5 The pro-étale topology
In this chapter we introduce the pro-étale site of a scheme and study its basic properties.
5.1 Preliminaries about the étale topology
Before going to the pro-étale case, we recall a few preliminary results about the étale site that we will need in the sequel. Let \(X\) be a scheme. By \(X_{\mathrm{ét}}\) we denote the étale site of \(X\) and by \(X^{\mathrm{aff}}_{\mathrm{ét}}\) the affine étale site. \(X^{\mathrm{aff}}_{\mathrm{ét}}\) consists of affine schemes étale over \(X\).
The inclusion functor \(X^{\mathrm{aff}}_{\mathrm{ét}} \to X_{\mathrm{ét}}\) is cover dense, i.e. for every \(Y\) in \(X_{\mathrm{ét}}\) there exists a cover \(\{ U_i \to Y\} \) with \(U_i\) in \(X^{\mathrm{aff}}_{\mathrm{ét}}\).
This is immediate, because open immersions are étale and étale is stable under composition.
5.2 Weakly étale morphisms of schemes
A map \(f \colon Y \to X\) is weakly étale if \(f\) is flat and \(\Delta _f\colon Y \to Y \times _{X} Y\) is flat.
\(f \colon Y \to X\) is weakly étale if and only if for every affine open \(U\) of \(Y\) and every affine open \(V\) of \(X\) with \(f(U) \subset V\), the induced ring homomorphism \(\Gamma (X, V) \to \Gamma (Y, U)\) is weakly étale.
Etale morphisms are weakly étale.
Any étale morphism is flat and its diagonal is an open immersion, hence flat.
This follows because flat is stable under composition and base change.
5.3 The site
Before we define the pro-étale site, we recall a few standard definitions.
A jointly surjective family \((f_i \colon Y_i \to X)_{i \in I}\) of morphisms of schemes is quasi-compact if for every affine open \(U\) of \(X\) there exist quasi-compact opens \(V_i\) in \(Y_i\) such that \(U = \bigcup _{i \in I} f_i(V_i)\).
Let \((f_i \colon Y_i \to X)_{i \in I}\) be a jointly-surjective family of morphisms of schemes. If \(f_i\) is flat and locally of finite presentation for every \(i \in I\), the family is quasi-compact.
In this case, every \(f_i\) is an open map, from which the claim easily follows.
Let \(\mathcal{P}\) be a morphism property of schemes and \(X\) be a scheme. The small \(\mathcal{P}\)-site of \(X\) is the category of \(X\)-schemes with structure morphism satisfying \(\mathcal{P}\) and with covers given by quasi-compact, jointly surjective families of morphisms satisfying \(\mathcal{P}\).
If we let \(\mathcal{P}\) to be flat morphisms of schemes, we obtain the fpqc site:
Let \(X\) be a scheme. The fqpc site of \(X\), denoted by \(X_{\mathrm{fpqc}}\), is the \(\mathcal{P}\)-site of \(X\) with \(\mathcal{P} = \text{flat}\).
For us, the main relevant property of the fqpc site is that it is subcanonical:
The fpqc site is subcanonical, i.e., every representable presheaf is a sheaf.
Analogous to the familiar criterion for fpqc sheaves, we have a generalization for the small \(\mathcal{P}\)-site of a scheme.
Let \(F\) be a presheaf on the small \(\mathcal{P}\)-site. Then \(F\) is a sheaf in the if and only if it is a sheaf in the Zariski topology and satisfies the sheaf property for \(\{ V \to U\} \) with \(V\), \(U\) affine and \(V \to U\) satisfying \(\mathcal{P}\).
We can now define the main object of this chapter:
Let \(X\) be a scheme. The pro-étale site of \(X\), denoted by \(X_{\mathrm{proét}}\), is the \(\mathcal{P}\)-site of \(X\) with \(\mathcal{P} = \text{weakly étale}\).
Despite the name proétale, Definition 5.12 does not mention pro-étale morphisms. The name is justified by Theorem 2.45 (see also Proposition 5.18).
Follows from 5.10, since every pro-étale sheaf is an fpqc-sheaf.
5.3.1 Affine covers
An object \(U\) of \(X_{\mathrm{proét}}\) is called a pro-étale affine if it can be written as \(U = \lim U_i\) over a small cofiltered diagram \(i \mapsto U_i\) of affine schemes étale over \(X\). The subcategory of pro-étale affines in \(X_{\mathrm{proét}}\) is denoted by \(X^{\mathrm{aff}}_{\mathrm{proét}}\).
The category \(X^{\mathrm{aff}}_{\mathrm{proét}}\) admits limits indexed by a connected diagram and the forgetful functor to \(\mathrm{Sch} / X\) preserves them. If \(X\) is affine, \(X^{\mathrm{aff}}_{\mathrm{proét}}\) has all small limits.
Let \(X\) be affine. The affine pro-étale site of \(X\) is the category \(X^{\mathrm{aff}}_{\mathrm{proét}}\). The covering families are the quasi-compact covers.
In other words, \(X^{\mathrm{aff}}_{\mathrm{proét}}\) carries the topology induced by the inclusion functor \(X^{\mathrm{aff}}_{\mathrm{proét}} \to X_{\mathrm{proét}}\). Although the objects of \(X_{\mathrm{proét}}\) are weakly étale \(X\)-schemes, by Theorem 2.45 the topology of \(X_{\mathrm{proét}}\) is still generated by objects in \(X^{\mathrm{aff}}_{\mathrm{proét}}\):
The inclusion \(X^{\mathrm{aff}}_{\mathrm{proét}} \to X_{\mathrm{proét}}\) is cover dense, i.e. for every object \(Y\) of \(X_{\mathrm{proét}}\), there exists a cover \(\{ U_i \to Y\} _{i \in I}\) with \(U_i\) in \(X^{\mathrm{aff}}_{\mathrm{proét}}\).
Let \(f\colon Y \to X\) be a weakly étale morphism and let \(\{ U_i \to X\} \) be an affine open cover of \(X\). If we prove the lemma for each \(U_i\) and the weakly étale morphism \(f^{-1}(U_i) \to U_i\), the lemma follows for \(f\) by taking the joint cover. Thus we may assume that \(X = \mathrm{Spec}(A)\) is affine.
Now let \(\{ U_i \to Y\} \) be an affine open cover. Again, if we prove the lemma for each \(U_i\) and the composition \(U_i \to Y \to X\), the claim follows by joining the covers. Hence we may assume that \(Y = \mathrm{Spec}(B)\) is affine.
We now have \(A \to B\) weakly étale, so by Theorem 2.45 there exists an ind-étale and faithfully flat \(B \to B'\) such that \(A \to B \to B'\) is ind-étale. The cover \(\{ \mathrm{Spec}(B’) \to \mathrm{Spec}(B) \} \) concludes the proof.
Restriction along the inclusion \(X^{\mathrm{aff}}_{\mathrm{proét}} \to X_{\mathrm{proét}}\) induces an equivalence of categories \(\mathrm{Shv}(X_{\mathrm{proét}}) \to \mathrm{Shv}(X^{\mathrm{aff}}_{\mathrm{proét}})\).
The functor \(X^{\mathrm{aff}}_{\mathrm{proét}} \to X_{\mathrm{proét}}\) is fully faithful and makes \(X^{\mathrm{aff}}_{\mathrm{proét}}\) a dense subsite of \(X_{\mathrm{proét}}\) by Proposition 5.18. Thus the claim follows from the comparison lemma for sites.
Restriction along the inclusion \(X^{\mathrm{aff}}_{\mathrm{proét}} \to X_{\mathrm{proét}}\) commutes with sheafification.
This follows immediately from 5.19.
We can also compare \(X^{\mathrm{aff}}_{\mathrm{proét}}\) to \(X^{\mathrm{aff}}_{\mathrm{ét}}\).
The canonical functor \(X^{\mathrm{aff}}_{\mathrm{ét}} \to X^{\mathrm{aff}}_{\mathrm{proét}}\) induces an equivalence of categories \(\mathrm{Pro}(X^{\mathrm{aff}}_{\mathrm{ét}}) \to X^{\mathrm{aff}}_{\mathrm{proét}}\).
By Proposition 4.4, it suffices to show that for every \(U\) in \(X^{\mathrm{aff}}_{\mathrm{ét}}\), the image of \(U\) in \(X^{\mathrm{aff}}_{\mathrm{proét}}\) is co-finitely-presentable. This follows from the fact that étale morphisms are finitely presented.
5.3.2 Repleteness
An important property of the pro-étale site is the following:
The category of sheaves on \(X_{\mathrm{proét}}\) has enough weakly contractible objects.
TBA.
The category of sheaves on \(X_{\mathrm{proét}}\) is replete.
5.4 Comparison with étale cohomology
Let \(X\) be a scheme. Since an étale map is also weakly étale by 5.4 we may define:
We denote by \(\nu = \nu _X\) the morphism of sites \(X_{\mathrm{proét}} \to X_{\mathrm{ét}}\) induced by the functor from \(X_{\mathrm{ét}}\) to \(X_{\mathrm{proét}}\) given by \((U \to X) \mapsto (U \to X)\).
The morphism of sites \(\nu \) is continuous.
This follows from the definitions, because every étale covering is a quasi-compact covering by 5.7.
We denote by \(\nu _{p}\colon \mathrm{PreShv}(X_{\mathrm{proét}}) \to \mathrm{PreShv}(X_{\mathrm{ét}})\) and \(\nu ^{p}\colon \mathrm{PreShv}(X_{\mathrm{ét}}) \to \mathrm{PreShv}(X_{\mathrm{proét}})\) the pushforward and pullback functors on the level of presheafs.
Let \(F\) be a presheaf on \(X_{\mathrm{ét}}\). Then \((\nu ^p F)|_{X^{\mathrm{aff}}_{\mathrm{proét}}}\) preserves filtered colimits.
Let \(U = \lim _{\lambda } U_{\lambda }\) be a cofiltered limit. By definition of \(X^{\mathrm{aff}}_{\mathrm{proét}}\) and because pro-pro-étale is pro-étale, we may assume that the \(U_{\lambda }\) are étale over \(X\). Then
where the \(V_{\mu }\) run over factorisations \(U = \lim U_{\lambda } \to V_{\mu } \to X\) where \(V_{\mu } \to X\) is étale. Since the \(U_{\lambda }\) are locally of finite presentation over \(X\), they are finitely presented in the categorical sense. Hence, for every such factorisation there exists a \(\lambda \) such that \(U = \lim U_{\lambda } \to V_{\mu }\) factors via \(U_{\lambda }\). This shows that the \(U_{\lambda }\) lie cofinal in all \(V_{\mu }\), so we obtain
The topology on \(X^{\mathrm{aff}}_{\mathrm{proét}}\) is generated by relatively pro-representable one-hypercovers with respect to the inclusion \(X^{\mathrm{aff}}_{\mathrm{ét}} \to X^{\mathrm{aff}}_{\mathrm{proét}}\).
The topology on \(X^{\mathrm{aff}}_{\mathrm{proét}}\) is generated by surjections \(V \to U\) in \(X^{\mathrm{aff}}_{\mathrm{proét}}\) and finite standard Zariski coverings. Both of these can be relatively pro-represented by Proposition 4.2 (see also Example 4.3).
Further denote by \(\nu _{*} \colon \mathrm{Shv}(X_{\mathrm{proét}}) \to \mathrm{Shv}(X_{\mathrm{ét}})\) and \(\nu ^* \colon \mathrm{Shv}(X_{\mathrm{ét}}) \to \mathrm{Shv}(X_{\mathrm{proét}})\) the corresponding pushforward and pullback functors. Since \(\nu ^*\) is exact, we have \(R \nu ^* = \nu ^*\) by abuse of notation and we simply write \(\nu ^*\) everywhere. In contrast to [ BS15 ] , we write \(R \nu _{*}\) for the derived functor of \(\nu _{*}\).
Let \(F\) be a sheaf on \(X_{\mathrm{ét}}\). Then \((\nu ^p F)|_{X^{\mathrm{aff}}_{\mathrm{proét}}}\) is a sheaf. In particular, \((\nu ^*F)|_{X^{\mathrm{aff}}_{\mathrm{proét}}} \cong (\nu ^p F)|_{X^{\mathrm{aff}}_{\mathrm{proét}}}\).
By Lemma 5.27 and Lemma 5.26, this follows from Proposition 4.19. Hence we have the following chain of isomorphisms
The second isomorphism is Corollary 5.20 and the last is the first part of this proposition.
Let \(F\) be a sheaf on \(X_{\mathrm{ét}}\) and \(U = \lim _i U_i\) in \(X^{\mathrm{aff}}_{\mathrm{proét}}\). Then \(\nu ^*F(U) = \operatorname {colim}_i F(U_i)\).
For any \(F \in \mathrm{Shv}(X_{\mathrm{ét}})\), the unit \(F \to \nu _{*} \nu ^{*} F\) is an isomorphism.
Follows formally from 5.30.
A sheaf \(F\) in \(\mathrm{Shv}(X_{\mathrm{proét}})\) is called classical if it lies in the essential image of \(\nu ^{*}\).
We can recognize the classical sheafs among all pro-étale sheafs by checking if they preserve affine filtered colimits:
A sheaf \(F\) in \(\mathrm{Shv}(X_{\mathrm{proét}})\) is classical if and only if for any \(U \in X^{\mathrm{aff}}_{\mathrm{proét}}\) with presentation \(U = \lim U_i\), it holds that \(F(U) = \operatorname {colim}F(U_i)\).
The only if part follows directly from Corollary 5.29. Conversely, let \(F\) be in \(\mathrm{Shv}(X_{\mathrm{proét}})\) and suppose \(F\) satisfies the formula in the statement. Then \(\nu ^* \nu _{*} F \to F\) is an isomorphism. Indeed, by Corollary 5.19, it suffices to check this after restricting to \(X^{\mathrm{aff}}_{\mathrm{proét}}\). Let \(U = \lim U_i\) be in \(X^{\mathrm{aff}}_{\mathrm{proét}}\). Then again by Corollary 5.29
Let \(K\) be in \(D^+(X_{\mathrm{ét}})\) and \(U = \lim _i U_i\) in \(X^{\mathrm{aff}}_{\mathrm{proét}}\). Then \(R \Gamma (U, \nu ^*K) = \operatorname {colim}_i R\Gamma (U_i, K)\).
TBA.
For any \(K\) in \(D^{+}(X_{\mathrm{ét}})\) the unit \(K \to R \nu _{*} \nu ^* K\) is an isomorphism.
TBA.
Let \(K\) be in \(D^{+}(X_{\mathrm{ét}})\). Then
By 5.35, we have \(R \Gamma (X_{\mathrm{ét}}, K) \cong R \Gamma (X_{\mathrm{ét}}, R \nu _{*}\nu ^*K)\). Since \(\nu _*\) maps injective sheaves to acyclic ones by general theory, we have \(R \Gamma (X_{\mathrm{ét}}, -) \circ \nu _{*} = R (\Gamma (X_{\mathrm{ét}}, -) \circ \nu _*) = R \Gamma (X_{\mathrm{proét}}, -)\). So the claim follows.
Denote by \(\Gamma _{\mathrm{cont}}(X, -)\colon \mathrm{Ab}(X_{\mathrm{ét}})^{\mathbb {N}} \to \mathrm{Ab}\) the functor
For an inverse system \((F_n)_{n \in \mathbb {N}}\), we call the cohomology groups \(H^i \mathrm{R} \Gamma _{\mathrm{cont}}\left(X, -\right)\left( \left( F_{n} \right)_{n \in \mathbb {N}} \right) \), denoted by \(H^i_{\mathrm{cont}}(X_{\mathrm{ét}}, (F_{n})_{n \in \mathbb {N}})\), the continuous étale cohomology of \(X\) with coefficients in \(\left( F_n \right)_{n \in \mathbb {N}}\).
Let \((F_n)_{n \in \mathbb {N}}\) be an inverse system of abelian sheaves on \(X_{\mathrm{ét}}\) with epimorphic transition maps. Then there exists a canonical identification
In particular, for every \(i \ge 0\) we have
We repeatedly use the fact that \(\Gamma \) commutes with inverse limits both on \(X_{\mathrm{ét}}\) and \(X_{\mathrm{proét}}\). Then we have