# Regular scheme

In algebraic geometry, a **regular scheme** is a scheme whose local rings are regular everywhere.[1] Every smooth scheme is regular, and every regular scheme over a perfect field is smooth.[2]

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

## References

- Hartshorne, Robin (1977),
*Algebraic Geometry*, Graduate Texts in Mathematics,**52**, Springer, p. 238, ISBN 9780387902449. - Demazure, Michel (1980),
*Introduction to algebraic geometry and algebraic groups*, North-Holland Mathematics Studies,**39**, North-Holland, Proposition 3.2, p. 168, ISBN 9780080871509.

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