Formally smooth map explained

is called formally smooth (from French: Formellement lisse) if it satisfies the following infinitesimal lifting property: , 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.

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

Smooth morphisms

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.^[1]

Non-example

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

\begin{matrix} X&\leftarrow&Spec\left(

	k[\varepsilon]
	(\varepsilon2)

\right)\\ \downarrow&&\downarrow\\ S&\leftarrow&Spec\left(

	k[\varepsilon]
	(\varepsilon3)

\right) \end{matrix}

where . For example, if

X=Spec\left(

	k[x,y]
	(xy)

\right)

and

Y=Spec(k)

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

	k[x,y]
	(xy)

\to

	k[\varepsilon]
	(\varepsilon2)

sending

\begin{align} x&\mapsto\varepsilon\\ y&\mapsto\varepsilon \end{align}

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

Spec\left(	k[\varepsilon]
	(\varepsilon3)

\right)\toX

is of the form

\begin{align} x&\mapsto\varepsilon+a\varepsilon²\\ y&\mapsto\varepsilon+b\varepsilon^{2 \end{align}}

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

• Dual number
• Smooth morphism
• Deformation theory

External links

• Formally smooth with smooth fibers, but not smooth https://mathoverflow.net/q/333596
• Formally smooth but not smooth https://mathoverflow.net/q/195

Notes and References

1. Web site: Lemma 37.11.7 (02H6): Infinitesimal lifting criterion—The Stacks project. stacks.math.columbia.edu. 2020-04-07.