Computing stalks: do direct limits behave like limits?

Question

Suppose that $X$ is a topological space with a sheaf of rings $\mathcal{O}_X$. In general, the stalk at a point $p \in X$ is the direct limit of the rings $\mathcal{O}_X(U)$ for all open sets $U$ containing $p$.

Here are two questions on computing stalks - I think both should be true, since a direct limit should be some sort of "limiting process", but that's far from convincing for me.

Can I compute the stalk of $\mathcal{O}_X$ at a point $p \in X$ by only limiting over basic open sets of $X$ containing $p$?

Can I compute the stalk of $\mathcal{O}_X$ at a point $p \in X$ by excluding some finite number of "large" open sets around $p$, and then limiting over the remaining open sets around $p$?

Answer 1

Yes. The general statement is the following: limit over a poset is equal to limit over its any coinitial subset. Formal proof is easy (hint: construct maps in both directions) and informally it's an analogue of "subsequence has the same limit as a sequence" theorem.

Answer 2

Basically, here's a way to think of the stalk that is more "down-to-earth" than direct limits. An element of the stalk $O_x$ is given by a pair $(f, U)$ where $f$ is a section over the open set $U$ and $U$ contains $x$. Two pairs $(f,U), (g,V)$ are considered equivalent if $f=g$ on a neighborhood $W$ of $x$ (contained in $U \cap V$).

With this definition, it's easy to see that what happens at $x$ doesn't depend on what happens on $F$, where $F$ is any closed set disjoint from $x$. The stalk is a purely local construction.

As for why this is equivalent to the direct limit: that's a direct corollary of how the construction works in most familiar categories with which one might define a sheaf (sets, groups, rings, etc.)

Answer 3

I think there is a missing word in Akhil Mathew's answer: and it's "filtered".

You can do that because stalks are filtered colimits (aka "direct limits").

For filtered colimits, $\varinjlim_i X_i$, you can take representatives of elements $x \in \varinjlim_i X_i$ for some $i$ belonging to the set of indexes $I$ (in our case, the open sets $U$). That is, you can find some $i$ and $x_i \in X_i$ that goes to your $x$ through the universal arrow $X_i \longrightarrow \varinjlim_i X_i$. For instance, every element of the stalk $O_{X,p}$ can be represented by a section $f \in O_X(U)$ for some open set $U$.

But this is not true for other kinds of colimits.

For instance, take the push-out of two arrows $f: A \longrightarrow B$ and $g: A \longrightarrow C$ in the category of, say, abelian groups. Elements of this push out $B \oplus_A C$ are classes of pairs $(b,c) \in B\oplus C$, where you quotient out elements of the form $(f(a), 0) - (0, g(a))$, for all $a\in A$. That is, $(f(a),0) = (0,g(a))$ in $B\oplus_A C$.

Elements of $B\oplus_A C$ cannot be represented, in general, by elements coming from just $B$ or $C$, which are of the form $(b,0)$ or $(0,c)$, respectively: so, for a general $(b,c) \in B\oplus_A C$ there is no $b \in B$, nor $c\in C$, that represents it.