In Vakil's notes on locally ringed spaces, he claimed that "we can't even make sense of the phrase of 'function vanishing' on ringed spaces in general. "
Could someone explain what this remark means (or how this notion is ill-defined in general ringed spaces)?
If $(X,\mathcal{O}_X)$ is a locally ringed space, we can define evaluation maps on functions as follows: $$f\in \mathcal{O}_X(U)\mapsto f_p\in \mathcal{O}_{X,p}\mapsto \overline{f}_p\in \mathcal{O}_{X,p}/\mathfrak{m}_p=\kappa(p).$$ Here, $p\in U$ and $U$ is an open set. Because $\mathfrak{m}_p$ is the unique maximal ideal of $\mathcal{O}_{X,p}$, we see that $\kappa(p)$ is independent of any choice but $p$ and is moreover a field. This is the residue field at $p$, and we call $\overline{f}_p$ the value of $f$ at $p$. Note that for a general ringed space this would not make sense. Then, by definition, $\mathfrak{m}_p$ is the set of germs of functions at $p$ that vanish at $p$.
Anyway, Vakil is saying that this can't really be defined for a general ringed space and so we don't have a good notion of vanishing at a point for such spaces.
