A space is called formally smooth if every morphisms into it has all possible infinitesimal extensions.
(If there is at most one extension per infinitesimal extension of with no guarantee of existence it is called a formally unramified morphism. If the thickenings exist uniquely, it is called a formally etale morphism).
Traditionally this has considered in the context of geometry over formal duals of rings and associative algebras. This we discuss in the section (Concrete notion). But generally the notion makes sense in any context of infinitesimal cohesion. This we discuss in the section General abstract notion.
General abstract notion
be an adjoint triple of functors with a full and faithful functor that preserves the terminal object.
We may think of this as exhibiting infinitesimal cohesion (see there for details, but notice that in the notation used there we have , and ).
We think of the objects of as cohesive spaces and of the objects of as such cohesive spaces possibly equipped with infinitesimal extension.
As a class of examples that is useful to keep in mind consider a Q-category of infinitesimal thickening of rings and let
be the corresponding Q-category of copresheaves.
For any such setup there is a canonical natural transformation
Details of this are in the section Adjoint quadruples at cohesive topos.
From this we get for every morphism in a canonical morphism
This appears as (KontsevichRosenberg, def. 5.1, prop. 18.104.22.168).
The dual notion, where the above morphism is a monomorphism is that of formally unramified morphism. If both conditions hold, hence if the above morphism is an isomorphism, one speaks of a formally étale morphism.
An object is called formally smooth if the morphism to the terminal object is formally smooth.
The object is formally smooth precisely if
is an effective epimorphism.
This appears as (KontsevichRosenberg, def. 5.3.2).
Formally smooth morphisms are closed under composition.
This appears as (KontsevichRosenberg, prop. 5.4).
Over commutative rings
Let be a field and let be the category of commutative associative algebras over . Write
for the presheaf topos over the opposite category . This is the context in which schemes and algebraic spaces over live.
A morphism in is formally smooth if it satisfies the infinitesimal lifting property: for every algebra and nilpotent ideal and morphism the induced map
This is due to (EGAIV 17.1.1)
An object is formally smooth in the concrete sense of def. 3 precisely if it is so in the abstract sense of def. 1.
This appears as (KontsevichRosenbergSpaces, 4.1).
Smoothness versus formal smoothness
For a morphism of schemes, and a point of , the following are equivalent
(i) is a smooth morphism of schemes at
(ii) is locally of finite presentation at and there is an open neighborhood of such that is formally smooth
(iii) is flat at , locally of finite presentation at and the sheaf of Kähler differentials is locally free in a neighborhood of
The relative dimension of at will equal the rank of the module of Kähler differentials.
This is (EGAIV 17.5.2 and 17.15.15)
Formally smooth scheme
A scheme , i.e. a scheme over the ground ring , is a formally smooth scheme if the corresponding morphism is a formally smooth morphism.
There is also an interpretation of formal smoothness via the formalism of Q-categories.
Over noncommutative algebras
Let be a field and let be the category of associative algebras over (not necessarily commutative). Let
be the Q-category of infinitesimal thickenings of -algebras (whose objects are surjective -algebra morphisms with nilpotent kernel). Notice that the presheaf topos
is the context in which noncommutative schemes live. Let be the copresheaf Q-category over .
Let be a morphism in such that is a separable algebra. Write for the corresponding morphism in .
This is formally smooth in the sense of def. 1 precisely if the -module
is a projective object in Mod.
In particular, setting we have that an object of the form is formally smooth according to def. 2 precisely if is projective. This is what in (CuntzQuillen) is called the condition for a quasi-free algebra.
formally smooth morphism and formally unramified morphism formally étale morphism.
The definition over commutative rings is in
The definition over noncommutative algebras is in
- J. Cuntz, D. Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8 (1995), 251–289.
The general abstract definition and its relation to the standard definitions is in
A. Ardizzoni, Separable functors and formal smoothness, J. K-Theory 1 (2008), no. 3, 535–582, math.QA/0407095, doi, MR2009k:16069
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)
Guillermo Cortiñas, The structure of smooth algebras in Kapranov’s framework for noncommutative geometry, J. of Algebra 281 (2004) 679-694, math.RA/0002177