---- > [!definition] Definition. ([[locally ringed space]]) > A **locally ringed space** is a [[ringed space]] $(X, \mathcal{O}_{X})$ whose [[(pre)sheaf stalk|stalks]] are [[local ring|local]] [[ring|rings]]. > > Locally ringed spaces are objects of the [[category]] $\mathsf{LRS}$, with morphisms being [[morphism of locally ringed spaces|morphisms of locally ringed spaces]]. > [!example] Key example. > The [[prime ideal|spectrum]] $\text{Spec }A$ of a [[ring]] $A$, together with its [[structure sheaf on a ring spectrum]] $\mathcal{O}_{\text{Spec }A}$, is a [[locally ringed space]]. Indeed, ($\text{Spec }A, \mathcal{O}_{\text{Spec } A}$) is a [[ringed space]] by construction, and the [[(pre)sheaf stalk|stalk]] $\mathcal{O}_{\text{Spec }A, \mathfrak{p}}$ equals $A_{\mathfrak{p}}$, [[local ring#^basic-example|which has a unique]] [[maximal ideal]] $\mathfrak{m}=\left\{ \frac{a}{s} : a \in \mathfrak{p}, s \notin \mathfrak{p} \right\}$. > $(\text{Spec } A, \mathcal{O}_{\text{Spec } A})$ is the model [[affine scheme]]. General [[scheme|schemes]] are [[locally ringed space|locally ringed spaces]], too: those that locally look like an $(\text{Spec } A, \mathcal{O}_{\text{Spec } A})$ for some $A$. ^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 > ```