---- > [!definition] Definition. ([[locally factorial scheme]]) > A [[scheme]] $(X, \mathcal{O}_{X})$ is said to be **locally factorial** if the [[local ring|local rings]]/[[(pre)sheaf stalk|stalks]] $\mathcal{O}_{X, x}$, $x \in X$, are all [[UFD|UFDs]]. ^definition > [!example] > [[regular scheme|Nonsingular]] varieties ('the nicest kind of variety') are locally factorial. This follows from the difficult theorem of Serre whch says that a [[regular local ring]] is a [[UFD]]. ^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 > ```