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)

Revised on February 8, 2013 12:08:29 by Urs Schreiber (