nLab
formally smooth morphism

Context

Étale morphisms

Geometry

=–

Contents

Idea

A space XX is called formally smooth if every morphisms YXY \to X into it has all possible infinitesimal extensions.

(If there is at most one extension per infinitesimal extension of YY 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

Definition

Let

Hu !u *u *H th \mathbf{H} \stackrel{\overset{u^*}{\hookrightarrow}}{\stackrel{\overset{u_*}{\leftarrow}}{\underset{u^!}{\to}}} \mathbf{H}_{th}

be an adjoint triple of functors with u *u^* 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 u *=i !u^* = i_!, u *=i *u_* = i^* and u !=i *u^! = i_*).

We think of the objects of H\mathbf{H} as cohesive spaces and of the objects of H th\mathbf{H}_{th} 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 (codϵdom):A¯A(cod \dashv \epsilon \dashv dom) : \bar A \to A of infinitesimal thickening of rings and let

((u *u *u !):H thH):=([dom,Set][ϵ,Set][codom,Set]:[A¯,Set][A,Set]) ((u^* \dashv u_* \dashv u^!) : \mathbf{H}_{th} \to \mathbf{H}) := ([dom,Set] \dashv [\epsilon, Set] \dashv [codom,Set] : [\bar A, Set] \to [A,Set])

be the corresponding Q-category of copresheaves.

For any such setup there is a canonical natural transformation

u *u !. u^* \to u^! \,.

Details of this are in the section Adjoint quadruples at cohesive topos.

From this we get for every morphism f:XYf : X \to Y in H\mathbf{H} a canonical morphism

(1)u *Xu *Y u !Yu !X. u^* X \to u^* Y \prod_{u^! Y} u^! X \,.
Definition

A morphism f:XYf : X \to Y in H\mathbf{H} is called formally smooth if (1) is an effective epimorphism.

This appears as (KontsevichRosenberg, def. 5.1, prop. 5.3.1.1).

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.

Definition

An object XHX \in \mathbf{H} is called formally smooth if the morphism X*X \to * to the terminal object is formally smooth.

Proposition

The object XX is formally smooth precisely if

u *Xu !X u^* X \to u^! X

is an effective epimorphism.

This appears as (KontsevichRosenberg, def. 5.3.2).

Properties

Proposition

Formally smooth morphisms are closed under composition.

This appears as (KontsevichRosenberg, prop. 5.4).

Concrete notion

Over commutative rings

Let kk be a field and let CAlk kCAlk_k be the category of commutative associative algebras over kk. Write

H=[CAlg k,Set] \mathbf{H} = [CAlg_k, Set]

for the presheaf topos over the opposite category CAlg k opCAlg_k^{op}. This is the context in which schemes and algebraic spaces over kk live.

Definition

A morphism f:XYf :X\to Y in H=[CAlg k,Set]\mathbf{H} = [CAlg_k, Set] is formally smooth if it satisfies the infinitesimal lifting property: for every algebra AA and nilpotent ideal IAI\subset A and morphism Spec(A)YSpec(A)\to Y the induced map

Hom Y(Spec(A),X)Hom Y(Spec(A/I),X) Hom_Y(Spec(A), X)\to Hom_Y(Spec(A/I),X)

is surjective.

This is due to (EGAIV 4{}_4 17.1.1)

Proposition

An object X[CAlg k,Set]X \in [CAlg_k, Set] 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 f:XYf:X\to Y of schemes, and xx a point of XX, the following are equivalent

(i) ff is a smooth morphism of schemes at xx

(ii) ff is locally of finite presentation at xx and there is an open neighborhood UXU\subset X of xx such that f| U:UYf|_U: U\to Y is formally smooth

(iii) ff is flat at xx, locally of finite presentation at xx and the sheaf of Kähler differentials Ω X/Y\Omega_{X/Y} is locally free in a neighborhood of xx

The relative dimension of ff at xx will equal the rank of the module of Kähler differentials.

This is (EGAIV 4{}_4 17.5.2 and 17.15.15)

Formally smooth scheme

A scheme SS, i.e. a scheme over the ground ring kk, is a formally smooth scheme if the corresponding morphism SSpec(k)S \to Spec(k) is a formally smooth morphism.

There is also an interpretation of formal smoothness via the formalism of Q-categories.

Over noncommutative algebras

Let kk be a field and let Alg kAlg_k be the category of associative algebras over kk (not necessarily commutative). Let

Alg k inf:A¯Alg k Alg_k^{inf} : \bar A \to Alg_k

be the Q-category of infinitesimal thickenings of kk-algebras (whose objects are surjective kk-algebra morphisms with nilpotent kernel). Notice that the presheaf topos

H:=[Alg k,Set] \mathbf{H} := [Alg_k, Set]

is the context in which noncommutative schemes live. Let H thQ\mathbf{H}_{th} \to \mathbf{Q} be the copresheaf Q-category over Alg k infAlg_k^{inf}.

Proposition

Let f:RSf : R \to S be a morphism in Alg kAlg_k such that RR is a separable algebra. Write Specf:SpecSSpecRSpec f : Spec S \to Spec R for the corresponding morphism in H=[Alg k,Set]\mathbf{H} = [Alg_k, Set].

This SpecfSpec f is formally smooth in the sense of def. 1 precisely if the S kS opS \otimes_k S^{op}-module

Ω S|R 1:=ker(R kRmultRfS) \Omega^1_{S|R} := ker ( R \otimes_k R \stackrel{mult}{\to} R \stackrel{f}{\to} S)

is a projective object in S kS opS \otimes_k S^{op}Mod.

In particular, setting R=kR = k we have that an object of the form SpecSSpec S is formally smooth according to def. 2 precisely if Ω 1(S|k)\Omega^1(S|k) is projective. This is what in (CuntzQuillen) is called the condition for a quasi-free algebra.

formally smooth morphism and formally unramified morphism \Rightarrow formally étale morphism.

cohesion

tangent cohesion

differential cohesion

References

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

See also

Revised on January 6, 2013 01:54:10 by Urs Schreiber (82.113.106.198)