Proof. As \(f\colon R\to A\) is of finite presentation, \(h\colon A\to B\) factors through \(h_i\colon A\to B_i\) for some \(i\) by [18]. We need to show that the composition \(\tilde{A}\to A\xrightarrow{h_i} B_i\) factors via \(\tilde{B}_i\hookrightarrow B_i\) for \(i\) large enough. By definition, \(\tilde{f}\colon \widetilde{R}\to \tilde{A}\) factors as a composition \(\widetilde{R}\to \widetilde{R}[t_1\ldots t_n]\twoheadrightarrow E\xrightarrow{\varphi} \tilde{A}\) with \(\varphi\) integral. As \(\widetilde{R}\to \widetilde{R}[t_1,\ldots,t_n]\) is of finite presentation, for \(i\) large enough, there is a map \(\nu\colon \widetilde{R}[t_1,\ldots t_n]\to \tilde{B}_i\) fitting in the following commutative diagram \[\xymatrix{ \tilde{R}[t_1,\ldots, t_n]\ar[r]\ar[rd]_-\nu &\tilde{A} \ar[r]& A \ar[d]^-{h_i} \\ &\tilde{B}_{i} \ar[r]& B_i}\] Since the composite \(\tilde{R}[t_1,\ldots, t_n]\to A\) factors through \(E\), the kernel of \(\widetilde{R}[x_1,\ldots x_n]\twoheadrightarrow E\) maps to zero in \(B_i\) via this composition. Since \(\tilde{B}_{i} \to B_{i}\) is injective, \(\nu\) induces a map \(E\to \tilde{B}_i\). As \(\tilde{A}\) is integral over \(E\), the image of \(\tilde{A}\) in \(B_{i}\) is integral over \(\tilde{B}_i\). Since \(\tilde{B}_i\) is integrally closed in \(B_i\), we conclude that the map \((h_i)_{|\tilde{A}}\colon \tilde{A}\to B_i\) factors via \(\tilde{h}_i\colon \tilde{A}\to \tilde{B}_{i}\). If \(\tilde{g}_i\) is ift, so is \(\tilde{h}_i\) by [13] ◻

Proof. We may take \(\overline{X}\) to be the integral closure of \(\overline{Y}\) in \(X\). More precisely, by assumption the composition \(j\circ f\) is quasi-compact and quasi separated and hence \((j\circ f)_*\mathcal{O}_X\) is a quasi-coherent \(\mathcal{O}_{\overline{Y}}\)-module. Let \(\mathcal{A}\) be the subsheaf which over an open \(V\subset \overline{Y}\) is given by \[\mathcal{A}(V)=\{s\in \mathcal{O}_X((j\circ f)^{-1}V)\mid s \text{ is integral over } \mathcal{O}_{\overline{Y}}(V)\}.\] Then \(\mathcal{A}\) is a quasi-coherent \(\mathcal{O}_{\overline{Y}}\)-algebra and \(j\circ f\) factors as \(X\xrightarrow{u}\overline{X}:=\operatorname{Spec}\mathcal{A}\xrightarrow{g} \overline{Y}\) with \(g\) integral. As \(f\) is integral we have \(g^{-1}(Y)=X\) by construction. ◻

Proof. By Lemma 5 we can factor \(f\) as \(X\xrightarrow{h} X'\xrightarrow{g} Y\) with \(h\) integral and \(g\) separated and of finite type. Let \(Z\subset X'\) denote the scheme-theoretic image of \(h\). As \(f\) is universally closed [18] yields that \(f\) factors as \(X\xrightarrow{h_1} Z\xrightarrow{g_1} Y\) with \(g_1\) proper and \(h_1\) integral and surjective. Now consider \(\mathcal{A}=h_{1*}\mathcal{O}_X\). It is an integral quasi-coherent \(\mathcal{O}_Z\)-algebra and can hence be written as a directed colimit \(\mathcal{A}= \operatornamewithlimits{\varinjlim}\mathcal{A}_i\) of its finite quasi-coherent \(\mathcal{O}_Z\)-subalgebras. Thus \(X= \operatornamewithlimits{\varprojlim}X_i\) is a directed limit with \(X_i=\operatorname{Spec}\mathcal{A}_i\) finite over \(Z\) and hence also finite transition maps \(\tau: X_j\to X_i\) which by construction are induced by the inclusions \(\mathcal{O}_{X_i}\hookrightarrow\tau_*\mathcal{O}_{X_j}\) inside \(\mathcal{A}\). Clearly, the composition \(X_i\to Z\to Y\) is proper. It remains to show that the projection \((X_{i})_U\to U\) is an isomorphism, for all \(i\).

The question is local on \(U\) and we can therefore assume \(U=\operatorname{Spec}R\) is affine. Hence \(X_U\) is affine as well and the projection induces an isomorphism \[R\xrightarrow{\simeq} A:=\mathcal{O}_{X}(X_U).\] By [18], the scheme \((X_i)_U\) is affine for \(i\) big enough, so by [20], \((X_i)_U\) is affine for all \(i\), therefore the map \(X_U\to (X_i)_U\) is induced by an integral ring extension \(A_i \hookrightarrow A\). Moreover, the section \(\sigma_i: U \xrightarrow{\simeq} X_U\to (X_{i})_U\) of the projection corresponds to a surjection of rings \(A_i\rightarrow\!\!\!\!\!\rightarrow R\) which factors via the isomorphism \(A\xrightarrow{\simeq} R\) and is therefore injective as well. This shows that \(\sigma_i\) is an isomorphism inverse to the projection. ◻

Proof. [lem:tame-Gal-clousre1]. Note that \(N^{sh}_{\tilde{w}}\) is a Galois hull of \(L^{sh}_w/K^{sh}_v\). Therefore we may assume \(K,L,N\) are strictly henselian valuation fields of characteristic \(p>0\). Thus if \((N,\tilde{w})/(K,v)\) is tame then \([N:K]=[N:L]\cdot [L:K]\) is prime to \(p\) and hence \((L,w)/(K,v)\) is tame as well. Now assume \((L,w)/(K,v)\) is tame. Denote by \(G_K\supset G_L\supset G_N\) the absolute Galois groups with respect to a fixed separable closure \(\bar{K}\) of \(K\), and by \(P\) the pro-\(p\)-Sylow subgroup of \(G_K\), which is a normal subgroup. The indices satisfy the following equality (of supernatural numbers) \[[G_K:G_L]\cdot [G_L:P\cap G_L]= [G_K:P]\cdot [P:P\cap G_L].\] As \(P\) is a normal subgroup of \(G_K\), the intersection \(P\cap G_L\) is a normal subgroup of \(G_L\) and we have an inclusion of profinite groups \(G_L/G_L\cap P\hookrightarrow G_K/P\). Hence \([G_K:P]\) and \([G_L:P\cap G_L]\) are prime to \(p\). By assumption \([G_K:G_L]=[L:K]\) is prime to \(p\) as well. Thus \([P:P\cap G_L]=1\), i.e., \(P=P\cap G_L\). The Galois hull of \(L/K\) is the composition field (inside \(\bar{K}\)) of all the \(\sigma(L)\), where \(\sigma\) runs through all the embeddings \(L\hookrightarrow\bar{K}\). Extending these \(\sigma\)’s to \(K\)-automorphisms of \(\bar{K}\), we find \(G_{\sigma(L)}=\sigma G_L\sigma^{-1}\). Hence \(G_N=\cap_{\sigma} \sigma G_L\sigma^{-1}\). As \(P\) is a normal subgroup of \(G_K\) it follows that \(P\) is contained in \(G_N\) as well. Thus \([G_K:P]=[G_K:G_N]\cdot [G_N:P]\) is prime to \(p\) and hence so is \([N:K]=[G_K:G_N]\).

[lem:tame-Gal-clousre2] follows from [lem:tame-Gal-clousre1] and the fact that \({K'}^t= K'\cdot K^t\), see [21]. ◻

Proof. We prove it for \({\operatorname{v\acute{e}t}}\). The proof for \(t\) is the same. Let \(F\) be a sheaf of sets on \((X,\tilde{X})_{{\operatorname{v\acute{e}t}}}\). For \((U,\tilde{U})\in (X,\tilde{X})_\tau\) and \(\tilde{U}'=\tilde{U}\times_{\tilde{X}} \tilde{X}'\), the map induced by projection \((U,\tilde{U}')\to (U,\tilde{U})\) and the map induced by the diagonal \(\delta: (U, \tilde{U}')\to (U,\tilde{U}'\times_{\tilde{U}} \tilde{U}')\) are modifications and thus are coverings in \((X,\tilde{X})_{{\operatorname{v\acute{e}t}}}\). Hence, \[\delta^*: F(U,\tilde{U}'\times_{\tilde{U}} \tilde{U}') \to F(U,\tilde{U}')\] is injective and we find \[\begin{align} F(U,\tilde{U}) &\cong {\rm eq}\left(F(U,\tilde{U}') \overset{pr_1^*}{\underset{pr_2^*}{\rightrightarrows}} F(U,\tilde{U}'\times_{\tilde{U}} \tilde{U}')\right)\\ &\cong {\rm eq}\left(F(U,\tilde{U}')\overset{id}{\underset{id}{\rightrightarrows}} F(U,\tilde{U}')\right)= F(U,\tilde{U}').\\ \end{align}\] Hence, the functor \[{\operatorname{\mathbf{Shv}}}((X,\tilde{X}')_{{\operatorname{v\acute{e}t}}})\to {\operatorname{\mathbf{Shv}}}((X,\tilde{X})_{{\operatorname{v\acute{e}t}}}),\quad G\mapsto \left((U,\tilde{U})\mapsto G(U, \tilde{U}\times_{\tilde{X}} \tilde{X}')\right)\] gives a quasi-inverse of the restiction functor \({\operatorname{\mathbf{Shv}}}((X,\tilde{X})_{{\operatorname{v\acute{e}t}}})\to {\operatorname{\mathbf{Shv}}}((X,\tilde{X}')_{{\operatorname{v\acute{e}t}}})\). ◻

Proof. First, we show that the result holds for \(\varphi_1\) a modification. This is classical (see [22]): let \(\mathcal{U}=(U,\tilde{U})\), \(\mathcal{V}=(V,\tilde{V})\), and \(\mathcal{Y}=(Y,\tilde{Y})\). As observed in Remark 12[rmk-def59XSt2], we can assume that \(Y\) dense in \(\tilde{Y}\). By Raynaud–Gruson [18], there exists a \(Y\)-admissible blow-up \(\tilde{T}\to \tilde{Y}\) such that the strict transform \(\tilde{V}'\) of \(\tilde{V}\) over \(\tilde{T}\) is flat of finite presentation over \(\tilde{T}\). Note that the map \(\tilde{V}'\to \tilde{T}\) factors as \(\tilde{V}'\xrightarrow{\alpha} \tilde{U}\times_{\tilde{Y}}\tilde{T}\xrightarrow{\beta} \tilde{T}\), where \(\beta\) is étale and \(\alpha\) is proper inducing an isomorphism over the dense open \(U=\tilde{U}\times_{\tilde{Y}}\tilde{T}\times_{\tilde{Y}} Y\). Moreover, \(\alpha\) is flat by [19] and thus an isomorphism by [19]. Hence, we get a morphism \(\tilde{U}\times_{\tilde{Y}}\tilde{T}\to \tilde{V}\), which is proper and an isomorphism over \(U\).

Let now \(\varphi_1\) be a quasi-modification. By Remark 10 (3), there exists \(\mathcal{V}_0=(U,\tilde{V}_0)\) such that \(\varphi_1\) factors as a composition of an integral birational morphism \(\mathcal{V}\to \mathcal{V}_0\) and a modification \(\mathcal{V}_0\to \mathcal{U}\). By the previous paragraph, there exists a modification \(\mathcal{T}'=(Y,\tilde{T}')\to \mathcal{Y}\) such that for \(\tilde{U}':=\tilde{U}\times_{\tilde{Y}} \tilde{T}'\), the map \(\mathcal{U}'=(U,\tilde{U}')\to \mathcal{U}\) factors through \(\mathcal{V}_0\), hence we have a commutative diagram\[\begin{tikzcd} \mathcal{V}\times_{\mathcal{V}_0}\mathcal{U}'\ar[r,"\varphi_1'"]\ar[d,"q_0"]&\mathcal{U}'\ar[rr,"\varphi_2'"]\ar[d]&&\mathcal{T}'\ar[d,"\psi_0"]\\ \mathcal{V}\ar[r]&\mathcal{V}_0\ar[r]&\mathcal{U}\ar[r]&\mathcal{Y} \end{tikzcd}\] with \(\varphi_1'\) integral birational, \(\varphi_2'\) strict étale and \(q_0\) and \(\psi_0\) modifications. Let \(\tilde{T}\) be the integral closure of \(\tilde{T}'\) in \(Y\) and put \(\tilde{W}:=\tilde{U}'\times_{\tilde{T}'}\tilde{T}\cong \tilde{U}\times_{\tilde{Y}} \tilde{T}\). Then, \(\tilde{W}\) is the integral closure of \(\tilde{U}'\) in \(U\) by [18] since \(\tilde{U}'\to \tilde{T}'\) is étale and \(U=\tilde{U}'\times_{\tilde{T}'} Y\). As \(\varphi_1'\) is integral birational, the \(\tilde{U}'\)-scheme \(\tilde{W}\) is isomorphic to the integral closure of \(\tilde{V}\times_{\tilde{V}_0} \tilde{U}'\) in \(U\). Hence \(\tilde{W}\to\tilde{U}'\) factors via \(\tilde{W}=(\tilde{V}\times_{\tilde{V}_0} \tilde{U}')^{\rm int}\to \tilde{V}\times_{\tilde{V}_0} \tilde{U}'\). Therefore we can take \(\mathcal{T}=(Y,\tilde{T})\) and this completes the proof of the lemma. ◻

Proof. By [24], \(I\) is flabby as a \({\operatorname{s\acute{e}t}}\) sheaf if and only if for any \(\mathcal{U}\in (X,\tilde{X})_\tau\) and a strict étale covering \(\mathcal{U}'\to \mathcal{U}\), the Čech complex\[0\to I(\mathcal{U})\to I(\mathcal{U}')\to I(\mathcal{U}'\times_{\mathcal{U}} \mathcal{U}')\to \cdots\] is exact. This implies that for any modification \(\mathcal{V}\to \mathcal{U}\), the Čech complex\[0\to I(\mathcal{V})\to I(\mathcal{V}\times_{\mathcal{U}} \mathcal{U}')\to I(\mathcal{V}\times_{\mathcal{U}} \mathcal{U}'\times_{\mathcal{U}} \mathcal{U}')\to \cdots\] is also exact. Noting that filtered colimits are exact, Lemma 17 together with Lemma 16 implies \[0\to a_{{\operatorname{v\acute{e}t}}}I(\mathcal{U})\to a_{{\operatorname{v\acute{e}t}}}I(\mathcal{U}')\to a_{{\operatorname{v\acute{e}t}}}I(\mathcal{U}'\times_{\mathcal{U}} \mathcal{U}')\to \cdots\] is exact. Noting \(a_{{\operatorname{v\acute{e}t}}}I(\mathcal{U})\simeq a_{{\operatorname{v\acute{e}t}}}I(\mathcal{V})\) for any modification \(\mathcal{V}\to \mathcal{U}\), this implies that \(a_{{\operatorname{v\acute{e}t}}}I\) is flabby on \((X,\tilde{X})_{{\operatorname{v\acute{e}t}}}\) again by [24]. ◻

Proof. This is very similar to [25]. Let \(F_{|(X,\tilde{X})_{{\operatorname{s\acute{e}t}}}}\to I^\bullet\) be a flabby resolution of the restriction of \(F\) to \((X,\tilde{X})_{{\operatorname{s\acute{e}t}}}\). By Lemma 18, this gives a flabby resolution \(F\to a_{{\operatorname{v\acute{e}t}}}I^{\bullet}\) on \((X,\tilde{X})_{{\operatorname{v\acute{e}t}}}\). Therefore, by Lemma 17\[H^i(\mathcal{U}_{{\operatorname{v\acute{e}t}}},F)= \pi_{-i} (a_{{\operatorname{v\acute{e}t}}}I^\bullet(\mathcal{U})) = \pi_{-i} (\operatornamewithlimits{\varinjlim}_{\mathcal{V}\to\mathcal{U}}I^\bullet(\mathcal{V})) = \operatornamewithlimits{\varinjlim}_{\mathcal{V}\to\mathcal{U}}H^i(\mathcal{V}_{{\operatorname{s\acute{e}t}}},F) = \operatornamewithlimits{\varinjlim}_{\mathcal{V}\to\mathcal{U}} H^i(\tilde{V}_{{\operatorname{\acute{e}t}}},\lambda^{\mathcal{V}}_*F).\] ◻

Proof. We prove the lemma only for the tame topology. The proof for the v-étale topology is the same. We can suppose \(\mathcal{Y}:=(\operatorname{Spec}(A),\operatorname{Spec}(\tilde{A}))\in (X,\tilde{X})_{\rm{affine},t}\). We use the same strategy of [27] (see [5]). Let \(I\) be the set of isomorphism classes of coverings \(\mathcal{U}\to\mathcal{Y}\) in \((X,\tilde{X})_{\rm{affine},t}\). For each \(i \in I\), pick a representative \((\operatorname{Spec}(B_i),\operatorname{Spec}(\tilde{B}_i))\to \mathcal{Y}\) and set \[\label{eq159lem:existence-acyclic} A_1 := \operatornamewithlimits{\varinjlim}_{J\subset I \;\mathrm{finite}} \bigotimes_{j\in J}B_j\qquad \tilde{A}_1 := \operatornamewithlimits{\varinjlim}_{J\subset I \;\mathrm{finite}} \bigotimes_{j\in J} \tilde{B}_j,\tag{5}\] where the tensor products are over \(A\) and \(\tilde{A}\), respectively. By construction, we can write \[\mathcal{Y}_1:=(\operatorname{Spec}(A_1),\operatorname{Spec}(\tilde{A}_1))=\varprojlim_{\lambda_1\in \Lambda_1} \mathcal{Y}_{\lambda_1} \;\text{ with } \mathcal{Y}_{\lambda_1}=(\operatorname{Spec}(A_{\lambda_1}),\operatorname{Spec}(\tilde{A}_{\lambda_1}))\] as a cofiltered limit of coverings \(\mathcal{Y}_{\lambda_1}\to \mathcal{Y}\) in \((X,\tilde{X})_{\rm{affine},t}\) such that for every covering \(\mathcal{U}\to \mathcal{Y}\) in \((X,\tilde{X})_t\), the map \(\mathcal{Y}_1 \to\mathcal{Y}\) factors through \(\mathcal{U}\). For each \(\lambda_1\in \Lambda_1\), let \(I_{\lambda_1}\) be the set of isomorphism classes of coverings \(\mathcal{U}\to\mathcal{Y}_{\lambda_1}\) in \((X,\tilde{X})_{\rm{affine},t}\) and apply the same construction as 5 to \((\mathcal{Y}_{\lambda_1},I_{\lambda_1})\) instead of \((\mathcal{Y},I)\) to get \((A_{2,\lambda_1}, \tilde{A}_{2,\lambda_1})\) instead of \((A_1,\tilde{A}_1)\) and put \(\mathcal{Y}_{2,\lambda_1}=(\operatorname{Spec}(A_{2,\lambda_1}), \operatorname{Spec}(\tilde{A}_{2,\lambda_1}))\). Then, for every covering \(\mathcal{U}\to \mathcal{Y}_{\lambda_1}\), the map \(\mathcal{Y}_{2,\lambda_1}\to \mathcal{Y}_{\lambda_1}\) factors through \(\mathcal{U}\). Put \[\label{eq259lem:existence-acyclic} A_2 :=\operatornamewithlimits{\varinjlim}_{J\subset \Lambda_1 \;\mathrm{finite}} \bigotimes_{\lambda_1\in J}(A_{2,\lambda_1}\otimes_{A_{\lambda_1}} A_1), \qquad \tilde{A}_2 :=\operatornamewithlimits{\varinjlim}_{J\subset \Lambda_1 \;\mathrm{finite}} \bigotimes_{\lambda_1\in J} (\tilde{A}_{2,\lambda_1}\otimes_{\tilde{A}_{\lambda_1}} \tilde{A}_1),\tag{6}\] where the tensor products are over \(A_1\) and \(\tilde{A}_1\), respectively. Noting \(\mathcal{Y}_{2,\lambda_1}\to \mathcal{Y}_{\lambda_1}\) and \(\mathcal{Y}_1\to \mathcal{Y}\) are cofiltered limits of coverings, we can write \[\mathcal{Y}_2:=(\operatorname{Spec}(A_2),\operatorname{Spec}(\tilde{A}_2))= \varprojlim_{\lambda_2\in \Lambda_2} \mathcal{Y}_{\lambda_2}\] as a cofiltered limit of coverings \(\mathcal{Y}_{\lambda_2}\to \mathcal{Y}\) in \((X,\tilde{X})_{\rm{affine},t}\) such that for every \(\lambda_1\in \Lambda_1\) and covering \(\mathcal{U}\to \mathcal{Y}_{\lambda_1}\) in \((X,\tilde{X})_t\), the map \(\mathcal{Y}_2 \to\mathcal{Y}_1\to \mathcal{Y}_{\lambda_1}\) factors through \(\mathcal{U}\). Iterating the construction, we get a sequence in \(\widetilde{(X,\tilde{X})_\tau}\) \[\cdots\to \mathcal{Y}_3\to \mathcal{Y}_2\to \mathcal{Y}_1\to \mathcal{Y}\,\text{ with }\; \mathcal{Y}_n=\varprojlim_{\lambda_n\in \Lambda_n} \mathcal{Y}_{\lambda_n}\] such that for every \(\lambda_n\in \Lambda_n\), \(\mathcal{Y}_{\lambda_n}\to \mathcal{Y}\) is a covering in \((X,\tilde{X})_{\rm{affine},t}\) and for every covering \(\mathcal{U}\to \mathcal{Y}_{\lambda_n}\) in \((X,\tilde{X})_t\), the map \(\mathcal{Y}_{n+1}\to \mathcal{Y}_n\to \mathcal{Y}_{\lambda_n}\) factors through \(\mathcal{U}\). Set \(\mathcal{W}=\varprojlim_n \mathcal{Y}_n\in \widetilde{(X,\tilde{X})_\tau}\). By the construction, \(\mathcal{W}\) is a cofiltered limit of tame coverings of \(\mathcal{Y}\). It remains to show that \(\mathcal{W}\) is tamely acyclic. Let \(\mathcal{V}\to \mathcal{U}\) be a covering in \((X,\tilde{X})_{\rm{affine},t}\) and \(\varphi:\mathcal{W}\to \mathcal{U}\) be a map in \(\widetilde{(X,\tilde{X})_\tau}\). By Lemma 3 (with \((f,\tilde{f})\) induced by \(\mathcal{U}\to\mathcal{X}\)), \(\varphi\) factors through \(\mathcal{Y}_{\lambda_n}\) for some \(\lambda_n\). Since \(\mathcal{Y}_{\lambda_n}\times_\mathcal{U}\mathcal{V}\to \mathcal{Y}_{\lambda_n}\) is a covering in \((X,\tilde{X})_t\), the map \(\mathcal{Y}_{n+1}\to\mathcal{Y}_n\to \mathcal{Y}_{\lambda_n}\) factors through \(\mathcal{Y}_{\lambda_n}\times_\mathcal{U}\mathcal{V}\) so that \(\varphi\) lifts to a map \(\mathcal{W}\to \mathcal{V}\). This completes the proof. ◻

Proof. Since \(\tilde{U}\) is quasi-compact and the open immersion \(U\hookrightarrow\tilde{U}\) is quasi-compact we find finitely many affine pairs \(\mathcal{U}_j=(U_j, \tilde{U}_j)\in (X,\tilde{X})_{{\rm affine}, t}\) such that \(U=\cup_j U_j\) and \(\tilde{U}=\cup_j \tilde{U}_j\) are open coverings. Thus the disjoint union \(\sqcup_j\mathcal{U}_j\) is an affine pair which tamely covers \(\mathcal{U}\). By Lemma 24 applied to this disjoint union, we find a morphism \(\mathcal{W}=(W,\tilde{W})\to \mathcal{U}\) in \(\widetilde{(X,\tilde{X})_t}\), which is a limit of tame coverings \(\mathcal{W}_i=(W_i, \tilde{W}_i)\to \mathcal{U}\) in \((X,\tilde{X})_{{\rm affine},t}\) with \(\mathcal{W}_i\in (X,\tilde{X})_{{\rm int},t}\), such that \(\mathcal{W}\) is tamely acyclic. Note that every connected component \(\mathcal{P}\) of \(\mathcal{W}\) is an \(x\)-local object over \(\mathcal{Y}\), for some points \(x\in Y\). Indeed, by Proposition 26, \(\mathcal{P}\) satisfies the conditions [def:x-local2] and [def:x-local3] of Definition 33, and it suffices to check that \(\mathcal{P}\) satisfies [def:x-local1]. Denote by \(x\) the image in \(Y\) of the closed point of \(P=\operatorname{Spec}S\). Then we get a natural local morphism \(\mathcal{O}_{Y,x}^h\to S\). It is ind-étale as \(S\) is a component of \(\operatornamewithlimits{\varinjlim}_i \mathcal{O}(W_i)\) and \(W_i\to U\) and \(U\to Y\) are étale.

Thus, \(\mathcal{W}^{n}\) (product over \(\mathcal{Y}\)) is a disjoint union of \(\mathcal{P}_1\times_{\mathcal{Y}} ... \times_{\mathcal{Y}} \mathcal{P}_n\), where \(\mathcal{P}_j\) are \(x_j\)-local objects for some \(x_j\in Y\). Hence, by Lemma 34 and Proposition 26, \(\mathcal{W}^{n}\) is tamely acyclic. Thus, for any tame covering \(\mathcal{V}\to \mathcal{U}^n\) the map \(\mathcal{W}^{n}\to \mathcal{U}^n\) factors via \(\mathcal{V}\). Noting \(\mathcal{W}_{i}^{n}\in (X,\tilde{X})_{{\rm int},t}\), Lemma 3 and Remark 4 imply that there is \(i_0\) and a map \(\mathcal{W}_{i_0}^{n}\to \mathcal{V}\) in \((X,\tilde{X})_{{\rm affine},t}\) that factors \(\mathcal{W}_{i_0}^{n}\to \mathcal{U}^n\), hence by choosing \(\mathcal{U}'= \mathcal{W}_{i_0}\) we conclude the proof. ◻

Proof. By Theorem 36, we have that \[\varinjlim_{\lambda\in \Lambda} H^i((\mathcal{U}_\lambda)_t, F) = \varinjlim_{\lambda\in \Lambda}\varinjlim_{\mathcal{V}_\lambda\to \mathcal{U}_{\lambda}}\pi_{-i}F(\mathcal{V}_\lambda^{\times_{\mathcal{U}_{\lambda}}\bullet}),\] where the colimit is indexed over tame covers of \(\mathcal{U}_{\lambda}\). By Lemma 28, this is equal to \[\label{eq159prop:tame-local-vanishing} \varinjlim_{\mathcal{V}\to \mathcal{U}} \pi_{-i}F(\mathcal{V}^{\times_{\mathcal{U}}\bullet}),\tag{13}\] where the colimit is indexed over tame covers of \(\mathcal{U}\) in the sense of Definition 9 and \(F\) is left Kan extended to \(\widetilde{ (X,\tilde{X})}_\tau\). By Remark 32, we can further suppose that \(\mathcal{V}\) are of the form \((V,\tilde{V})=(\operatorname{Spec}(B),\operatorname{Spec}(\tilde{B}))\) with \(A\to B\) is a finite étale map of henselian local rings associated to the residue field extension \(L/K\) which is Galois and tame with respect to \(v\) and \(\tilde{B}=B\times_L \mathcal{O}_w\) with \(w\) the valuation on \(L\) extending \(v\). Set \[B_n:=B^{\otimes_{A} n}, \quad V_n:=\operatorname{Spec}B_n \quad\text{and} \quad \tilde{B}_n:=\tilde{B}^{\otimes_{\tilde{A}}n },\quad \tilde{V}_n:=\operatorname{Spec}\tilde{B}_n.\] Let \(B_n^{int}\) be the integral closure of \(\tilde{B}_n\) in \(B_n\) and set \(\tilde{V}_n^{int}=\operatorname{Spec}\tilde{B}_n^{int}\). As \((V_n,\tilde{V}_n^{int})\to (V_n,\tilde{V}_n)\) is a quasi-modification, see Remark 22, the desired vanishing holds once we know that the following complex is exact \[\label{prop:tame-local-vanishing1} 0\to F(U,\tilde{U})\to F(V,\tilde{V}^{\rm int})\to F(V_2,\tilde{V}^{\rm int}_2)\to F(V_3,\tilde{V}_3^{\rm int})\to\cdots.\tag{14}\] By [32] the ring \(\mathcal{O}_w\) is also the integral closure of \(\mathcal{O}_v\) in \(L\). Hence the Galois group \({\rm Gal}(L/K)\) is equal to the decomposition group \({\rm Aut}(\mathcal{O}_w/\mathcal{O}_v)\). Moreover, as the category of finite separable field extensions of \(K\) is equivalent to the category of finite local étale \(A\)-algebras, we can identify the \(A\)-algebra automorphisms of \(B\) with \({\rm Gal}(L/K)\). Hence \(G:={\rm Gal}(L/K)={\rm Aut}(B/A)={\rm Aut}(\tilde{B}/\tilde{A})\). As in [24] the isomorphism \(B_2\to \prod_{\sigma\in G} B\), \(b_0\otimes b_1\mapsto (\sigma(b_0)b_1)_\sigma\) and induction give the isomorphism for \(n\ge 2\) \[\varphi_n: B_n\to \prod_{(\sigma_0,\ldots, \sigma_{n-2})\in G^{n-1}}B\] with \[\varphi_n(b_0\otimes \ldots \otimes b_{n-1})_{(\sigma_0,\ldots, \sigma_{n-2})}= (\sigma_{n-2}\cdots \sigma_0)(b_0)\cdot (\sigma_{n-2}\cdots \sigma_1)(b_1)\cdots \sigma_{n-2}(b_{n-2})\cdot b_{n-1}.\] As \(\tilde{B}\) is integral over \(\tilde{A}\), so is \(\tilde{B}_n\), hence \(\tilde{B}_n^{\rm int}\) is the integral closure of \(\tilde{A}\) in \(B_n\) and thus \(\varphi_n\) restricts to an isomorphism \[\tilde{B}_n^{\rm int}\to \prod_{(\sigma_0,\ldots,\sigma_{n-2})} \tilde{B}.\] We thus find isomorphisms \[(V_n,\tilde{V}_n^{\rm int})\cong (V,\tilde{V})\times G^{n-1}\] as in [24] and can therefore identify the cohomology of 14 with Galois cohomology \[\pi_{-i}F(\mathcal{V}_\lambda^{\times_{\mathcal{U}_{\lambda}}\bullet})= H^i(G, F(V,\tilde{V})).\] This vanishes as \(F(V,\tilde{V})\) is a \(\mathbb{Z}_{(p)}\)-module and the order of \(G\) is invertible in \(\mathbb{Z}_{(p)}\) by tameness. ◻

Proof. Note that \(j^{\mathcal{V}}\) is the composition of the morphism of sites \[\nu:(V,\tilde{V})_t \to (V,\tilde{V})_{{\operatorname{v\acute{e}t}}}\] corresponding to the inclusion functor and the morphism of sites \[\lambda^{\mathcal{V}}\colon (V,\tilde{V})_{{\operatorname{v\acute{e}t}}}\to \tilde{V}_{{\operatorname{\acute{e}t}}}\] defined by the functor \(\tilde{V}_{{\operatorname{\acute{e}t}}} \to (V,\tilde{V})_{{\operatorname{v\acute{e}t}}}:\tilde{W}/\tilde{V}\mapsto (\tilde{W}\times_{\tilde{V}} V,\tilde{W})\). By Lemma 20, we have \[H^i(\mathcal{U}_{t},F) = H^i(\mathcal{U}_{{\operatorname{v\acute{e}t}}},R\nu_*F) = \operatornamewithlimits{\varinjlim}_{\mathcal{V}\to \mathcal{U}} H^i(\tilde{V}_{{\operatorname{\acute{e}t}}},\lambda^{\mathcal{V}}_*R\nu_*F)= \operatornamewithlimits{\varinjlim}_{\mathcal{W}\to \mathcal{U}} H^i(\tilde{V}_{{\operatorname{\acute{e}t}}},\lambda^{\mathcal{W}}_*R\nu_*F).\] Hence, it suffices to show \(R^i\nu_*F=0\) for \(i\ge 1\). By Remark 30, this follows from Proposition 37 and the fact that the assumptions of loc. cit. are satisfied by condition [thm:tame-vs-etale1]. This completes the proof. ◻

Proof. We start by showing that \(F_\beta\) is a presheaf. Let \((u,\tilde{u}): (U',\tilde{U}')\to (U,\tilde{U})\) be a morphism in \((X,\tilde{X})_t\) and take \(a\in F_{\beta}(U,\tilde{U})\). Let \((y,v,\varepsilon)\in \mathrm{Spa}(U',\tilde{U}')\). Set \(x:=u(y)\in U\) and denote by \(v_x\) the restriction of \(v\) to \(k(x)\). Note that \(\mathcal{O}_{v_x}=\mathcal{O}_v\cap k(x)\), the intersection taken inside \(k(y)\). Hence \(\varepsilon: \operatorname{Spec}\mathcal{O}_v\to \tilde{U}'\to \tilde{U}\) factors uniquely via a map \(\varepsilon_x:\operatorname{Spec}\mathcal{O}_{v_x}\to \tilde{U}\). We obtain a point \((x, v_x,\varepsilon_x)\in \mathrm{Spa}(U,\tilde{U})\). By definition there exists a finite tame extension \((L,w)/(k(x),v_x)\) such that \(a_L\in F_w\). Denote by \(L_1\) the composition field of \(L\) and \(k(y)\) in a separable closure of \(k(x)\) and choose a valuation \(v_1\) on \(L_1\) extending \(v\). Then the extension \((L_1,v_1)/(k(y),v)\) is tame, by Lemma 8[lem:tame-Gal-clousre2]. Now \(u^*(a)_{L_1}\), the image of \(u^*(a)\) under \(F(U')\to F(\mathcal{O}^h_{U',L_1})\), is equal to the image of the pullback of \(a_L\) under \(F(\mathcal{O}^h_{U,L})\to F(\mathcal{O}^h_{U',L_1})\). As \(a_L\in F_w\) we find \(u^*(a)_{L_1}\in F_{w_1}\) by [beta1] in 40. This shows \(u^*(a)\in F_\beta(U',\tilde{U}')\). Hence \(F_\beta\) is a presheaf.

As \(F\) is an étale sheaf on \(X\), \(F_\beta\) will be a sheaf on \((X,\tilde{X})_t\) if we show the following: Let \(\{(U_i,\tilde{U}_i)\to (U,\tilde{U})\}_{i\in I}\) be a tame covering in \((X,\tilde{X})_t\) and let \(a\in F(U)\), then \[\label{prop:constr-tame-sheaves1} a_{|U_i}\in F_{\beta}(U_i,\tilde{U}_i), \quad \text{for all }i \, \Longrightarrow\, a\in F_\beta(U,\tilde{U}).\tag{17}\] Let \((x,v,\varepsilon)\in \mathrm{Spa}(U,\tilde{U})\). By definition of tame covering we find an index \(i\) and a point \((y,w,\varepsilon')\in \mathrm{Spa}(U_i,\tilde{U}_i)\) over \((x,v,\varepsilon)\) such that \((k(y),w)/(k(x),v)\) is a finite tame extension. As \(a_{|U_i}\in F_{\beta}(U_i, \tilde{U}_i)\) we find a finite tame extension \((L,w_1)/(k(y), w)\) such that \((a_{|U_i})_L\in F_{w_1}\). Hence, we get 17 . ◻

Proof. By Lemma 13 we can assume \(\tilde{U}=\tilde{U}^{\rm int}\). We notice that both sides of the first equality in the statement are contained in \(\mathcal{O}(U)\) and that this "\(\supset\)" inclusion holds by definition of the left hand side. As both sides are sheaves on \(\tilde{U}\) it suffices to check the other inclusion for \(\tilde{U}\) affine. As an affine open cover \(U=\cup_i U_i\) induces an affine open cover \(\tilde{U}=\cup_i \tilde{U}_i^{\rm int}\) we can assume \((U,\tilde{U})=(\operatorname{Spec}A, \operatorname{Spec}\tilde{A})\) with \(\tilde{A}\) integrally closed in \(A\). Let \(x\in A\setminus \tilde{A}\).

Admitting the claim, we also have \(w(x)<0\) for all (tame) extensions \((L,w)/(K,v)\) of valuation fields and \(x\) cannot lie in \(\mathcal{O}^t(U,\tilde{U})\), which completes the proof of the lemma.

We prove the claim. Denote by \(C=\tilde{A}[1/x]\) the subring of \(A_x\) generated by the image of \(\tilde{A}\) and \(1/x\). As \(x\) is not integral over \(\tilde{A}\) it follows from [32], that \(A_x\) is not the zero-ring and that there exists a maximal ideal \(\mathfrak{m}\subset C\) such that \(1/x\in \mathfrak{m}\) and \(\mathfrak{m}\cap \tilde{A}\) is a maximal ideal of \(\tilde{A}\). Consider the localization \(C_\mathfrak{m}\) of \(C\). By the flatness of \(C_\mathfrak{m}\) over \(C\), the inclusion \(C\hookrightarrow A_x\) induces an injection of rings \(C_\mathfrak{m}=C\otimes_C C_\mathfrak{m}\hookrightarrow A_x\otimes_C C_{\mathfrak{m}}\). As \(C_{\mathfrak{m}}\) is not the zero ring, the ring \(A_x\otimes_C C_{\mathfrak{m}}\) is not zero and hence has a prime ideal; it corresponds to a prime ideal \(\mathfrak{p}\) of \(A\), which does not contain \(x\) and has empty intersection with \(C\setminus \mathfrak{m}\). Set \(\mathfrak{p}_0:=C\cap \mathfrak{p}A_x\). By construction we have \(\mathfrak{p}_0\subset \mathfrak{m}\), and in fact this inclusion is strict as else we would have \(1/x\in \mathfrak{p}A_x\) since \(1/x\in \mathfrak{m}\). Thus, \(C/\mathfrak{p}_0\) is not a field. Set \(K_0:={\operatorname{Frac}}(C/\mathfrak{p}_0)\subset K:={\operatorname{Frac}}(A_x/\mathfrak{p}A_x)={\operatorname{Frac}}(A/\mathfrak{p})\). Let \(v_0\) be a valuation on \(K_0\) such that \(C/\mathfrak{p}_0\subset \mathcal{O}_{v_0}\subset K_0\) and that \(\mathfrak{m}_{v_0}\cap C/\mathfrak{p}_0\) is the image of \(\mathfrak{m}\) in \(C/\mathfrak{p}_0\). Let \(v\) be a valuation on \(K\) extending \(v_0\). Thus we obtain a commutative diagram \[\begin{tikzcd} \tilde{A}\ar[r]\ar[d] & C/\mathfrak{p}_0\ar[r]\ar[d] & \mathcal{O}_{v_0}\ar[r] & \mathcal{O}_v\ar[d]\\ A\ar[r] & A_x/\mathfrak{p}A_x\ar[rr] & & K. \end{tikzcd}\] Note \(v(x)=v_0(x)<0\) since \(1/x\in \mathfrak{m}\) so that its image in \(C/\mathfrak{p}_0\) is in \(\mathfrak{m}_{v_0}\). This completes the proof of the claim.

For the last statement, we observe that as \(\tilde{U}\) is Nagata and \(U\) is normal the integral closure \(\tilde{U}^{int}\) is finite over \(\tilde{U}\) and hence is locally noetherian again. As \(U\) is normal so is \(\tilde{U}^{int}\). We may therefore assume that \(\tilde{U}^{int}\) is noetherian, integral and normal. Hence \[\mathcal{O}^t(U,\tilde{U})=\mathcal{O}(\tilde{U}^{\mathrm{int}})=\bigcap_{x\in {\tilde{U}}^{int,(1)}}\mathcal{O}_{\tilde{U}^{int},x}= \mathcal{O}^{div}(U,\tilde{U}),\] where \(\tilde{U}^{int, (1)}\) is the set of 1-codimensional points in \(U^{int}\) and where the last equality follows from \(\mathcal{O}^{div}(U,\tilde{U})=\mathcal{O}^{div}(U,\tilde{U}^{int})\) and the definition of \(\mathcal{O}^{div}\), see [para:example-tame-sheaves1465]. ◻

Proof. This is a version of [37] for the big de Rham–Witt complex. Note that by the Leibniz rule [11] we have \[\label{lem:dRW-quotient0}J_S^q= \mathbb{W}_S(I)\cdot \mathbb{W}_S\Omega^q_A+d(\mathbb{W}_S(I))\cdot \mathbb{W}_S\Omega^{q-1}_A\tag{20}\] We use this presentation of \(J^q_S\) in the following. We get a functor from truncation sets to anticommutative graded rings \(S\mapsto \mathbb{W}_S\Omega^*_A/ J_S^*\) and as in loc. cit. it suffices to show that the differential \(d\), Frobenius \(F_n\), and Verschiebung \(V_n\) on \(\mathbb{W}_{-}\Omega^*_A\) induce well-defined maps on \(\mathbb{W}_{-}\Omega^*_A/ J_{-}^*\). Indeed, in this case the natural surjection of Witt complexes \(\mathbb{W}_{-}\Omega^*_A\to \mathbb{W}_{-}\Omega^*_{A/I}\) induces a surjection of Witt-complexes \(\mathbb{W}_{-}\Omega^*_A/J_{-}^*\to \mathbb{W}_{-}\Omega^*_{A/I}\) and the universal property of the de Rham–Witt complex yields the inverse map.

Every element in \(J_S^*\) can be written as a sum of elements \[V_{n}([a])\cdot \beta \quad \text{and} \quad dV_n([a])\cdot \beta,\] where \(\beta\in \mathbb{W}_S\Omega^*_A\), \(a\in I\), \([a]\in \mathbb{W}_{S/n}(I)\) denotes the multiplicative lift of \(a\), and \(V_n\) is the \(n\)th Verschiebung. By [11] we have \[\label{lem:dRW-quotient1}d(V_n([a])\beta)= dV_n([a]) \beta+ V_n([a]) d\beta\tag{21}\] and \[d(dV_n([a])\beta)= {\operatorname{dlog}}[-1]dV_n([a])\beta- dV_n([a]) d\beta= -dV_n([a])({\operatorname{dlog}}[-1]\beta+d\beta).\] Hence \(d(J^*_S)\subset J^{*+1}_S\). As \(F_m: \mathbb{W}_S\Omega^*_A\to \mathbb{W}_{S/m}\Omega^*_A\) (\(m\ge 1\)) is a ring map which restricts to a map \(\mathbb{W}_S(I)\to\mathbb{W}_{S/m}(I)\) the containment \(F_m(J^*_S)\subset J^*_{S/m}\) follows from \[\begin{align} F_mdV_n([a]) &= idF_{m_1}V_{n_1}([a])+jF_{m_1}V_{n_1}d([a]) +(c-1){\operatorname{dlog}}[-1] F_{m_1}V_{n_1}([a]) \label{lem:dRW-quotient2}\\ &= idV_{n_1}([a^{m_1}])+jV_{n_1}([a]^{m_1-1}d[a]) + (c-1) V_{n_1}([a]^{m_1}){\operatorname{dlog}}[-1] \notag\\ &= idV_{n_1}([a^{m_1}]) +jV_{n_1}([a]^{m_1-1})dV_{n_1}([a]) +(c-1-jn_1+j)V_{n_1}([a]^{m_1}){\operatorname{dlog}}[-1], \notag \end{align}\tag{22}\] where \(c={\rm gcd}(m,n)\), \(m=cm_1\), \(n=cn_1\), and \(i,j\in \mathbb{Z}\) satisfy \(im+nj=c\). Here the first equality is the third equation in [11], the second equality uses \(V_{r}F_{s}=F_{s}V_{r}\), for \((r,s)=1\), \(F_{s}([a])= [a]^{s}\), and \(F_s(d[a])=[a]^{s-1}d[a]\), and the third equality uses \(F_rdV_r([a])=d[a]+ (r-1){\operatorname{dlog}}[-1]\cdot [a]\), \(F_r({\operatorname{dlog}}[-1])={\operatorname{dlog}}[-1]\), and \(V_r(F_r(x)y)=xV_r(y)\), see [11]. (If \(m_1=1\) we can choose \(i=1\) and \(j=0\) and deduce from the second equality that \(F_mdV_n([a])\) lies in \(J_{S/m}^1\).) It remains to show \(V_m(J^*_{S})\subset J^*_T\) for any truncation set \(T\) with \(T/m=S\). From \(\eqref{lem:dRW-quotient1}\) and \(V_md=mdV_m\) we see that it suffices to show \(V_m(a\beta)\in J^*_T\), for any \(a\in \mathbb{W}_S(I)\). As by construction of the big de Rham–Witt complex a form \(\beta\) is a sum of elements \(b_0d b_1\cdots db_{q}\) with \(b_i\in \mathbb{W}_S(A)\) and since \(a b_0\in \mathbb{W}_S(I)\) it suffices to consider elements \(\beta=db_1\cdots db_q\). As \(db_i= F_mdV_m(b_i)- (m-1)b_i{\operatorname{dlog}}[-1]\) and \({\operatorname{dlog}}[-1]\cdot{\operatorname{dlog}}[-1]=0\) [11] we find \[\begin{align} adb_1\cdots db_q & =a F_m\left(dV_m(b_1)\cdots dV_m(b_q)\right)\\ & -(m-1)\sum_{i=1}^q ab_i F_m\left(dV_m(b_1)\cdots \underbrace{{\operatorname{dlog}}[-1]}_{i\text{th spot}}\cdots dV_m(b_q)\right). \end{align}\] Thus \(V_m(adb_1\cdots db_q)\in J_T^*\) follows from the formula \(V_m(xF_m(y))=V_m(x)y\). This completes the proof of the lemma. ◻

Proof. By assumption \(\mathcal{O}_v=\tilde{A}/\mathfrak{m}_A\to \tilde{B}/\mathfrak{m}_A \tilde{B}\) is an étale local homomorphism and it follows from the arguments in [21] that \(\tilde{B}/\mathfrak{m}_A \tilde{B}=\mathcal{O}_w\) is a valuation ring, which is unramified over \(\mathcal{O}_v\) by [21]. In particular, \(\mathcal{O}_w\) is the integral closure of \(\mathcal{O}_v\) in \(L\) and hence \(L=K\otimes_{\mathcal{O}_v}\mathcal{O}_w= A\otimes_{\tilde{A}} \tilde{B}/\mathfrak{m}_A \tilde{B}\). It follows that \(B=A\otimes_{\tilde{A}}\tilde{B}\) is a local ring with maximal ideal \(A\otimes_{\tilde{A}}\mathfrak{m}_A\tilde{B}=\mathfrak{m}_A B\), where we use for the last equality that \(A=\tilde{A}_{\mathfrak{m}_A}\) is a localization of \(\tilde{A}\). On the other hand as \(\tilde{B}\) is flat over \(\tilde{A}\) and \(\mathfrak{m}_A\) is an ideal of \(\tilde{A}\) we have \(\mathfrak{m}_A\otimes_{\tilde{A}} \tilde{B}=\mathfrak{m}_A\tilde{B}\). Thus \(\mathfrak{m}_A B=\mathfrak{m}_A\tilde{B}\). The commutative diagram with exact rows \[\xymatrix{ 0\ar[r] & \mathfrak{m}_A \tilde{B}\ar@{=}[d]\ar[r] & \tilde{B}\ar[r]\ar[d] & \tilde{B}/\mathfrak{m}_A\tilde{B}\ar@{=}[d]\ar[r] & 0\\ 0\ar[r] & \mathfrak{m}_A B\ar[r] & B\times_L \mathcal{O}_w \ar[r] & \mathcal{O}_w\ar[r] & 0 }\] therefore yields \(\tilde{B}=B\times_L \mathcal{O}_w\).

As already noted in 46, the ring \(\mathbb{W}_S(\tilde{B})\) is an étale \(\mathbb{W}_S(\tilde{A})\) algebra. It is also faithfully flat if we show that \(\operatorname{Spec}\mathbb{W}_S(\tilde{B})\to \operatorname{Spec}\mathbb{W}_S(\tilde{A})\) is surjective. Let \(p={\rm char}(\mathcal{O}_v/\mathfrak{m}_v)\). Then \(\tilde{A}\) is a \(\mathbb{Z}_{(p)}\)-algebra and hence it suffices to show that \(\operatorname{Spec}W_n(\tilde{B})\to \operatorname{Spec}W_n(\tilde{A})\) is surjective for all \(n\ge 1\), e.g., [11]. By [34] there is an integral ring map \(W_n(\tilde{A})\to W_{n-1}(\tilde{A})\times \tilde{A}\) with nilpotent kernel and which is functorial in \(\tilde{A}\). Thus the induced map \(\operatorname{Spec}(W_{n-1}(\tilde{A})\times \tilde{A})\to \operatorname{Spec}\, W_n(\tilde{A})\) is surjective, and similarly with \(\tilde{B}\) instead of \(\tilde{A}\). As \(\operatorname{Spec}(W_{n-1}(\tilde{B})\times \tilde{B}) \to \operatorname{Spec}(W_{n-1}(\tilde{A})\times \tilde{A})\) is surjective by induction, we see that \(\operatorname{Spec}W_n(\tilde{B})\to\operatorname{Spec}W_n(\tilde{A})\) is surjective as well.

Finally, we show that 24 is an isomorphism. Consider the composition \[\mathbb{W}_S(\tilde{B})\otimes_{\mathbb{W}_S(\tilde{A})} \mathbb{W}_S\Omega^q_{\tilde{A}}(\log) \to \mathbb{W}_S(\tilde{B})\otimes_{\mathbb{W}_S(\tilde{A})} \mathbb{W}_S\Omega^q_{A}\xrightarrow{\simeq} \mathbb{W}_S(B)\otimes_{\mathbb{W}_S(A)} \mathbb{W}_S\Omega^q_A \xrightarrow{\simeq}\mathbb{W}_S\Omega^q_B\] where the first map is injective as \(\mathbb{W}_S(\tilde{B})\) is flat over \(\mathbb{W}_S(\tilde{A})\), the second map is an isomorphism as \(\mathbb{W}_S(B)=\mathbb{W}_S(\tilde{B}\otimes_{\tilde{A}} A)=\mathbb{W}_S(\tilde{B})\otimes_{\mathbb{W}_S(\tilde{A})} \mathbb{W}_S(A)\), see [34], and the last map is an isomorphism by étale base change, see [11]. Hence 24 is injective and by étale base change for \(\tilde{B}/\tilde{A}\) it remains to show that the elements \({\operatorname{dlog}}[b]\), for \(b\in B^\times\), are in the image of 24 for \(q=1\). Let \(\bar{b}\) be the image of \(b\) in \(\mathcal{O}_w\). As \(\mathcal{O}_w/\mathcal{O}_V\) is unramified we find \(a\in A^\times\) and \(u\in \tilde{B}^\times\) such that \(\bar{b}= \bar{a} \bar{u}\). Thus \(b=au(1+c)\), for some \(c\in \mathfrak{m}_B\). Hence \[{\operatorname{dlog}}[b] = {\operatorname{dlog}}[a] +{\operatorname{dlog}}[u(1+c)].\] Clearly the first summand lies in the image of 24 and as \(u(1+c)\in \tilde{B}^\times\) the second summand lies in \(\mathbb{W}_S\Omega^1_{\tilde{B}}\) and therefore also in the image of 24 by étale base change for \(\tilde{B}/\tilde{A}\). This completes the proof. ◻

Proof. For the first statement it suffices to show \(J_S^q\subset \mathbb{W}_S\Omega^q_{\tilde{A}}(\log)\). Thus it remains to show 25 . This "\(\supset\)" inclusion follows immediately from Lemma 47. As the right hand side in 25 is closed under \(d\) (by the Leibniz rule and the formula \(dda={\operatorname{dlog}}[-1]da\)) it remains by Lemma 47 to show \[\label{lem:ses-log-dRW01}\mathbb{W}_S(\mathfrak{m}_A)\cdot \mathbb{W}_S\Omega^q_A\subset \mathbb{W}_S(\mathfrak{m}_A)\cdot\mathbb{W}_S\Omega^q_{\tilde{A}}(\log)+ d(\mathbb{W}_S(\mathfrak{m}_A)\cdot\mathbb{W}_S\Omega^{q-1}_{\tilde{A}}(\log))=:I^q_S.\tag{26}\] We prove this by induction over \(q\), the case \(q=0\) being trivial. For \(q=1\), we prove for later use the stronger statement \[\label{lem:ses-log-dRW1}\mathbb{W}_S(\mathfrak{m}_A)\cdot \mathbb{W}_S\Omega^1_A\subset \mathbb{W}_S(\mathfrak{m}_A)\cdot \mathbb{W}_S\Omega^1_{\tilde{A}}(\log)+ d\mathbb{W}^0_S(\mathfrak{m}_A)=:H^1_S\subset I^1_S,\tag{27}\] where \(\mathbb{W}_S^0(\mathfrak{m}_A)=\mathrm{ker}(\mathbb{W}_S(\mathfrak{m}_A)\to \mathfrak{m}_A)\) is the kernel of the restriction map. It suffices to show that any element of the form \(V_m([a])dV_n([b])\) with \(a\in \mathfrak{m}_A\), \(b\in A\), and \(n,m\in S\), lies in \(H^1_S\). As \(\mathfrak{m}_A\subset \tilde{A}\), we may assume \(b\in A\setminus \mathfrak{m}_A=A^\times\). We have \[\begin{align} V_m([a])dV_n([b])& = V_m([a]F_m dV_n([b]))\\ &= iV_m([a]dV_{n_1}([b]^{m_1}))+jV_m([a]V_{n_1}([b]^{m_1-1}d[b]))\\ &\phantom{=} + (c-1) V_m([a]V_{n_1}([b]^{m_1}){\operatorname{dlog}}[-1]), \end{align}\] where \(c={\rm gcd}(m,n)\), \(m=cm_1\), \(n=cn_1\), and \(i,j\in \mathbb{Z}\) satisfy \(im+nj=c\). Here the second equality follows from the second equality in 22 (with \(a\) there replaced by \(b\) here). The third summand \[(c-1)V_m([a]V_{n_1}([b]^{m_1}){\operatorname{dlog}}[-1])= (c-1)V_{mn_1}([a^{n_1}b^{m_1}]){\operatorname{dlog}}[-1]\] clearly is contained in \(H^1_S\), as is the second summand \[jV_m([a]V_{n_1}([b]^{m_1-1}d[b]))= jV_{mn_1}([a^{n_1}b^{m_1}]){\operatorname{dlog}}[b].\] If \(n_1=1\), then we can take \(j=1\) and \(i=0\) and hence when considering the first summand we can assume \(n_1\ge 2\). Using the Leibniz rule and \(V_md=mdV_m\) we can rewrite the first summand as \[iV_m([a]dV_{n_1}([b]^{m_1}))= imdV_{mn_1}([a^{n_1}b^{m_1}])- iV_m(V_{n_1}([b]^{m_1}) d[a])\] and it remains to show that \(V_m(V_{n_1}([b]^{m_1}) d[a])\) (\(n_1\ge 2\)) is contained in \(H^1_S\). We can write \[[a]=[a+1]-[1] +V_r(\alpha),\] for some \(r>1\) in \(S\) and \(\alpha\in \mathbb{W}_{S/r}(\mathfrak{m}_A)\). (Note that \(\alpha\) lies in \(\mathbb{W}_{S/r}(\mathfrak{m}_A)\) as \([a]+[1]-[a+1]\) maps to zero in \(\mathbb{W}_S(A/\mathfrak{m}_A)\).) Thus \[\begin{align} V_m(V_{n_1}([b]^{m_1}) d[a]) & = V_{mn_1}([b]^{m_1}[a]^{n_1-1} d[a])\label{lem:ses-log-dRW2}\\ & = V_{mn_1}([b]^{m_1}[a]^{n_1-1}[a+1]){\operatorname{dlog}}[a+1] + V_{mn_1}([b]^{m_1}[a]^{n_1-1} dV_r(\alpha)).\notag \end{align}\tag{28}\] As \(n_1\ge 2\), the first summand in this last equality is contained in \(H^1_S\). Using \[dV_r(\alpha)= F_{mn_1}dV_{mn_1r}(\alpha)- (mn_1-1){\operatorname{dlog}}[-1]V_r(\alpha),\] see [11], we rewrite the second summand of the last equality in 28 as follows \[\begin{gather} V_{mn_1}([b]^{m_1}[a]^{n_1-1} dV_r(\alpha))\\ = V_{mn_1}([b^{m_1}a^{n_1-1}])dV_{mn_1r}(\alpha) - (mn_1-1)V_{mn_1r}\left([b^{m_1r}a^{(n_1-1)r}]\alpha\right){\operatorname{dlog}}[-1], \end{gather}\] which hence lies in \(H^1_S\) as well. This completes the proof of claim 27 .

Now we show 26 for \(q\ge 1\), i.e., for \(a\in \mathbb{W}_S(\mathfrak{m})\) and \(\beta\in \mathbb{W}_S\Omega^q_A\) \[a\beta\in I^q_S.\] We may assume that \(\beta=\beta_1\beta_{q-1}\) with \(\beta_1\in \mathbb{W}_S\Omega^1_A\) and \(\beta_{q-1}=db_1\cdots db_{q-1}\), \(b_i\in \mathbb{W}_S(A)\). By 27 we find \(a_i, a_i'\in \mathbb{W}_S(\mathfrak{m}_A)\), and \(\gamma_i\in \mathbb{W}_S\Omega^1_{\tilde{A}}(\log )\) such that \[a\beta=a\beta_1\beta_{q-1}= \sum_i a_i'\gamma_i\beta_{q-1} + \sum_i d(a_i)\beta_{q-1}.\] Thus by induction it remains to show \(dadb_1\cdots db_{q-1}\in I^q_S\), for \(a\in \mathbb{W}_S(\mathfrak{m}_A)\). By the Leibniz rule and the formula \(ddx={\operatorname{dlog}}[-1]dx\) we find an element \(\gamma\in \mathbb{W}_S\Omega^{q-1}_A\) such that \[dadb_1\cdots db_{q-1}= d\left(a db_1\cdots db_{q-1}\right)+ a\gamma{\operatorname{dlog}}[-1],\] which lies in \(I^q_S\) by induction. This completes the proof of the lemma. ◻

Proof. First note that by étale base change \(\mathbb{W}_S\Omega^q_K\to \mathbb{W}_S\Omega^q_L\) is injective. Hence both sides of the statement are contained in \(\mathbb{W}_S\Omega^q_L\) and clearly the right-hand side is contained in the left-hand side. The statement is trivially true for \(q=0\).

We consider the case \(q=1\). Let \(p\) be the characteristic exponent of the residue field \(\mathcal{O}_v/\mathfrak{m}_v\). As \((L,w)/(K,v)\) is a tame extension, [21] yields: \(L=K[b_1,\ldots, b_k]\) with \(b_i^{r_i}=a_i\in K^\times\), \(r_i\ge 1\) and \((p,r_i)=1\), and there is an isomorphism of the quotient of the valuation groups \(\Gamma_L/\Gamma_K\) onto \(\mathbb{Z}/r_1\mathbb{Z}\oplus\ldots\oplus \mathbb{Z}/r_k\mathbb{Z}\), which sends \(w(b_i)\) to \((0,\ldots, 0,1, 0,\ldots, 0)\) with 1 sitting at position \(i\). Up to replacing \(b_i\) and \(a_i\) by their inverse we can assume \(b_i\in \mathcal{O}_w\) and \(a_i\in \mathcal{O}_v\). Set \(\Lambda:= \prod_{j=1}^k \{0,1,\ldots, r_j\}\). We can write any \(z\in L\) as \(z=\sum_I x_I b^I\), where \(I=(i_1,\ldots, i_k)\in \Lambda\), \(x_I\in K\), and \(b^I=b_1^{i_1}\cdots b_k^{i_k}\). If \(z\in \mathcal{O}_w\), then \(c_I:=x_I b^I\in \mathcal{O}_w\), for all \(I\). Indeed, we have \(w(c_I)= v(x_I)+ \sum_j i_jw(b_j)\) in \(\Gamma_L\) which maps to the residue class of \((i_1,\ldots, i_k)\) under \(\Gamma_L\to \Gamma_L/\Gamma_K\cong \bigoplus_j \mathbb{Z}/r_j\mathbb{Z}\). Hence all \(w(c_I)\), for \(I\) as above, are pairwise different and therefore \(0\le w(z)=\min_I\{w(c_I)\}\). Thus any element in \(\mathcal{O}_w\) can be written as a sum of elements of the form \(xb^I\), with \(x\in K\) and \(I\in \Lambda\) such that \(xb^I\in \mathcal{O}_w\). By induction over the length of \(S\) we see that any element in \(\mathbb{W}_S(\mathcal{O}_w)\) can be written as a sum of elements \(V_n([xb^I])\) with \(xb^I\) as above.

Hence to prove the statement in the case \(q=1\) it suffices to show that the elements

1. \(V_m([xb^I])dV_n([y b^J])\), for \(x,y\in K^\times\), \(I,J\in \Lambda\) with \(xb^I, y b^J\in \mathcal{O}_w\), \(n,m\in S\), and

2. \(\alpha\cdot {\operatorname{dlog}}[c]\), for \(c\in L^\times\), \(\alpha\in \mathbb{W}_S(\mathcal{O}_w)\),

are contained in \[\label{lem:DRW-val-tame-bc3}\mathbb{W}_S(\mathcal{O}_w)\cdot \mathbb{W}_S\Omega^1_{\mathcal{O}_v}(\log )+ d\mathbb{W}_S(\mathcal{O}_w).\tag{30}\] We start by considering the elements of type [lem:DRW-val-tame-bc1]. Set \(\alpha=xb^I\), \(\beta=yb^J\) and \(e={\rm gcd}(m,n)\) and write \(m=em_1\), \(n=en_1\), and let \(i,j\in \mathbb{Z}\) with \(im+jn=e\). We have \[V_m([\alpha])dV_n([\beta]) = V_m([\alpha]F_mdV_n([\beta])).\] Using the second equality in 22 to rewrite \(F_mdV_n([\beta])\) and using further the relations \(V_md=mdV_m\), \(F_n(d[a])=[a]^n{\operatorname{dlog}}[a]\), \(V_n(a)b=V_n(aF_n(b))\) and the Leibniz rule we find that the above element is equal to the following element in \(\mathbb{W}_S\Omega^1_L\) \[imdV_{mn_1}([\gamma])-iV_{mn_1}([\gamma]){\operatorname{dlog}}[\alpha] + jV_{mn_1}([\gamma]){\operatorname{dlog}}[\beta]+(e-1)V_{mn_1}([\gamma]){\operatorname{dlog}}[-1],\] where \(\gamma=\alpha^{n_1}\beta^{m_1}\in \mathcal{O}_w\). As \(r_j\) is invertible in \(K\) we have moreover \[{\operatorname{dlog}}[\alpha]={\operatorname{dlog}}[x]+\sum_{j=1}^k i_j {\operatorname{dlog}}[b_j]= {\operatorname{dlog}}[x]+\sum_{j=1}^k \frac{i_j}{r_j} {\operatorname{dlog}}[a_j] \in \mathbb{W}_S\Omega^1_{\mathcal{O}_v}(\log )\] and similarly with \({\operatorname{dlog}}[\beta]\). Thus the elements of type [lem:DRW-val-tame-bc1] are contained in 30 . Let \(\alpha\cdot{\operatorname{dlog}}[c]\) be an element of type [lem:DRW-val-tame-bc2]. Let \(r=r_1\cdots r_k\), which is invertible in \(K\). Then \(w(c^r)\) is mapped to zero in \(\Gamma_L/\Gamma_K=\bigoplus_{j=1}^k \mathbb{Z}/r_j\mathbb{Z}\) and thus we find a unit \(u\in \mathcal{O}_w^\times\) and an element \(x\in K^\times\) with \(c^r=ux\). Thus \[\label{lem:DRW-val-tame-bc4}\alpha\cdot {\operatorname{dlog}}[c]=\tfrac{1}{r}\alpha\cdot{\operatorname{dlog}}[u]+ \tfrac{1}{r}\alpha\cdot{\operatorname{dlog}}[x].\tag{31}\] Here the second summand clearly is contained in 30 and the first summand is contained in \(\mathbb{W}_S\Omega^1_{\mathcal{O}_w}\), hence is a sum of elements of type [lem:DRW-val-tame-bc1] and therefore lies in 30 as well. This completes the proof in the case \(q=1\).

For the case \(q\ge 1\), we denote by \(R^q_S\) the right hand side of 29 . Note \(R^q_S\subset \mathbb{W}_S\Omega^q_{\mathcal{O}_w}(\log )\) and \(dR^{q-1}_S\subset R^q_S\), as follows from the Leibniz rule and the formula \(ddx= {\operatorname{dlog}}[-1]dx\). From this latter formula and 23 it also follows that any element in \(\mathbb{W}_S\Omega^{q-1}_{\mathcal{O}_w}(\log )\), which is a product of elements of the form \(db\), with \(b\in\mathbb{W}_S(\mathcal{O}_w)\), and \({\operatorname{dlog}}[c]\), with \(c\in L^\times\), can be written as \(d(\beta)+ \beta'{\operatorname{dlog}}[-1]\) with \(\beta, \beta'\in \mathbb{W}_S\Omega^{q-2}_{\mathcal{O}_w}(\log)\). Thus any element in \(\mathbb{W}_S\Omega^{q}_{\mathcal{O}_w}(\log )\) is a sum of elements \[\alpha d\beta\quad \text{and}\quad \alpha \beta'{\operatorname{dlog}}[-1]\quad \text{with }\alpha\in \mathbb{W}_S\Omega^1_{\mathcal{O}_w}(\log), \beta, \beta'\in \mathbb{W}_S\Omega^{q-2}_{\mathcal{O}_w}(\log ).\] We show by induction over \(q\) that these elements are in \(R^q_S\). By induction we have \(\alpha\beta'\in R_S^{q-1}\), hence \(\alpha\beta'{\operatorname{dlog}}[-1]\in R^q_S\). By the Leibniz rule \(\alpha d\beta= -d(\alpha\beta)+ d(\alpha)\beta\) and as \(d(\alpha\beta)\in dR^{q-1}_S\) by induction, we are left to consider the element \(d(\alpha)\beta\). By the case \(q=1\) we know that \(\alpha\) is contained in 30 and hence is a sum of elements \(b \gamma + db'\), with \(b,b'\in \mathbb{W}_S(\mathcal{O}_w)\) and \(\gamma\in \mathbb{W}_S\Omega^1_{\mathcal{O}_v}(\log )\). Thus \(d(\alpha)\beta\) is a sum of elements \[d(\gamma) b\beta - \gamma d(b)\beta+ {\operatorname{dlog}}[-1] d(b')\beta.\] By induction \(b\beta\in R_S^{q-2}\) and \(d(b)\beta\), \(d(b')\beta\in R_S^{q-1}\). Hence \(d(\alpha)\beta\) is contained in \(R^q_S\). ◻

Proof. As \(\mathcal{O}_w/\mathcal{O}_v\) is flat, by [32], the map \[\mathcal{O}_w\otimes_{\mathcal{O}_v} \Omega^q_{\mathcal{O}_v}(\log)\to \mathcal{O}_w\otimes_{\mathcal{O}_v} \Omega^q_K= L\otimes_K\Omega^q_K=\Omega^q_L\] is injective, hence it suffices to show \(\Omega^q_{\mathcal{O}_w}(\log)\subset \mathcal{O}_w\cdot \Omega^q_{\mathcal{O}_v}(\log)\). To this end we show \(db\), \({\operatorname{dlog}}\,c\in \mathcal{O}_w\cdot \Omega^q_{\mathcal{O}_v}(\log)\), for all \(b\in \mathcal{O}_w\) and \(c\in L^\times\). Concerning the elements of type \(db\) note that as explained in the beginning of the proof of Lemma 52 (and with the notation from there) it suffices to consider elements \(b= x b_1^{i_1}\cdots b_k^{i_k}\in \mathcal{O}_w\), with \(x\in K\) and \(\sqrt[r_i]{b_i}=a_i\in K^\times\). In this case \[db= b\cdot{\operatorname{dlog}}\, b= b\cdot{\operatorname{dlog}}\, x + \sum_j \tfrac{i_j}{r_j}b\cdot{\operatorname{dlog}}\,a_j\] clearly lies in \(\mathcal{O}_w\cdot \Omega^q_{\mathcal{O}_v}(\log)\). For \({\operatorname{dlog}}\, c\) the argument around 31 applies. ◻

Proof. Denote by \(R^q_S\) the right-hand side of the equality in the statement. The image of \(\mathbb{W}_S\Omega^q_{\tilde{B}}(\log )\) under the natural map \(\mathbb{W}_S\Omega^q_B\to \mathbb{W}_S\Omega^q_L\) is by definition equal to \(\mathbb{W}_S\Omega^q_{\mathcal{O}_w}(\log )\), which by Lemma 52 and our assumption is also equal to the image of \(R^q_S\) in \(\mathbb{W}_S\Omega^q_{L}\). By Lemma 51 it suffices to show that the group \[\mathbb{W}_S(\mathfrak{m}_B)\cdot \mathbb{W}_S\Omega^q_B+ d(\mathbb{W}_S(\mathfrak{m}_B)\cdot \mathbb{W}_S\Omega^{q-1}_B)\] is contained in \(R^q_S\), where \(\mathfrak{m}_B=\mathfrak{m}_A B\) is the maximal ideal of \(B\). As \(R^q_S\) is closed under the differential \(d\) it suffices to show \[\label{prop:DRW-tame-bc1}\mathbb{W}_S(\mathfrak{m}_B)\cdot \mathbb{W}_S\Omega^q_B\subset R^q_S.\tag{33}\] As we have a surjection \(\mathbb{W}_S(\mathfrak{m}_A\otimes_A B)\to \mathbb{W}_S(\mathfrak{m}_B)\), [34] yields \[\label{prop:DRW-tame-bc2}\mathbb{W}_S(\mathfrak{m}_B)=\mathbb{W}_S(\mathfrak{m}_A)\cdot \mathbb{W}_S(B).\tag{34}\] By étale base change [11] \[\mathbb{W}_S\Omega^q_B=\mathbb{W}_{S}(B)\otimes_{\mathbb{W}_S(A)}\mathbb{W}_S\Omega^q_A.\] Thus \[\begin{align} \mathbb{W}_S(\mathfrak{m}_B)\cdot \mathbb{W}_S\Omega^q_B & =\mathbb{W}_S(B)\cdot \mathbb{W}_S(\mathfrak{m}_A)\cdot\mathbb{W}_S\Omega^q_A\\ & \subset \mathbb{W}_S(B)\cdot \left(\mathbb{W}_S(\mathfrak{m}_A)\cdot \mathbb{W}_S\Omega^q_{\tilde{A}}(\log) + d(\mathbb{W}_S(\mathfrak{m}_A)\cdot \mathbb{W}_S\Omega^{q-1}_{\tilde{A}}(\log))\right) \\ &\subset \mathbb{W}_S(\mathfrak{m}_B)\cdot \mathbb{W}_S\Omega^q_{\tilde{A}}(\log) + \mathbb{W}_S(B)\cdot d(\mathbb{W}_S(\mathfrak{m}_A)\cdot \mathbb{W}_S\Omega^{q-1}_{\tilde{A}}(\log))\\ &\subset R^q_S + \mathbb{W}_S(\mathfrak{m}_A) \cdot d(\mathbb{W}_S(B))\cdot \mathbb{W}_S\Omega^{q-1}_{\tilde{A}}(\log), \end{align}\] where we use 34 for the equality, 25 for the first inclusion, 34 for the second inclusion, and an application of the Leibniz rule together with 34 for the last inclusion. It remains to show \[\label{prop:DRW-tame-bc3} \mathbb{W}_S(\mathfrak{m}_A)\cdot \mathbb{W}_S\Omega^1_B\subset R^1_S.\tag{35}\] We prove this by induction over the cardinality of \(S\). For \(S=\{1\}\) this amounts by étale base change to show that for \(a\in \mathfrak{m}_A\), \(a_0\in A\) and \(b\in B\) the element \(\alpha:=abda_0\) is in \(\mathfrak{m}_B\cdot \Omega^1_{\tilde{A}}(\log)\). But this is immediate if \(a_0\in \mathfrak{m}_A\subset\tilde{A}\) and if \(a_0\in A^\times\), this follows from \(\alpha=aba_0{\operatorname{dlog}}a_0\). Now we assume 35 is proven for all truncation sets of smaller cardinality than \(S\). First observe that \(V_n: \mathbb{W}_{S/n}\Omega^1_{\tilde{B}}(\log)\to \mathbb{W}_S\Omega^1_{\tilde{B}}(\log)\), \(n\in S\), induces a map \[V_n: R^1_{S/n}\to R^1_S.\] Indeed, any element in \(R^1_{S/n}\) can be written as sum of elements \[\begin{gather} b_0da_0 + b_1{\operatorname{dlog}}[a_1]+ db_2 \\ = b_0 F_ndV_n(a_0)-(n-1)b_0a_0F_n({\operatorname{dlog}}[-1])+ b_1 F_n({\operatorname{dlog}}[a_1]) + db_2, \end{gather}\] where \(b_i\in \mathbb{W}_{S/n}(\tilde{B})\), \(a_0\in \mathbb{W}_{S/n}(\tilde{A})\), \(a_1\in A^\times\), see [11] for the equality. By [11] this is under \(V_n\) mapped to \[V_n(b_0) dV_n(a_0)-(n-1)V_n(b_0a_0){\operatorname{dlog}}[-1]+ V_n(b_1){\operatorname{dlog}}[a_1]+ ndV_n(b_2),\] which lies in \(R^1_S\). We show the inclusion 35 . By étale base change, 27 , and 34 it remains to show \[\mathbb{W}_S(B)\cdot d\mathbb{W}^0_S(\mathfrak{m}_A)\subset R^1_S,\] where \(\mathbb{W}^0_S(\mathfrak{m}_A)=\operatorname{Ker}(\mathbb{W}_S(\mathfrak{m}_A)\to\mathfrak{m}_A)\). An element in the left hand side can be written as a sum of elements \[bdV_n(a)= dV_n(F_n(b)a)- V_n(a F_ndb),\] for \(b\in \mathbb{W}_S(B)\), \(a\in\mathbb{W}_{S/n}(\mathfrak{m}_A)\), and \(1<n\in S\). As \(F_n(b)a\in \mathbb{W}_{S/n}(\mathfrak{m}_B)\), we find \(dV_n(F_n(b)a)\in R^1_S\). By induction \(aF_ndb\in \mathbb{W}_{S/n}(\mathfrak{m}_A)\cdot \mathbb{W}_{S/n}\Omega^1_B\subset R^1_{S/n}\), and hence \(V_n(a F_ndb)\in R^1_S\). This completes the proof of the proposition. ◻

Proof. In case \(A\) is noetherian the ring \(\tilde{B}\) is a flat \(\tilde{A}\)-module as follows from the argument in [14] and the fact that in the situation at hand we have \(B=\tilde{B}\otimes_{\tilde{A}} A\) and \(\tilde{B}\) is \(\tilde{A}\)-torsion free. In the same way as in the beginning of the proof of Lemma 53 we see that the natural maps \(\tilde{B}\otimes_{\tilde{A}} \Omega^q_{\tilde{A}}(\log)\to \Omega^q_{\tilde{B}}(\log)\) is injective. Thus it remains to show the first statement in which we allow \(A\) also to be non-noetherian.

By Proposition 54 it suffices to show \(d(\tilde{B})\subset \Omega^1_{\tilde{B}}(\log)\subset \tilde{B}\cdot \Omega^1_{\tilde{A}}(\log)\). By Lemmas 53 and 51 and since \(\mathfrak{m}_B=\mathfrak{m}_A\cdot B\) it remains to show that the group \[\mathfrak{m}_A\cdot \Omega^1_{\tilde{B}}(\log)+ d(\mathfrak{m}_A\cdot\tilde{B})= \mathfrak{m}_A\Omega^1_{\tilde{B}}(\log)+\tilde{B}\cdot d\mathfrak{m}_A\] is contained in \(\tilde{B}\cdot\Omega^1_{\tilde{A}}(\log)\). This is clear for \(\tilde{B}\cdot d\mathfrak{m}_A\). By étale base change for \(\Omega^1_B\) any element in \(\mathfrak{m}_A\Omega^1_{\tilde{B}}(\log)\) is a sum of elements of the form \(abda_0\) with \(a\in\mathfrak{m}_A\), \(b\in B\), \(a_0\in A\). If \(a_0\in \mathfrak{m}_A\), then \(abda_0 \in \tilde{B}\cdot \Omega^1_{\tilde{A}}(\log)\); if \(a_0\in A^\times\), then \(ab d a_0=ab a_0{\operatorname{dlog}}\, a_0\in \tilde{B}\cdot\Omega^1_{\tilde{A}}(\log)\). Hence the statement. ◻

Proof. This "\(\supset\)" inclusion follows in both cases directly from the definition and the equality \[\mathbb{W}_S\Omega^q_{\mathcal{O}^h_{\tilde{X}, k(x), v}}(log)= \mathbb{W}_S\Omega^q_{\mathcal{O}^h_{X,x}}\times_{\mathbb{W}_S\Omega^q_{k(x)}} \mathbb{W}_S\Omega^q_{\mathcal{O}_v}(\log),\] see Lemma 51. For the other inclusion, let \(a\in \mathbb{W}_S\Omega^{q,t}(X,\tilde{X})\) (resp. \(\mathbb{W}_S\Omega^{q,div}(X,\tilde{X})\)) and take \((x,v,\varepsilon)\in \mathrm{Spa}(X,\tilde{X})\) (resp. \(\mathrm{Spa}(X,\tilde{X})^{div}\)). By Definition 40 and Lemma 8 there exists a finite tame Galois extension \((L,w)/(K=k(x),v)\), such that \(a_L\), the pullback of \(a\) along \(\operatorname{Spec}\mathcal{O}_{X,L}^h\to X\), lies in \(\mathbb{W}_S\Omega^q_{\mathcal{O}^h_{\tilde{X}, L,w}}(\log)\). Let \(L^{sh}\) and \(K^{sh}\) be the strict henselizations, so that \(L^{sh}/K^{sh}\) is a finite Galois extension of degree \(r\), which is prime to the residue characteristic of \(K^{sh}\). Thus \(\mathcal{O}^h_{X, K^{sh}}\to \mathcal{O}^h_{X, L^{sh}}\) is a finite étale extension as in 55 and thus \[a_{K^{sh}}= \operatorname{Tr}(\tfrac{1}{r} a_{L^{sh}})\in \mathbb{W}_S\Omega^q_{\mathcal{O}^h_{\tilde{X}, K^{sh},v}}(\log).\] By Lemma 49, the map \(\mathbb{W}_S(\mathcal{O}^h_{\tilde{X}, K,v})\to \mathbb{W}_S(\mathcal{O}^h_{\tilde{X}, K^{sh},v})\) is faithfully flat and we have \[\mathbb{W}_S\Omega^q_{\mathcal{O}^h_{\tilde{X}, K^{sh},v}}(\log)= \mathbb{W}_S(\mathcal{O}^h_{\tilde{X}, K^{sh},v})\otimes_{\mathbb{W}_S(\mathcal{O}^h_{\tilde{X}, K,v})} \mathbb{W}_S\Omega^q_{\mathcal{O}^h_{\tilde{X}, K,v}}(\log),\] thus descent yields \(a_K\in \mathbb{W}_S\Omega^q_{\mathcal{O}^h_{\tilde{X}, K,v}}(\log)\). As this is true for all \((x,v,\varepsilon)\in \mathrm{Spa}(X,\tilde{X})\), the element \(a\) is contained in the respective right hand sides of the statement. ◻

Proof. This "\(\subset\)" inclusion holds by Proposition 56 and Lemma 51. For the other inclusion, let \((x,v,\varepsilon)\in \mathrm{Spa}(U,\tilde{U})\). It corresponds to ring maps \(A\to k(x)\) and \(\tilde{A}\to \mathcal{O}_v\), which induce a morphism \(\mathbb{W}_S\Omega^q_{\tilde{A}}(\log)\to \mathbb{W}_S\Omega^q_{\mathcal{O}_v}(\log )\). Hence this inclusion \(\mathbb{W}_S\Omega^q_{\tilde{A}}(\log)\subset \mathbb{W}_S\Omega^{q,t}(U,\tilde{U})\) follows from Proposition 56. ◻

Proof. We may assume \(\tilde{X}\) is integral. Clearly, we have the inclusions \[\Omega^{q}_{\tilde{X}/T}(\log D)\subset j_*\Omega^{q,t}_{/T} \subset j_*\Omega^{q, div}_{/T}\] and the equality \[\Omega^{q}_{\tilde{X}/T}(\log D)= \operatorname{Ker}\left(\Omega^q_{X/T}\to \bigoplus_{x\in D^{(0)}} \Omega^q_{\eta/T}/\Omega^q_{\tilde{X}, x/T}(\log D)\right),\] where \(\eta\in \tilde{X}\) is the generic point. Thus it remains to show: Let \(R\) be a ring and \((K,v)\) a discrete valuation field, such that \(\mathcal{O}_v\) is an \(R\)-algebra. Let \(\alpha\in \Omega^q_{K/R}\) and assume there exists a finite tame Galois extension \((L,w)/(K,v)\) such that \(\alpha_L\in \Omega^q_{\mathcal{O}_w}(\log)\). Then \(\alpha\in \Omega^q_{\mathcal{O}_v/R}(\log)\).

To this end, we may assume by étale descent that \((K,v)\) and \((L,w)\) are strictly henselian. We have a surjection \(\Omega^q_{\mathcal{O}_w}(\log )\rightarrow\!\!\!\!\!\rightarrow\Omega^q_{\mathcal{O}_w/R}(\log )\) and hence by Lemma 53 \[\Omega^q_{\mathcal{O}_w/R}(\log)= \mathcal{O}_w\cdot \Omega^q_{\mathcal{O}_v/R}(\log).\] Hence the trace from 55 for \(S=\{1\}\) also induces a trace for relative differentials \[\operatorname{Tr}: \Omega^q_{\mathcal{O}_w/R}(\log)\to \Omega^q_{\mathcal{O}_v/R}(\log ).\] Denoting by \(e=[L:K]\) the degree, which is prime to the residue characteristic of \(\mathcal{O}_v\), we obtain \(\alpha= \frac{1}{e}\operatorname{Tr}(\alpha_L)\in \Omega^q_{\mathcal{O}_v/R}(\log )\). This completes the proof. ◻

Proof. For definiteness, we set \[W_n\Omega^q_{\tilde{X}}(\log D):= W_n\Lambda^q_{(\tilde{X}, \mathcal{M})/(\operatorname{Spec}k, {\rm triv})}\] with the notation from [40], where \(\mathcal{M}= j_*\mathcal{O}_X^\times\cap \mathcal{O}_{\tilde{X}}\) is the log structure defined by \(D\) and we equip the base \(\operatorname{Spec}k\) with the trivial log structure. If \(U\) is a smooth \(k\)-scheme and \(x\in U\) is a 1-codimensional point such that its closure \(D_x\) in \(U\) is smooth, and \(L\) is the fraction field of \(\mathcal{O}_{U,x}\) (resp. the fraction field of its strict henselization at a geometric point \(\bar{x}\) over \(x\)) and \(w\) is the discrete valuation defined by \(x\) on \(L\), then it follows from étale base change and [40] that the Zariski stalk at \(x\) (resp. the étale stalk at \(\bar{x}\)) of \(W_n\Omega^q_{U}(\log D_x)\) is equal to \(W_n\Omega^q_{\mathcal{O}_w}(\log )\). Furthermore, it follows from the explanations in [40] and [35] that \(W_n\Omega^q_{\tilde{X}}(\log D)\) is a successive extension of locally free \(\mathcal{O}_{\tilde{X}}\)-modules and hence is Cohen–Macaulay. Thus \(W_n\Omega^q_{\tilde{X}}(\log D)=j_*W_n\Omega^{q,div}\), by Proposition 56 with \(S=\{1, p,\ldots, p^{n-1}\}\). Thus also \(j_*W_n\Omega^{q,t}\subset W_n\Omega^q_{\tilde{X}}(\log D)\). On the other hand it follows from the local description of \(W_n\Omega^q_{\tilde{X}}(\log D)\), given by [40] and étale base change, that the pullback of a local section \(\alpha\in W_n\Omega^q_{\tilde{X}}(\log D)\) along \((\operatorname{Spec}L, \operatorname{Spec}\mathcal{O}_w)\to (X,\tilde{X})\), with \((L,w)\) a valuation field, lies in \(W_n\Omega^q_{\mathcal{O}_w}(\log)\). Hence the inclusion \(W_n\Omega^q_{\tilde{X}}(\log D)\subset j_*W_n\Omega^{q,t}\). ◻

Proof. For the first statement, observe that by Proposition 59 we have a natural map \[W_n\Omega^{q}_{\tilde{X}}(\log D)=j_*W_n\Omega^{q,t}\to Rj_*W_n\Omega^{q,t}.\] It remains to show that this yields an isomorphism after applying \(H^i(\tilde{X}_{{\operatorname{Zar}}}, -)\). By Theorem 38 we have \[H^i((X,\tilde{X})_t, W_n\Omega^{q,t})= \varinjlim_{\mathcal{Y}\to \mathcal{X}} H^i(\tilde{Y}_{{\operatorname{\acute{e}t}}}, j^{\mathcal{Y}}_*W_n\Omega^{q,t}),\] where the direct limit is over all modifications \(\mathcal{Y}=(X,\tilde{Y})\to \mathcal{X}\). Under [res2] modifications as in [res2] are cofinal in the cofiltered category of all modifications to \(\mathcal{X}\), see, e.g., the argument in the proof of [18]. Using further that étale- and Zariski cohomology of \(W_n\Omega^q_{\tilde{X}}(\log D)\) coincide, we obtain \[H^i((X,\tilde{X})_t, W_n\Omega^{q,t})= \varinjlim_{\mathcal{Y}\to \mathcal{X}} H^i(\tilde{Y}_{{\operatorname{Zar}}}, W_n\Omega^{q}_{\tilde{Y}}(\log D')),\] where the limit is over modifications induced by morphisms \(\pi:\tilde{Y}\to \tilde{X}\) as in [res2] and and where \(D':=\pi^{-1}(D)_{\rm red}\). The pullback \[\pi^*: H^i(\tilde{X}_{{\operatorname{Zar}}}, W_n\Omega^{q}_{\tilde{X}}(\log D ))\to H^i(\tilde{X}_{{\operatorname{Zar}}}, W_n\Omega^q_{\tilde{Y}}(\log D'))\] is an isomorphism by [41] and the fact that with the notation from loc. cit. the Nisnevich sheaf given by \(U\mapsto \underline{\omega}^{\operatorname{\mathbf{CI}}}W_n\Omega^q(U,D_{|U})\) is equal to \(W_n\Omega^q_{\tilde{X}}(\log D)\). For \(q=0\), this follows from [42], for \(q\ge 1\) this holds by [43] or the proof of [4]. This yields the first statement.

Considering the total complex of the double complex \((\ldots \to R\Gamma((X,\tilde{X})_t, W_n\Omega^{q,t})\xrightarrow{d} R\Gamma((X,\tilde{X})_t, W_n\Omega^{q+1,t})\to \ldots)\) and applying \(R\operatornamewithlimits{\varprojlim}\) yields the second statement. ◻

Proof. This is the tame version of [45]. Indeed, \(\iota_r^q\) (resp. \(F^q_r\)) is nilpotent in degrees \(\le r_1\) (resp. \(\ge r\)) and hence \(\iota_r-F_r\) is an isomorphism of pro-complexes in degrees \(\neq r\). It remains to show that \[{\rm id}-F: W_{\mathbin{\ThisStyle{\vcenter{ \scalebox{.6}{\SavedStyle\bullet}}}}}\Omega^{r,t}\to W_{\mathbin{\ThisStyle{\vcenter{ \scalebox{.6}{\SavedStyle\bullet}}}}} \Omega^{r,t}\] is surjective on \((X,\tilde{X})\) with kernel \(W\Omega^{q,t}_{\log}\). By Proposition 26, Remark 30, and Proposition 57 it suffices to show that the sequence \[\label{thm:syn-vs-log1}0\to W_{\mathbin{\ThisStyle{\vcenter{ \scalebox{.6}{\SavedStyle\bullet}}}}}\Omega^{r}_{A, \log}\to W_{\mathbin{\ThisStyle{\vcenter{ \scalebox{.6}{\SavedStyle\bullet}}}}}\Omega^{r}_{\tilde{A}}(\log)\xrightarrow{{\rm id} -F} W_{\mathbin{\ThisStyle{\vcenter{ \scalebox{.6}{\SavedStyle\bullet}}}}}\Omega^r_{\tilde{A}}(\log)\to 0\tag{38}\] is exact for henselian local rings \((A,\mathfrak{m})\) with a valuation \(v\) on the residue field \(K=A/\mathfrak{m}\), such that \(A\) is ind-étale over \(X\), \((K,v)\) is tamely closed, and \(\tilde{A}=A\times_K \mathcal{O}_v\). As 38 is a subsequence of the sequence \[0\to W_{\mathbin{\ThisStyle{\vcenter{ \scalebox{.6}{\SavedStyle\bullet}}}}}\Omega^{r}_{A, \log}\to W_{\mathbin{\ThisStyle{\vcenter{ \scalebox{.6}{\SavedStyle\bullet}}}}}\Omega^{r}_{A} \xrightarrow{{\rm id} -F} W_{\mathbin{\ThisStyle{\vcenter{ \scalebox{.6}{\SavedStyle\bullet}}}}}\Omega^r_{A},\] which is exact by [35], we get that the sequence 38 is left exact as well. As \(\tilde{A}\) is strictly henselian, the map \(W_{\mathbin{\ThisStyle{\vcenter{ \scalebox{.6}{\SavedStyle\bullet}}}}}\Omega^{r}_{\tilde{A}}\xrightarrow{{\rm id} -F} W_{\mathbin{\ThisStyle{\vcenter{ \scalebox{.6}{\SavedStyle\bullet}}}}}\Omega^r_{\tilde{A}}\) is surjective, by loc. cit. Thus the formula \(Fd\log[x]={\operatorname{dlog}}[x]\) yields the right exactness of 38 . ◻

Proof. We denote by \(\mathbb{Z}(r)_X\) Bloch’s complex of Zariski sheaves \(U\mapsto z^r(U,*)[-2r]\) [46], which is an avatar of the motivic complex of weight \(r\). Let \(i:Y=\tilde{X}\setminus X\hookrightarrow\tilde{X}\) be the closed immersion, and denote by \(\sigma: X\hookrightarrow\tilde{X}\) the open immersion and by \(c={\rm codim(Y,\tilde{X})}\) the codimension. Consider the following composition of maps between pro-complexes \[\frac{\mathbb{Z}(r)_{\tilde{X}}}{i_*\mathbb{Z}(r)_{Y}}\to \sigma_* \mathbb{Z}(r)_X\to \sigma_*\mathbb{Z}/p^{\mathbin{\ThisStyle{\vcenter{ \scalebox{.6}{\SavedStyle\bullet}}}}}(r)_X\cong \sigma_*W_{\mathbin{\ThisStyle{\vcenter{ \scalebox{.6}{\SavedStyle\bullet}}}}}\Omega^r_{X,\log}[-r]=j_*W_{\mathbin{\ThisStyle{\vcenter{ \scalebox{.6}{\SavedStyle\bullet}}}}}\Omega^{r,t}_{\log}[-r]\to Rj_*\mathbb{Z}/p^{\mathbin{\ThisStyle{\vcenter{ \scalebox{.6}{\SavedStyle\bullet}}}}}(r)^t,\] where the isomorphism in the middle is [47], the equality holds by definition of \(W_n\Omega^{q,t}_{\log}\), and the last map is induced by Theorem 64. Applying \(R\operatornamewithlimits{\varprojlim}R\Gamma(\tilde{X}, -)\) and using \(R\Gamma(\tilde{X}_{{\operatorname{Zar}}}, \frac{\mathbb{Z}(r)_{\tilde{X}}}{i_*\mathbb{Z}(r)_{Y}}) \cong \Gamma(X,\mathbb{Z}(r)_X)\), which holds by [46] together with [48], we get the first map in 39 . The second map is the composition \[H^i((X,\tilde{X})_t, \mathbb{Z}_p(r)^t)\to H^i((X,\tilde{X}), R\operatornamewithlimits{\varprojlim}\mathcal{N}^r W_n\Omega^{\bullet,t})\; \xrightarrow{\iota_r} H^i((X,\tilde{X}), R\operatornamewithlimits{\varprojlim}W_n\Omega^{\bullet,t}).\] This completes the construction of the cycle map. The other statements follow immediately from this construction. ◻

Proof. We start by showing this \(\supset\) inclusion. Let \(a\in \underline{\omega}^{\operatorname{\mathbf{CI}}}F(\tilde{X},D)\subset F(X)\). Let \((x,w,\varepsilon)\in \mathrm{Spa}(X,\tilde{X})\). It suffices to show that \(a_{k(x)}\) the pullback of \(a\) to \(F(k(x))\) lies in \(\log F(w)\) as defined in 40 . But for \(v\in \Phi_w\) we have a natural map \(\operatorname{Spec}\mathcal{O}_v^h\to \tilde{X}\) and it follows from [41], that this map induces a morphism \(\underline{\omega}^{\operatorname{\mathbf{CI}}}F(\tilde{X},D)\to \underline{\omega}^{\operatorname{\mathbf{CI}}}F(\mathcal{O}_v^h, \mathfrak{m}_v^{-1})\). Hence \(a_{k(x)}\in \log F(w)\). Now the other inclusion. Let \(a\in F_{\beta^{\rm rec}_{\log}}(X,\tilde{X})\). By [41] it suffices to show that for each generic point \(\eta\) of \(D\) the pullback of \(a\) to \(F(K^h_{\eta})\), where \(K=k(\tilde{X})\), lies in \(\underline{\omega}^{\operatorname{\mathbf{CI}}}F(\mathcal{O}_{\tilde{X},\eta}^h, \mathfrak{m}_{\eta}^{-1})\). Let \(v\) be the discrete valuation on \(K\) corresponding to \(\eta\), giving rise to the point \((\eta, v,\varepsilon)\in \mathrm{Spa}(X,\tilde{X})\), where \(\varepsilon:\operatorname{Spec}\mathcal{O}_{\tilde{X},\eta}\to \tilde{X}\) is the natural map. By definition of \(F_{\beta^{\rm rec}_{\log}}\) we find a finite tame extension \((L, w)/(K,v)\) such that \[a_{L}\in \log F_{w}=\log F(w)=\operatorname{Ker}\left(F(L)\to \frac{F(L_w^h)}{\underline{\omega}^{\operatorname{\mathbf{CI}}}F(\mathcal{O}_{w}^h, \mathfrak{m}_w^{-1})}\right),\] where we use that \(w\) is already a discrete valuation of geometric type over \(k\) on \(L\) and hence \(\Phi_w=\{w\}\). We have a finite morphism of pro-modulus pairs (see [42]) \(f:(\operatorname{Spec}\mathcal{O}_w^{sh}, e\cdot s_w)\to (\operatorname{Spec}\mathcal{O}_{\tilde{X}, \eta}^{sh}, \eta)\), where \(s_w\) denotes the closed point and \(e\) is the ramification index, which is also equal to the degree of \(f\). The transfer structure thus induces a pushforward \[f_*: \underline{\omega}^{\operatorname{\mathbf{CI}}}F(\mathcal{O}_w^{sh}, \mathfrak{m}_w^{-1}) \subset \underline{\omega}^{\operatorname{\mathbf{CI}}}F(\mathcal{O}_w^{sh}, \mathfrak{m}_w^{-e})\to \underline{\omega}^{\operatorname{\mathbf{CI}}}F(\mathcal{O}_{\tilde{X},\eta}^{sh}, \mathfrak{m}_\eta^{-1}).\] By tameness, \((e,p)=1\) and hence \[a_{K_\eta^{sh}}= \tfrac{1}{e} f_*(a_{L_w^{sh}})\in \underline{\omega}^{\operatorname{\mathbf{CI}}}F(\mathcal{O}_{\tilde{X},\eta}^{sh}, \mathfrak{m}_\eta^{-1}).\] Thus \(a_{K_{\eta}^h}\in \underline{\omega}^{\operatorname{\mathbf{CI}}}F(\mathcal{O}_{\tilde{X},\eta}^h, \mathfrak{m}_{\eta}^{-1})\), by 41 , which completes the proof. ◻

Proof. Under [res2] the modifications as in [res2] are cofinal in the cofiltered category of all modifications to \(\mathcal{X}\), see, e.g., the argument in the proof of [18]. Thus \[\label{thm:tamecohRSC1}H^i((X,\tilde{X})_t, F_{\beta_{\log}^{\rm rec}}) =\varinjlim_{\mathcal{Y}\to \mathcal{X}} H^i(\tilde{Y}_{{\operatorname{\acute{e}t}}}, j_*^{\mathcal{Y}}F_{\beta_{\log}^{\rm rec}}) = \varinjlim_{\mathcal{Y}\to \mathcal{X}} H^i(\tilde{Y}_{{\operatorname{Nis}}}, F_{(\tilde{Y},D)}),\tag{42}\] where the direct limit is over all modifications \(\mathcal{Y}=(X,\tilde{Y})\to \mathcal{X}\) as in [res2]. The first equality holds by Theorem 38, and the second equality by assumptions [thm:tamecohRSCa], [thm:tamecohRSCb], and Lemma 66. Thus it remains to prove the colimit on the right is constant. Let \(\pi:\tilde{Y}\to \tilde{X}\) be a morphism as in [res2] and set \(D':=\pi^{-1}(D)_{\rm red}\). In this case the pullback \[\pi^*: H^i(\tilde{X}_{{\operatorname{Nis}}}, F_{(\tilde{X},D)})\to H^i(\tilde{Y}_{{\operatorname{Nis}}}, F_{(\tilde{Y}, D')})\] is an isomorphism by [41], hence the statement. ◻

Proof. We need to show that \(R^qu_*u^*G = 0\) for \(q>0\). Indeed, if this holds, we have that the map \(G\to Ru_*u^*G\) is an equivalence, therefore\[R\Gamma((X/S)_t,G) \simeq R\Gamma((X/S)_t,Ru_*u^*G)\simeq R\Gamma((X,\tilde{X})_t,u^*G).\] Let \((\overline{x},\overline{v})\) be a tame point of \(X\) in the sense of [5], then\[(R^qu_*u^*G)_{(\overline{x},\overline{v})} = \operatornamewithlimits{\varinjlim}_{U\to X} H^q((U,\tilde{U})_t,u^*G),\] where the colimit on the right hand side is indexed along all étale neighborhoods \(U\to X\) of \(\overline{x}\) which are tame at \((\overline{x},\overline{v})\), and \(\tilde{U}\to \tilde{X}\) a choice of a Nagata compactification of \(U\to \tilde{X}\). We can always suppose that \(U\) is affine, therefore the map \(U\to S\) is quasi-projective. By Chow’s lemma and the control of the birational locus of Raynaud and Gruson (see [20]), we can choose \(\tilde{U}\) quasi-projective over \(S\). Hence \(\tilde{U}\) satisfies the property that every finite set of points is contained in an affine open. Thus by Theorem 36 \[\operatornamewithlimits{\varinjlim}_{U\to X} H^q((U,\tilde{U})_t,u^*G) \cong \operatornamewithlimits{\varinjlim}_{U\to X} \check H^{q}((U,\tilde{U})_t,u^*G),\] Recall that \(u^*G\) is the sheafification of the presheaf \((V,\tilde{V})\mapsto G(V)\). Since Čech cohomology of presheaves is invariant under taking sheafification by Theorem 36, we have that\[\operatornamewithlimits{\varinjlim}_{U\to X} \check H^{q}((U,\tilde{U})_t,u^*G) \cong \operatornamewithlimits{\varinjlim}_{(V,\tilde{V})\to (U,\tilde{U})}\operatornamewithlimits{\varinjlim}_{U\to X} \pi_{-q}G(V^{\times \bullet}),\] where we can suppose that \((V,\tilde{V})\to (U,\tilde{U})\) ranges over all coverings in \((X,\tilde{X})_{{\rm affine},t}\) by Lemma 15, and moreover by Claim 69 that factors through \((V,\overline{V})\to (U,\tilde{U})\) in \((X,\tilde{X})_{\rm suc}\). Since \(\pi_{-q}G(V^{\times \bullet})\) does not depend on \(\tilde{V}\), we can in fact suppose that \(\tilde{V}=\overline{V}\), therefore we have that\[\operatornamewithlimits{\varinjlim}_{U\to X} \check H^{q}((U,\tilde{U})_t,u^*G)\cong \operatornamewithlimits{\varinjlim}_{(V,\overline{V})\to (U,\tilde{U})}\operatornamewithlimits{\varinjlim}_{U\to X} \pi_{-q}G(V^{\times \bullet}),\] where \((V,\tilde{V})\to (U,\tilde{U})\) ranges over all coverings in \((X,\tilde{X})_{suc,t}\) with \(V\) affine. By the valuative criterion (as in the proof of Proposition 70) this implies that \(V\to U\) is a covering in \((X/S)_t\), and every covering \(V\to U\) in \((X/S)_t\) with \(V\) affine induces a covering \((V,\overline{V})\to (U,\tilde{U})\) in \((X,\tilde{X})_{suc,t}\) by the choice of a Nagata compactification, therefore \[\operatornamewithlimits{\varinjlim}_{(V,\overline{V})\to (U,\tilde{U})}\operatornamewithlimits{\varinjlim}_{U\to X} \pi_{-q}G(V^{\times \bullet})\cong \operatornamewithlimits{\varinjlim}_{V\to U}\operatornamewithlimits{\varinjlim}_{U\to X} \pi_{-q}G(V^{\times \bullet}),\] where on the right \(\{V\to U\}\) is the cofiltered set of coverings in \((X/S)_t\) and \(\{U\to X\}\) is the cofiltered set of tame neighborhoods of \((\overline{x},\overline{v})\), therefore by [5] it is\[\operatornamewithlimits{\varinjlim}_{U\to X} \check{H}^q((U/S)_t,G) = H^q((\operatorname{Spec}(\mathcal{O}_{X,(\overline{x},\overline{v})}^t)/S)_t,G) = 0.\] ◻

Proof. The second statement follows directly from the first and Theorem 72. As \(h^0_{\mathbf{A}^1}(F)\) is a Zariski sheaf it suffices to show the equality of the first statement when evaluated on an affine étale \(X\)-scheme \(V\). By [res1] we find a smooth compactification \(\overline{V}\) of \(V\) with simple normal crossing complement \(D_V=(\overline{V}\setminus V)_{{\operatorname{red}}}\). By Proposition 70 together with Lemma 66 we have \[\label{lem:tameRSCA1-1}u_*(F_{\beta^{\rm rec}_{\log}})(V)= F_{\beta^{\rm rec}_{\log}}(V,\overline{V})= {\underline{\omega}}^{\operatorname{\mathbf{CI}}}F(\overline{V}, D_V).\tag{46}\] By [41] for each map \(\operatorname{Spec}\mathcal{O}_v^h\to \overline{V}\), with \(\mathcal{O}^h_v\) a henselian discrete valuation ring of geometric type over \(k\), we get an induced morphism \(\underline{\omega}^{\operatorname{\mathbf{CI}}}F(\overline{V},D_V)\to \underline{\omega}^{\operatorname{\mathbf{CI}}}F(\mathcal{O}_v^h, \mathfrak{m}_v^{-1})\). Therefore, the right hand side of 46 is precisely the subset of \(F(V)\) consisting of the sections with all the motivic conductors \(\le 1\). Thus by [42], \[\label{lem:tameRSCA1-2}{\underline{\omega}}^{\operatorname{\mathbf{CI}}}F(\overline{V}, D_V)= h^0_{\mathbf{A}^1}(F)(V),\tag{47}\] which yields the statement. ◻