Coherent étale Tate sections and root trivialization on a slice

Choose $\mathcal{B}$ and the resolutions under [A1]. The formulas defining $M_j$ use the same presheaves and local sheafification operations as [F1], now on $\mathcal{B}$. Restricting these formulas to $T_{\mathrm{e}\mathrm{t}}$ gives exactly its specified coefficient sheaves: every étale cover of an étale $T$-scheme is again composed of étale $T$-schemes, so the matching families and common covering refinements used in sheafification are the same. The same argument applies to restrictions to $\mathcal{B}\mathrm{/}X$. Thus restrictions of the tensor sheafifications are the stated tensor sheafifications. By [F3], restriction of $I_j^{\bullet }$ to $T_{\mathrm{e}\mathrm{t}}$ is a K-injective resolution of ${\mu }_n^{\otimes j}$; evaluating it at $T$ gives $I_j^{\bullet }(T)$, hence the actual derived sections of [F2]. The differentials and restrictions of $I_j^{\bullet }$ form one strict complex-valued presheaf. For $f:T^{\mathrm{\prime }}\to T$, the transition map of [F3] is $f_{small}^{-1}(I_j{\mathrm{\mid }}_{T_{\mathrm{e}\mathrm{t}}})\to I_j{\mathrm{\mid }}_{T_{\mathrm{e}\mathrm{t}}^{\mathrm{\prime }}}$, induced by restrictions along $U{\times }_T{T}^{\mathrm{\prime }}\to U$. Together with the coaugmentation it computes the derived pullback with its corresponding coefficient transition. Its map on sections is precisely $I_j^{\bullet }(f)$. The composition identity in [F3] gives compatible diagrams for every string of arrows; these diagrams exist before localization. The derived-inclusion construction of [F2] makes comparisons between resolutions natural cochain homotopy equivalences, so restriction to ${\mathcal{C}}_F$ preserves them and their homotopies. This proves resolution independence for the whole presheaf, rather than only its evaluated objects. For $j=0$ choose exactly the constant model of [F2] on this same site; alternatively the same resolution comparison identifies the models. Transport along the representative functor of [F1] handles schemes outside the literal skeleton.

On $\mathcal{B}\mathrm{/}X$, the chosen root pulls back to every object. The formulas $a\mapsto {\zeta }^a$ and $1\mapsto {\zeta }^{\otimes j}$ in [F1] define a single morphism $M_0{\mathrm{\mid }}_X\to M_j{\mathrm{\mid }}_X$ of abelian sheaves. It is an isomorphism: the coefficient supplier's local root decomposition uses open covers of each $k$-scheme and its sheafified tensor operations, hence applies to this big slice as well as its small sites. In particular its local inverses are compatible with every restriction and glue uniquely; for $n=1$ both sheaves are zero. By [F3], $I_0^{\bullet }{\mathrm{\mid }}_X$ and $I_j^{\bullet }{\mathrm{\mid }}_X$ remain K-injective resolutions. Apply the derived-inclusion functor of [F2] on $\mathcal{B}\mathrm{/}X$ to the coefficient isomorphism. Computation of maps between these resolutions represents its image by a cochain map, with a cochain homotopy inverse since the coefficient map is invertible in the derived category. Restrict these maps and their homotopies to ${\mathcal{C}}_F\mathrm{/}X$. They give the required equivalence $r_X{C}_0\simeq r_X{C}_j$ in derived presheaves. On every small étale site they represent precisely the root coefficient map by the restriction comparison of step 1.1. Since the map and its inverse homotopies were maps of complexes of sheaves on one slice site, their restrictions are natural for all slice arrows and retain all composition compatibilities. This supplies the claimed coherent coefficient equivalence with only the root over $X$.