Smooth scheme

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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 is smooth for any algebraic closure 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 is smooth of relative dimension n.


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}}[citation needed] }}

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 of smooth schemes are:


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


See also