## 38.34 The h topology

For us, loosely speaking, an h sheaf is a sheaf for the Zariski topology which satisfies the sheaf property for surjective proper morphisms of finite presentation, see Lemma 38.34.16. However, it may be worth pointing out that the definition of the h topology on the category of schemes depends on the reference.

Voevodsky initially defined an h covering to be a finite collection of finite type morphisms which are jointly universally submersive (Morphisms, Definition 29.24.1). See [Definition 3.1.2, Voevodsky]. This definition works best if the underlying category of schemes is restricted to all schemes of finite type over a fixed Noetherian base scheme. In this setting, Voevodsky relates h coverings to ph coverings. The ph topology is generated by Zariski coverings and proper surjective morphisms. See Topologies, Section 34.8 for more information.

In Topologies, Section 34.10 we study the V topology. A quasi-compact morphism $X \to Y$ defines a V covering, if any specialization of points of $Y$ is the image of a specialization of points in $X$ and the same is true after any base change (Topologies, Lemma 34.10.13). In this case $X \to Y$ is universally submersive (Topologies, Lemma 34.10.14). It turns out the notion of a V covering is a good replacement for “families of morphisms with fixed target which are jointly universally submersive” when working with non-Noetherian schemes.

Our approach will be to first prove the equivalence between ph covers and V coverings for (possibly infinite) families of morphisms which are locally of finite presentation. We will then use these families as our notion of h coverings in the Stacks project. For Noetherian schemes and finite families these coverings match those in Voevodsky's definition, see Lemma 38.34.3. On the category of schemes of finite presentation over a fixed quasi-compact and quasi-separated scheme $S$ these coverings determine the same topology as the one in [Definition 2.7, Witt-Grass].

Lemma 38.34.1. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be a family of morphisms of schemes with fixed target with $f_ i$ locally of finite presentation for all $i$. The following are equivalent

1. $\{ X_ i \to X\}$ is a ph covering, and

2. $\{ X_ i \to X\}$ is a V covering.

Proof. Let $U \subset X$ be affine open. Looking at Topologies, Definitions 34.8.4 and 34.10.7 it suffices to show that the base change $\{ X_ i \times _ X U \to U\}$ can be refined by a standard ph covering if and only if it can be refined by a standard V covering. Thus we may assume $X$ is affine and we have to show $\{ X_ i \to X\}$ can be refined by a standard ph covering if and only if it can be refined by a standard V covering. Since a standard ph covering is a standard V covering, see Topologies, Lemma 34.10.3 it suffices to prove the other implication.

Assume $X$ is affine and assume $\{ f_ i : X_ i \to X\} _{i \in I}$ can be refined by a standard V covering $\{ g_ j : Y_ j \to X\} _{j = 1, \ldots , m}$. For each $j$ choose an $i_ j$ and a morphism $h_ j : Y_ j \to X_{i_ j}$ such that $g_ j = f_{i_ j} \circ h_ j$. Since $Y_ j$ is affine hence quasi-compact, for each $j$ we can find finitely many affine opens $U_{j, k} \subset X_{i_ j}$ such that $\mathop{\mathrm{Im}}(h_ j) \subset \bigcup U_{j, k}$. Then $\{ U_{j, k} \to X\} _{j, k}$ refines $\{ X_ i \to X\}$ and is a standard V covering (as it is a finite family of morphisms of affines and it inherits the lifting property for valuation rings from the corresponding property of $\{ Y_ j \to X\}$). Thus we reduce to the case discussed in the next paragraph.

Assume $\{ f_ i : X_ i \to X\} _{i = 1, \ldots , n}$ is a standard V covering with $f_ i$ of finite presentation. We have to show that $\{ X_ i \to X\}$ can be refined by a standard ph covering. Choose a generic flatness stratification

$X = S \supset S_0 \supset S_1 \supset \ldots \supset S_ t = \emptyset$

as in Lemma 38.21.4 for the finitely presented morphism

$\coprod \nolimits _{i = 1, \ldots , n} f_ i : \coprod \nolimits _{i = 1, \ldots , n} X_ i \longrightarrow X$

of affines. We are going to use all the properties of the stratification without further mention. By construction the base change of each $f_ i$ to $U_ k = S_ k \setminus S_{k + 1}$ is flat. Denote $Y_ k$ the scheme theoretic closure of $U_ k$ in $S_ k$. Since $U_ k \to S_ k$ is a quasi-compact open immersion (see Properties, Lemma 28.24.1), we see that $U_ k \subset Y_ k$ is a quasi-compact dense (and scheme theoretically dense) open immersion, see Morphisms, Lemma 29.6.3. The morphism $\coprod _{k = 0, \ldots , t - 1} Y_ k \to X$ is finite surjective, hence $\{ Y_ k \to X\}$ is a standard ph covering and hence a standard V covering (see above). By the transitivity property of standard V coverings (Topologies, Lemma 34.10.5) it suffices to show that the pullback of the covering $\{ X_ i \to X\}$ to each $Y_ k$ can be refined by a standard V covering. This reduces us to the case described in the next paragraph.

Assume $\{ f_ i : X_ i \to X\} _{i = 1, \ldots , n}$ is a standard V covering with $f_ i$ of finite presentation and there is a dense quasi-compact open $U \subset X$ such that $X_ i \times _ X U \to U$ is flat. By Theorem 38.30.7 there is a $U$-admissible blowup $X' \to X$ such that the strict transform $f'_ i : X'_ i \to X'$ of $f_ i$ is flat. Observe that the projective (hence closed) morphism $X' \to X$ is surjective as $U \subset X$ is dense and as $U$ is identified with an open of $X'$. After replacing $X'$ by a further $U$-admissible blowup if necessary, we may also assume $U \subset X'$ is scheme theoretically dense (see Remark 38.30.1). Hence for every point $x \in X'$ there is a valuation ring $V$ and a morphism $g : \mathop{\mathrm{Spec}}(V) \to X'$ such that the generic point of $\mathop{\mathrm{Spec}}(V)$ maps into $U$ and the closed point of $\mathop{\mathrm{Spec}}(V)$ maps to $x$, see Morphisms, Lemma 29.6.5. Since $\{ X_ i \to X\}$ is a standard V covering, we can choose an extension of valuation rings $V \subset W$, an index $i$, and a morphism $\mathop{\mathrm{Spec}}(W) \to X_ i$ such that the diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[d] \ar[rr] & & X_ i \ar[d] \\ \mathop{\mathrm{Spec}}(V) \ar[r] & X' \ar[r] & X }$

is commutative. Since $X'_ i \subset X' \times _ X X_ i$ is a closed subscheme containing the open $U \times _ X X_ i$, since $\mathop{\mathrm{Spec}}(W)$ is an integral scheme, and since the induced morphism $h : \mathop{\mathrm{Spec}}(W) \to X' \times _ X X_ i$ maps the generic point of $\mathop{\mathrm{Spec}}(W)$ into $U \times _ X X_ i$, we conclude that $h$ factors through the closed subscheme $X'_ i \subset X' \times _ X X_ i$. We conclude that $\{ f'_ i : X'_ i \to X'\}$ is a V covering. In particular, $\coprod f'_ i$ is surjective. In particular $\{ X'_ i \to X'\}$ is an fppf covering. Since an fppf covering is a ph covering (More on Morphisms, Lemma 37.48.7), we can find a standard ph covering $\{ Y_ j \to X'\}$ refining $\{ X'_ i \to X\}$. Say this covering is given by a proper surjective morphism $Y \to X'$ and a finite affine open covering $Y = \bigcup Y_ j$. Then the composition $Y \to X$ is proper surjective and we conclude that $\{ Y_ j \to X\}$ is a standard ph covering. This finishes the proof. $\square$

Here is our definition.

Definition 38.34.2. Let $T$ be a scheme. A h covering of $T$ is a family of morphisms $\{ f_ i : T_ i \to T\} _{i \in I}$ such that each $f_ i$ is locally of finite presentation and one of the equivalent conditions of Lemma 38.34.1 is satisfied.

For Noetherian schemes we recover Voevodsky's notion.

Lemma 38.34.3. Let $X$ be a Noetherian scheme. Let $\{ X_ i \to X\} _{i \in I}$ be a finite family of finite type morphisms. The following are equivalent

1. $\coprod _{i \in I} X_ i \to X$ is universally submersive (Morphisms, Definition 29.24.1), and

2. $\{ X_ i \to X\} _{i \in I}$ is an h covering.

Proof. The implication (2) $\Rightarrow$ (1) follows from the more general Topologies, Lemma 34.10.14 and our definition of h covers. Assume $\coprod X_ i \to X$ is universally submersive. We will show that $\{ X_ i \to X\}$ can be refined by a ph covering; this will suffice by Topologies, Lemma 34.8.7 and our definition of h coverings. The argument will be the same as the one used in the proof of Lemma 38.34.1.

Choose a generic flatness stratification

$X = S \supset S_0 \supset S_1 \supset \ldots \supset S_ t = \emptyset$

as in Lemma 38.21.4 for the finitely presented morphism

$\coprod \nolimits _{i = 1, \ldots , n} f_ i : \coprod \nolimits _{i = 1, \ldots , n} X_ i \longrightarrow X$

We are going to use all the properties of the stratification without further mention. By construction the base change of each $f_ i$ to $U_ k = S_ k \setminus S_{k + 1}$ is flat. Denote $Y_ k$ the scheme theoretic closure of $U_ k$ in $S_ k$. Since $U_ k \to S_ k$ is a quasi-compact open immersion (all schemes in this paragraph are Noetherian), we see that $U_ k \subset Y_ k$ is a quasi-compact dense (and scheme theoretically dense) open immersion, see Morphisms, Lemma 29.6.3. The morphism $\coprod _{k = 0, \ldots , t - 1} Y_ k \to X$ is finite surjective, hence $\{ Y_ k \to X\}$ is a ph covering. By the transitivity property of ph coverings (Topologies, Lemma 34.8.8) it suffices to show that the pullback of the covering $\{ X_ i \to X\}$ to each $Y_ k$ can be refined by a ph covering. This reduces us to the case described in the next paragraph.

Assume $\coprod X_ i \to X$ is universally submersive and there is a dense open $U \subset X$ such that $X_ i \times _ X U \to U$ is flat for all $i$. By Theorem 38.30.7 there is a $U$-admissible blowup $X' \to X$ such that the strict transform $f'_ i : X'_ i \to X'$ of $f_ i$ is flat for all $i$. Observe that the projective (hence closed) morphism $X' \to X$ is surjective as $U \subset X$ is dense and as $U$ is identified with an open of $X'$. After replacing $X'$ by a further $U$-admissible blowup if necessary, we may also assume $U \subset X'$ is dense (see Remark 38.30.1). Hence for every point $x \in X'$ there is a discrete valuation ring $A$ and a morphism $g : \mathop{\mathrm{Spec}}(A) \to X'$ such that the generic point of $\mathop{\mathrm{Spec}}(A)$ maps into $U$ and the closed point of $\mathop{\mathrm{Spec}}(A)$ maps to $x$, see Limits, Lemma 32.15.1. Set

$W = \mathop{\mathrm{Spec}}(A) \times _ X \coprod X_ i = \coprod \mathop{\mathrm{Spec}}(A) \times _ X X_ i$

Since $\coprod X_ i \to X$ is universally submersive, there is a specialization $w' \leadsto w$ in $W$ such that $w'$ maps to the generic point of $\mathop{\mathrm{Spec}}(A)$ and $w$ maps to the closed point of $\mathop{\mathrm{Spec}}(A)$. (If not, then the closed fibre of $W \to \mathop{\mathrm{Spec}}(A)$ is stable under generalizations, hence open, which contradicts the fact that $W \to \mathop{\mathrm{Spec}}(A)$ is submersive.) Say $w' \in \mathop{\mathrm{Spec}}(A) \times _ X X_ i$ so of course $w \in \mathop{\mathrm{Spec}}(A) \times _ X X_ i$ as well. Let $x'_ i \leadsto x_ i$ be the image of $w' \leadsto w$ in $X' \times _ X X_ i$. Since $x'_ i \in X'_ i$ and since $X'_ i \subset X' \times _ X X_ i$ is a closed subscheme we see that $x_ i \in X'_ i$. Since $x_ i$ maps to $x \in X'$ we conclude that $\coprod X'_ i \to X'$ is surjective! In particular $\{ X'_ i \to X'\}$ is an fppf covering. But an fppf covering is a ph covering (More on Morphisms, Lemma 37.48.7). Since $X' \to X$ is proper surjective, we conclude that $\{ X'_ i \to X\}$ is a ph covering and the proof is complete. $\square$

The following lemma and [Theorem 8.4, rydh_descent] shows our definition agrees with (or at least is closely related to) the definition in the paper by David Rydh. We restrict to affine base for simplicity.

Lemma 38.34.4. Let $X$ be an affine scheme. Let $\{ X_ i \to X\} _{i \in I}$ be an h covering. Then there exists a surjective proper morphism

$Y \longrightarrow X$

of finite presentation (!) and a finite affine open covering $Y = \bigcup _{j = 1, \ldots , m} Y_ j$ such that $\{ Y_ j \to X\} _{j = 1, \ldots , m}$ refines $\{ X_ i \to X\} _{i \in I}$.

Proof. By assumption there exists a proper surjective morphism $Y \to X$ and a finite affine open covering $Y = \bigcup _{j = 1, \ldots , m} Y_ j$ such that $\{ Y_ j \to X\} _{j = 1, \ldots , m}$ refines $\{ X_ i \to X\} _{i \in I}$. This means that for each $j$ there is an index $i_ j \in I$ and a morphism $h_ j : Y_ j \to X_{i_ j}$ over $X$. See Definition 38.34.2 and Topologies, Definition 34.8.4. The problem is that we don't know that $Y \to X$ is of finite presentation. By Limits, Lemma 32.13.2 we can write

$Y = \mathop{\mathrm{lim}}\nolimits Y_\lambda$

as a directed limit of schemes $Y_\lambda$ proper and of finite presentation over $X$ such that the morphisms $Y \to Y_\lambda$ and the the transition morphisms are closed immersions. Observe that each $Y_\lambda \to X$ is surjective. By Limits, Lemma 32.4.11 we can find a $\lambda$ and quasi-compact opens $Y_{\lambda , j} \subset Y_\lambda$, $j = 1, \ldots , m$ covering $Y_\lambda$ and restricting to $Y_ j$ in $Y$. Then $Y_ j = \mathop{\mathrm{lim}}\nolimits Y_{\lambda , j}$. After increasing $\lambda$ we may assume $Y_{\lambda , j}$ is affine for all $j$, see Limits, Lemma 32.4.13. Finally, since $X_ i \to X$ is locally of finite presentation we can use the functorial characterization of morphisms which are locally of finite presentation (Limits, Proposition 32.6.1) to find a $\lambda$ such that for each $j$ there is a morphism $h_{\lambda , j} : Y_{\lambda , j} \to X_{i_ j}$ whose restriction to $Y_ j$ is the morphism $h_ j$ chosen above. Thus $\{ Y_{\lambda , j} \to X\}$ refines $\{ X_ i \to X\}$ and the proof is complete. $\square$

We return to the development of the general theory of h coverings.

Lemma 38.34.5. An fppf covering is a h covering. Hence syntomic, smooth, étale, and Zariski coverings are h coverings as well.

Proof. This is true because in an fppf covering the morphisms are required to be locally of finite presentation and because fppf coverings are ph covering, see More on Morphisms, Lemma 37.48.7. The second statement follows from the first and Topologies, Lemma 34.7.2. $\square$

Lemma 38.34.6. Let $f : Y \to X$ be a surjective proper morphism of schemes which is of finite presentation. Then $\{ Y \to X\}$ is an h covering.

Lemma 38.34.7. 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 presentation for all $i$. The following are equivalent

1. $\{ T_ i \to T\} _{i \in I}$ is an h covering,

2. there is an h covering which refines $\{ T_ i \to T\} _{i \in I}$, and

3. $\{ \coprod _{i \in I} T_ i \to T\}$ is an h covering.

Proof. This follows from the analogous statement for ph coverings (Topologies, Lemma 34.8.7) or from the analogous statement for V coverings (Topologies, Lemma 34.10.8). $\square$

Next, we show that our notion of an h covering satisfies the conditions of Sites, Definition 7.6.2.

Lemma 38.34.8. Let $T$ be a scheme.

1. If $T' \to T$ is an isomorphism then $\{ T' \to T\}$ is an h covering of $T$.

2. If $\{ T_ i \to T\} _{i\in I}$ is an h covering and for each $i$ we have an h covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is an h covering.

3. If $\{ T_ i \to T\} _{i\in I}$ is an h covering and $T' \to T$ is a morphism of schemes then $\{ T' \times _ T T_ i \to T'\} _{i\in I}$ is an h covering.

Proof. Follows immediately from the corresponding statement for either ph or V coverings (Topologies, Lemma 34.8.8 or 34.10.9) and the fact that the class of morphisms which are locally of finite presentation is preserved under base change and composition. $\square$

Next, we define the big h sites we will work with in the Stacks project. It makes sense to read the general discussion in Topologies, Section 34.2 before proceeding.

Definition 38.34.9. A big h site is any site $\mathit{Sch}_ h$ as in Sites, Definition 7.6.2 constructed as follows:

1. Choose any set of schemes $S_0$, and any set of h coverings $\text{Cov}_0$ among these schemes.

2. As underlying category take any category $\mathit{Sch}_\alpha$ constructed as in Sets, Lemma 3.9.2 starting with the set $S_0$.

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

See the remarks following Topologies, Definition 34.3.5 for motivation and explanation regarding the definition of big sites.

Definition 38.34.10. Let $T$ be an affine scheme. A standard h covering of $T$ is a family $\{ f_ i : T_ i \to T\} _{i = 1, \ldots , n}$ with each $T_ i$ affine, with $f_ i$ of finite presentation satisfying either of the following equivalent conditions: (1) $\{ U_ i \to U\}$ can be refined by a standard ph covering or (2) $\{ U_ i \to U\}$ is a V covering.

The equivalence of the conditions follows from Lemma 38.34.1, Topologies, Definition 34.8.4, and Lemma 34.8.7.

Before we continue with the introduction of the big h site of a scheme $S$, let us point out that the topology on a big h site $\mathit{Sch}_ h$ is in some sense induced from the h topology on the category of all schemes.

Lemma 38.34.11. Let $\mathit{Sch}_ h$ be a big h site as in Definition 38.34.9. Let $T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_ h)$. Let $\{ T_ i \to T\} _{i \in I}$ be an arbitrary h covering of $T$.

1. There exists a covering $\{ U_ j \to T\} _{j \in J}$ of $T$ in the site $\mathit{Sch}_ h$ which refines $\{ T_ i \to T\} _{i \in I}$.

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

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

Proof. Omitted. Hint: this is exactly the same as the proof of Topologies, Lemma 34.8.10. $\square$

Definition 38.34.12. Let $S$ be a scheme. Let $\mathit{Sch}_ h$ be a big h site containing $S$.

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

2. The big affine h site of $S$, denoted $(\textit{Aff}/S)_ h$, is the full subcategory of $(\mathit{Sch}/S)_ h$ whose objects are affine $U/S$. A covering of $(\textit{Aff}/S)_ h$ is any covering $\{ U_ i \to U\}$ of $(\mathit{Sch}/S)_ h$ which is a standard h covering.

We explicitly state that the big affine h site is a site.

Lemma 38.34.13. Let $S$ be a scheme. Let $\mathit{Sch}_ h$ be a big h site containing $S$. Then $(\textit{Aff}/S)_ h$ is a site.

Proof. Reasoning as in the proof of Topologies, Lemma 34.4.9 it suffices to show that the collection of standard h coverings satisfies properties (1), (2) and (3) of Sites, Definition 7.6.2. This is clear since for example, given a standard h covering $\{ T_ i \to T\} _{i\in I}$ and for each $i$ a standard h covering $\{ T_{ij} \to T_ i\} _{j \in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a h covering (Lemma 38.34.8), $\bigcup _{i\in I} J_ i$ is finite and each $T_{ij}$ is affine. Thus $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a standard h covering. $\square$

Lemma 38.34.14. Let $S$ be a scheme. Let $\mathit{Sch}_ h$ be a big h site containing $S$. The underlying categories of the sites $\mathit{Sch}_ h$, $(\mathit{Sch}/S)_ h$, and $(\textit{Aff}/S)_ h$ have fibre products. In each case the obvious functor into the category $\mathit{Sch}$ of all schemes commutes with taking fibre products. The category $(\mathit{Sch}/S)_ h$ has a final object, namely $S/S$.

Proof. For $\mathit{Sch}_ h$ 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 $U, V, W \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_ h)$. The fibre product $V \times _ U W$ in $\mathit{Sch}_ h$ 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 $(\mathit{Sch}/S)_ h$. This proves the result for $(\mathit{Sch}/S)_ h$. If $U, V, W$ are affine, so is $V \times _ U W$ and hence the result for $(\textit{Aff}/S)_ h$. $\square$

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

Lemma 38.34.15. Let $S$ be a scheme. Let $\mathit{Sch}_ h$ be a big h site containing $S$. The functor $(\textit{Aff}/S)_ h \to (\mathit{Sch}/S)_ h$ is cocontinuous and induces an equivalence of topoi from $\mathop{\mathit{Sh}}\nolimits ((\textit{Aff}/S)_ h)$ to $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_ h)$.

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 $u : (\textit{Aff}/S)_ h \to (\mathit{Sch}/S)_ h$. Being cocontinuous follows because any h covering of $T/S$, $T$ affine, can be refined by a standard h covering for example by Lemma 38.34.4. Hence (1) holds. We see $u$ is continuous simply because a standard h covering is a h 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 h covering). $\square$

Lemma 38.34.16. Let $\mathcal{F}$ be a presheaf on $(\mathit{Sch}/S)_ h$. Then $\mathcal{F}$ is a sheaf if and only if

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

2. if $f : V \to U$ is proper, surjective, and of finite presentation, 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

1. 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 covering in $(\mathit{Sch}/S)_ h$. 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 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 $V_ j \to U_ j$ of finite presentation and a finite affine open covering $V_ j = \bigcup V_{jk}$ such that $\{ V_{jk} \to U_ j\}$ refines $\{ T_ i \times _ T U_ j \to U_ j\}$. See Lemma 38.34.4. 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 $(\mathit{Sch}/S)_ h$ which is proper, surjective, and of finite presentation. 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 38.34.17. Let $\mathit{Sch}_ h$ be a big h site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_ h$. The functor

$u : (\mathit{Sch}/T)_ h \longrightarrow (\mathit{Sch}/S)_ h, \quad V/T \longmapsto V/S$

is cocontinuous, and has a continuous right adjoint

$v : (\mathit{Sch}/S)_ h \longrightarrow (\mathit{Sch}/T)_ h, \quad (U \to S) \longmapsto (U \times _ S T \to T).$

They induce the same morphism of topoi

$f_{big} : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/T)_ h) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_ h)$

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

Lemma 38.34.18. Given schemes $X$, $Y$, $Y$ in $(\mathit{Sch}/S)_ h$ 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 38.34.17. $\square$

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