nLab smooth scheme

Idea

Grothendieck developed in EGA a number of notions of smoothness for a scheme and, more generally, for a morphism of schemes. For algebraic varieties over a field, one already had a classical notion of a nonsingular variety.

Definition

For schemes

A scheme of finite type over a field $k$ is smooth if after extension of scalars from $k$ to the algebraic closure $\bar{k}$ it becomes a regular scheme, i.e. the stalks of its structure sheaf are regular local rings in the sense of commutative algebra.

A relative version of a smooth scheme is the notion of smooth morphism of schemes.

For affine schemes

Specifically, a finitely presented commutative associative algebra $A$ over a field $k$ is smooth if either of the following equivalent conditions holds

• the $A$-module $\Omega^1_k(A)$ of Kähler differential forms is a projective object in $A Mod$;

• with $A$ regarded as an $A$-bimodule in the evident way, it has a projective resolution.

Related concepts

• Bondal-Orlov reconstruction theorem

References

For commutative $k$-algebras a discussion is for instance around theorem 9.1.2 in

• Victor Ginzburg, Lectures on noncommutative geometry (arXiv:math/0506603)