I am trying to solve the question about espace etale of a presheaf on Hartshrone (Chap II ex 1.13):

Given a presheaf $\mathcal{F}$, we define a topological space $$\mathrm{Spe(\mathcal{F})}=\coprod_{p\in X}\mathcal{F}_p$$ with the strongest topology such that for all the maps $$\bar{s}:U\rightarrow \mathrm{Spe(\mathcal{F})}$$ which sends $P$ to the germ $s_P$ for all $U$, and all $s\in\mathcal{F}(U)$ are continuous.

Show that for any $U\subset X$, the associated sheaf $\mathcal{F}^+(U)$ is the set of continuous sections of $\mathrm{Spe(\mathcal{F})}$ over $U$.

My try:

Here the only difficult part is to show any continuous section $\bar{s}$ of $\mathrm{Spe(\mathcal{F})}$ over $U$ is an element in $\mathcal{F}^+(U)$, i.e.

  1. $\bar{s}(P)\in \mathcal{F}_P$ for any $P\in U$: this is clear from the definition of section;
  2. for every $P\in U$, there exists a neighborhood $V$ and $t\in \mathcal{F}(V)$ such that for every $Q\in V$ we have $s(Q)=t_Q$.

I am struggling about the second part. Naturally, if we take a point $P\in U$, then $s(P)=\langle V,t\rangle$ for some open subset $V\subset X$ and $t\in \mathcal{F}(U)$. I don't know how to continue.

It seems that the only thing we can use here is $\bar{s}$ is continuous. So the expected neighborhood might be of the form $\bar{s}^{-1}(W)$ for some open set in $\mathrm{Spe(\mathcal{F})}$.

Any hints and answers are welcome!

Remark: Someone did some relevant work here, but I think there is a mistake at the third equivalence $$\cdots\iff \bar t(P) = \bar s (P) \iff \langle V,t\rangle_P=\langle U,s\rangle_P \iff \cdots$$ I think $\bar{s}(P)=t_Q$ cannot imply $\langle V,t\rangle_P=\langle U,s\rangle_P$ because we do not know what $s$ is.

However, if the above is true, then it will solve the problem. So probably I missed something obvious.


For every $P\in U$, we need to find a neighborhood $V$ and $t\in \mathcal{F}(V)$ such that for every $Q\in V$ we have $s(Q)=t_Q$.

Since $\bar{s}(P)\in \mathcal{F}_P$, so we can write $\bar{s}(P)=\langle V, t\rangle$ for some neighborhood $V$ of $P$ and $t\in\mathcal{F}(V)$. Then we have a "standard section" sending each point $Q$ to $t_Q$, the germ of $t$ at the point $Q$: $$\bar{t}:V\rightarrow \mathrm{Spe}(\mathcal F)$$

Consider the restriction of $\bar{t}$ on $U\cap V$, then $\bar{t}(U\cap V)$ is open since $\bar{t}^{-1}(\bar{t}(U\cap V))=U\cap V$ is open. Then take the neighbohood of $P$ as $$\bar{s}^{-1}(\bar{t}(U\cap V)).$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.