Differential dynamic microscopy

2013-11-28T15:30:52Z

143.167.229.18:

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 ''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}}

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Differential dynamic microscopy