smooth morphism of schemes


This entry is about smooth morphisms of schemes. There are many notions of smoothness in algebra and algebraic geometry, many under the name of (such and such) regularity (nonsingularity); even in EGA there are something like 11 notions, one of which is called the smooth morphism of schemes. In the literature there are sometimes even small variations of the latter (e.g. whether we allow globally varying dimension of the smooth morphism or not). In nnLab a prominent role is played by the formal smoothness, which is weaker than smoothness.


Smooth morphism is a relativization of the notion of a smooth scheme.


A morphism f:XYf:X\to Y of schemes is smooth if


Smoothness of a morphism is a higher dimensional analogue of the notion of a morphism being étale (which is a smooth morphism of relative dimension 00), but stronger than the notion of formal smoothness.

Smoothness versus formal smoothness

For a morphism f:XYf:X\to Y of schemes, and xx a point of XX, the following are equivalent

(i) ff is a smooth morphism at xx

(ii) ff is locally of finite presentation at xx and there is an open neighborhood UXU\subset X of xx such that f| U:UYf|_U: U\to Y is formally smooth

(iii) ff is flat at xx, locally of finite presentation at xx and the sheaf of Kähler differentials Ω X/Y\Omega_{X/Y} is locally free in a neighborhood of xx

The relative dimension of ff at xx will equal the rank of the module of Kähler differentials.

This is (EGAIV 4{}_4 17.5.2 and 17.15.15)

A smooth morphism of relative dimension 0 is an étale morphism.

See also formally smooth morphism.


  • Arthur Ogus, Smooth morphisms (pdf)

  • Peter Bruin, Smooth morphisms (ps)

Last revised on May 25, 2015 at 17:48:23. See the history of this page for a list of all contributions to it.