nLab étale topos

Definition

In the context of the geometry of schemes there is a traditional notion of étale morphism of schemes and an étale topos is a category of sheaves on the étale site of a scheme, consisting of covers by such étale morphisms. This traditional notion we discuss in

• Étale topos of a scheme.

More abstractly, given that étale morphisms of schemes may be characterized as modal morphisms with respect to an infinitesimal shape modality, one can consider étale toposes in every context of differential cohesion. This we discuss in

• Étale topos of a differentially cohesive object

Étale topos of a scheme

An étale topos is the sheaf topos over an étale site, hence over a site whose “open subsets” are étale morphisms into the base space. The intrinsic cohomology of an étale (∞,1)-topos is étale cohomology.

More generally there is the pro-étale topos over a pro-étale site, which is a bit better behaved. In particular the intrinsic cohomology of a pro-étale (∞,1)-topos includes the Weil cohomology theory ℓ-adic cohomology.

Generally, given that an étale morphism of schemes is a formally étale morphism subject to a size constraint on its fibers – for an actual étale morphism of schemes the fibers are finite sets in the suitable sense (formal duals to étale algebras) while for a pro-étale morphism of schemes they are pro-objects of such fibers – in a suitable ambient context (“differential cohesion”) one can drop all finiteness conditions and consider just opens given by formally étale morphisms as encoded by an infinitesimal shape modality. This we discuss below.

Étale topos of a differentially cohesive object

We discuss how in differential cohesion $\mathbf{H}_{th}$ every object $X$ canonically induces its étale topos $Sh_{\mathbf{H}_{th}}(X)$.

For $X \in \mathbf{H}_{th}$ any object in a differential cohesive $\infty$-topos, we formulate

1. the (∞,1)-topos denoted $\mathcal{X}$ or $Sh_\infty(X)$ of (∞,1)-sheaves over $X$, or rather of formally étale maps into $X$;

2. the structure (∞,1)-sheaf $\mathcal{O}_{X}$ of $X$.

The resulting structure is essentially that discussed (Lurie, Structured Spaces) if we regard $\mathbf{H}_{th}$ equipped with its formally étale morphisms, (def.), as a (large) geometry for structured (∞,1)-toposes.

One way to motivate this is to consider structure sheaves of flat differential forms. To that end, let $G \in Grp(\mathbf{H}_{th})$ a differential cohesive ∞-group with de Rham coefficient object $\flat_{dR}\mathbf{B}G$ and for $X \in \mathbf{H}_{th}$ any differential homotopy type, the product projection

$$X \times \flat_{dR} \mathbf{B}G \to X$$

regarded as an object of the slice (∞,1)-topos $(\mathbf{H}_{th})_{/X}$ almost qualifies as a “bundle of flat $\mathfrak{g}$-valued differential forms” over $X$: for $U \to X$ an cover (a 1-epimorphism) regarded in $(\mathbf{H}_{th})_{/X}$, a $U$-plot of this product projection is a $U$-plot of $X$ together with a flat $\mathfrak{g}$-valued de Rham cocycle on $X$.

This is indeed what the sections of a corresponding bundle of differential forms over $X$ are supposed to look like – but only if $U \to X$ is sufficiently spread out over $X$, hence sufficiently étale. Because, on the extreme, if $X$ is the point (the terminal object), then there should be no non-trivial section of differential forms relative to $U$ over $X$, but the above product projection instead reproduces all the sections of $\flat_{dR} \mathbf{B}G$.

In order to obtain the correct cotangent-like bundle from the product with the de Rham coefficient object, it needs to be restricted to plots out of suficiently étale maps into $X$. In order to correctly test differential form data, “suitable” here should be “formally”, namely infinitesimally. Hence the restriction should be along the full inclusion

$$(\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X}$$

of the formally étale maps (see def.) into $X$. Since on formally étale covers the sections should be those given by $\flat_{dR}\mathbf{B}G$, one finds that the corresponding “cotangent bundle” must be the coreflection along this inclusion. The following proposition establishes that this coreflection indeed exists.

Definition

For $X \in \mathbf{H}_{th}$ any object, write

$$(\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}$$

for the full sub-(∞,1)-category of the slice (∞,1)-topos over $X$ on those maps into $X$ which are formally étale, (see def.).

We also write $FEt_{\mathbf{X}}$ or $Sh_{\mathbf{H}}(X)$ for $(\mathbf{H}_{th})_{/X}^{fet}$.

Proposition

The inclusion $\iota$ of def. is both reflective as well as coreflective, hence it fits into an adjoint triple of the form

$$(\mathbf{H}_{th})_{/X}^{fet} \stackrel{\overset{L}{\leftarrow}}{\stackrel{\overset{\iota}{\hookrightarrow}}{\underset{Et}{\leftarrow}}} (\mathbf{H}_{th})_{/X} \,.$$

Proof

By the general discussion at reflective factorization system, the reflection is given by sending a morphism $f \colon Y \to X$ to $X \times_{\mathbf{\Pi}_{inf}(X)} \mathbf{\Pi}_{inf}(Y) \to Y$ and the reflection unit is the left horizontal morphism in

$$\array{ Y &\to& X \times_{\mathbf{\Pi}_{inf}(Y)} \mathbf{\Pi}_{inf}(Y) &\to& \mathbf{\Pi}_{inf}(Y) \\ & \searrow & \downarrow^{} && \downarrow^{\mathrlap{\mathbf{\Pi}_{inf}(f)}} \\ && X &\to& \mathbf{\Pi}_{inf}(X) } \,.$$

Therefore $(\mathbf{H}_{th})_{/X}^{fet}$, being a reflective subcategory of a locally presentable (∞,1)-category, is (as discussed there) itself locally presentable. Hence by the adjoint (∞,1)-functor theorem it is now sufficient to show that the inclusion preserves all small (∞,1)-colimits in order to conclude that it also has a right adjoint (∞,1)-functor.

So consider any diagram (∞,1)-functor $I \to (\mathbf{H}_{th})_{/X}^{fet}$ out of a small (∞,1)-category. Since the inclusion of $(\mathbf{H}_{th})_{/X}^{fet}$ is full, it is sufficient to show that the $(\infty,1)$-colimit over this diagram taken in $(\mathbf{H}_{th})_{/X}$ lands again in $(\mathbf{H}_{th})_{/X}^{fet}$ in order to have that $(\infty,1)$-colimits are preserved by the inclusion. Moreover, colimits in a slice of $\mathbf{H}_{th}$ are computed in $\mathbf{H}_{th}$ itself (this is discussed at slice category - Colimits).

Therefore we are reduced to showing that the square

$$\array{ \underset{\to_i}{\lim} Y_i &\to& \mathbf{\Pi}_{inf} \underset{\to_i}{\lim} Y_i \\ \downarrow && \downarrow \\ X &\to& \mathbf{\Pi}_{inf}(X) }$$

is an (∞,1)-pullback square. But since $\mathbf{\Pi}_{inf}$ is a left adjoint it commutes with the $(\infty,1)$-colimit on objects and hence this diagram is equivalent to

$$\array{ \underset{\to_i}{\lim} Y_i &\to& \underset{\to_i}{\lim} \mathbf{\Pi}_{inf} Y_i \\ \downarrow && \downarrow \\ X &\to& \mathbf{\Pi}_{inf}(X) } \,.$$

This diagram is now indeed an (∞,1)-pullback by the fact that we have universal colimits in the (∞,1)-topos $\mathbf{H}_{th}$, hence that on the left the component $Y_i$ for each $i \in I$ is the (∞,1)-pullback of $\mathbf{\Pi}_{inf}(Y_i) \to \mathbf{\Pi}_{inf}(X)$, by assumption that we are taking an $(\infty,1)$-colimit over formally étale morphisms.

Proposition

The $\infty$-category $(\mathbf{H}_{th})_{/X}^{fet}$ is an (∞,1)-topos and the canonical inclusion into $(\mathbf{H}_{th})_{/X}$ is a geometric embedding.

Proof

By prop. the inclusion $(\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X}$ is reflective with reflector given by the $(\mathbf{\Pi}_{inf}-equivalences , \mathbf{\Pi}_{inf}-closed)$ factorization system. Since $\mathbf{\Pi}_{inf}$ is a right adjoint and hence in particular preserves (∞,1)-pullbacks, the $\mathbf{\Pi}_{inf}$-equivalences are stable under pullbacks. By the discussion at stable factorization system this is the case precisely if the corresponding reflector preserves finite (∞,1)-limits. Hence the embedding is a geometric embedding which exhibits a sub-(∞,1)-topos inclusion.

Definition

For $X \in \mathbf{H}_{th}$ we speak of

$$\mathcal{X} \coloneqq Sh_{\mathbf{H}_{th}}(X) \coloneqq (\mathbf{H}_{th})_{/X}^{fet}$$

also as the (petit) (∞,1)-topos of $X$, or the étale topos of $X$.

Write

$$\mathcal{O}_X \colon \mathbf{H}_{th} \stackrel{(-) \times X}{\to} (\mathbf{H}_{th})_{/X} \stackrel{Et}{\to} (\mathbf{H}_{th})_{/X}^{fet}$$

for the composite (∞,1)-functor that sends any $A \in \mathbf{H}_{th}$ to the etalification, prop. , of the projection $A \times X \to X$.

We call $\mathcal{O}_X$ the structure sheaf of $X$.

Remark

For $X, A \in \mathbf{H}_{th}$ and for $U \to X$ a formally étale morphism in $\mathbf{H}_{th}$ (hence like an open subset of $X$), we have that

$$\begin{aligned} \mathcal{O}_{X}(A)(U) & \coloneqq Sh_{\mathbf{H}_{th}}(X)( U , \mathcal{O}_{X}(A) ) \\ & \coloneqq Sh_{\mathbf{H}_{th}}(X)( U , Et(X \times A) ) \\ & \simeq (\mathbf{H}_{th})_{/X}(U, X \times A) \\ & \simeq \mathbf{H}_{th}(U,A) \\ & \simeq A(U) \end{aligned} \,,$$

where we used the ∞-adjunction $(\iota \dashv Et)$ of prop. and the (∞,1)-Yoneda lemma.

This means that $\mathcal{O}_{X}(A)$ behaves as the sheaf of $A$-valued functions over $X$.

Since $\mathcal{O}_{X}$ is right adjoint to the forgetful functor

$$Sh_{\mathbf{H}}(X) \simeq (\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X} \stackrel{\underset{X}{\sum}}{\to} \mathbf{H}_{th}$$

it preserves (∞,1)-limits. Therefore this is a structure sheaf which exhibits $Sh_{\mathbf{H}_{th}}(X)$ as a structured (∞,1)-topos over $\mathbf{H}_{th}$ regarded as a (large) geometry (for structured (∞,1)-toposes), with the formally étale morphisms being the “admissible morphisms”.

Example

Let $G \in Grp(\mathbf{H}_{th})$ be an ∞-group and write $\flat_{dR} \mathbf{B}G \in \mathbf{H}_{th}$ for the corresponding de Rham coefficient object.

Then

$$\mathcal{O}_X(\flat_{dR}\mathbf{B}G) \in Sh_{\mathbf{H}}(X)$$

we may call the $G$-valued flat cotangent sheaf of $X$.

Remark

For $U \in \mathbf{H}_{th}$ a test object (say an object in a (∞,1)-site of definition, under the Yoneda embedding) a formally étale morphism $U \to X$ is like an open map/open embedding. Regarded as an object in $(\mathbf{H}_{th})_{/X}^{fet}$ we may consider the sections over $U$ of the cotangent bundle as defined above, which in $\mathbf{H}_{th}$ are diagrams

$$\array{ U &&\to&& \mathcal{O}_X(\flat_{dR}\mathbf{B}G) \\ & \searrow && \swarrow \\ && X } \,.$$

By the fact that $Et(-)$ is right adjoint, such diagrams are in bijection to diagrams

$$\array{ U &&\to&& X \times \flat_{dR} \mathbf{B}G \\ & \searrow && \swarrow \\ && X }$$

where we are now simply including on the left the formally étale map $(U \to X)$ along $(\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}$.

In other words, the sections of the $G$-valued flat cotangent sheaf $\mathcal{O}_X(\flat_{dR}\mathbf{B}G)$ are just the sections of $X \times \flat_{dR}\mathbf{B}G \to X$ itself, only that the domain of the section is constrained to be a formally é patch of $X$.

But then by the very nature of $\flat_{dR}\mathbf{B}G$ it follows that the flat sections of the $G$-valued cotangent bundle of $X$ are indeed nothing but the flat $G$-valued differential forms on $X$.

Proposition

For $X \in \mathbf{H}_{th}$ an object in a differentially cohesive $\infty$-topos, then its petit structured $\infty$-topos $Sh_{\mathbf{H}_{th}}(X)$, according to def. , is locally ∞-connected.

Proof

We need to check that the composite

$$\infty Grpd \stackrel{Disc}{\longrightarrow} \mathbf{H}_{th} \stackrel{(-) \times X}{\longrightarrow} (\mathbf{H}_{th})_{/X} \stackrel{L}{\longrightarrow} Sh_{\mathbf{H}}(X)$$

preserves (∞,1)-limits, so that it has a further left adjoint. Here $L$ is the reflector from prop. . Inspection shows that this composite sends an object $A \in \infty Grpd$ to $\mathbf{\Pi}_{inf}(Disc(A)) \times X \to X$:

$$\array{ \mathbf{\Pi}_{inf}(Disc(A)) \times X &\longrightarrow& \mathbf{\Pi}_{inf}(Disc(A) \times X) & \simeq \mathbf{\Pi}_{inf}(Disc(A)) \times \mathbf{\Pi}_{inf}(X) \\ \downarrow &{}^{(pb)}& \downarrow \\ X &\longrightarrow& \mathbf{\Pi}_{inf}(X) } \,.$$

By the discussion at slice (∞,1)-category – Limits and colimits an (∞,1)-limit in the slice $(\mathbf{H}_{th})_{/X}$ is computed as an (∞,1)-limit in $\mathbf{H}$ of the diagram with the slice cocone adjoined. By right adjointness of the inclusion $Sh_{\mathbf{H}}(X) \hookrightarrow (\mathbf{H}_{th})_{/X}$ the same is then true for $Sh_{\mathbf{H}}(X) \coloneqq (\mathbf{H}_{th})_{/X}^{et}$.

Now for $A \colon J \to \infty Grpd$ a diagram, it is taken to the diagram $j \mapsto \mathbf{\Pi}_{inf}(Disc(A_j)) \times X \to X$ in $Sh_{\mathbf{H}}(X)$ and so its $\infty$-limit is computed in $\mathbf{H}$ over the diagram locally of the form

$$\array{ X \times \mathbf{\Pi}_{inf}(Disc(A_{j})) &&\longrightarrow&& X \times \mathbf{\Pi}_{inf}(Disc(A_{j'})) \\ & \searrow && \swarrow \\ && X } \simeq \array{ X \times \mathbf{\Pi}_{inf}(Disc(A_{j})) &&\longrightarrow&& X \times \mathbf{\Pi}_{inf}(Disc(A_{j'})) \\ & \searrow && \swarrow \\ && X \times \ast } \,.$$

Since $\infty$-limits commute with each other this limit is the product of

1. $\underset{\leftarrow}{\lim}_j \mathbf{\Pi}_{inf}(Disc(A_j))$

2. $\underset{\leftarrow}{\lim}_{J \star \Delta^0} X$ (over the co-coned diagram constant on $X$).

For the first of these, since the infinitesimal shape modality $\mathbf{\Pi}_{inf}$ is in particular a right adjoint (with left adjoint the reduction modality), and since $Disc$ is also right adjoint by cohesion, we have a natural equivalence

$$\underset{\leftarrow}{\lim}_j \mathbf{\Pi}_{inf}(Disc(A_j)) \simeq \mathbf{\Pi}_{inf}(Disc(\underset{\leftarrow}{\lim}_j(A_j))) \,.$$

For the second, the $\infty$-limit over an $\infty$-category $J \star \Delta^0$ of a functor constant on $X$ is

$$\begin{aligned} \underset{\leftarrow}{\lim}_{J \star \Delta^0} X & \simeq \underset{\leftarrow}{\lim}_{J \star \Delta^0} [\ast, X] \\ & \simeq [\underset{\rightarrow}{\lim}_{J \star \Delta^0} \ast, X] \\ & \simeq [{\vert {J \star \Delta^0}\vert}, X] \\ & \simeq [\ast, X] \simeq X \end{aligned} \,,$$

where the last line follows since ${J \star \Delta^0}$ has a terminal object and hence contractible geometric realization.

In conclusion this shows that $\infty$-limits are preserved by $L \circ (-)\times X\circ Disc$.

Properties

Sheaf condition and examples of étale sheaves

Proposition

For $X$ a scheme, and $A \in PSh(X_{et})$ a presheaf, for checking the sheaf condition it is sufficient to check descent on the following two kinds of covers in the étale site

1. jointly surjective collections of open immersions of schemes;

2. single surjective/étale morphisms between affine schemes

(all over $X$).

(Tamme, II Lemma (3.1.1), Milne, prop. 6.6)

Proof (sketch)

Since covers by standard open immersions of schemes in the Zariski topology are also étale morphisms of schemes and étale covers, we may take any étale cover $\{Y_i \to Y\}$ over $X$, find an Zariski cover $\{U_i \to X\}$ of $X$, pull back the original cover to that and in turn cover the pullbacks themselves by Zariski covers. The result is still a cover and is so by a collection of open immersions of schemes. Now using compactness assumptions we find finite subcovers of all these covers. This makes their disjoint union be a single morphisms of affines.

Proposition

For $Z \to X$ any scheme over a scheme $X$, the induced presheaf on the étale site

$$(U_Y \to X) \mapsto Hom_X(U_Y, Z)$$

is a sheaf.

This is due to (Grothendieck, SGA1 exp. XIII 5.3) A review is in (Tamme, II theorem (3.1.2), Milne, 6.2).

Proof

By prop. we are reduced to showing that the represented presheaf satisfies descent along collections of open immersions and along surjective maps of affines. For the first this is clear (it is Zariski topology-descent). For the second case of a faithfully flat cover of affines $Spec(B) \to Spec(A)$ it follows with the exactness of the correspomnding Amitsur complex. See there for details.

Remark

This map from $X$-schemes to sheaves on $X_{et}$ is not injective, different $X$-schemes may represent the same sheaf on $X_{et}$. Unique representatives are given by étale schemess over $X$.

(e.g. Tamme, II theorem 3.1)

We consider some examples of sheaves of abelian groups induced by prop. from group schemes over $X$.

Example

The additive group over $X$ is the group scheme

$$\mathbb{G}_a \coloneqq Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X \,.$$

By the universal property of the pullback, the corresponding sheaf $(\mathbb{G}_a)_X$ is given by the assignment

$$\begin{aligned} (\mathbb{G}_a)_X(U_X \to X) & = Hom_X(U_X, Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X) \\ & = Hom(U_X, Spec(\mathbb{Z}[t])) \\ & = Hom(\mathbb{Z}[t], \Gamma(U_X, \mathcal{O}_{U_X})) \\ & = \Gamma(U_X, \mathcal{O}_{U_X}) \end{aligned} \,.$$

In other words, the sheaf represented by the additive group is the abelian sheaf underlying the structure sheaf of $X$.

Similarly one finds

Example

The multiplicative group over $X$

$$\mathbb{G}_m \coloneqq Spec(\mathbb{Z}[t,t^{-1}]) \times_{Spec(\mathbb{Z})} X$$

represents the sheaf $(\mathbb{G}_m)_X$ given by

$$(\mathbb{G}_m)_X(U_X) \mapsto \Gamma(U_X, \mathcal{O}_{U_X})^\times \,.$$

(e.g. Tamme, II, 3)

Base change and sheaf cohomology

Definition

For $f \colon X \longrightarrow Y$ a homomorphism of schemes, there is induced a functor on the categories underlying the étale site

$$f^{-1} \;\colon\; Y_{et} \longrightarrow X_{et}$$

given by sending an object $U_Y \to Y$ to the fiber product/pullback along $f$

$$f^{-1} \colon (U_Y \to Y) \mapsto (X \times_Y U_Y \to X) \,.$$

Proposition

The morphism in def. is a morphism of sites and hence induces a geometric morphism between the étale toposes

$$(f^\ast \dashv f_\ast) \;\colon\; Sh(X_{et}) \stackrel{\overset{f^\ast}{\leftarrow}}{\underset{f_\ast}{\longrightarrow}} Sh(Y_{et}) \,.$$

Here the direct image is given on a sheaf $\mathcal{F} \in Sh(X_{et})$ by

$$f_\ast \mathcal{F} \;\colon\; (U_Y \to Y) \mapsto \mathcal{F}(f^{-1}(U_Y)) = \mathcal{F}(X \times_X U_Y)$$

while the inverse image is given on a sheaf $\mathcal{F} \in Sh_(Y_{et})$ by

$$f^\ast \mathcal{F} \;\colon\; (U_X \to X) \mapsto \underset{\underset{U_X \to f^{-1}(U_Y)}{\longrightarrow}}{\lim} \mathcal{F}(U_Y) \,.$$

By the discussion at morphisms of sites – Relation to geometric morphisms. See also for instance (Tamme I 1.4).

Definition

For $X_{et}$ an étale site, write $\mathcal{D}(X_{et})$ for the derived category of the abelian category $Ab(Sh(X_{et}))$ of abelian sheaves on $X$.

Proposition

The $q$th derived functor $R^q f_\ast$ of the direct image functor of def. sends $\mathcal{F} \in Ab(Sh(X_{et}))$ to the sheafification of the presheaf

$$(U_Y \to Y) \mapsto H^q(X \times_Y U_Y, \mathcal{F}) \,,$$

where on the right we have the degree $q$ abelian sheaf cohomology group with coefficients in the given $\mathcal{F}$ (étale cohomology).

By the discussion at direct image and at abelian sheaf cohomology. See e.g. (Tamme, II (1.3.4), Milne prop. 12.1).

Remark

For $O_X \stackrel{f^{-1}}{\leftarrow} O_Y \stackrel{g^{-1}}{\leftarrow} O_Z$ two composable morphisms of sites, the Leray spectral sequence for the corresponding direct images exists and is of the form

$$E^{p,q}_2 = R^p f_\ast(R^q g_\ast(\mathcal{F})) \Rightarrow E^{p+q} = R^{p+q}(g f)_\ast(\mathcal{F}) \,.$$

For the special case that $S_Z = \ast$ and $g^{-1}$ includes an étale morphism $U_Y \to Y$ this yields

$$E^{p,q}_2 = H^p(U_Y, R^q f_\ast \mathcal{F}) \Rightarrow E^{p+q} = H^{p+q}(U_Y \times_Y X , \mathcal{F}) \,.$$

Quasi-coherent modules

Proposition

For $X$ a scheme and $N$ a quasicoherent module over its structure sheaf $\mathcal{O}_X$, then this induces an abelian sheaf on the étale site by

$$N_{et} \;\colon\; (U_X \to X) \mapsto \Gamma(U_Y, N \otimes_{\mathcal{O}_X} \mathcal{O}_{U_X}) \,.$$

(e.g. Tamme, II 3.2.1)

Relation to Zariski topos

Remark

A cover in the Zariski topology on schemes is an open immersion of schemes and hence is in particular an étale morphism of schemes. Hence the étale site is finer than the Zariski site and so every étale sheaf is a Zariski sheaf, but not necessarily conversely. Expressed in a different way, the étale topos is a subtopos of the Zariski topos.

For more see at étale cohomology – Properties – Relation to Zariski cohomology.

As a classifying topos

The étale topos over the big étale site of commutative rings is the classifying topos for strict local rings.

Related concepts

• basics of étale cohomology

• étale homotopy

• étale cohomology

References

Etale topos of a schemes

• Günter Tamme, section II 1 of Introduction to Étale Cohomology

• James Milne, section 7 of Lectures on Étale Cohomology

Etale topos of a differentially cohesive object

• Urs Schreiber, differential cohomology in a cohesive topos (arXiv:1310.7930)