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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.

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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 $$\bar{s}^{-1}(\bar{t}(U\cap V)).$$

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