formally smooth scheme



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

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

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


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

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


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)

Last revised on February 8, 2013 at 12:08:29. See the history of this page for a list of all contributions to it.