Definition of restriction maps of schemes

Question

I understand that for Spec $A$ the restriction map is defined in a natural way. Given $V \subset U \subset Spec \ A = X$ open sets, for $f \in O_X (U)$ we define $f|_V$ by restricting the domain to $V$. Now scheme is defined by locally ringed space where every point has an affine open neighborhood. I am guessing that one can deduce that the restrictio maps on schemes also is just given by restricting the domains... how can I show this? thank you.

Answer 1

Let $X$ be a arbitrary topological space such that $(X,\mathcal{O}_X)$ is a scheme then $\mathcal{O}_X$ is just a sheaf such that $\mathcal{O}_X(U)$ is a ring for all open subset $U$ of $X$ so the restriction maps for the scheme $(X,\mathcal{O}_X)$ are defined exactly the same as how we define for sheaves .