Sheaves on small étale sites (possibly to be extended)

We follow Milne's book. I think Patrick Kelley for summarizing this material before when we studied together.

We have explained the definition of a site in this posting. The target example is $\textbf{Et}_{X},$ the site of étale maps into a scheme $X,$ whose maps between them are (étale) maps over $X.$

A site is a category $\mathcal{C}$ together with coverings for each of its object. A presheaf valued in $\textbf{Set}$ on $\mathcal{C}$ is a functor $\mathscr{F} : \mathcal{C}^{\mathrm{op}} \rightarrow \textbf{Set}.$ We may change the target category $\textbf{Set}$ into other categories to define analogous definitions. Maps between two presheaves are given by natural transformations, and thus, presheaves on $\mathcal{C}$ form a category.

A presheaf $\mathscr{F} : \mathcal{C}^{\mathrm{op}} \rightarrow \textbf{Set}$ is said to be a sheaf if it satsifies the following two extra axioms for every object $U$ of $\mathscr{C}.$

1. Given any covering $\{U_{i} \rightarrow U\}_{i \in I}$ of $U$ and $s, t \in \mathscr{F}(U),$ if $s|_{U_{i}} = t|_{U_{i}}$ for all maps $U_{i} \rightarrow U$ in the covering, then $s = t.$
2. Given any covering $\{U_{i} \rightarrow U\}_{i \in I}$ of $U$ and $s_{i} \in \mathscr{F}(U_{i})$ for each map $U_{i} \rightarrow U$ in the covering, if $s_{i}|_{U_{i} \times_{U} U_{j}} = s_{j}|_{U_{i} \times_{U} U_{j}}$ for all maps $U_{i} \rightarrow U$ and $U_{j} \rightarrow U$ in the covering, then there is $s \in \mathscr{F}(U)$ such that $s|_{U_{i}} = s_{i}$ for all maps $U_{i} \rightarrow U$ in the covering.

Note that sheaves on $\mathcal{C}$ form a full subcategory of the category of presheaves on $\mathcal{C}.$ We now focus on the case $\mathcal{C} = \textbf{Et}_{X}$ for the following proposition, which we learn from Milne (II. Proposition 1.5). We write $\textbf{Zar}_{X}$ to mean the small Zariski site of a scheme $X.$ We will take the values of sheaves in $\textbf{Ab}.$

Proposition. Let $\mathscr{F}$ be a presheaf on $\textbf{Et}_{X}.$ Then $\mathscr{F}$ is a sheaf if and only if the following two properties are satisfied.

1. For any object $U$ of $\textbf{Et}_{X},$ the restriction of $\mathscr{F}$ to $\textbf{Zar}_{U}$ is a sheaf.
2. For any covering $\{V \rightarrow U\},$ consisting of a single map of $\textbf{Et}_{X}$ such that $V$ and $U$ are affine, the sequence $\mathscr{F}(U) \rightarrow \mathscr{F}(V) \rightrightarrows \mathscr{F}(V \times_{U} V)$ given by restrictions is an equalizer.

Sketch of proof. It is evident that sheaf axioms of $\mathscr{F}$ implies the two given conditions, noting that open embeddings are of étale. 

For the converse, our goal is to check that the sequence $$\mathscr{F}(U) \rightarrow \prod_{i \in I}\mathscr{F}(U_{i}) \rightrightarrows \prod_{i,j \in I}\mathscr{F}(U_{i} \times_{U} U_{j})$$ given by the restrictions is an equalizer.

Fix any covering $\{U_{i} \rightarrow U\}_{i \in I}$ in $\textbf{Et}_{X}$ and consider $V := \bigsqcup_{i \in I}U_{i}.$ The first condition implies that we have $$\mathscr{F}(V) \simeq \prod_{i \in I}\mathscr{F}(U_{i})$$ induced by the restrictions. Note that we have $$V \times_{U} V \simeq \bigsqcup_{i, j \in I} (U_{i} \times_{U} U_{j}),$$ given by showing that the right-hand side satisfies the universal property of the left-hand side. Having this in mind, one may check the our goal reduces to checking that the sequence $$\mathscr{F}(U) \rightarrow \mathscr{F}(V) \rightrightarrows \mathscr{F}(V \times_{U} V)$$ is an equalizer. The second condition tells us that this holds when $I$ is finite and each $U_{i}$ for $i \in I$ and $U$ are affine.

For the general case, denote by $\pi : V \rightarrow U$ be the morphism induced by maps $U_{i} \rightarrow U$ from the fixed covering of $U.$ We may write $U = \bigcup_{j \in J} W_{j},$ where each $W_{j} \subset U$ is affine open. Then $\pi^{-1}(W_{j}) \subset V$ is open, so we may write $$\pi^{-1}(W_{j}) = \bigcup_{k \in K_{j}} W_{j,k},$$ where $W_{j,k} \subset V$ are affine opens. Note that $\{W_{j,k} \rightarrow W_{j}\}_{k \in K_{j}}$ forms an étale cover of $W_{j},$ where the maps are given by $$W_{j,k} \hookrightarrow \pi^{-1}(U_{j}) \xrightarrow{\pi} W_{j}.$$ Since $\pi(W_{j,k}) \subset W_{j}$ is open and $W_{j}$ is quasi-compact, we may assume that $K_{j}$ is finite for such étale covering of $W_{j}.$ We have $$V = \bigcup_{j \in J}\pi^{-1}(W_{j}) = \bigcup_{j \in J, k \in K_{j}} W_{j,k}.$$

So far, we know that the following sequences are equalizers:

• $\mathscr{F}(U) \rightarrow \prod_{j \in J}\mathscr{F}(W_{j}) \rightrightarrows \prod_{j,j' \in J}\mathscr{F}(W_{j} \cap W_{j'})$ by the first condition;
• $\mathscr{F}(V) \rightarrow \prod_{j \in J}\prod_{k \in J_{j}}\mathscr{F}(W_{j,k}) \rightrightarrows \prod_{j,j' \in J}\prod_{k \in K_{j}, k' \in K_{j'}}\mathscr{F}(W_{j,k} \cap W_{j',k'})$ by the first condition;
• $\mathscr{F}(W_{j}) \rightarrow \prod_{k \in K_{j}}\mathscr{F}(W_{j,k}) \rightrightarrows \prod_{k,l \in K_{j}} \mathscr{F}(W_{j,k} \times_{U} W_{j,l})$ since $K_{j}$ is a finite set and $W_{j}, W_{j,k}$ are affines.

We are now ready to show that $$\mathscr{F}(U) \rightarrow \mathscr{F}(V) \rightrightarrows \mathscr{F}(V \times_{U} V)$$ is an equalizer. Given any $s \in \mathscr{F}(U),$ suppose that $s|_{V} = 0 \in \mathscr{F}(V).$ This means that $s|_{W_{j,k}} = 0 \in \mathscr{F}(W_{j,k}),$ for all $j \in J$ and $k \in K_{j}.$ Hence, we have $s|_{W_{j}} = 0 \in \mathscr{F}(W_{j})$ for all $j \in J,$ which implies that $s = 0 \in \mathscr{F}(U).$ This already shows that $\mathscr{F}$ is a separated presheaf (meaning a presheaf with the first condition for being a sheaf), and we will use this observation soon.

Denote by $p : V \times_{U} V \rightarrow V$ the projection on the first component and $q : V \times_{U} V \rightarrow V$ the second. Given any $u \in \mathscr{F}(V),$ suppose that $$p^{*}u = q^{*}u\in \mathscr{F}(V \times_{U} V).$$ Then $$(p^{*}u)|_{W_{j,k} \times_{U} W_{j,l}} = (q^{*}u)|_{W_{j,k} \times_{U} W_{j,l}} \in \mathscr{F}(W_{j,k} \times_{U} W_{j,l})$$ for all $j \in J$ and $k, l \in K_{j}.$ Define $(t_{j,k}) \in \prod_{k \in K_{j}}\mathscr{F}(W_{j,k})$ by $$t_{j,k}|_{W_{j,k} \times_{U} W_{j,l}} := (p^{*}u)|_{W_{j,k} \times_{U} W_{j,l}}$$ and $$t_{j,l}|_{W_{j,k} \times_{U} W_{j,l}} := (q^{*}u)|_{W_{j,k} \times_{U} W_{j,l}}.$$ Then we have a unique $t_{j} \in \mathscr{F}(W_{j})$ such that $t_{j}|_{W_{j,k}} = t_{j,k}$ for all $j, k.$

We will be done as soon as we can show that $t_{j}|_{W_{j} \cap W_{j'}} = t_{j'}|_{W_{j} \cap W_{j'}}$ for all $j, j' \in J.$ The reason is that this would give us a unique $t \in \mathscr{F}(U)$ such that $t|_{W_{j}} = t_{j}$ for all $j \in J$ and since $$(t|_{V})|_{W_{j,k}} = (t_{j})|_{W_{j,k}} = t_{j,k} = u|_{W_{j,k}}$$ for all $j, k,$ as we can check by applying one more map, we must have $t|_{V} = u,$ as desired.

To check $t_{j}|_{W_{j} \cap W_{j'}} = t_{j'}|_{W_{j} \cap W_{j'}},$ it is enough to check $$t_{j}|_{W_{j,k} \cap W_{j',k'}} = t_{j'}|_{W_{j,k} \cap W_{j',k'}},$$ where $k \in K_{k}$ and $k' \in K_{j'}.$ This is the same as checking $$t_{j,k}|_{W_{j,k} \cap W_{j',k'}} = t_{j',k'}|_{W_{j,k} \cap W_{j',k'}},$$ which we already know. This finishes the proof. $\Box$