---- > [!definition] Definition. ([[morphism of locally ringed spaces]]) > A **morphism of [[locally ringed space|locally ringed spaces]]** $f:(X, \mathcal{O}_{X}) \to (Y, \mathcal{O}_{Y})$ is a [[morphism of ringed spaces|morphism]] $(f, f^{\sharp})$ of [[ringed space|ringed spaces]] for which the induced [[ring homomorphism]] on [[(pre)sheaf stalk|stalks]] > >$\begin{align} f^{\sharp}_{p}: \mathcal{O}_{Y,f(p)} & \to \mathcal{O}_{X, p} \\ [U, \varphi] & \mapsto [f ^{-1} (U), f^{\sharp}_{U}(\varphi)] \end{align}$ is a [[homomorphism of local rings|local homomorphism]] for all $p$. > Notationally, we often just wite $f:X \to Y$ to represent the *whole* morphism. ^definition > [!basicexample] Key example. > For $A$ a [[commutative ring]], $(\text{Spec }A, \mathcal{O}_{\text{Spec }A})$ is a [[locally ringed space]]. Indeed, the [[structure sheaf on a ring spectrum]] [[(pre)sheaf stalk|stalks]] are of the form $A_{\mathfrak{p}}$ for $\mathfrak{p} \in \text{Spec }A$, which is a [[local ring]]: its unique [[maximal ideal]] is $\mathfrak{m}=\left\{ \frac{a}{s}: a \in \mathfrak{p}, s \notin \mathfrak{p} \right\},$ that is, the [[extension of an ideal|extension]] of $\mathfrak{p}$ under the [[localization|localization map]] $a \mapsto \frac{a}{1}$. This is often (slightly misleadingly if the [[localization|localization map]] is not [[injection|injective]]) denoted $\mathfrak{p}A_{\mathfrak{p}}$. ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```