(\mathit{Sch}/S)_{ph}$. 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 $(\mathit{Sch}/S)_{ph}$ 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 $\mathcal{F}(V)$. $\square$

Next, we establish some relationships between the topoi associated to these sites.

Lemma 34.8.16. Let $\mathit{Sch}_{ph}$ be a big ph site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{ph}$. The functor

is cocontinuous, and has a continuous right adjoint

They induce the same morphism of topoi

We have $f_{big}^{-1}(\mathcal{G})(U/T) = \mathcal{G}(U/S)$. 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 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 $V/S$ we have $\mathop{Mor}\nolimits _ S(u(U), V) = \mathop{Mor}\nolimits _ T(U, V \times _ S T)$ as desired. Thus we may apply Sites, Lemmas 7.22.1 and 7.22.2 to get the formula for $f_{big, *}$.
$\square$

Lemma 34.8.17. Given schemes $X$, $Y$, $Y$ in $(\mathit{Sch}/S)_{ph}$ 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 34.8.16.
$\square$

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