# Smooth scheme

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*.

## 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=
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## 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

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

## References

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