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(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos



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

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 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 H th\mathbf{H}_{th} every object XX canonically induces its étale topos Sh H th(X)Sh_{\mathbf{H}_{th}}(X).

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

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

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

The resulting structure is essentially that discussed (Lurie, Structured Spaces) if we regard H th\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 GGrp(H th)G \in Grp(\mathbf{H}_{th}) a differential cohesive ∞-group with de Rham coefficient object dRBG\flat_{dR}\mathbf{B}G and for XH thX \in \mathbf{H}_{th} any differential homotopy type, the product projection

X× dRBGX X \times \flat_{dR} \mathbf{B}G \to X

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

This is indeed what the sections of a corresponding bundle of differential forms over XX are supposed to look like – but only if UXU \to X is sufficiently spread out over XX, hence sufficiently étale. Because, on the extreme, if XX is the point (the terminal object), then there should be no non-trivial section of differential forms relative to UU over XX, but the above product projection instead reproduces all the sections of dRBG\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 XX. In order to correctly test differential form data, “suitable” here should be “formally”, namely infinitesimally. Hence the restriction should be along the full inclusion

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

of the formally étale maps (see def.) into XX. Since on formally étale covers the sections should be those given by dRBG\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.


For XH thX \in \mathbf{H}_{th} any object, write

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

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

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


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

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

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

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

Therefore we are reduced to showing that the square

lim iY i Π inflim iY i X Π inf(X) \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 Π inf\mathbf{\Pi}_{inf} is a left adjoint it commutes with the (,1)(\infty,1)-colimit on objects and hence this diagram is equivalent to

lim iY i lim iΠ infY i X Π inf(X). \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 H th\mathbf{H}_{th}, hence that on the left the component Y iY_i for each iIi \in I is the (∞,1)-pullback of Π inf(Y i)Π inf(X)\mathbf{\Pi}_{inf}(Y_i) \to \mathbf{\Pi}_{inf}(X), by assumption that we are taking an (,1)(\infty,1)-colimit over formally étale morphisms.


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


By prop. the inclusion (H th) /X fet(H th) /X(\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X} is reflective with reflector given by the (Π infequivalences,Π infclosed)(\mathbf{\Pi}_{inf}-equivalences , \mathbf{\Pi}_{inf}-closed) factorization system. Since Π inf\mathbf{\Pi}_{inf} is a right adjoint and hence in particular preserves (∞,1)-pullbacks, the Π inf\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.


For XH thX \in \mathbf{H}_{th} we speak of

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

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


𝒪 X:H th()×X(H th) /XEt(H th) /X fet \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 AH thA \in \mathbf{H}_{th} to the etalification, prop. , of the projection A×XXA \times X \to X.

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


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

𝒪 X(A)(U) Sh H th(X)(U,𝒪 X(A)) Sh H th(X)(U,Et(X×A)) (H th) /X(U,X×A) H th(U,A) A(U), \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 (ιEt)(\iota \dashv Et) of prop. and the (∞,1)-Yoneda lemma.

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

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

Sh H(X)(H th) /X fet(H th) /XXH th 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 H th(X)Sh_{\mathbf{H}_{th}}(X) as a structured (∞,1)-topos over H th\mathbf{H}_{th} regarded as a (large) geometry (for structured (∞,1)-toposes), with the formally étale morphisms being the “admissible morphisms”.


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


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

we may call the GG-valued flat cotangent sheaf of XX.


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

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

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

U X× dRBG X \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 (UX)(U \to X) along (H th) /X fet(H th) /X(\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}.

In other words, the sections of the GG-valued flat cotangent sheaf 𝒪 X( dRBG)\mathcal{O}_X(\flat_{dR}\mathbf{B}G) are just the sections of X× dRBGXX \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 XX.

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


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


We need to check that the composite

GrpdDiscH th()×X(H th) /XLSh H(X) \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 LL is the reflector from prop. . Inspection shows that this composite sends an object AGrpdA \in \infty Grpd to Π inf(Disc(A))×XX\mathbf{\Pi}_{inf}(Disc(A)) \times X \to X:

Π inf(Disc(A))×X Π inf(Disc(A)×X) Π inf(Disc(A))×Π inf(X) (pb) X Π inf(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 (H th) /X(\mathbf{H}_{th})_{/X} is computed as an (∞,1)-limit in H\mathbf{H} of the diagram with the slice cocone adjoined. By right adjointness of the inclusion Sh H(X)(H th) /XSh_{\mathbf{H}}(X) \hookrightarrow (\mathbf{H}_{th})_{/X} the same is then true for Sh H(X)(H th) /X etSh_{\mathbf{H}}(X) \coloneqq (\mathbf{H}_{th})_{/X}^{et}.

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

X×Π inf(Disc(A j)) X×Π inf(Disc(A j)) XX×Π inf(Disc(A j)) X×Π inf(Disc(A j)) X×*. \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. lim jΠ inf(Disc(A j))\underset{\leftarrow}{\lim}_j \mathbf{\Pi}_{inf}(Disc(A_j))

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

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

lim jΠ inf(Disc(A j))Π inf(Disc(lim j(A j))). \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 JΔ 0J \star \Delta^0 of a functor constant on XX is

lim JΔ 0X lim JΔ 0[*,X] [lim JΔ 0*,X] [|JΔ 0|,X] [*,X]X, \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 JΔ 0{J \star \Delta^0} has a terminal object and hence contractible geometric realization.

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


Sheaf condition and examples of étale sheaves


For XX a scheme, and APSh(X et)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 XX).

(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 iY}\{Y_i \to Y\} over XX, find an Zariski cover {U iX}\{U_i \to X\} of XX, 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.


For ZXZ \to X any scheme over a scheme XX, the induced presheaf on the étale site

(U YX)Hom X(U Y,Z) (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).


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)Spec(A)Spec(B) \to Spec(A) it follows with the exactness of the correspomnding Amitsur complex. See there for details.


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

(e.g. Tamme, II theorem 3.1)

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


The additive group over XX is the group scheme

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

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

(𝔾 a) X(U XX) =Hom X(U X,Spec([t])× Spec()X) =Hom(U X,Spec([t])) =Hom([t],Γ(U X,𝒪 U X)) =Γ(U X,𝒪 U X). \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 XX.

Similarly one finds


The multiplicative group over XX

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

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

(𝔾 m) X(U X)Γ(U X,𝒪 U X) ×. (\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


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

f 1:Y etX et f^{-1} \;\colon\; Y_{et} \longrightarrow X_{et}

given by sending an object U YYU_Y \to Y to the fiber product/pullback along ff

f 1:(U YY)(X× YU YX). f^{-1} \colon (U_Y \to Y) \mapsto (X \times_Y U_Y \to X) \,.

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

(f *f *):Sh(X et)f *f *Sh(Y et). (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 Sh(X et)\mathcal{F} \in Sh(X_{et}) by

f *:(U YY)(f 1(U Y))=(X× XU Y) 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 Sh (Y et)\mathcal{F} \in Sh_(Y_{et}) by

f *:(U XX)limU Xf 1(U Y)(U Y). 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).


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


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

(U YY)H q(X× YU Y,), (U_Y \to Y) \mapsto H^q(X \times_Y U_Y, \mathcal{F}) \,,

where on the right we have the degree qq 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).


For O Xf 1O Yg 1O ZO_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 2 p,q=R pf *(R qg *())E p+q=R p+q(gf) *(). 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=*S_Z = \ast and g 1g^{-1} includes an étale morphism U YYU_Y \to Y this yields

E 2 p,q=H p(U Y,R qf *)E p+q=H p+q(U Y× YX,). 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


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

N et:(U XX)Γ(U Y,N 𝒪 X𝒪 U X). 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


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.


Etale topos of a schemes

Etale topos of a differentially cohesive object

Last revised on June 11, 2022 at 10:58:33. See the history of this page for a list of all contributions to it.