Sheaf condition for finite coverings

Question

I'm interested in Proposition 3.5 in Milne's book "Etale Cohomology," which says that a presheaf on a noetherian site is a sheaf if it satisfies the sheaf axiom with respect to finite coverings. By the way, a site is noetherian if every covering has a finite subcovering (e.g. open sets in a noetherian space). Milne proves it by an elementwise calculation, but I'm wondering if there's an abstract nonsense-type argument which works for, say, sheaves with values in an arbitrary cocomplete category. I feel like there's some kind of cofinality argument lurking around which I can't quite formulate.

Answer 1

First of all, you have to formulate the sheaf axiom correctly, i.e. with sieves. For ease of notation I will work with the standard site of a topological space – it will be clear how to do this for a general site.

Recall that a sieve (of open sets) is a collection $\mathfrak{U}$ of open sets with the following property: if $U \in \mathfrak{U}$ and $U' \subseteq U$, then $U' \in \mathfrak{U}$. A sieve $\mathfrak{U}$ covers $V$ if $V = \bigcup_{U \in \mathfrak{U}} U$. The sheaf condition on a presheaf $F$ can then be stated as follows:

• If $\mathfrak{U}$ is a sieve that covers $V$, then the diagram $$F (V) \to \prod_{U \in \mathfrak{U}} F (U) \rightrightarrows \prod_{U \in \mathfrak{U}} \prod_{U' \subseteq U} F (U')$$ is an equaliser, where one of the arrows is defined by projection and the other by restriction.

It is straightforward to check that this condition is equivalent to the usual one. The next step is to prove the following:

• If $F$ satisfies the sheaf condition for the sieve $\mathfrak{U}'$ and $\mathfrak{U}' \subseteq \mathfrak{U}$, then $F$ satisfies the sheaf condition for $\mathfrak{U}$.

Finally, observe that in a noetherian topological space, every covering sieve contains a finitely generated covering sieve. Thus, it suffices to check the sheaf condition on finitely generated covering sieves, or equivalently, finite covers.