Morphism of schemes

Remarks

Example: a ring homomorphism $\varphi \colon A \to B$ gives a morphism $\operatorname{Spec} B \to \operatorname{Spec} A$ with $f(\mathfrak{q}) = \varphi^{-1}(\mathfrak{q})$. On stalks it is the map $A_{\varphi^{-1}(\mathfrak q)} \to B_{\mathfrak q}$, which sends the maximal ideal into $\mathfrak{q} B_{\mathfrak q}$, so it is local. Non-example: let $R$ be a discrete valuation ring with fraction field $K$. Let $f$ send the single point of $\operatorname{Spec} K$ to the closed point of $\operatorname{Spec} R$, and let $f^\#$ come from the inclusion $R \to K$. This is a morphism of ringed spaces, but it fails clause 3. The stalk map $R \to K$ is not local, since it sends the maximal ideal of $R$ into the units of $K$.