formally étale morphism of schemes



Étale morphisms




The notion of formally étale morphism between schemes.


We first state the traditional

Then we give the abstract

Explicitly in components


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

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

there is a unique diagonal morphism

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

that makes both triangles commute.

(e.g. Stacks Project 57.9, 57.12)

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

Characterization by reduction/infinitesimal shape


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

Red:CRing fin extCRing fin 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:CRing finCRing fin ext i \;\colon\; CRing_{fin} \hookrightarrow CRing_{fin}^{ext}

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


There is an adjoint triple of idempotent (co-)monads

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

where the left adjoint comonad RedRed 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).


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

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

This exhibits CRing finCRing_{fin} as a reflective subcategory of CRing fin extCRing_{fin}^{ext}.

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

Via Kan extension this adjoint pair induces an adjoint quadruple of functors on categories of presheaves

PSh(CRing fin op)Red *Red *i *=Red !i !PSh((CRing fin ext) op). 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 RedRed coincides with the reduction functor on representables is a standard property of left Kan extension (see here for details).


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


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

SpecA infSpecA SpecR infSpecR \array{ Spec A &\longrightarrow& \int_{inf} Spec A \\ \downarrow && \downarrow \\ Spec R &\longrightarrow& \int_{inf} Spec R }

is a pullback square.


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

CRing(A,B) CRing(A,B/I) CRing(R,B) CRing(R,B/I). \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:


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

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

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

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


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


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

Last revised on November 27, 2013 at 12:34:12. See the history of this page for a list of all contributions to it.