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 an algebraic 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, an algebraic scheme X over k is smooth of dimension n if and only if $X\to \operatorname {Spec} k$ is smooth of relative dimension n.

Properties

A smooth scheme is connected if and only if it is irreducible. A connected smooth scheme is normal.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}} }}

Generic smoothness

A scheme X is said to be generically smooth of dimension n over k if X contains an 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.

Examples

Examples of smooth schemes are:

• A nonsingular variety.
• A linear algebraic group.

References

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

See also

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

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