The Stacks project

Proof. This is clear from the definitions and the fact that an open immersion is an étale morphism, see Morphisms, Lemma 29.36.9. $\square$

Proof. Omitted. $\square$

Proof. Omitted. $\square$

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 Lemma 34.4.3 the refinement $\{ T_{ij} \to T\} _{i \in I, j \in J_ i}$ is an étale 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 $\mathit{Sch}_{\acute{e}tale}$. But then $T_ i$ as a finite type scheme over $W_ i$ is isomorphic to an object $V_ i$ of $\mathit{Sch}_{\acute{e}tale}$ 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 $\mathit{Sch}_{\acute{e}tale}$ 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 $\mathit{Sch}_{\acute{e}tale}$ by Sets, Lemma 3.9.9, and another application of Sets, Lemma 3.11.1 gives what we want. $\square$

Proof. Let us show that $S_{\acute{e}tale}$ 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 $(\mathit{Sch}/S)_{\acute{e}tale}$ is a site, it suffices to prove that given any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{\acute{e}tale}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$ we also have $U_ i \in \mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$. This follows from the definitions as the composition of étale morphisms is an étale morphism.

Let us show that $(\textit{Aff}/S)_{\acute{e}tale}$ is a site. Reasoning as above, it suffices to show that the collection of standard étale coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 7.6.2. This is clear since for example, given a standard étale covering $\{ T_ i \to T\} _{i\in I}$ and for each $i$ we have a standard étale covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a standard étale covering because $\bigcup _{i\in I} J_ i$ is finite and each $T_{ij}$ is affine.

We omit the proof that $S_{affine, étale}$ is a site. $\square$

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

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 $u : (\textit{Aff}/S)_{\acute{e}tale}\to (\mathit{Sch}/S)_{\acute{e}tale}$. Being cocontinuous just means that any étale covering of $T/S$, $T$ affine, can be refined by a standard étale covering of $T$. This is the content of Lemma 34.4.4. Hence (1) holds. We see $u$ is continuous simply because a standard étale covering is a étale 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. $\square$

Proof. Omitted. Hint: compare with the proof of Lemma 34.4.11. $\square$

Proof. Denote the functor $u : T_{\acute{e}tale}\to (\mathit{Sch}/S)_{\acute{e}tale}$. In other words, given an étale morphism $j : U \to T$ corresponding to an object of $T_{\acute{e}tale}$ we set $u(U \to T) = (f \circ j : U \to S)$. This functor commutes with fibre products, see Lemma 34.4.10. Let $a, b : U \to V$ be two morphisms in $T_{\acute{e}tale}$. In this case the equalizer of $a$ and $b$ (in the category of schemes) is

which is a fibre product of schemes étale over $T$, hence étale over $T$. Thus $T_{\acute{e}tale}$ has equalizers 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. $\square$

Proof. In this case the functor $u : S_{\acute{e}tale}\to (\mathit{Sch}/S)_{\acute{e}tale}$, in addition to the properties seen in the proof of Lemma 34.4.13 above, also is fully faithful and transforms the final object into the final object. The lemma follows from Sites, Lemma 7.21.8. $\square$

Proof. The functor $u$ is cocontinuous, continuous and commutes with fibre products and equalizers (details omitted; compare with the proof of Lemma 34.4.13). 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 $f_{big, *}$. $\square$

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).

The functor $u : S_{\acute{e}tale}\to T_{\acute{e}tale}$, $u(U \to S) = (U \times _ S T \to T)$ transforms coverings into coverings and commutes with fibre products, see Lemma 34.4.3 (3) and 34.4.10. Moreover, both $S_{\acute{e}tale}$, $T_{\acute{e}tale}$ have final objects, namely $S/S$ and $T/T$ and $u(S/S) = T/T$. Hence by Sites, Proposition 7.14.7 the functor $u$ corresponds to a morphism of sites $T_{\acute{e}tale}\to S_{\acute{e}tale}$. This in turn gives rise to the morphism of topoi, see Sites, Lemma 7.15.2. The description of the pushforward is clear from these references.

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 functors $U \mapsto U \times _ S T$.

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

Proof. This follows from the simple description of pushforward and pullback for the functors on the big sites from Lemma 34.4.16. For the functors on the small sites this follows from the description of the pushforward functors in Lemma 34.4.17. $\square$

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 $(\mathit{Sch}/T)_{\acute{e}tale}$ to the sheaf $U' \mapsto \mathcal{F}(U' \times _{S'} T')$ on $S'_{\acute{e}tale}$ (use Lemmas 34.4.13 and 34.4.16). The second equality can be proved in the same manner or can be deduced from the very general Sites, Lemma 7.28.1. $\square$

Proof. This lemma follows from a purely sheaf theoretic statement discussed in Sites, Remark 7.26.7. We also give a direct proof in this case.

Given a sheaf $\mathcal{F}$ on $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$ 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 $U \to T$ in $T_{\acute{e}tale}$ see Lemma 34.4.13. Hence given $f : T' \to T$ over $S$ and $U \to T$ we get a canonical map $\mathcal{F}_ T(U) = \mathcal{F}(U/S) \to \mathcal{F}(U \times _ T T'/S) = \mathcal{F}_{T'}(U \times _ T T')$ 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 a map $c'_ f : \mathcal{F}_ T \to f_{small, *}\mathcal{F}_{T'}$ where $u : T_{\acute{e}tale}\to T'_{\acute{e}tale}$ is the base change functor associated to $f$. By adjunction of $f_{small, *}$ (see Sites, Section 7.13) with $f_{small}^{-1}$ this is the same as a map $c_ f : f_{small}^{-1}\mathcal{F}_ T \to \mathcal{F}_{T'}$. It is clear that $c'_{f \circ g}$ is the composition of $c'_ f$ and $f_{small, *}c'_ g$, since composition of restriction maps of $\mathcal{F}$ gives restriction maps, and this gives the desired relationship among $c_ f$, $c_ g$ and $c_{f \circ g}$.

Conversely, given a system $(\mathcal{F}_ T, c_ f)$ as in the lemma we may define a presheaf $\mathcal{F}$ on $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$ 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 a map $\mathcal{F}_ T \to f_{small, *}\mathcal{F}_{T'}$ 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. $\square$