Definition 34.3.1. Let $T$ be a scheme. A *Zariski covering of $T$* is a family of morphisms $\{ f_ i : T_ i \to T\} _{i \in I}$ of schemes such that each $f_ i$ is an open immersion and such that $T = \bigcup f_ i(T_ i)$.

## 34.3 The Zariski topology

This defines a (proper) class of coverings. Next, we show that this notion satisfies the conditions of Sites, Definition 7.6.2.

Lemma 34.3.2. Let $T$ be a scheme.

If $T' \to T$ is an isomorphism then $\{ T' \to T\} $ is a Zariski covering of $T$.

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

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

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

Lemma 34.3.3. Let $T$ be an affine scheme. Let $\{ T_ i \to T\} _{i \in I}$ be a Zariski covering of $T$. Then there exists a Zariski covering $\{ U_ j \to T\} _{j = 1, \ldots , m}$ which is a refinement of $\{ T_ i \to T\} _{i \in I}$ such that each $U_ j$ is a standard open of $T$, see Schemes, Definition 26.5.2. Moreover, we may choose each $U_ j$ to be an open of one of the $T_ i$.

**Proof.**
Follows as $T$ is quasi-compact and standard opens form a basis for its topology. This is also proved in Schemes, Lemma 26.5.1.
$\square$

Thus we define the corresponding standard coverings of affines as follows.

Definition 34.3.4. Compare Schemes, Definition 26.5.2. Let $T$ be an affine scheme. A *standard Zariski covering* of $T$ is a Zariski covering $\{ U_ j \to T\} _{j = 1, \ldots , m}$ with each $U_ j \to T$ inducing an isomorphism with a standard affine open of $T$.

Definition 34.3.5. A *big Zariski site* is any site $\mathit{Sch}_{Zar}$ as in Sites, Definition 7.6.2 constructed as follows:

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

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

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

It is shown in Sites, Lemma 7.8.8 that, after having chosen the category $\mathit{Sch}_\alpha $, the category of sheaves on $\mathit{Sch}_\alpha $ does not depend on the choice of coverings chosen in (3) above. In other words, the topos $\mathop{\mathit{Sh}}\nolimits (\mathit{Sch}_{Zar})$ only depends on the choice of the category $\mathit{Sch}_\alpha $. It is shown in Sets, Lemma 3.9.9 that these categories are closed under many constructions of algebraic geometry, e.g., fibre products and taking open and closed subschemes. We can also show that the exact choice of $\mathit{Sch}_\alpha $ does not matter too much, see Section 34.12.

Another approach would be to assume the existence of a strongly inaccessible cardinal and to define $\mathit{Sch}_{Zar}$ to be the category of schemes contained in a chosen universe with set of coverings the Zariski coverings contained in that same universe.

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

Lemma 34.3.6. Let $\mathit{Sch}_{Zar}$ be a big Zariski site as in Definition 34.3.5. Let $T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{Zar})$. Let $\{ T_ i \to T\} _{i \in I}$ be an arbitrary Zariski covering of $T$. There exists a covering $\{ U_ j \to T\} _{j \in J}$ of $T$ in the site $\mathit{Sch}_{Zar}$ which is tautologically equivalent (see Sites, Definition 7.8.2) to $\{ T_ i \to T\} _{i \in I}$.

**Proof.**
Since each $T_ i \to T$ is an open immersion, we see by Sets, Lemma 3.9.9 that each $T_ i$ is isomorphic to an object $V_ i$ of $\mathit{Sch}_{Zar}$. The covering $\{ V_ i \to T\} _{i \in I}$ is tautologically equivalent to $\{ T_ i \to T\} _{i \in I}$ (using the identity map on $I$ both ways). 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 $\mathit{Sch}_{Zar}$ by Sets, Lemma 3.11.1.
$\square$

Definition 34.3.7. Let $S$ be a scheme. Let $\mathit{Sch}_{Zar}$ be a big Zariski site containing $S$.

The

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

*small Zariski site of $S$*, which we denote $S_{Zar}$, is the full subcategory of $(\mathit{Sch}/S)_{Zar}$ whose objects are those $U/S$ such that $U \to S$ is an open immersion. A covering of $S_{Zar}$ is any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{Zar}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{Zar})$.The

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

It is not completely clear that the small Zariski site and the big affine Zariski site are sites. We check this now.

Lemma 34.3.8. Let $S$ be a scheme. Let $\mathit{Sch}_{Zar}$ be a big Zariski site containing $S$. Both $S_{Zar}$ and $(\textit{Aff}/S)_{Zar}$ are sites.

**Proof.**
Let us show that $S_{Zar}$ is a site. It is a category with a given set of families of morphisms with fixed target. Thus we have to show properties (1), (2) and (3) of Sites, Definition 7.6.2. Since $(\mathit{Sch}/S)_{Zar}$ is a site, it suffices to prove that given any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{Zar}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{Zar})$ we also have $U_ i \in \mathop{\mathrm{Ob}}\nolimits (S_{Zar})$. This follows from the definitions as the composition of open immersions is an open immersion.

Let us show that $(\textit{Aff}/S)_{Zar}$ is a site. Reasoning as above, it suffices to show that the collection of standard Zariski coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 7.6.2. Let $R$ be a ring. Let $f_1, \ldots , f_ n \in R$ generate the unit ideal. For each $i \in \{ 1, \ldots , n\} $ let $g_{i1}, \ldots , g_{in_ i} \in R_{f_ i}$ be elements generating the unit ideal of $R_{f_ i}$. Write $g_{ij} = f_{ij}/f_ i^{e_{ij}}$ which is possible. After replacing $f_{ij}$ by $f_ i f_{ij}$ if necessary, we have that $D(f_{ij}) \subset D(f_ i) \cong \mathop{\mathrm{Spec}}(R_{f_ i})$ is equal to $D(g_{ij}) \subset \mathop{\mathrm{Spec}}(R_{f_ i})$. Hence we see that the family of morphisms $\{ D(g_{ij}) \to \mathop{\mathrm{Spec}}(R)\} $ is a standard Zariski covering. From these considerations it follows that (2) holds for standard Zariski coverings. We omit the verification of (1) and (3). $\square$

Lemma 34.3.9. Let $S$ be a scheme. Let $\mathit{Sch}_{Zar}$ be a big Zariski site containing $S$. The underlying categories of the sites $\mathit{Sch}_{Zar}$, $(\mathit{Sch}/S)_{Zar}$, $S_{Zar}$, and $(\textit{Aff}/S)_{Zar}$ have fibre products. In each case the obvious functor into the category $\mathit{Sch}$ of all schemes commutes with taking fibre products. The categories $(\mathit{Sch}/S)_{Zar}$, and $S_{Zar}$ both have a final object, namely $S/S$.

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

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

Lemma 34.3.10. Let $S$ be a scheme. Let $\mathit{Sch}_{Zar}$ be a big Zariski site containing $S$. The functor $(\textit{Aff}/S)_{Zar} \to (\mathit{Sch}/S)_{Zar}$ is a special cocontinuous functor. Hence it induces an equivalence of topoi from $\mathop{\mathit{Sh}}\nolimits ((\textit{Aff}/S)_{Zar})$ to $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar})$.

**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 $u : (\textit{Aff}/S)_{Zar} \to (\mathit{Sch}/S)_{Zar}$. Being cocontinuous just means that any Zariski covering of $T/S$, $T$ affine, can be refined by a standard Zariski covering of $T$. This is the content of Lemma 34.3.3. Hence (1) holds. We see $u$ is continuous simply because a standard Zariski covering is a Zariski 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.
$\square$

Let us check that the notion of a sheaf on the small Zariski site corresponds to notion of a sheaf on $S$.

Lemma 34.3.11. The category of sheaves on $S_{Zar}$ is equivalent to the category of sheaves on the underlying topological space of $S$.

**Proof.**
We will use repeatedly that for any object $U/S$ of $S_{Zar}$ the morphism $U \to S$ is an isomorphism onto an open subscheme. Let $\mathcal{F}$ be a sheaf on $S$. Then we define a sheaf on $S_{Zar}$ by the rule $\mathcal{F}'(U/S) = \mathcal{F}(\mathop{\mathrm{Im}}(U \to S))$. For the converse, we choose for every open subscheme $U \subset S$ an object $U'/S \in \mathop{\mathrm{Ob}}\nolimits (S_{Zar})$ with $\mathop{\mathrm{Im}}(U' \to S) = U$ (here you have to use Sets, Lemma 3.9.9). Given a sheaf $\mathcal{G}$ on $S_{Zar}$ we define a sheaf on $S$ by setting $\mathcal{G}'(U) = \mathcal{G}(U'/S)$. To see that $\mathcal{G}'$ is a sheaf we use that for any open covering $U = \bigcup _{i \in I} U_ i$ the covering $\{ U_ i \to U\} _{i \in I}$ is combinatorially equivalent to a covering $\{ U_ j' \to U'\} _{j \in J}$ in $S_{Zar}$ by Sets, Lemma 3.11.1, and we use Sites, Lemma 7.8.4. Details omitted.
$\square$

From now on we will not make any distinction between a sheaf on $S_{Zar}$ or a sheaf on $S$. We will always use the procedures of the proof of the lemma to go between the two notions. Next, we establish some relationships between the topoi associated to these sites.

Lemma 34.3.12. Let $\mathit{Sch}_{Zar}$ be a big Zariski site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{Zar}$. The functor $T_{Zar} \to (\mathit{Sch}/S)_{Zar}$ is cocontinuous and induces a morphism of topoi

For a sheaf $\mathcal{G}$ on $(\mathit{Sch}/S)_{Zar}$ we have the formula $(i_ f^{-1}\mathcal{G})(U/T) = \mathcal{G}(U/S)$. The functor $i_ f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes with fibre products and equalizers.

**Proof.**
Denote the functor $u : T_{Zar} \to (\mathit{Sch}/S)_{Zar}$. In other words, given and open immersion $j : U \to T$ corresponding to an object of $T_{Zar}$ we set $u(U \to T) = (f \circ j : U \to S)$. This functor commutes with fibre products, see Lemma 34.3.9. Moreover, $T_{Zar}$ has equalizers (as any two morphisms with the same source and target are the same) and $u$ commutes with them. It is clearly cocontinuous. It is also continuous as $u$ transforms coverings to coverings and commutes with fibre products. Hence the lemma follows from Sites, Lemmas 7.21.5 and 7.21.6.
$\square$

Lemma 34.3.13. Let $S$ be a scheme. Let $\mathit{Sch}_{Zar}$ be a big Zariski site containing $S$. The inclusion functor $S_{Zar} \to (\mathit{Sch}/S)_{Zar}$ satisfies the hypotheses of Sites, Lemma 7.21.8 and hence induces a morphism of sites

and a morphism of topoi

such that $\pi _ S \circ i_ S = \text{id}$. Moreover, $i_ S = i_{\text{id}_ S}$ with $i_{\text{id}_ S}$ as in Lemma 34.3.12. In particular the functor $i_ S^{-1} = \pi _{S, *}$ is described by the rule $i_ S^{-1}(\mathcal{G})(U/S) = \mathcal{G}(U/S)$.

**Proof.**
In this case the functor $u : S_{Zar} \to (\mathit{Sch}/S)_{Zar}$, in addition to the properties seen in the proof of Lemma 34.3.12 above, also is fully faithful and transforms the final object into the final object. The lemma follows.
$\square$

Definition 34.3.14. In the situation of Lemma 34.3.13 the functor $i_ S^{-1} = \pi _{S, *}$ is often called the *restriction to the small Zariski site*, and for a sheaf $\mathcal{F}$ on the big Zariski site we denote $\mathcal{F}|_{S_{Zar}}$ this restriction.

With this notation in place we have for a sheaf $\mathcal{F}$ on the big site and a sheaf $\mathcal{G}$ on the big site that

Moreover, we have $(i_{S, *}\mathcal{G})|_{S_{Zar}} = \mathcal{G}$ and we have $(\pi _ S^{-1}\mathcal{G})|_{S_{Zar}} = \mathcal{G}$.

Lemma 34.3.15. Let $\mathit{Sch}_{Zar}$ be a big Zariski site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{Zar}$. 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 (details omitted; compare with proof of Lemma 34.3.12). 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 34.3.16. Let $\mathit{Sch}_{Zar}$ be a big Zariski site. Let $f : T \to S$ be a morphism in $\mathit{Sch}_{Zar}$.

We have $i_ f = f_{big} \circ i_ T$ with $i_ f$ as in Lemma 34.3.12 and $i_ T$ as in Lemma 34.3.13.

The functor $S_{Zar} \to T_{Zar}$, $(U \to S) \mapsto (U \times _ S T \to T)$ is continuous and induces a morphism of topoi

\[ f_{small} : \mathop{\mathit{Sh}}\nolimits (T_{Zar}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (S_{Zar}). \]The functors $f_{small}^{-1}$ and $f_{small, *}$ agree with the usual notions $f^{-1}$ and $f_*$ is we identify sheaves on $T_{Zar}$, resp. $S_{Zar}$ with sheaves on $T$, resp. $S$ via Lemma 34.3.11.

We have a commutative diagram of morphisms of sites

\[ \xymatrix{ T_{Zar} \ar[d]_{f_{small}} & (\mathit{Sch}/T)_{Zar} \ar[d]^{f_{big}} \ar[l]^{\pi _ T} \\ S_{Zar} & (\mathit{Sch}/S)_{Zar} \ar[l]_{\pi _ S} } \]so that $f_{small} \circ \pi _ T = \pi _ S \circ f_{big}$ as morphisms of topoi.

We have $f_{small} = \pi _ S \circ f_{big} \circ i_ T = \pi _ S \circ i_ f$.

**Proof.**
The equality $i_ f = f_{big} \circ i_ T$ follows from the equality $i_ f^{-1} = i_ T^{-1} \circ f_{big}^{-1}$ which is clear from the descriptions of these functors above. Thus we see (1).

Statement (2): See Sites, Example 7.14.2.

Part (3) follows because $\pi _ S$ and $\pi _ T$ are given by the inclusion functors and $f_{small}$ and $f_{big}$ by the base change functor $U \mapsto U \times _ S T$.

Statement (4) follows from (3) by precomposing with $i_ T$. $\square$

In the situation of the lemma, using the terminology of Definition 34.3.14 we have: for $\mathcal{F}$ a sheaf on the big Zariski site of $T$

This equality is clear from the commutativity of the diagram of sites of the lemma, since restriction to the small Zariski site of $T$, resp. $S$ is given by $\pi _{T, *}$, resp. $\pi _{S, *}$. A similar formula involving pullbacks and restrictions is false.

Lemma 34.3.17. Given schemes $X$, $Y$, $Y$ in $(\mathit{Sch}/S)_{Zar}$ and morphisms $f : X \to Y$, $g : Y \to Z$ we have $g_{big} \circ f_{big} = (g \circ f)_{big}$ and $g_{small} \circ f_{small} = (g \circ f)_{small}$.

**Proof.**
This follows from the simple description of pushforward and pullback for the functors on the big sites from Lemma 34.3.15. For the functors on the small sites this is Sheaves, Lemma 6.21.2 via the identification of Lemma 34.3.11.
$\square$

Lemma 34.3.18. Let $\mathit{Sch}_{Zar}$ be a big Zariski site. Consider a cartesian diagram

in $\mathit{Sch}_{Zar}$. Then $i_ g^{-1} \circ f_{big, *} = f'_{small, *} \circ (i_{g'})^{-1}$ and $g_{big}^{-1} \circ f_{big, *} = f'_{big, *} \circ (g'_{big})^{-1}$.

**Proof.**
Since the diagram is cartesian, we have for $U'/S'$ that $U' \times _{S'} T' = U' \times _ S T$. Hence both $i_ g^{-1} \circ f_{big, *}$ and $f'_{small, *} \circ (i_{g'})^{-1}$ send a sheaf $\mathcal{F}$ on $(\mathit{Sch}/T)_{Zar}$ to the sheaf $U' \mapsto \mathcal{F}(U' \times _{S'} T')$ on $S'_{Zar}$ (use Lemmas 34.3.12 and 34.3.16). The second equality can be proved in the same manner or can be deduced from the very general Sites, Lemma 7.28.1.
$\square$

We can think about a sheaf on the big Zariski site of $S$ as a collection of “usual” sheaves on all schemes over $S$.

Lemma 34.3.19. Let $S$ be a scheme contained in a big Zariski site $\mathit{Sch}_{Zar}$. A sheaf $\mathcal{F}$ on the big Zariski site $(\mathit{Sch}/S)_{Zar}$ is given by the following data:

for every $T/S \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{Zar})$ a sheaf $\mathcal{F}_ T$ on $T$,

for every $f : T' \to T$ in $(\mathit{Sch}/S)_{Zar}$ a map $c_ f : f^{-1}\mathcal{F}_ T \to \mathcal{F}_{T'}$.

These data are subject to the following conditions:

given any $f : T' \to T$ and $g : T'' \to T'$ in $(\mathit{Sch}/S)_{Zar}$ the composition $c_ g \circ g^{-1}c_ f$ is equal to $c_{f \circ g}$, and

if $f : T' \to T$ in $(\mathit{Sch}/S)_{Zar}$ is an open immersion then $c_ f$ is an isomorphism.

**Proof.**
Given a sheaf $\mathcal{F}$ on $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar})$ we set $\mathcal{F}_ T = i_ p^{-1}\mathcal{F}$ where $p : T \to S$ is the structure morphism. Note that $\mathcal{F}_ T(U) = \mathcal{F}(U'/S)$ for any open $U \subset T$, and $U' \to T$ an open immersion in $(\mathit{Sch}/T)_{Zar}$ with image $U$, see Lemmas 34.3.11 and 34.3.12. Hence given $f : T' \to T$ over $S$ and $U, U' \to T$ we get a canonical map $\mathcal{F}_ T(U) = \mathcal{F}(U'/S) \to \mathcal{F}(U'\times _ T T'/S) = \mathcal{F}_{T'}(f^{-1}(U))$ where the middle is the restriction map of $\mathcal{F}$ with respect to the morphism $U' \times _ T T' \to U'$ over $S$. The collection of these maps are compatible with restrictions, and hence define an $f$-map $c_ f$ from $\mathcal{F}_ T$ to $\mathcal{F}_{T'}$, see Sheaves, Definition 6.21.7 and the discussion surrounding it. It is clear that $c_{f \circ g}$ is the composition of $c_ f$ and $c_ g$, since composition of restriction maps of $\mathcal{F}$ gives restriction maps.

Conversely, given a system $(\mathcal{F}_ T, c_ f)$ as in the lemma we may define a presheaf $\mathcal{F}$ on $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar})$ by simply setting $\mathcal{F}(T/S) = \mathcal{F}_ T(T)$. As restriction mapping, given $f : T' \to T$ we set for $s \in \mathcal{F}(T)$ the pullback $f^*(s)$ equal to $c_ f(s)$ (where we think of $c_ f$ as an $f$-map again). The condition on the $c_ f$ guarantees that pullbacks satisfy the required functoriality property. We omit the verification that this is a sheaf. It is clear that the constructions so defined are mutually inverse. $\square$

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