---- > [!definition] Definition. ([[regular scheme]]) > A [[scheme]] $(X, \mathcal{O}_{X})$ is said to be **regular**, or **nonsingular**, if it is [[locally Noetherian scheme|locally Noetherian]] with [[local ring|local]] [[ring|rings]] all [[regular local ring|regular]]: $\mathcal{O}_{X, x}$ is [[regular local ring|regular]] for all $x \in X$. > We call $X$ **regular in codimension $N$** if $\mathcal{O}_{X ,x}$ is [[regular local ring|regular]] whenver $\text{dim }\mathcal{O}_{X ,x}=N$. ^definition > [!basicexample] If we are looking at regularity in [[codimension of a closed subspace|codimension]] $1$:![[CleanShot 2025-03-06 at [email protected]]] ---- #### ---- #### 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 > ```