Formally smooth map

In algebraic geometry and commutative algebra, a ring homomorphism f : A → B {\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 N ⊆ C {\displaystyle N\subseteq C} , any A-algebra homomorphism B → C / N {\displaystyle B\to C/N} may be lifted to an A-algebra map B → C {\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 f : X → S {\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 k [ ε ] / ( ε 3 ) → k [ ε ] / ( ε 2 ) {\displaystyle k[\varepsilon ]/(\varepsilon ^{3})\to k[\varepsilon ]/(\varepsilon ^{2})} the infinitesimal lifting criterion can be described using the commutative square

X ← Spec ( k [ ε ] ( ε 2 ) ) ↓ ↓ S ← Spec ( k [ ε ] ( ε 3 ) ) {\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 X , S ∈ S c h / S {\displaystyle X,S\in Sch/S} . For example, if

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

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

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

sending

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

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

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

is of the form

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

and x y ↦ ε 2 + ( a + b ) ε 3 = ε 2 {\displaystyle xy\mapsto \varepsilon ^{2}+(a+b)\varepsilon ^{3}=\varepsilon ^{2}} , there cannot be an infinitesimal lift since this is non-zero, hence X ∈ S c h / k {\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