# From Presheaf to Sheaf

In Hartshorne's Algebraic Geometry is written that "A sheaf is roughly speaking a presheaf whose sections (i.e. elements of $\mathcal{F}(U)$ for open subset $U$) are determined by local data". What does it means? What is the local data?

After this remark Robin Hartshorne gave a definition of sheaf ("...a sheaf is a presheaf satisfying certain extra condition...") which I understood but did not feel.

Thanks a lot!

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local data: restrictions of a section to the elements of an open cover of $U$. –  wildildildlife Jul 8 '12 at 15:41

For a presheaf, given two sections $s,t\in\mathcal{F}(U)$, you can have that they agree in every single neighbourhood, yet be different. That is, if $U_i$ is an open cover of $U$ and $s|_{U_i}=t|_{U_i}$ for each $i$ then if you have a presheaf, it is possible that $s\neq t$, even if this condition held for every single open cover of $U$. In a sheaf, the local data (being the sets in the cover) actually determines the section uniquely.
The other half of being a sheaf says that your sections may be glued together. That is, suppose you have sections $s_i$ over open sets $U_i$, such that the restriction of $s_i$ and $s_j$ to $U_i\cap U_j$ agree for all $i,j$. We would like to be able to glue these sections together, which we can do only if you have a sheaf. If you have a sheaf, then you are guaranteed that all of your sections $s_i$ are just the restrictions of some section of $\mathcal F(\bigcup_i U_i)$.
Your opening paragraph isn't quite right. At the very least, you need to insist that $V \neq U$, although I'm guessing you intended an even stronger condition, such as a collection of $V$'s that form a non-trivial open cover of U. –  Hurkyl Jul 8 '12 at 15:07