Regular scheme

Not to be confused with regular embedding.

In algebraic geometry, a regular scheme is a locally Noetherian scheme whose local rings are regular everywhere.^[1][2] Every smooth scheme is regular, and every regular scheme of finite type over a perfect field is smooth.^[3]

For an example of a regular scheme that is not smooth, see Geometrically regular ring § Examples.

See also

• Étale morphism
• Dimension of an algebraic variety
• Glossary of scheme theory
• Smooth completion