Regular scheme

{{distinguish|regular embedding}}

In algebraic geometry, a regular scheme is a locally Noetherian scheme whose local rings are regular everywhere.{{citation|last=Hartshorne|first=Robin|title=Algebraic Geometry|url=https://books.google.com/books?id=3rtX9t-nnvwC&pg=PA238|volume=52|page=238|year=1977|series=Graduate Texts in Mathematics|publisher=Springer|isbn=9780387902449|authorlink=Robin Hartshorne}}. Note that the cited definition that Hartshorne gives is slightly misleading. A locally Noetherian scheme is regular if all its local rings are regular, but it is not the case for schemes which are not locally Noetherian. See the cited Stacks Project page for more details.{{Cite web|title=Section 28.9 (02IR): Regular schemes|url=https://stacks.math.columbia.edu/tag/02IR|access-date=2022-02-18|website=stacks.math.columbia.edu|publisher=The Stacks project}} Every smooth scheme is regular, and every regular scheme of finite type over a perfect field is smooth.{{citation|title=Introduction to algebraic geometry and algebraic groups|volume=39|series=North-Holland Mathematics Studies|publisher=North-Holland|first=Michel|last=Demazure|authorlink=Michel Demazure|year=1980|isbn=9780080871509|at=Proposition 3.2, p. 168|url=https://books.google.com/books?id=RDKRyP00aoMC&pg=PA168}}.

For an example of a regular scheme that is not smooth, see {{section link|Geometrically regular ring#Examples}}.