nLab formally smooth scheme

Idea

A formally smooth scheme is a scheme $X$ with the property that every map into it admits an infinitesimal lift :

for any “test space”, i.e. an affine scheme $U=Spec(A)$ with underlying reduced scheme $Red(U)$ (obtained from $U$ by removing all infinitesimal directions, compare formal scheme) every map $Red(U) \to X$ lifts to a map $U \to X$.

Similarly, one can talk about formally smooth algebraic spaces, algebraic stacks and so on.

Definition

A scheme $X$ over some ground field $k$, given by a morphism $X \to Spec(k)$, is formally smooth if that morphism is a formally smooth morphism, as described there.

If the scheme $X$ is regarded as an absolute (as opposed to relative) scheme, then the morphism $X \to Spec(\mathbb{Z})$ is required to be a formally smooth morphism for $X$ to be formally smooth.

(Nowadays people will also start to consider schemes over the field with one element…).

General definition

There is an interpretation of formal smoothness of (set-valued) functors as an epipresheaf condition with respect to Sasha Rosenberg’s generalization of Grothendieck topologies, so-called Q-category formalism, namely the Q-category of nilpotent (infinitesimal) thickenings. For more on this see formally smooth morphism.

References

Discussion of formally smooth schemes in the general context of formally smooth morphisms is at

• Maxim Kontsevich, Alexander Rosenberg, Noncommutative spaces, preprint MPI-2004-35 (ps)

• T. Brzeziński, Notes on formal smoothness, in: Modules and Comodules (series Trends in Mathematics). T Brzeziński, JL Gomez Pardo, I Shestakov, PF Smith (eds), Birkhäuser, Basel, 2008, pp. 113-124 (doi, arXiv:0710.5527)