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Formally smooth map

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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

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Smooth morphisms

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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

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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

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References

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  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.
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