The Stacks project

Example 6.11.5. Let $X$ be a topological space. Let $A_ x$ be a set for each $x \in X$. Consider the sheaf $\mathcal{F} : U \mapsto \prod _{x\in U} A_ x$ of Example 6.7.5. We would just like to point out here that the stalk $\mathcal{F}_ x$ of $\mathcal{F}$ at $x$ is in general not equal to the set $A_ x$. Of course there is a map $\mathcal{F}_ x \to A_ x$, but that is in general the best you can say. For example, suppose $x = \mathop{\mathrm{lim}}\nolimits x_ n$ with $x_ n \not= x_ m$ for all $n \not= m$ and suppose that $A_ y = \{ 0, 1\} $ for all $y \in X$. Then $\mathcal{F}_ x$ maps onto the (infinite) set of tails of sequences of $0$s and $1$s. Namely, every open neighbourhood of $x$ contains almost all of the $x_ n$. On the other hand, if every neighbourhood of $x$ contains a point $y$ such that $A_ y = \emptyset $, then $\mathcal{F}_ x = \emptyset $.

Comments (3)

Comment #429 by JuanPablo on

"if each neighbourhood of has infinitely many points, and each has exactly two elements, then has infinitely many elements"

I think this statement is false.

Take for ease of notation . Take an infinite set and take a non-principal ultrafilter of , and take the open sets of to be exactly those sets which are in . Then for any point , any neighbourhood of contains infinitely many points.

But has four points, one is given by the section constant with for all , another is given by such that iff . The other two are by interchanging and for this two.

This are the only four because for any with we can define and so that , and as is an ultrafilter then either or , this with or correspond to the four elements.

(There seems to be a problem in my computer with the subscripts in the preview)

Comment #430 by on

The preview should now be fine, there was a silly mistake in the code. I will let Johan handle the math :).

Comment #431 by on

Yes, very good! How bizarre to get 4 elements! I replaced it with the statement: with and all sets have two elements then the stalks are infinite. The fix is here. Stop by my office if you are in NY and I'll give you a Stacks project mug, if I have't run out by then.

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