Smooth scheme

In algebraic geometry, a smooth scheme X of dimension n over an algebraically closed field k is an algebraic scheme that is regular and has dimension n. More generally, an algebraic scheme over a field k is said to be smooth if $X\times _{k}{\overline {k}}$ is smooth for any algebraic closure ${\overline {k}}$ of k.

If k is perfect, then a scheme over k is smooth if and only if it is regular.

There is also a notion of a "smooth morphism" between schemes, and the above definition coincides with it. That is, a scheme X over k is smooth of dimension n if and only if $X\to \operatorname {Spec} k$ is smooth of relative dimension n.

Generic smoothness

A scheme X is said to be generically smooth of dimension n over k if X contains a open dense subset that is smooth of dimension n over k. Any integral scheme over a perfect field (in particular an algebraically closed field) is generically smooth.

References

• D. Gaitsgory's notes on flatness and smoothness at http://www.math.harvard.edu/~gaitsgde/Schemes_2009/BR/SmoothMaps.pdf
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

See also

• Étale morphism
• Dimension (scheme)
• Glossary of scheme theory