4 The pro-étale topology
4.1 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.
4.2 The site
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 \(\mathcal{P}\).
Let \(X\) be a scheme. The fqpc site of \(X\), denoted by \(X_{\mathrm{fpqc}}\), is the \(\mathcal{P}\)-site of \(X\) with \(\mathcal{P} = \text{flat}\).
Let \(F\) be a presheaf on the category of schemes. Then \(F\) is a sheaf in the fpqc topology 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 \(V \to U\) faithfully flat.
The fpqc site is subcanonical, i.e., every representable presheaf is a sheaf.
TBA.
Let \(X\) be a scheme. The pro-étale site of \(X\), denoted by \(X_{\mathrm{proét}}\), is the \(\mathcal{P}\)-site of \(X\) with \(\mathcal{P} = \text{weakly étale}\).
Follows from 4.10, since every pro-étale sheaf is an fpqc-sheaf.
4.2.1 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 \(X^{\mathrm{aff}}_{\mathrm{proét}}\) with the covering families are the quasi-compact covers.
The topos \(\mathrm{Shv}(X_{\mathrm{et}})\) is generated by the affine étale site of \(X\), i.e., the full subcategory of \(X_{\mathrm{et}}\) consisting of affine, étale schemes over \(X\).
TBA.
The topos \(\mathrm{Shv}(X_{\mathrm{proét}})\) is generated by \(X^{\mathrm{aff}}_{\mathrm{proét}}\).
TBA.
4.2.2 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.
4.3 Comparison with étale cohomology
Let \(X\) be a scheme. Since an étale map is also weakly étale by 4.3 we may define:
We denote by \(\nu = \nu _X\) the morphism of sites \(X_{\mathrm{proét}} \to X_{\mathrm{et}}\) induced by the functor from \(X_{\mathrm{et}}\) 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 4.6.
We denote by \(\nu _{p}\colon \mathrm{PreShv}(X_{\mathrm{proét}}) \to \mathrm{PreShv}(X_{\mathrm{et}})\) and \(\nu ^{p}\colon \mathrm{PreShv}(X_{\mathrm{et}}) \to \mathrm{PreShv}(X_{\mathrm{proét}})\) the pushforward and pullback functors on the level of presheafs.
Let \(F\) be a presheaf on \(X_{\mathrm{et}}\). 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.
Further denote by \(\nu _{*} \colon \mathrm{Shv}(X_{\mathrm{proét}}) \to \mathrm{Shv}(X_{\mathrm{et}})\) and \(\nu ^* \colon \mathrm{Shv}(X_{\mathrm{et}}) \to \mathrm{Shv}(X_{\mathrm{proét}})\) 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 \(X_{\mathrm{et}}\). Then \((\nu ^p F)|_{X^{\mathrm{aff}}_{\mathrm{proét}}}\) is a sheaf. In particular, \((\nu ^*F)|_{X^{\mathrm{aff}}_{\mathrm{proét}}} \cong (\nu ^p F)|_{X^{\mathrm{aff}}_{\mathrm{proét}}}\).
Let \(F\) be a sheaf on \(X_{\mathrm{et}}\) and \(U = \lim _i U_i\) in \(X^{\mathrm{aff}}_{\mathrm{proét}}\). Then \(\nu ^*F(U) = \operatorname {colim}_i F(U_i)\).
For any \(F \in \mathrm{Shv}(X_{\mathrm{et}})\), the unit \(F \to \nu _{*} \nu ^{*} F\) is an isomorphism.
Follows formally from 4.27.
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 4.26. 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 Lemma 4.18, we may check this on sections in \(X^{\mathrm{aff}}_{\mathrm{proét}}\). Let \(U = \lim U_i\) be in \(X^{\mathrm{aff}}_{\mathrm{proét}}\). Then again by Corollary 4.26
Note that the proof of relies on the highly nontrivial Lemma 4.18.
Let \(K\) be in \(D^+(X_{\mathrm{et}})\) 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 \(D^{+}(X_{\mathrm{et}})\) the unit \(K \to R \nu _{*} \nu ^* K\) is an isomorphism.
TBA.
Let \(K\) be in \(D^{+}(X_{\mathrm{et}})\). Then
By 4.33, we have \(R \Gamma (X_{\mathrm{et}}, K) \cong R \Gamma (X_{\mathrm{et}}, R \nu _{*}\nu ^*K)\). Since \(\nu _*\) maps injective sheaves to acyclic ones by general theory, we have \(R \Gamma (X_{\mathrm{et}}, -) \circ \nu _{*} = R (\Gamma (X_{\mathrm{et}}, -) \circ \nu _*) = R \Gamma (X_{\mathrm{proét}}, -)\). So the claim follows.
Denote by \(\Gamma _{\mathrm{cont}}(X, -)\colon \mathrm{Ab}(X_{\mathrm{et}})^{\mathbb {N}} \to \mathrm{Ab}\) 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 \(H^i_{\mathrm{cont}}(X_{\mathrm{et}}, (F_{n})_{n \in \mathbb {N}})\), 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 \(X_{\mathrm{et}}\) 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 \(X_{\mathrm{et}}\) and \(X_{\mathrm{proét}}\). Then we have