smooth scheme



Grothendieck developed in EGA a number of notions of smoothness for a scheme and, more generally, for a morphism of schemes. For algebraic varieties over a field, one already had a classical notion of a nonsingular variety.


For schemes

A scheme of finite type over a field kk is smooth if after extension of scalars from kk to the algebraic closure k¯\bar{k} it becomes a regular scheme, i.e. the stalks of its structure sheaf are regular local rings in the sense of commutative algebra.

A relative version of a smooth scheme is the notion of smooth morphism of schemes.

For affine schemes

Specifically, a finitely presented commutative associative algebra AA over a field kk is smooth if either of the following equivalent conditions holds


For commutative kk-algebras a discussion is for instance around theorem 9.1.2 in

Last revised on November 20, 2012 at 20:18:29. See the history of this page for a list of all contributions to it.