nLab
formally smooth morphism

We should distinguish

• Cuntz–Quillen formal smoothness in noncommutative setup (see quasi-free algebra)

• the Grothendieck’s notion of formal smoothness for commutative rings, schemes, morphisms (or even more general functors ${\mathrm{Scheme}}^{\mathrm{op}}\to \mathrm{Set}$) which will be discussed here.

In EGA IV, Grothendieck defines formal smoothness in the generality of relative setup: for morphisms. Formal smoothness is weaker than smoothness, and there are characterizations when formally smooth morphisms are smooth.

Definition (EGAIV${}_4$ 17.1.1)

A morphism $f:X\to Y$ is formally smooth if it satisfies the infinitesimal lifting property: for every ring $A$ and nilpotent ideal $I\subset A$ and morphism $\mathrm{Spec}(A)\to Y$ the induced map

$${\mathrm{Hom}}_Y(\mathrm{Spec}(A),X)\to {\mathrm{Hom}}_Y(\mathrm{Spec}(A/I),X)$$Hom_Y(Spec(A), X)\to Hom_Y(Spec(A/I),X)

is surjective.

Smoothness versus formal smoothness (EGAIV${}_4$ 17.5.2 and 17.15.15)

For a morphism $f:X\to Y$ of schemes, and $x$ a point of $X$, the following are equivalent

(i) $f$ is smooth at $x$

(ii) $f$ is locally of finite presentation at $x$ and there is an open neighborhood $U\subset X$ of $x$ such that $f{\mid }_U:U\to Y$ is formally smooth

(iii) $f$ is flat at $x$, locally of finite presentation at $x$ and the sheaf of Kähler differentials ${\Omega }_{X/Y}$ is locally free in a neighborhood of $x$

The relative dimension of $f$ at $x$ will equal the rank of the module of Kähler differentials.

For EGAIV${}_4$ (Publ. IHÉS, 32 (1967), p. 5-361) see numdam.

formally smooth scheme

A scheme $S$, i.e. a scheme over the ground ring $k$, is a formally smooth scheme if the corresponding morphism $S\to \mathrm{Spec}(k)$ is a formally smooth morphism.

There is also an interpretation of formal smoothness via the formalism of Q-categories.