In algebraic geometry and commutative algebra, a ring homomorphism is called formally smooth (from French: Formellement lisse) if it satisfies the following infinitesimal lifting property:

Suppose B is given the structure of an A-algebra via the map f. Given a commutative A-algebra, C, and a nilpotent ideal , any A-algebra homomorphism may be lifted to an A-algebra map . If moreover any such lifting is unique, then f is said to be formally étale.[1][2]

Formally smooth maps were defined by Alexander Grothendieck in Éléments de géométrie algébrique IV.

For finitely presented morphisms, formal smoothness is equivalent to usual notion of smoothness.

Examples

edit

Smooth morphisms

edit

All smooth morphisms   are equivalent to morphisms locally of finite presentation which are formally smooth. Hence formal smoothness is a slight generalization of smooth morphisms.[3]

Non-example

edit

One method for detecting formal smoothness of a scheme is using infinitesimal lifting criterion. For example, using the truncation morphism   the infinitesimal lifting criterion can be described using the commutative square

 

where  . For example, if

  and  

then consider the tangent vector at the origin   given by the ring morphism

 

sending

 

Note because  , this is a valid morphism of commutative rings. Then, since a lifting of this morphism to

 

is of the form

 

and  , there cannot be an infinitesimal lift since this is non-zero, hence   is not formally smooth. This also proves this morphism is not smooth from the equivalence between formally smooth morphisms locally of finite presentation and smooth morphisms.

See also

edit

References

edit
  1. ^ Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie". Publications Mathématiques de l'IHÉS. 20: 5–259. doi:10.1007/bf02684747. MR 0173675.
  2. ^ Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS. 32: 5–361. doi:10.1007/bf02732123. MR 0238860.
  3. ^ "Lemma 37.11.7 (02H6): Infinitesimal lifting criterion—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-04-07.
edit