# Some facts about sheafification functor on étale site

I'm studying the book Etale cohomology and the Weil conjecture by Freitag, Kiehl and I have some questions on the subchapter introducing the machinery associating to an étale presheaf a sheaf (that is the "sheafification" procedure in étale world): see pages 11-13.

Let $$A$$ be a commutative, unital, Noetherian ring and $$\operatorname{Et}(A)$$ the category of étale extensions of $$A$$ (étale means that the extensions $$A \to B$$ are flat, unramified $$A$$-algebras of finite type).
We follow the notation from the book and call as an étale presheaf (of set, abelian groups,...; for sake of simplicity say sets) a covariant functor

$$\mathcal{F}: \operatorname{Et}(A) \to (Set), (Ab),...$$

Now we want to associate to $$\mathcal{F}$$ another étale presheaf $$\widetilde{\mathcal{F}}$$ as follows:
Let $$B \in \operatorname{Et}(A)$$ and $$\mathcal{B}:= (B \to B_i, i \in I)$$ an étale cover of $$B$$, that's a finite family $$(B \to B_i, i \in I)$$ of étale maps , when the images of $$\operatorname{Spec} B_i$$ cover all elements of $$\operatorname{Spec} B$$.

(These form naturaly a category; a morphism $$\mathcal{B} \to \mathcal{B}'$$ between

$$\mathcal{B}= (B \to B_i, i \in I), \mathcal{B}'= (B \to B_j', j \in J)$$

is given by a map $$\sigma:J \to I$$ inducing $$B_{\sigma(j)} \to B_j'$$ (datails can be worked out easily).

Now we associate to an etale cover $$\mathcal{B}= (B \to B_i)$$ the object $$\mathcal{F}(\mathcal{B})$$ defined as set of all $$I$$-tuples $$(s_i)_{i \in I}$$ with $$s_i \in \mathcal{F}(B_i)$$ such that all $$s_i$$ fulfil the compatibility condition,
that is for every $$i,j$$ should hold

$$f^l_i(s_i)= f^r_j(s_j),$$

where the involved maps are $$f^l_i: \mathcal{F}(B_i) \to \mathcal{F}(B_i \otimes_B B_j)$$ (respectively $$f^r_j: \mathcal{F}(B_j) \to \mathcal{F}(B_i \otimes_B B_j)$$) are naturally induced by family of canonical maps $$l_i: B_i \to B_i \otimes_B B_j, b_i \mapsto b_i \otimes 1$$,
(resp $$r_j: B_i \to B_i \otimes_B B_j, b_i \mapsto b_i \otimes 1$$ )

We define the presheaf $$\widetilde{\mathcal{F}}$$ as direct limit over all coverings of $$B$$

$$\tilde{\mathcal{F}}(B):= \varinjlim_{\mathcal{B}} \mathcal{F}(\mathcal{B})$$

(details like checking thats this is a functor etc. I omit here)
Now we come to my Questions. The book states some very important facts without giving a reference. And I'm looking for proofs of several statements there.

Question 1a: It is stated (p. 12) that for every $$B \in \operatorname{Et}(A)$$ and etale cover $$\mathcal{B}= (B \to B_i)$$ the induced map $$\widetilde{\mathcal{F}}(B) \to \widetilde{\mathcal{F}}(\mathcal{B})$$ is injective (note that $$\widetilde{\mathcal{F}}$$ is in general still a presheaf). How can it be proved?

Question 1b: Futhermore it is claimed that if for all $$B$$ and $$\mathcal{B}= (B \to B_i)$$ the maps

$$\mathcal{F}(B) \to \mathcal{F}(\mathcal{B})$$

are injective, then $$\widetilde{\mathcal{F}}$$ is already an étale sheaf. Why that's true? (Rmk.: combined with statement of Question 1a this implies that the application of double tilde $$\widetilde{\widetilde{\mathcal{F}}}$$ gives a sheaf)

Question 2: Why is the double tilde functor (= sheafification as we saw above modulo the two questions) $$\widetilde{\widetilde{\mathcal{F}}}$$ is exact? That is if we apply if to short exact sequence $$0 \to \mathcal{A} \to \mathcal{B} \to \mathcal{C} \to 0$$ of étale presheaves we obtain a ses of sheaves?
(recall, that by definition a sequence $$0 \to \mathcal{A} \to \mathcal{B} \to \mathcal{C} \to 0$$ of presheaves on site $$\operatorname{Et}(A)$$ is ses iff $$0 \to \mathcal{A}(B) \to \mathcal{B}(B) \to \mathcal{C}(B) \to 0$$ is ses for every $$B \in \operatorname{Et}(A)$$.)
Is $$\widetilde{\mathcal{F}}$$ exact as functor between etale presheaves?

A good trick for answering questions of this type is that there are multiple foundational resources in étale cohomology. If a detail isn't explained in one you're reading, you can quickly check another one - the notation should hopefully be similar enough that you can transfer the proof over. In particular, the Stacks project tends to give a lot of details, and I found an answer to question 1 and a partial answer to question 2 there. (You should probably try to solve them as an exercise before doing this.)

The way I would say it.

Question 1a: Suppose we are given two elements of $$\widetilde{\mathcal F}(B)$$ whose image in $$\widetilde{\mathcal F}(\mathcal B)$$ is equal. We must show they are equal. By assumption, the two elements arise from covers $$\mathcal A, \mathcal C$$ of $$\mathcal B$$ and families of sections on these covers satisfying a compatibility condition. Their pullbacks to $$\widetilde{\mathcal F}(\mathcal B)$$ are given by the pullbacks of those families of sections to $$\mathcal B \times_{B} \mathcal A$$ and $$\mathcal B\times_B \mathcal C$$ respectively. If those are equal, then by the definition of direct limit, there is a common refinement where they are equal. But this common refinement is also a common refinement of $$\mathcal A$$ and $$\mathcal C$$ where our two families of sections are equal, so again by the definition of directly limit, the two starting elements of $$\mathcal F(B)$$ are equal.

Question 1b: Well, one must show that given a cover $$U_1,\dots, U_n$$ of $$B$$, the map from sections of $$\widetilde{\mathcal F}$$ on $$B$$ to compatible sections on $$U_1,\dots, U_n$$ is an isomorphism. So let's construct an inverse map. Given a compatible section of $$\widetilde{\mathcal F}$$ on a cover $$U_1,\dots, U_n$$, we obtain for each $$i$$ a cover $$\mathcal U_i$$ of $$U_i$$ and a compatible family of sections of $$\mathcal F$$ on that cover. We would like to make $$\bigcup_i \mathcal U_i$$ a cover of $$B$$ and this family of sections a compatible family. The only issue is that when we pass to an intersection $$\mathcal U_i \times \mathcal U_j$$, the pullbacks of the sections from $$U_i$$ and $$U_j$$ are only necessarily equal in $$\widetilde{\mathcal F}$$, not in $$\mathcal F$$. But given that $$\mathcal F \to \widetilde{\mathcal F}$$ is injective, we get the desired compatibility also in $$\mathcal F$$.

Question 2: No, $$\tilde{F}$$ is not exact as a functor between etale presheaves. To make a counterexample, take any exact sequence $$0 \to \mathcal A \to \mathcal B \to \mathcal C \to 0$$ of sheaves whose sequence of global sections $$0 \to H^0( X, \mathcal A) \to H^0(X,\mathcal B)\to H^0(X,\mathcal C)$$ is not right exact. Let $$\mathcal C'$$ be the quotient of $$\mathcal B$$ by $$\mathcal A$$ as presheaves, i.e. $$\mathcal C'(U) = \mathcal B(U)/\mathcal A(U)$$.

Then $$0 \to \mathcal A \to \mathcal B \to \mathcal C' \to 0$$ is a short exact sequence of presheaves. Its sheafification is simply $$\mathcal A \to \mathcal B \to \mathcal C$$, which is a short exact sequence of sheaves, but not a short exact sequence of presheaves, since its global sections are not short exact.

Instead, the proof that sheafification is exact makes use of the fact that $$\widetilde{\widetilde{\mathcal F}}$$ is a left adjoint and therefore right exact, without proving right exactness for $$\widetilde{\mathcal F}$$.

• Thank you very much for the answer. A remark: when you proved right exactness of $\widetilde{\widetilde{\mathcal F}}$ you exploted it's left adjointness (which $\widetilde{\mathcal F}$ not has). On the other hand following the proof in stacks.math.columbia.edu/tag/00WJ the proof of left exactness uses esentially just the category theoretically fact that finite limits commute with filtered colimits. But this can be also applied directly to $\widetilde{\mathcal F}$ because of it's construction. So left exactness works unproblematically already for $\widetilde{\mathcal F}$, right? Aug 5, 2021 at 19:54
• @katalaveino I believe so, yes. (And it's even not so hard to check directly, without calling on the category-theoretic fact, by expanding the definitions and going step by step.) Aug 5, 2021 at 19:57