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Contents

Idea

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

(If there is at most one extension per infinitesimal extension of $Y$ 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

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

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

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

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

be the corresponding Q-category of copresheaves.

For any such setup there is a canonical natural transformation

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

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

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

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

A morphism $f : X \to Y$ in $\mathbf{H}$ 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 $X \in \mathbf{H}$ is called formally smooth if the morphism $X \to *$ to the terminal object is formally smooth.

Proposition

The object $X$ is formally smooth precisely if

$u^* X \to u^! X$

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 $k$ be a field and let $CAlk_k$ be the category of commutative associative algebras over $k$. Write

$\mathbf{H} = [CAlg_k, Set]$

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

Definition

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

$Hom_Y(Spec(A), X)\to Hom_Y(Spec(A/I),X)$

is surjective.

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

Proposition

An object $X \in [CAlg_k, Set]$ 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 $f:X\to Y$ of schemes, and $x$ a point of $X$, the following are equivalent

(i) $f$ is a smooth morphism of schemes at $x$

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

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

The relative dimension of $f$ at $x$ will equal the rank of the module of Kähler differentials.

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

Formally smooth scheme

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

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

Over noncommutative algebras

Let $k$ be a field and let $Alg_k$ be the category of associative algebras over $k$ (not necessarily commutative). Let

$Alg_k^{inf} : \bar A \to Alg_k$

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

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

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

Proposition

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

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

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

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

In particular, setting $R = k$ we have that an object of the form $Spec S$ is formally smooth according to def. 2 precisely if $\Omega^1(S|k)$ 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

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

References

The definition over commutative rings is in

• EGAIV${}_4$, Publ. IHÉS 32 (1967), p. 5-361, numdam

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