Reference request: Sheaf for the Zariski topology

Question

In my course notes page 62 I've read the following: let $(E,0)$ be an elliptic curve over an arbitrary scheme $S$, then $U\rightarrow\ker(\,0^*_U:\operatorname{Pic}(E_U)\rightarrow \operatorname{Pic}(U))$ is a sheaf for the Zariski topology.

What is "sheaf for the Zariski topology" ? So far I know what is a sheaf and what is Zariski topology. I'll be thankful for any references in this subject.

Answer 1

You can define a sheaf $\cal F$ with values in a category $\mathbf{C}$ on any topological space $X$ by letting ${\cal F}(U)$ be an object in $\mathbf{C}$ for any open set $U\subset X$ so that the usual axioms are met.

E.g. see this Wikipedia entry.

To fix ideas you may think that $\mathbf{C}$ is the category of groups, so that ${\cal F}(U)$ is a group for every open $U\subset X$ and the restriction maps $$ \rho_{U,V}:{\cal F}(V)\longrightarrow{\cal F}(U) $$ will be homomorphisms for every inclusion $U\subset V$ of open sets.

Answer 2

Quite simply, a sheaf is defined on open sets and the Zariski topology is a choice of open sets. So a "sheaf for the Zariski topology" is an assignment of groups/modules/rings etc. (Picard groups in your case) to Zariski-open sets of the top space in a way that compatible sections glue uniquely.