At certain point of the proof of the theorem in Hartshorne's book, he mentioned the following in which I don't understand:

  1. In Step 3, he said we can reduce our consideration to a sheaf $\mathscr{F}$ of abelian group which is generated by a single section over certain open set $U\subset X$ ($X$ is irreducible topological space). Then in that case $\mathscr{F}$ $\textbf{is a quotient of the sheaf}$ $\textbf{Z}_U$ (where $\textbf{Z}$ is a constant sheaf on $X$).

Question: I couldn't see how the statement is true.

  1. Assume the things mentioned in (1) is correct, then we will be able to write an exact sequence of $\mathscr{F}$ as $$0\to\mathscr{R}\to\textbf{Z}_U\to\mathscr{F}\to 0$$ where $\mathscr{R}$ is of course, the kernel of the map from $\textbf{Z}_U$ to $\mathscr{F}$. Then for any $x\in U$, $\mathscr{R}_x$ is a subgroup of $\textbf{Z}$ and we have certain $d\in\textbf{Z}$ which is the least positive integer which occurs in any of the groups $\mathscr{R}_x$, no problem. But the problem comes when he said there exist another open subset $V\subseteq U$ such that $\mathscr{R}|_V\cong d\cdot\textbf{Z}|_V$ as a subsheaf of $\textbf{Z}|_V$.

Question: Explicitly why there is such an existence?

Thank you in advance and please let me know if there is any unclear point in my question.


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