I am just learning about locally ringed space. Let $(X, O_X)$ be a locally ringed space. It (the notes I am reading) says that: The $O_X,p$ is a local ring for each $p \in X$. Let $m_p$ be the unique maximal ideal of $O_X,p$. Functions on an open subset $U$ has values for each point in $p \in U$, taking values in the residue field $O_X,p/m_p$.
Is that mean if $f \in \Gamma(U,O_X)$ then the value of $f$ at point $p \in U$ is precisely $[(f,U)_p] + m_p$? (where $[(f,U)_p]$ is the germ of $f$, $[]$ denotes the equivalence relation.) I was a bit confused, because it didn't say so explicitly. Could I possibly verify this with someone? Thanks!
Yes, the "value" of a section $f\in\Gamma(U,\mathscr{O}_X)$ at $x\in U$ is defined to be the image of $f$ in the residue field $k(x)=\mathscr{O}_{X,x}/\mathfrak{m}_x$.