Formally smooth map 📖 Wikipedia

In algebraic geometry and commutative algebra, a ring homomorphism ${\displaystyle f:A\to B}$ 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 ${\displaystyle N\subseteq C}$, any A-algebra homomorphism ${\displaystyle B\to C/N}$ may be lifted to an A-algebra map ${\displaystyle B\to C}$. 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.

All smooth morphisms ${\displaystyle f:X\to S}$ are equivalent to morphisms locally of finite presentation which are formally smooth. Hence formal smoothness is a slight generalization of smooth morphisms.^[3]

One method for detecting formal smoothness of a scheme is using infinitesimal lifting criterion. For example, using the truncation morphism ${\displaystyle k[\varepsilon ]/(\varepsilon ^{3})\to k[\varepsilon ]/(\varepsilon ^{2})}$ the infinitesimal lifting criterion can be described using the commutative square

${\displaystyle {\begin{matrix}X&\leftarrow &{\text{Spec}}\left({\frac {k[\varepsilon ]}{(\varepsilon ^{2})}}\right)\\\downarrow &&\downarrow \\S&\leftarrow &{\text{Spec}}\left({\frac {k[\varepsilon ]}{(\varepsilon ^{3})}}\right)\end{matrix}}}$

where ${\displaystyle X,S\in Sch/S}$. For example, if

${\displaystyle X={\text{Spec}}\left({\frac {k[x,y]}{(xy)}}\right)}$ and ${\displaystyle Y={\text{Spec}}(k)}$

then consider the tangent vector at the origin ${\displaystyle (0,0)\in X(k)}$ given by the ring morphism

${\displaystyle {\frac {k[x,y]}{(xy)}}\to {\frac {k[\varepsilon ]}{(\varepsilon ^{2})}}}$

sending

${\displaystyle {\begin{aligned}x&\mapsto \varepsilon \\y&\mapsto \varepsilon \end{aligned}}}$

Note because ${\displaystyle xy\mapsto \varepsilon ^{2}=0}$, this is a valid morphism of commutative rings. Then, since a lifting of this morphism to

${\displaystyle {\text{Spec}}\left({\frac {k[\varepsilon ]}{(\varepsilon ^{3})}}\right)\to X}$

is of the form

${\displaystyle {\begin{aligned}x&\mapsto \varepsilon +a\varepsilon ^{2}\\y&\mapsto \varepsilon +b\varepsilon ^{2}\end{aligned}}}$

and ${\displaystyle xy\mapsto \varepsilon ^{2}+(a+b)\varepsilon ^{3}=\varepsilon ^{2}}$, there cannot be an infinitesimal lift since this is non-zero, hence ${\displaystyle X\in Sch/k}$ 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.

• Dual number
• Smooth morphism
• Deformation theory

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