# Concerning Hartshorne's proof of the Vanishing Theorem of Grothendieck (Hartshorne III 2.7)

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.