nLab smooth morphism of schemes

This entry is about smooth morphisms of schemes. There are many notions of smoothness in algebra and algebraic geometry, many under the name of (such and such) regularity (nonsingularity); even in EGA there are something like 11 notions, one of which is called the smooth morphism of schemes. In the literature there are sometimes even small variations of the latter (e.g. whether we allow globally varying dimension of the smooth morphism or not). In $n$Lab a prominent role is played by the formal smoothness, which is weaker than smoothness.

Idea

Smooth morphism is a relativization of the notion of a smooth scheme.

Definition

A morphism $f:X\to Y$ of schemes is smooth if

• it is flat

• and finitely presented (cf. relativization in algebraic geometry)

• and if all the fibers $f^{-1}(y)$ where $y\in Y$ are smooth schemes over the corresponding residue fields $k_y$.

Properties

Smoothness of a morphism is a higher dimensional analogue of the notion of a morphism being étale (which is a smooth morphism of relative dimension $0$), but stronger than the notion of formal smoothness.

Smoothness versus formal smoothness

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

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

(ii) $f$ is locally of finite presentation at $x$ and there is an open neighborhood $U\subset X$ of $x$ such that $f|_U: U\to Y$ 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.

This is (EGAIV${}_4$ 17.5.2 and 17.15.15)

Related concepts

A smooth morphism of relative dimension 0 is an étale morphism.

See also formally smooth morphism.

• The lisse-étale site of a scheme $X$ consists of smooth morphisms $U \to X$.

References

• Arthur Ogus, Smooth morphisms (pdf)

• Peter Bruin, Smooth morphisms (ps)