Smooth scheme

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

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

References

• D. Gaitsgory's notes on flatness and smoothness at http://www.math.harvard.edu/~gaitsgde/Schemes_2009/BR/SmoothMaps.pdf

See also

• Dimension (scheme)
• Glossary of scheme theory