Contents

# Contents

## Idea

The notion of formally étale morphism between schemes.

## Definition

Then we give the abstract

### Explicitly in components

###### Definition

A morphism of commutative rings $R \hookrightarrow A$ is called formally étale if for every ring $B$ and for every nilpotent ideal $I \subset B$ and for every commuting diagram of the form

$\array{ B/I &\leftarrow& A \\ \uparrow && \uparrow \\ B &\leftarrow& R }$

there is a unique diagonal morphism

$\array{ B/I &\leftarrow& A \\ \uparrow &\swarrow& \uparrow \\ B &\leftarrow& R }$

that makes both triangles commute.

By formal duality and locality this yields a notion of formally étale morphisms of affine varieties and of schemes.

### Characterization by reduction/infinitesimal shape

###### Definition

Write $CRing_{fin}$ for the category of finitely generated commutative rings and write $CRing_{fin}^{ext}$ for the category of infinitesimal ring extensions. Write

$Red \;\colon\; CRing_{fin}^{ext} \longrightarrow CRing_{fin}$

for the functor which sends an infinitesimal ring extension to the underlying commutative ring (in the maximal case this sends a commutative ring to its reduced ring, whence the name of the functor), and write

$i \;\colon\; CRing_{fin} \hookrightarrow CRing_{fin}^{ext}$

for the full subcategory inclusion that regards a ring as the trivial infinitesimal extension over itself.

###### Proposition

$(Red \dashv \int_{inf} \dashv \flat_{inf}) \;\colon\; PSh((CRing_{fin}^{ext})^{op}) \longrightarrow PSh((CRing_{fin}^{ext})^{op})$

where the left adjoint comonad $Red$ is given on representables by the reduction functor of def. (followed by the inclusion).

This statement and the following prop. is a slight paraphrase of an observation due to (Kontsevich-Rosenberg 04).

###### Proof

The functors from def. form an adjoint pair $(Red \dashv i)$ because an extension element can only map to an extension element; so for $\widehat R \to R$ an infinitesimal ring extension of $R = Red(\widehat R)$, and for $S$ a commutative ring with $i(S) = (S \to S)$ its trivial extension, there is a natural isomorphism

$Hom_{CRing_{fin}^{ext}}(\widehat R, i(S)) \simeq Hom_{CRing_{fin}}(R,S) \,.$

This exhibits $CRing_{fin}$ as a reflective subcategory of $CRing_{fin}^{ext}$.

$(Red \dashv i) \;\colon\; CRing_{fin} \stackrel{\overset{Red}{\leftarrow}}{\underset{i}{\hookrightarrow}} CRing_{fin}^{ext} \,.$

$PSh(CRing_{fin}^{op}) \stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^\ast = Red_!}{\leftarrow}}{\stackrel{\overset{Red^\ast}{\hookrightarrow}}{\underset{Red_\ast}{\leftarrow}}}} PSh((CRing_{fin}^{ext})^{op}) \,.$

The adjoint triple to be shown is obtained from composing these adjoints pairwise.

That $Red$ coincides with the reduction functor on representables is a standard property of left Kan extension (see here for details).

###### Remark

These considerations make sense in the general abstract context of “differential cohesion” where the adjoint triple of prop. would be called:

(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality).

Due to the full subcategory inclusion $i_!$ in the proof of prop. we may equivalently regard presheaves on $(CRing_{fin})^{op}$ (e.g. schemes) as presheaves on $(CRing_{fin}^{ext})^{op}$ (e.g. formal schemes). This is what we do implicitly in the following.

###### Proposition

A morphism $f \;\colon\; Spec A \to Spec R$ in $CRing_{fin}^{op} \hookrightarrow PSh(CRing_{fin}^{op})$ is formally étale, def. , precisely if it is $\int_{inf}$-modal relative $Spec R$, hence if the naturality square of the infinitesimal shape modality-unit

$\array{ Spec A &\longrightarrow& \int_{inf} Spec A \\ \downarrow && \downarrow \\ Spec R &\longrightarrow& \int_{inf} Spec R }$

is a pullback square.

###### Proof

Evaluated on $I \hookrightarrow R \to R/I \in CRing_{fin}^{ext}$ any object, by the Yoneda lemma and the $(Red \dashv \int_{inf})$-adjunction the naturality square becomes

$\array{ CRing(A,B) &\longrightarrow& CRing(A,B/I) \\ \downarrow && \downarrow \\ CRing(R,B) &\longrightarrow& CRing(R,B/I) } \,.$

in Set. Chasing elements through this shows that this is a pullback precisely if the condition in def. holds.

The basic stability property of étale morphisms, which we need in the following, immediately follows from this characterization:

###### Proposition

For $\stackrel{f}{\to} \stackrel{g}{\to}$ two composable morphisms, then

1. if $f$ and $g$ are both (formally) étale, then so is their composite $g \circ f$;

2. if $g$ and $g\circ f$ are (formally) étale, then so is $f$;

3. the pullback of a (formally) étale morphism along any morphism is again (formally) étale.

###### Proof

With prop. this is equivalently the statement of the pasting law for pullback diagrams.

## Properties

### Relation to étale morphism

Formally étale morphisms of schemes which are in addition locally of finite presentation are equivalently étale morphisms of schemes.

Relaxing this finiteness condition yields the notion of weakly étale morphisms.

étale morphism$\Rightarrow$ pro-étale morphism $\Rightarrow$ weakly étale morphism $\Rightarrow$ formally étale morphism

The traditional formulation is for instance in

The characterization via a (reduction modality $\dashv$ infinitesimal shape modality) is more or less explicit in

The formalization as differential cohesion is discussed in