I was reading some notes on Algebraic Geometry and it says, "Suppose $O_X$ is a sheaf on rings on a topological space $X$ (i.e., a sheaf on $X$ with values in the category of rings)."
What does this "sheaf on $X$ with values in the category of rings" mean? Does it mean given an open set $U$ in X, $O_X(U)$ is some ring (in the category of rings)? Thanks!
Are you comfortable with category theory? A sheaf of rings $\mathcal{F}$ on a topological space $X$ is a functor from the category of open subsets of $X$, where the morphisms are inclusions, to the category of rings where the objects are rings and the morphisms are ring homomorphism, satisfying the standard sheaf axioms. In particular, for an open subset $U$ we have $\mathcal{F}(U)$ is a ring and if you have $V \hookrightarrow U$, both open subsets of $X$, then the induced morphism $\mathcal{F}(U) \rightarrow \mathcal{F}(V)$ is a ring homomorphism.