Coherent derived étale sections for the constant and Tate coefficients

Construct $M_0,M_3$ by [F3]. For any scheme $V$ represented in $\mathcal{B}$, an étale object over $V$ has the same étale covering families and further covering refinements whether computed in $V_{\mathrm{e}\mathrm{t}}$ or in $\mathcal{B}$ (with the cofinal covering convention of [F2]). Consequently the locally represented-section construction of sheafification commutes with restriction here, including the sheafifications defining the tensor cube. Thus $r_V{M}_0={\underset{‾}{\mathbb{Z}\mathrm{/}p}}_V$ and $r_V{M}_3={\mu }_{p,V}^{\otimes 3}$ canonically, with their actual coefficient restrictions. Choose functorially $M[0]\to I(M)$ on $\mathcal{B}$ by [A1, F1]. For either coefficient set $C_M(V)=I(M)(V)$; its maps are the restriction chain maps $I(M)(f)$. This is a strict contravariant functor to complexes, functorial in $M$. By [F1, F2], $r_{V}I(M)$ is a K-injective resolution of $r_{V}M$, and $I(M)(V)=(r_{V}I(M))(V)$ computes its small-site derived sections. For $f:W\to V$, apply [F2]'s transition construction degreewise to this resolution. The square over $c_f:f_{\mathrm{e}\mathrm{t}}^{-1}{r}_{V}M\to r_{W}M$ commutes, and $f_{\mathrm{e}\mathrm{t}}^{-1}$ is exact. Its right adjoint $f_{\mathrm{e}\mathrm{t},\ast }$ preserves K-injectives: maps from an acyclic complex into $f_{\mathrm{e}\mathrm{t},\ast }I$ identify, including cochain homotopies, with maps from its acyclic inverse image into $I$, and are therefore nullhomotopic. The adjoint transition $r_{V}I(M)\to f_{\mathrm{e}\mathrm{t},\ast }{r}_{W}I(M)$ thus computes $r_{V}M\to R{f}_{\mathrm{e}\mathrm{t},\ast }{r}_{W}M$. Evaluation at $V$ gives $I(M)(V)\to I(M)(W)$ by the pushforward formula in [F2]. The canonical derived pullback followed by the coefficient map is therefore represented by the restriction chain map $I(M)(f)$. Composition of these maps holds already on complexes, by the transition identity in [F2]. Localization into the enhanced target of [F4] retains the full diagram and coefficient transformations, producing $E_0,E_3$ with the asserted actual pullbacks.

Restrict the two big sheaves and their resolutions to $\mathcal{B}\mathrm{/}U$. By [F1, F2] these remain resolutions by K-injectives, and their evaluation diagrams are exactly the restricted ambient diagrams. On this big slice the locally defined rule $1\mapsto {\zeta }^{\otimes 3}$ gives a morphism $M_0{\mathrm{\mid }}_U\to M_3{\mathrm{\mid }}_U$: pullback of the base-field section preserves the formula and the sheaf tensor relations. For every object $(X\to U)$ of $\mathcal{B}\mathrm{/}U$, the coefficient identifications in step 1.1, applied to the underlying scheme $X$, identify the restriction of this morphism to $X_{\mathrm{e}\mathrm{t}}$ with $1\mapsto {\zeta }^{\otimes 3}$ between the constant sheaf $\mathbb{Z}\mathrm{/}p$ and ${\mu }_{p,X}^{\otimes 3}$. The chosen-root assertion of Roots of unity and étale Tate twists in [F4] applies to every such $k$-scheme $X$, including those that are not qcqs: $p$ is invertible in $k$ and $\zeta$ has order $p$, and the induced small sites have the open restrictions and covering refinements used in that construction by [F2]. Thus every one of these small-site morphisms is an isomorphism. Evaluating at the terminal object $X\mathrm{/}X$ of each small site shows that every component of the big-sheaf morphism is an isomorphism. The componentwise inverses are natural, by the naturality equations for the original morphism, so they define an inverse big-sheaf morphism. On the qcqs subcategory $\mathcal{C}\mathrm{/}U$, the same formula is the pullback-compatible coefficient family of A chosen root trivializes the actual Tate coefficient on the slice in [F4]. Apply a functorial resolution $J$ on $\mathcal{B}\mathrm{/}U$ to these two inverse morphisms. It gives inverse maps between the evaluation diagrams of $J(M_0{\mathrm{\mid }}_U)$ and $J(M_3{\mathrm{\mid }}_U)$. To compare with the ambient resolutions, use the natural zigzags $I(M){\mathrm{\mid }}_U\to J(I(M){\mathrm{\mid }}_U)\leftarrow J(M{\mathrm{\mid }}_U)$. Both arrows are quasi-isomorphisms between K-injectives: the first is the resolution map and the second comes from the original resolution by functoriality and the natural quasi-isomorphisms to $J$. By [F1] they are homotopy equivalences and remain so after evaluation at any object. These are zigzags of entire evaluation diagrams on the slice. By [F4] they give coherent natural equivalences to the restricted ambient diagrams $j^{\ast }{E}_0,j^{\ast }{E}_3$. Transport the root transformation of the $J$-models and its inverse through these equivalences. This constructs the asserted transformation between the restricted ambient diagrams without an ambient coefficient morphism. The same comparison applied to two choices of functorial resolution proves independence up to coherent natural equivalence; all maps just constructed are the derived maps computed in step 1.1.