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In [[algebraic geometry]], a '''smooth scheme''' ''X'' of dimension ''n'' over an [[algebraically closed field]] ''k'' is an algebraic scheme<ref>By "algebraic scheme" we mean a scheme of finite type over a field.</ref> that is [[Glossary_of_scheme_theory#Properties_of_schemes|regular]] and has dimension ''n''. More generally, an algebraic scheme over a field ''k'' is said to be smooth if <math>X \times_k \overline{k}</math> is smooth for any algebraic closure <math>\overline{k}</math> of ''k''.
 
If ''k'' is [[perfect field|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 <math>X \to \operatorname{Spec} k</math> is smooth of relative dimension&nbsp;''n''.
 
== Properties ==
A smooth scheme is connected if and only if it is irreducible. A connected smooth scheme is [[normal scheme|normal]].{{citation needed|date=March 2012}}
 
== 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]].
 
== Notes ==
{{reflist}}
 
== References ==
* [[Dennis Gaitsgory|D. Gaitsgory]]'s notes on flatness and smoothness at http://www.math.harvard.edu/~gaitsgde/Schemes_2009/BR/SmoothMaps.pdf
*{{Hartshorne AG}}
 
== See also ==
*[[Étale morphism]]
*[[Dimension (scheme)]]
*[[Glossary of scheme theory]]
*[[Smooth completion]]
 
[[Category:Algebraic geometry]]
[[Category:Scheme theory| ]]
 
 
{{geometry-stub}}

Latest revision as of 16:30, 28 November 2013

In algebraic geometry, a smooth scheme X of dimension n over an algebraically closed field k is an algebraic scheme[1] that is regular and has dimension n. More generally, an algebraic scheme over a field k is said to be smooth if X×kk‾ is smooth for any algebraic closure 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→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.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

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:

Notes

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References

See also




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  1. ↑ By "algebraic scheme" we mean a scheme of finite type over a field.
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