Hello,

I have two questions, the first less important.

Let $X$ be a scheme, $x \in X$ a schematic point.

What is an elegant way of defining/characterizing the map $\operatorname{Spec}(O_{X,x}) \to X$?

This map can be defined by choosing affine neighbourhood, building the map and then showing independence; maybe there is a better way? For example a characterization which will make sense for locally ringed spaces (but existence perhaps will not be in that generality).

Now, suppose furthermore that $X$ is integral, $x \in X$ is a closed point, $U=X-\{x\}$. We denote by $K$ the field of rational functions on $X$, i.e. stalk at the generic point.

$$ \begin{matrix} \operatorname{Spec}(K) & \to & U \\\\ \downarrow & & \downarrow \\\\ \operatorname{Spec}(O_{X,x}) & \to & X \end{matrix} $$

Is it true that this diagram is cartesian and co-cartesian?

If one can require more to get the result then it is also interesting (say pushout is in the category of schemes of certain type, or $X$ is a non-singular curve over a field...).

Thanks, Sasha