Proof. By construction, \(N^{\bullet}(K,L;\FF_2)\) is a cup-\(i\) subcomplex of the functorial cup-\(i\) complex \(N^{\bullet}(K;\FF_2)\), and \(\Sq^r_{(K,L)}\) is defined by the same cochain-level formula as in the absolute case. Consequently, the standard proofs of naturality, the Cartan formula, the Adem relations, the unstable identities, and the identification \(\Sq^1=\beta\) apply verbatim to \(N^{\bullet}(K,L;\FF_2)\); see [3]. ◻

Proof. By Isaksen’s equivalence \[(\Pro(\sSet))^{[1]}\simeq \Pro(\sSet^{[1]})\] for the arrow category \([1]\), a level representation of \(f\) is the same thing as a diagram in \(\sSet^{[1]}\) representing the corresponding pro-object of \(\Pro(\sSet^{[1]})\); see [14]. Any two such presentations admit a common coinitial refinement by the standard reindexing formalism for pro-categories, and coinitial reindexing gives isomorphic pro-objects; see [12].

If \[\mu\colon T'\to T\] is coinitial, then \[\mu^{\op}\colon T^{\op}\to (T')^{\op}\] is cofinal, so restriction along \(\mu\) does not change the filtered colimit defining \(C^{\bullet}(f;\FF_2)\). After passing to a common coinitial refinement, the two resulting diagrams of simplicial pairs are levelwise identified, hence the associated filtered systems of relative cochain complexes are canonically isomorphic. It follows that \(C^{\bullet}(f;\\FF_2)\) and \(H^*(f;\FF_2)\) depend only on \(f\), up to canonical isomorphism.

The same argument applies to the Steenrod operations, since the classical relative Steenrod squares on simplicial pairs are natural in maps of pairs. The usual formal identities hold levelwise, and therefore pass to filtered colimits. ◻

Proof. We use the derived form of the Barnea–Schlank–Chough comparison: for every Grothendieck topos \(\mathcal{R}\) occurring below, there is a natural equivalence \[\chi_{\mathcal{R}}\colon C^\bullet(h(\mathcal{R});\Lambda) \xrightarrow{\;\sim\;} R\Gamma(\mathcal{R},\underline{\Lambda}_{\mathcal{R}})\] in \(\mathcal{D}(\mathbf{Z})\), whose effect on cohomology is the inverse of the comparison isomorphism of Proposition [prop:chough95comp]. The naturality of \(h(-)\) gives commutative squares in \(\mathcal{D}(\mathbf{Z})\) \[\begin{tikzcd}[column sep=large] C^\bullet(h(\mathcal{T});\Lambda) \arrow[r] \arrow[d,"\chi_{\mathcal{T}}"'] & C^\bullet(h(\mathcal{Z});\Lambda) \arrow[d,"\chi_{\mathcal{Z}}"] \\ R\Gamma(\mathcal{T},\underline{\Lambda}_{\mathcal{T}}) \arrow[r] & R\Gamma(\mathcal{Z},\underline{\Lambda}_{\mathcal{Z}}) \end{tikzcd}\] and \[\begin{tikzcd}[column sep=large] C^\bullet(h(\mathcal{T});\Lambda) \arrow[r] \arrow[d,"\chi_{\mathcal{T}}"'] & C^\bullet(h(\mathcal{U});\Lambda) \arrow[d,"\chi_{\mathcal{U}}"] \\ R\Gamma(\mathcal{T},\underline{\Lambda}_{\mathcal{T}}) \arrow[r] & R\Gamma(\mathcal{U},\underline{\Lambda}_{\mathcal{U}}). \end{tikzcd}\]

By the definition of relative cochains for a map of pro-simplicial sets, the object \(C^\bullet(h(i);\Lambda)\) is the homotopy fiber of the restriction morphism \[C^\bullet(h(\mathcal{T});\Lambda) \longrightarrow C^\bullet(h(\mathcal{Z});\Lambda).\] On the topos side, the recollement triangle \[j_!\underline{\Lambda}_{\mathcal{U}} \longrightarrow \underline{\Lambda}_{\mathcal{T}} \longrightarrow i_*\underline{\Lambda}_{\mathcal{Z}} \longrightarrow\] gives, after applying \(R\Gamma(\mathcal{T},-)\), a fiber sequence \[R\Gamma_c(\mathcal{U}/\mathcal{T};\underline{\Lambda}_{\mathcal{U}}) \longrightarrow R\Gamma(\mathcal{T},\underline{\Lambda}_{\mathcal{T}}) \longrightarrow R\Gamma(\mathcal{Z},\underline{\Lambda}_{\mathcal{Z}}).\] Thus \[R\Gamma_c(\mathcal{U}/\mathcal{T};\underline{\Lambda}_{\mathcal{U}}) \simeq \operatorname{Fib}\!\left( R\Gamma(\mathcal{T},\underline{\Lambda}_{\mathcal{T}}) \longrightarrow R\Gamma(\mathcal{Z},\underline{\Lambda}_{\mathcal{Z}}) \right).\] The first commutative square above therefore induces an equivalence on fibers, \[\alpha_c\colon C^\bullet(h(i);\Lambda) \xrightarrow{\;\sim\;} R\Gamma_c(\mathcal{U}/\mathcal{T};\underline{\Lambda}_{\mathcal{U}}).\]

The same argument applied to the open immersion gives the second comparison. Functoriality follows from the functoriality of the absolute comparison morphisms \(\chi_{\mathcal{R}}\), of the recollement triangles, and of the fiber construction in \(\mathcal{D}(\mathbf{Z})\). ◻

Proof. The functors \[R\Gamma_{\acute et}(\Spec(\mathbf{Z}),-), \qquad v^*, \qquad R\Gamma_{\acute et}(\Spec(\mathbf{C}),-)\] are exact. The functors \[(-)^{hG_{\mathbf{R}}}, \qquad (-)^{tG_{\mathbf{R}}}\] are exact on \(\mathcal{D}(\mathbf{Z})^{G_{\mathbf{R}}}\): homotopy fixed points are a limit functor between stable \(\infty\)-categories, and the Tate construction is the cofiber of the norm transformation from homotopy orbits to homotopy fixed points. Hence \[R\Gamma_{\mathrm{fin}}, \quad R\Gamma_\infty, \quad \widehat R\Gamma_\infty\] are exact.

The morphisms \[\delta^{\mathrm{ord}}_{\mathcal{G}}, \qquad \delta^{\mathrm{mod}}_{\mathcal{G}}\] are functorial in \(\mathcal{G}^\bullet\). Thus they define exact functors from \(\mathcal{D}^b((\Spec\mathbf{Z})_{\acute et},\mathbf{Z})\) to the arrow category \[\operatorname{Fun}(\Delta^1,\mathcal{D}(\mathbf{Z})).\] Since this arrow category is stable and the fiber functor is exact, both \[R\Gamma_{c} \qquad\text{and}\qquad \widehat R\Gamma_{\acute et}(\Spec(\mathbf{Z}),-)\] are exact. ◻

Proof. Let \(\tau=(12)\in \Sigma_2\). By the defining relation for the binary cells of an Adem–Cartan operad, one has \[d e_i=e_{i-1}+\tau e_{i-1} \qquad (i\geq 0);\] see [20]. Since \(\theta_2\) is a morphism of complexes, for homogeneous \(x,y\in A^\bullet\) we obtain \[\begin{align} d(x\cup_i y) &= d\,\theta_2(e_i\otimes x\otimes y) \\ &= \theta_2(de_i\otimes x\otimes y) + \theta_2(e_i\otimes dx\otimes y) + \theta_2(e_i\otimes x\otimes dy) \\ &= \theta_2(e_{i-1}\otimes x\otimes y) + \theta_2(\tau e_{i-1}\otimes x\otimes y) + dx\cup_i y + x\cup_i dy \\ &= x\cup_{i-1}y + y\cup_{i-1}x + dx\cup_i y + x\cup_i dy. \end{align}\] This proves \((1)\).

If \(f\colon A^\bullet\to B^\bullet\) is a morphism of non-unital \(\bar{\mathcal{O}}\)-algebras, then \[f(x\cup_i y) = f\bigl(\theta_2(e_i\otimes x\otimes y)\bigr) = \theta_2(e_i\otimes f(x)\otimes f(y)) = f(x)\cup_i f(y),\] which proves \((2)\).

Let \(x\in Z^n(A^\bullet)\). Taking \(y=x\) and \(i=n-r\) in \((1)\), and using \(dx=0\), gives \[d(x\cup_{n-r}x)=0.\] Thus \(x\cup_{n-r}x\) is a cocycle. It remains to show independence of the representative. Let \(x'=x+da\) with \(dx=0\), and set \(k:=n-r\). A direct computation using \((1)\) gives \[x'\cup_k x' + x\cup_k x = d\bigl(a\cup_k x + x\cup_k a + a\cup_k da + a\cup_{k-1}a\bigr).\] Hence \(x'\cup_k x'\) and \(x\cup_k x\) define the same class in cohomology. This proves \((3)\). ◻

Proof. By Berger–Fresse, for every simplicial set \(K\) the normalized cochain complex \[N^\bullet(K;\FF_2)\] is functorially an algebra over the Barratt–Eccles operad \(E\); see [21]. Let \[u\colon L\hookrightarrow K\] be a simplicial inclusion. Since the induced restriction morphism \[u^*\colon N^\bullet(K;\FF_2)\longrightarrow N^\bullet(L;\FF_2)\] is a morphism of \(E\)-algebras, its kernel \[N^\bullet(K,L;\FF_2) = \ker\!\bigl(u^*\colon N^\bullet(K;\FF_2)\to N^\bullet(L;\FF_2)\bigr)\] is stable under all operations of strictly positive arity. Hence \[N^\bullet(K,L;\FF_2)\] is naturally an algebra over the reduced operad \(\bar E\), i.e. a non-unital \(E_\infty\)-algebra.

Now let \[f\colon A\longrightarrow B\] be a morphism in \(\Pro(\sSet)\), represented on a common small cofiltered index category by a natural transformation of simplicial sets. By construction, the relative complex \[C^\bullet(f;\FF_2)\] is the filtered colimit of the relative normalized cochain complexes of the associated levelwise mapping-cylinder pairs; see §2.1.2. By the preceding paragraph, each levelwise relative cochain complex is a \(\bar E\)-algebra, and the transition morphisms are morphisms of \(\bar E\)-algebras. Since filtered colimits in \(\Ch(\FF_2)\) commute with finite tensor products, the \(\bar E\)-action passes to the colimit. Functoriality is inherited from the functoriality of the mapping-cylinder construction and of the Berger–Fresse operad action.

Applying this to the morphisms \[h(i) \qquad\text{and}\qquad h(i_{\mathbf{C}})\] gives the claim. ◻

Proof. Let \[i_{\mathbf{R}}\colon (\partial_jX)_{\mathbf{R}}\hookrightarrow \bar X_{\mathbf{R}}\] be the base change of \(i\) to \(\Spec(\mathbf{R})\). Harpaz and Schlank associate to every finite-type \(\mathbf{R}\)-scheme \(Y\) a relative étale homotopy type \[\Et_{/\mathbf{R}}(Y),\] functorial in \(Y\), which is a pro-object in the homotopy category of simplicial \(G_{\mathbf{R}}\)-sets; see [22]. Applying this construction to the morphism \(i_{\mathbf{R}}\) yields an arrow \[\Et_{/\mathbf{R}}\bigl((\partial_jX)_{\mathbf{R}}\bigr) \longrightarrow \Et_{/\mathbf{R}}(\bar X_{\mathbf{R}})\] in the corresponding pro-homotopy category. Forgetting the \(G_{\mathbf{R}}\)-action, Harpaz–Schlank identify the resulting underlying pro-simplicial set with the absolute étale homotopy type of the geometric fiber; see [22]. Barnea and Schlank show that the relative étale homotopy type admits a lift from the homotopy category to an honest pro-category of simplicial \(G_{\mathbf{R}}\)-sets; see [15]. We therefore obtain a representative \[\widetilde{h}(i_{\mathbf{C}})\colon \mathcal{A}_j\longrightarrow \mathcal{B}_j\] of \(h(i_{\mathbf{C}})\) in \(\Pro(\sSet^{G_{\mathbf{R}}})\).

After reindexing source and target on a common small cofiltered category, as in §2.1.2, we may assume that \(\widetilde{h}(i_{\mathbf{C}})\) is level-wise represented by a natural transformation of simplicial \(G_{\mathbf{R}}\)-sets. Form the associated relative cochain complex \[\widetilde{C}^\bullet_{\mathbf{C},j}(X;\FF_2)\] by the mapping-cylinder construction of §2.1.2. By Proposition [prop:finite-and-geometric-boundary-einfty], the induced action on relative cochains is through automorphisms of non-unital \(\bar E\)-algebras. Passing to the filtered colimit yields a naive \(G_{\mathbf{R}}\)-action on \[\widetilde{C}^\bullet_{\mathbf{C},j}(X;\FF_2)\] by automorphisms of non-unital \(E_\infty\)-algebras.

Finally, \(\widetilde{h}(i_{\mathbf{C}})\) and the chosen representative of \(h(i_{\mathbf{C}})\) determine the same arrow of \(\Ho(\Pro(\sSet))\). The independence statement for the relative cochain construction of §2.1.2 therefore gives a canonical quasi-isomorphism \[\widetilde{C}^\bullet_{\mathbf{C},j}(X;\FF_2) \xrightarrow{\sim} C^\bullet_{\mathbf{C},j}(X;\FF_2).\] This proves the claim. ◻

Proof. By Propositions [prop:finite-and-geometric-boundary-einfty] and [prop:geometric-boundary-gr-equivariant-refinement], the geometric-boundary complex \[C^\bullet_{\mathbf{C},j}(X;\mathbb{F}_2)\] is a non-unital \(E_\infty\)-algebra object of \[\operatorname{Fun}(BG_{\mathbf{R}},\mathcal{D}(\mathbb{F}_2)).\] Moreover, \[\rho_j^*\colon \operatorname{triv} \bigl(C^\bullet_{c,\mathrm{fin},j}(X;\mathbb{F}_2)\bigr) \longrightarrow C^\bullet_{\mathbf{C},j}(X;\mathbb{F}_2)\] is a morphism of non-unital \(E_\infty\)-algebras in \(\operatorname{Fun}(BG_{\mathbf{R}},\mathcal{D}(\mathbb{F}_2))\).

The constant-action functor \[\operatorname{triv}\colon \mathcal{D}(\mathbb{F}_2) \longrightarrow \operatorname{Fun}(BG_{\mathbf{R}},\mathcal{D}(\mathbb{F}_2))\] is symmetric monoidal. Its right adjoint \[(-)^{hG_{\mathbf{R}}}\colon \operatorname{Fun}(BG_{\mathbf{R}},\mathcal{D}(\mathbb{F}_2)) \longrightarrow \mathcal{D}(\mathbb{F}_2)\] is therefore lax symmetric monoidal. Hence it carries non-unital \(E_\infty\)-algebras to non-unital \(E_\infty\)-algebras. Applying this to \(C^\bullet_{\mathbf{C},j}(X;\mathbb{F}_2)\) gives the canonical non-unital \(E_\infty\)-structure on \[C^\bullet_{R,j}(X;\mathbb{F}_2) = C^\bullet_{\mathbf{C},j}(X;\mathbb{F}_2)^{hG_{\mathbf{R}}}.\] Under the induced adjunction on non-unital \(E_\infty\)-algebras, the morphism \(\rho_j^*\) corresponds to \[u^{\mathrm{ord}}_{X,j}\colon C^\bullet_{c,\mathrm{fin},j}(X;\mathbb{F}_2) \longrightarrow C^\bullet_{R,j}(X;\mathbb{F}_2).\] Thus \(u^{\mathrm{ord}}_{X,j}\) is a morphism of non-unital \(E_\infty\)-algebras.

By the lax symmetric monoidality of the Tate construction, \[(-)^{tG_{\mathbf{R}}}\colon \operatorname{Fun}(BG_{\mathbf{R}},\mathcal{D}(\mathbb{F}_2)) \longrightarrow \mathcal{D}(\mathbb{F}_2),\] the object \[C^\bullet_{\infty,j}(X;\mathbb{F}_2) = C^\bullet_{\mathbf{C},j}(X;\mathbb{F}_2)^{tG_{\mathbf{R}}}\] inherits a canonical non-unital \(E_\infty\)-algebra structure. Moreover, the canonical natural transformation \[\operatorname{can}\colon (-)^{hG_{\mathbf{R}}} \longrightarrow (-)^{tG_{\mathbf{R}}}\] is lax symmetric monoidal. Therefore its value on \(C^\bullet_{\mathbf{C},j}(X;\mathbb{F}_2)\), \[\vartheta_{X,j} = \operatorname{can}_{C^\bullet_{\mathbf{C},j}(X;\mathbb{F}_2)} \colon C^\bullet_{R,j}(X;\mathbb{F}_2) \longrightarrow C^\bullet_{\infty,j}(X;\mathbb{F}_2),\] is a morphism of non-unital \(E_\infty\)-algebras.

Finally, \[v_{X,j} = \vartheta_{X,j}\circ u^{\mathrm{ord}}_{X,j}\] is a composition of morphisms of non-unital \(E_\infty\)-algebras, and is therefore again a morphism of non-unital \(E_\infty\)-algebras. ◻

Proof. Let \[\bar s\colon \bar X\longrightarrow \Spec(\mathbf{Z})\] be the structural morphism of the chosen compactification, and let \[\bar s_{\mathbf{C}}\colon \bar X_{\mathbf{C}}\longrightarrow \Spec(\mathbf{C})\] be its base change. Since \(s=\bar s\circ j\) and \(\bar s\) is proper, there is a canonical identification \[Rs_!\FF_2 \simeq R\bar s_*j_!\FF_2 .\]

For the finite-boundary term, Proposition [prop:relative-comparison], applied to the open–closed decomposition \[X\subset \bar X, \qquad i\colon \partial_jX\hookrightarrow \bar X,\] gives an equivalence \[C^\bullet_{c,\mathrm{fin},j}(X;\FF_2) = C^\bullet(h(i);\FF_2) \xrightarrow{\;\sim\;} R\Gamma_{\acute et}(\bar X,j_!\FF_2).\] Composing with \[R\Gamma_{\acute et}(\bar X,j_!\FF_2) \simeq R\Gamma_{\acute et}(\Spec(\mathbf{Z}),R\bar s_*j_!\FF_2) \simeq R\Gamma_{\acute et}(\Spec(\mathbf{Z}),Rs_!\FF_2) = R\Gamma_{c,\mathrm{fin}}(X,\FF_2)\] gives the desired equivalence \[\alpha_{c,\mathrm{fin},j}\colon C^\bullet_{c,\mathrm{fin},j}(X;\FF_2) \xrightarrow{\;\sim\;} R\Gamma_{c,\mathrm{fin}}(X,\FF_2).\]

We next construct the real-boundary comparisons. Applying Proposition [prop:relative-comparison] after base change to \(\Spec(\mathbf{C})\) gives a comparison equivalence \[C^\bullet_{\mathbf{C},j}(X;\FF_2) = C^\bullet(h(i_{\mathbf{C}});\FF_2) \xrightarrow{\;\sim\;} R\Gamma_{\acute et}(\bar X_{\mathbf{C}},j_{\mathbf{C},!}\FF_2).\] By Proposition [prop:geometric-boundary-gr-equivariant-refinement] and by the functoriality of the comparison for maps of open–closed decompositions, this equivalence is \(G_{\mathbf{R}}\)-equivariant.

By proper base change, \[v^*Rs_!\FF_2 \simeq v^*R\bar s_*j_!\FF_2 \simeq R\bar s_{\mathbf{C},*}j_{\mathbf{C},!}\FF_2 .\] Hence \[R\Gamma_{\acute et}(\bar X_{\mathbf{C}},j_{\mathbf{C},!}\FF_2) \simeq R\Gamma_{\acute et} \!\left( \Spec(\mathbf{C}),v^*Rs_!\FF_2 \right) = M_\infty(X,\FF_2)\] as objects of \[\operatorname{Fun}(BG_{\mathbf{R}},\mathcal{D}(\FF_2)).\] Thus we obtain a \(G_{\mathbf{R}}\)-equivariant equivalence \[\alpha_{\mathbf{C},j}\colon C^\bullet_{\mathbf{C},j}(X;\FF_2) \xrightarrow{\;\sim\;} M_\infty(X,\FF_2).\]

Applying homotopy fixed points gives \[\alpha_{R,j}:= \alpha_{\mathbf{C},j}^{hG_{\mathbf{R}}}\colon C^\bullet_{\mathbf{C},j}(X;\FF_2)^{hG_{\mathbf{R}}} \xrightarrow{\;\sim\;} M_\infty(X,\FF_2)^{hG_{\mathbf{R}}}.\] By Definitions 18 and 15, this is precisely an equivalence \[C^\bullet_{R,j}(X;\FF_2) \xrightarrow{\;\sim\;} R\Gamma_\infty(X,\FF_2).\]

Similarly, applying the Tate construction gives \[\alpha_{\infty,j}:= \alpha_{\mathbf{C},j}^{tG_{\mathbf{R}}}\colon C^\bullet_{\mathbf{C},j}(X;\FF_2)^{tG_{\mathbf{R}}} \xrightarrow{\;\sim\;} M_\infty(X,\FF_2)^{tG_{\mathbf{R}}},\] i.e. \[C^\bullet_{\infty,j}(X;\FF_2) \xrightarrow{\;\sim\;} \widehat R\Gamma_\infty(X,\FF_2).\]

It remains to verify compatibility with the boundary morphisms. The morphism \[\rho_j^*\colon \operatorname{triv} \bigl(C^\bullet_{c,\mathrm{fin},j}(X;\FF_2)\bigr) \longrightarrow C^\bullet_{\mathbf{C},j}(X;\FF_2)\] is induced by the Cartesian square \[\begin{tikzcd} (\partial_jX)_{\mathbf{C}} \arrow[r] \arrow[d,"i_{\mathbf{C}}"'] & \partial_jX \arrow[d,"i"] \\ \bar X_{\mathbf{C}} \arrow[r] & \bar X . \end{tikzcd}\] By the functoriality of the comparison equivalences above, the square \[\begin{tikzcd}[column sep=large] \operatorname{triv} C^\bullet_{c,\mathrm{fin},j}(X;\FF_2) \arrow[r,"\rho_j^*"] \arrow[d,"\operatorname{triv}(\alpha_{c,\mathrm{fin},j})"'] & C^\bullet_{\mathbf{C},j}(X;\FF_2) \arrow[d,"\alpha_{\mathbf{C},j}"] \\ \operatorname{triv} R\Gamma_{c,\mathrm{fin}}(X,\FF_2) \arrow[r] & M_\infty(X,\FF_2) \end{tikzcd}\] commutes in \[\operatorname{Fun}(BG_{\mathbf{R}},\mathcal{D}(\FF_2)),\] where the lower horizontal morphism is restriction to the geometric point at infinity.

Passing to the right adjoint \[(-)^{hG_{\mathbf{R}}}\] and using the definition of \[u^{\mathrm{ord}}_{X,j}\] as the adjoint of \(\rho_j^*\), we obtain the commutative square \[\begin{tikzcd}[column sep=large] C^\bullet_{c,\mathrm{fin},j}(X;\FF_2) \arrow[r,"u^{\mathrm{ord}}_{X,j}"] \arrow[d,"\alpha_{c,\mathrm{fin},j}"'] & C^\bullet_{R,j}(X;\FF_2) \arrow[d,"\alpha_{R,j}"] \\ R\Gamma_{c,\mathrm{fin}}(X,\FF_2) \arrow[r,"\delta^{\mathrm{ord}}_{X,\FF_2}"'] & R\Gamma_\infty(X,\FF_2). \end{tikzcd}\] This proves compatibility with the ordinary real-boundary map.

The compatibility with the ordinary-to-Tate map is the naturality of \[\operatorname{can}\colon (-)^{hG_{\mathbf{R}}} \longrightarrow (-)^{tG_{\mathbf{R}}}\] applied to the \(G_{\mathbf{R}}\)-equivariant equivalence \[\alpha_{\mathbf{C},j}\colon C^\bullet_{\mathbf{C},j}(X;\FF_2) \xrightarrow{\;\sim\;} M_\infty(X,\FF_2).\]

Finally, \[v_{X,j} = \vartheta_{X,j}\circ u^{\mathrm{ord}}_{X,j}, \qquad \delta^{\mathrm{mod}}_{X,\FF_2} = \delta^{\mathrm{Tate}}_{X,\FF_2}\circ \delta^{\mathrm{ord}}_{X,\FF_2}.\] Thus the outer square comparing \(v_{X,j}\) with \(\delta^{\mathrm{mod}}_{X,\FF_2}\) commutes in \(\mathcal{D}(\FF_2)\), equivalently the two arrows are identified in \(\operatorname{Fun}(\Delta^1,\mathcal{D}(\FF_2))\).

Taking fibers and using the modified boundary triangle of Remark [rem:ordinary-modified-boundary-triangles] gives \[\operatorname{Fib}(v_{X,j}) \simeq \operatorname{Fib}(\delta^{\mathrm{mod}}_{X,\FF_2}) = \widehat R\Gamma_c(X,\FF_2).\] ◻

Proof. Write \[i\colon \partial_jX\hookrightarrow \bar X, \qquad i'\colon \partial_{j'}X\hookrightarrow \bar X'\] for the complementary closed immersions. Since \[\bar\pi^{-1}(X)=X,\] the morphism \(\bar\pi\) induces morphisms of open–closed decompositions \[(\bar X',\partial_{j'}X)\longrightarrow(\bar X,\partial_jX)\] and, after base change, \[(\bar X'_{\mathbf{C}},(\partial_{j'}X)_{\mathbf{C}}) \longrightarrow (\bar X_{\mathbf{C}},(\partial_jX)_{\mathbf{C}}) .\] By functoriality of relative pro-simplicial cochains, these give morphisms of non-unital \(E_\infty\)-algebras \[\bar\pi^*_{c,\mathrm{fin}}\colon C^\bullet_{c,\mathrm{fin},j}(X;\FF_2) \longrightarrow C^\bullet_{c,\mathrm{fin},j'}(X;\FF_2),\] and \[\bar\pi^*_{\mathbf{C}}\colon C^\bullet_{\mathbf{C},j}(X;\FF_2) \longrightarrow C^\bullet_{\mathbf{C},j'}(X;\FF_2).\] After passing to a common \(G_{\mathbf{R}}\)-equivariant refinement, the second morphism is a morphism in \[\operatorname{Fun}(BG_{\mathbf{R}},\mathcal{D}(\FF_2)).\] Applying the lax symmetric monoidal functors \[(-)^{hG_{\mathbf{R}}}, \qquad (-)^{tG_{\mathbf{R}}},\] gives morphisms of non-unital \(E_\infty\)-algebras \[\bar\pi^*_{R}\colon C^\bullet_{R,j}(X;\FF_2) \longrightarrow C^\bullet_{R,j'}(X;\FF_2),\] and \[\bar\pi^*_{\infty}\colon C^\bullet_{\infty,j}(X;\FF_2) \longrightarrow C^\bullet_{\infty,j'}(X;\FF_2).\] By functoriality of the adjunction \[\operatorname{triv}\dashv (-)^{hG_{\mathbf{R}}}\] and by naturality of \[\operatorname{can}\colon (-)^{hG_{\mathbf{R}}}\longrightarrow (-)^{tG_{\mathbf{R}}},\] these morphisms are compatible with \[u^{\mathrm{ord}}_{X,j}, \qquad \vartheta_{X,j}, \qquad v_{X,j}.\] Thus they induce morphisms of non-unital \(E_\infty\)-algebras on the two fiber models \[\bar\pi^*_c\colon C^\bullet_{c,j}(X;\FF_2) \longrightarrow C^\bullet_{c,j'}(X;\FF_2),\] and \[\widehat{\bar\pi}^{\,*}_c\colon \widehat C^\bullet_{c,j}(X;\FF_2) \longrightarrow \widehat C^\bullet_{c,j'}(X;\FF_2).\]

It remains only to identify the induced maps on arithmetic cohomology. Since \(\bar\pi\) is proper and restricts to the identity on \(X\), extension by zero satisfies \[j_!\FF_2 \xrightarrow{\;\sim\;} R\bar\pi_*j'_!\FF_2\] in \(\mathcal{D}(\bar X_{\acute et},\FF_2)\), and similarly after base change to \(\Spec(\mathbf{C})\). By the functoriality of Proposition [prop:relative-comparison] and Theorem 4, the maps \[\bar\pi^*_{c,\mathrm{fin}}, \qquad \bar\pi^*_{R}, \qquad \bar\pi^*_{\infty}\] identify with the identity maps on \[R\Gamma_{c,\mathrm{fin}}(X,\FF_2), \qquad R\Gamma_\infty(X,\FF_2), \qquad \widehat R\Gamma_\infty(X,\FF_2).\] Taking fibers of the corresponding identity squares identifies \[\bar\pi^*_c \quad\text{and}\quad \widehat{\bar\pi}^{\,*}_c\] with the identity maps on \[R\Gamma_c(X,\FF_2) \qquad\text{and}\qquad \widehat R\Gamma_c(X,\FF_2),\] respectively.

Thus all five comparison morphisms induce the identity on the corresponding arithmetic cohomology groups. Since they are morphisms of non-unital \(E_\infty\)-algebras, after passing to common dg Adem–Cartan models they commute with the operations \[[u]\longmapsto [u\cup_{n-r}u].\] Because the induced maps on cohomology are the identity, the operations transported from the models attached to \(j\) and \(j'\) coincide. ◻

Proof. Let \[j_1\colon X\hookrightarrow \bar X_1, \qquad j_2\colon X\hookrightarrow \bar X_2\] be dense compactifications. By [11], the category of compactifications of \(X\) is cofiltered, and the full subcategory of dense compactifications is initial. Hence there exists a dense compactification \[j_3\colon X\hookrightarrow \bar X_3\] dominating both \(j_1\) and \(j_2\). Applying Lemma 1 to the two domination morphisms \[\bar X_3\longrightarrow \bar X_1, \qquad \bar X_3\longrightarrow \bar X_2\] shows that the operations attached to \(j_1\) and to \(j_2\) both agree with those attached to \(j_3\). Hence they agree with each other.

It remains only to remove the auxiliary choices for a fixed compactification \[j\colon X\hookrightarrow \bar X.\] The choices of representatives of pro-simplicial arrows, cofiltered reindexings, and \(G_{\mathbf{R}}\)-equivariant refinements admit common refinements. The identity morphism of \(\bar X\), after passing to such a common refinement, gives comparison morphisms between the corresponding compactification-based models. By the same argument as in Lemma 1, with \[\bar\pi=\id_{\bar X},\] these comparison morphisms are morphisms of non-unital \(E_\infty\)-algebras and identify with the identity maps on \[R\Gamma_{c,\mathrm{fin}}(X,\FF_2), \qquad R\Gamma_\infty(X,\FF_2), \qquad \widehat R\Gamma_\infty(X,\FF_2), \qquad R\Gamma_c(X,\FF_2), \qquad \widehat R\Gamma_c(X,\FF_2).\] Thus they induce the identity on the five cohomology theories above.

Finally, any two dg Adem–Cartan representatives of the same non-unital \(E_\infty\)-algebra may be compared after passing to a common dg operadic refinement. The resulting comparison morphisms are morphisms of cup-\(i\) complexes, hence commute with the operations \[[u]\longmapsto [u\cup_{n-r}u]\] by Proposition [prop:generalized-cupi-from-nonunital-einfty]. Since the comparison maps induce the identity on cohomology, the resulting Steenrod operations are independent of all auxiliary choices.

The notation without the compactification subscript is therefore well-defined. ◻

Proof. We first prove proper naturality. Let \[f\colon Y\longrightarrow X\] be proper, and write \[s_X\colon X\to \Spec(\mathbf{Z}), \qquad s_Y=s_X\circ f\colon Y\to \Spec(\mathbf{Z}).\] Since \(f\) is proper, \(Rf_!=Rf_*\). The adjunction unit \[\FF_2\longrightarrow Rf_*\FF_2=Rf_!\FF_2\] induces a morphism \[\eta_f\colon Rs_{X,!}\FF_2 \longrightarrow Rs_{Y,!}\FF_2\] in \[\mathcal{D}((\Spec\mathbf{Z})_{\acute et},\FF_2).\] Applying the functors of Definition 15 and Remark [rem:ordinary-modified-boundary-triangles] gives morphisms \[R\Gamma_{c,\mathrm{fin}}(f),\qquad R\Gamma_\infty(f),\qquad \widehat R\Gamma_\infty(f),\] and, by taking fibers of the ordinary and modified boundary morphisms, \[R\Gamma_c(f),\qquad \widehat R\Gamma_c(f).\] Passing to cohomology gives canonical pullback maps on the five theories \[H^*_{c,\mathrm{fin}},\quad H^*_\infty,\quad \widehat H^*_\infty,\quad H^*_c,\quad \widehat H^*_c .\]

Choose a dense compactification \[j_X\colon X\hookrightarrow \bar X.\] By [11], there exist a dense compactification \[j_Y\colon Y\hookrightarrow \bar Y\] and a proper morphism \[\bar f\colon \bar Y\longrightarrow \bar X\] such that \[\bar f\circ j_Y=j_X\circ f, \qquad \bar f^{-1}(X)=Y.\] The induced morphisms of open–closed decompositions, and their base changes to \(\Spec(\mathbf{C})\), give morphisms of non-unital \(E_\infty\)-algebras \[f^*_{c,\mathrm{fin},j}\colon C^\bullet_{c,\mathrm{fin},j_X}(X;\FF_2) \longrightarrow C^\bullet_{c,\mathrm{fin},j_Y}(Y;\FF_2),\] and \[f^*_{\mathbf{C},j}\colon C^\bullet_{\mathbf{C},j_X}(X;\FF_2) \longrightarrow C^\bullet_{\mathbf{C},j_Y}(Y;\FF_2)\] in the \(G_{\mathbf{R}}\)-equivariant setting after passing to a common refinement. Applying the lax symmetric monoidal functors \[(-)^{hG_{\mathbf{R}}}, \qquad (-)^{tG_{\mathbf{R}}},\] and using the naturality of \[\operatorname{can}\colon (-)^{hG_{\mathbf{R}}}\to(-)^{tG_{\mathbf{R}}},\] gives a commutative diagram of non-unital \(E_\infty\)-algebras \[\begin{tikzcd}[column sep=large,row sep=large] C^\bullet_{c,\mathrm{fin},j_X}(X;\FF_2) \arrow[r,"u^{\mathrm{ord}}_{X,j_X}"] \arrow[rr,"v_{X,j_X}",bend left=18] \arrow[d,"f^*_{c,\mathrm{fin},j}"'] & C^\bullet_{R,j_X}(X;\FF_2) \arrow[r,"\vartheta_{X,j_X}"] \arrow[d,"f^*_{R,j}"'] & C^\bullet_{\infty,j_X}(X;\FF_2) \arrow[d,"f^*_{\infty,j}"] \\ C^\bullet_{c,\mathrm{fin},j_Y}(Y;\FF_2) \arrow[r,"u^{\mathrm{ord}}_{Y,j_Y}"'] \arrow[rr,"v_{Y,j_Y}"',bend right=18] & C^\bullet_{R,j_Y}(Y;\FF_2) \arrow[r,"\vartheta_{Y,j_Y}"'] & C^\bullet_{\infty,j_Y}(Y;\FF_2). \end{tikzcd}\] Taking fibers in \[\operatorname{CAlg}^{\mathrm{nu}}_{E_\infty}(\mathcal{D}(\FF_2))\] gives morphisms of non-unital \(E_\infty\)-algebras \[f^*_{c,j}\colon C^\bullet_{c,j_X}(X;\FF_2) \longrightarrow C^\bullet_{c,j_Y}(Y;\FF_2),\] and \[\widehat f^*_{c,j}\colon \widehat C^\bullet_{c,j_X}(X;\FF_2) \longrightarrow \widehat C^\bullet_{c,j_Y}(Y;\FF_2).\]

By the functoriality of Proposition [prop:relative-comparison] and Theorem 4, these five compactification-based morphisms identify with the five canonical arithmetic pullback morphisms above. Since they are morphisms of non-unital \(E_\infty\)-algebras, after choosing dg Adem–Cartan models they are morphisms of cup-\(i\) complexes. Therefore their induced maps on cohomology commute with the operations \[[u]\longmapsto [u\cup_{n-r}u].\] Transporting through the comparison equivalences and using Theorem 5, proper naturality follows for all five theories.

It remains to prove the Cartan formula, the Adem relations, and instability. Fix \(X\) and a dense compactification \[j\colon X\hookrightarrow \bar X.\] The five compactification-based models \[C^\bullet_{c,\mathrm{fin},j}(X;\FF_2),\quad C^\bullet_{R,j}(X;\FF_2),\quad C^\bullet_{\infty,j}(X;\FF_2),\quad C^\bullet_{c,j}(X;\FF_2),\quad \widehat C^\bullet_{c,j}(X;\FF_2)\] are non-unital \(E_\infty\)-algebras in \(\mathcal{D}(\FF_2)\). Choose dg Adem–Cartan models for them. By Remark [rem:einfty-operads-are-adem-cartan] and Proposition [prop:generalized-cupi-from-nonunital-einfty], their cohomologies carry the extended Steenrod operations given by \[[u]\longmapsto [u\cup_{n-r}u], \qquad u\in Z^n.\] The Cartan formula, Adem relations, and instability relations hold on these model cohomologies by the Adem–Cartan formalism. Transporting the identities through the comparison equivalences, and using Theorem 5, gives the stated identities on \[H^*_{c,\mathrm{fin}}(X;\FF_2),\quad H^*_\infty(X;\FF_2),\quad \widehat H^*_\infty(X;\FF_2),\quad H^*_c(X;\FF_2),\quad \widehat H^*_c(X;\FF_2).\] ◻

Proof. Fix a dense compactification \[j\colon X\hookrightarrow \bar X .\] By Theorem 5, it suffices to work with the compactification-based models attached to \(j\).

The finite-boundary model \[C^\bullet_{c,\mathrm{fin},j}(X;\FF_2) = C^\bullet(h(i);\FF_2)\] is a filtered colimit of relative normalized cochains of simplicial pairs. The ordinary real-boundary model \[C^\bullet_{R,j}(X;\FF_2) = C^\bullet_{\mathbf{C},j}(X;\FF_2)^{hG_{\mathbf{R}}}\] is computed by the Borel construction on a \(G_{\mathbf{R}}\)-equivariant relative pro-simplicial set model. Finally, \[C^\bullet_{c,j}(X;\FF_2) = \Fib\!\left( C^\bullet_{c,\mathrm{fin},j}(X;\FF_2) \xrightarrow{u^{\mathrm{ord}}_{X,j}} C^\bullet_{R,j}(X;\FF_2) \right)\] is represented by relative cochains of the induced map from the Borel real-boundary model to the finite-boundary model. Thus all three ordinary models are obtained, after filtered colimits, from relative normalized cochains of simplicial pairs. Proposition [prop:classic95sq95properties](4)–(5) therefore gives \[\Sq^0=\id, \qquad \Sq^1=\beta\] on their cohomology. Transporting through the comparison equivalences of Theorem 4 and the ordinary boundary fiber sequence gives the identities on \[H^*_{c,\mathrm{fin}}(X;\FF_2), \qquad H^*_\infty(X;\FF_2), \qquad H^*_c(X;\FF_2).\]

It remains to treat the Tate and modified theories. Set \[M:=M_\infty(X,\FF_2).\] By definition, \[R\Gamma_\infty(X,\FF_2)=M^{hG_{\mathbf{R}}}, \qquad \widehat R\Gamma_\infty(X,\FF_2)=M^{tG_{\mathbf{R}}},\] and \[\delta^{\mathrm{Tate}}_{X,\FF_2} = \operatorname{can}_M\colon M^{hG_{\mathbf{R}}} \longrightarrow M^{tG_{\mathbf{R}}}.\] The cofiber sequence defining the Tate construction, \[M_{hG_{\mathbf{R}}} \xrightarrow{\,N_M\,} M^{hG_{\mathbf{R}}} \xrightarrow{\operatorname{can}_M} M^{tG_{\mathbf{R}}},\] and the homotopy-orbit spectral sequence \[E_2^{-p,q} = H_p\!\left(G_{\mathbf{R}},H^q(M)\right) \Longrightarrow H^{q-p}\!\left(M_{hG_{\mathbf{R}}}\right), \qquad p\ge0,\] show that \[H^m(M_{hG_{\mathbf{R}}})=0 \qquad(m>N_X).\] Hence \[H^n(\operatorname{can}_M)\colon H^n_\infty(X;\FF_2) \xrightarrow{\;\sim\;} \widehat H^n_\infty(X;\FF_2) \qquad(n>N_X).\] Since \[\operatorname{can}\colon (-)^{hG_{\mathbf{R}}}\longrightarrow (-)^{tG_{\mathbf{R}}}\] is natural and lax symmetric monoidal, it commutes with the Steenrod operations. The Bockstein is functorial for the same comparison. Therefore the identities already proved on \(H^*_\infty(X;\FF_2)\) transfer to \[\widehat H^n_\infty(X;\FF_2) \qquad(n>N_X).\]

The ordinary-to-modified compact-support morphism \[\gamma_X\colon R\Gamma_c(X,\FF_2) \longrightarrow \widehat R\Gamma_c(X,\FF_2)\] is induced by the identity on \(R\Gamma_{c,\mathrm{fin}}(X,\FF_2)\) and by \(\operatorname{can}_M\) on the real boundary. Comparing the long exact cohomology sequences of the ordinary and modified boundary fiber sequences gives \[H^n(\gamma_X)\colon H^n_c(X;\FF_2) \xrightarrow{\;\sim\;} \widehat H^n_c(X;\FF_2) \qquad(n>N_X+1),\] because the real-boundary comparison is an isomorphism in degrees \(n-1\) and \(n\). Moreover, \(\gamma_X\) is induced by a morphism of non-unital \(E_\infty\)-fiber models, and therefore commutes with Steenrod operations; the Bockstein is again functorial. The identities on \(H^*_c(X;\FF_2)\) consequently transfer to \[\widehat H^n_c(X;\FF_2) \qquad(n>N_X+1).\] ◻

Proof. For (i), this is immediate from Definition 21 and the standard properties of Steenrod squares: one has \(\Sq^0=\mathrm{id}\), hence \[w_0(E)=\phi_E^{-1}\bigl(\Sq^0(s_{X/E})\bigr)=\phi_E^{-1}(s_{X/E})=1,\] and the instability relation \(\Sq^j(x)=0\) for \(j>\deg(x)\) applied to \(x=s_{X/E}\in H^{2r}_{X,\mathrm{\acute{e}t}}(E;\mu_2^{\otimes r})\) gives \(w_j(E)=0\) for \(j>2r\); see [8].

For (ii), let \(\widetilde{f}\colon f^*E\to E\) be the canonical morphism of total spaces. Since the square defined by the zero sections is cartesian, the compatibility of Gysin morphisms (equivalently, of Thom classes) with base change implies \[\widetilde{f}^*(s_{X/E})=s_{Y/f^*E};\] see [23]. On the other hand, étale Steenrod squares commute with pullback; see [8]. Therefore \[f^*w_j(E) =\phi_{f^*E}^{-1}\bigl(\widetilde{f}^*\Sq^j(s_{X/E})\bigr) =\phi_{f^*E}^{-1}\bigl(\Sq^j(\widetilde{f}^*s_{X/E})\bigr) =\phi_{f^*E}^{-1}\bigl(\Sq^j(s_{Y/f^*E})\bigr) =w_j(f^*E).\]

(iii) is exactly [8]. ◻

Proof of Theorem 9. Let \(x\in \widehat H_c^*(X;\mu_2^{\otimes d+1})\). Our goal is to show: \[\int_X \widehat{\mathrm{\Sq}}_c(x) = \int_X \bigl(x\cdot \mathrm{\Sq}^{-1}(w^{\text{\'{e}t}}(\tau_f))\cdot f^*v_B\bigr).\]

By the projection formula and the defining property of \(v_B\), we have \[\int_X \bigl(x\cdot \mathrm{\Sq}^{-1}(w^{\text{\'{e}t}}(\tau_f))\cdot f^*v_B\bigr) = \int_B \bigl(\widehat{f_{*,c}}(x\cdot \mathrm{\Sq}^{-1}(w^{\text{\'{e}t}}(\tau_f)))\cdot v_B\bigr) = \int_B \widehat{\mathrm{\Sq}}_c\bigl(\widehat{f_{*,c}}(x\cdot \mathrm{\Sq}^{-1}(w^{\text{\'{e}t}}(\tau_f)))\bigr).\] Theorem 10 and the mixed Cartan formula give \[\begin{align} \widehat{\mathrm{\Sq}}_c\bigl(\widehat{f_{*,c}}(x\cdot \mathrm{\Sq}^{-1}(w^{\text{\'{e}t}}(\tau_f)))\bigr)&= \widehat{f_{*,c}}\Bigl(\widehat{\mathrm{\Sq}}_c\bigl(x\cdot \mathrm{\Sq}^{-1}(w^{\text{\'{e}t}}(\tau_f))\bigr)\cdot w^{\text{\'{e}t}}(\nu_f)\Bigr)\\ &= \widehat{f_{*,c}}\bigl(\widehat{\mathrm{\Sq}}_c(x)\cdot w^{\text{\'{e}t}}(\tau_f)\cdot w^{\text{\'{e}t}}(\nu_f)\bigr)\\ &= \widehat{f_{*,c}}\bigl(\widehat{\mathrm{\Sq}}_c(x)\bigr), \end{align}\] Therefore \[\int_X \bigl(x\cdot \mathrm{\Sq}^{-1}(w^{\text{\'{e}t}}(\tau_f))\cdot f^*v_B\bigr) = \int_B \widehat{f_{*,c}}\bigl(\widehat{\mathrm{\Sq}}_c(x)\bigr) = \int_X \widehat{\mathrm{\Sq}}_c(x),\] which completes the proof. ◻

Proof of Theorem 10. Following Benoist, we decompose \(f\) into (see [11]): \[Y\xhookrightarrow{g} \mathbb{P}^n_X\xrightarrow{p} X,\] where \(g\) is a regular closed immersion, and \(p\) is the structure morphism. Riou’s compatibility of Gysin morphisms with composition (see Remark [rem:riou-composition]), together with the mixed Cartan formula (See Remark [rem:mixed-cartan-formulas]) for \(\widehat{\Sq}_c\), the multiplicativity of \(w^{\mathrm{\acute et}}\), and the additivity of virtual normal bundles, gives: \[\begin{align} \widehat{\Sq}_c(\widehat{f}_{*,c}x) &=\widehat{\Sq}_c(\widehat{p}_{*,c}\widehat{g}_{*,c}x)\\ &=\widehat{p}_{*,c}\Bigl(\widehat{\Sq}_c(\widehat{g}_{*,c}x)\cdot w^{\mathrm{\acute et}}(\nu_p)\Bigr)\\ &=\widehat{p}_{*,c}\Bigl(\widehat{g}_{*,c}\bigl(\widehat{\Sq}_c(x)\cdot w^{\mathrm{\acute et}}(\nu_g)\bigr)\cdot w^{\mathrm{\acute et}}(\nu_p)\Bigr)\\ &=\widehat{f}_{*,c}\bigl(\widehat{\Sq}_c(x)\cdot w^{\mathrm{\acute et}}(\nu_g)\cdot g^*w^{\mathrm{\acute et}}(\nu_p)\bigr)\\ &=\widehat{f}_{*,c}\bigl(\widehat{\Sq}_c(x)\cdot w^{\mathrm{\acute et}}(\nu_f)\bigr), \end{align}\] which is the desired formula.

Therefore, it suffices to prove Theorem 10 in the two special families of morphisms: regular closed immersions, and projective bundles.

Case 1: regular closed immersions. Let \[g\colon Z\hookrightarrow T\] be a regular closed immersion of codimension \(d\).

For any \(M^{\bullet}\in D^+(T_{\mathrm{\acute et}},\Lambda)\), set \[\widehat{R}\Gamma_{c,Z}(T,M^{\bullet}) := \widehat{R}\Gamma_{c}\!\bigl(T,Rg_*Rg^!M^{\bullet}\bigr),\qquad R\Gamma_{\text{\'{e}t},Z}(T,M^{\bullet}) := R\Gamma_{\text{\'{e}t}}\!\bigl(T,Rg_*Rg^!M^{\bullet}\bigr).\] Since \(g\) is proper, we have \[\widehat{R}\Gamma_{c,Z}(T,M^{\bullet}) = \widehat{R}\Gamma_{c}\!\bigl(Z,Rg^!M^{\bullet}\bigr),\qquad R\Gamma_{\text{\'{e}t},Z}(T,M^{\bullet}) = R\Gamma_{\text{\'{e}t}}\!\bigl(Z,Rg^!M^{\bullet}\bigr).\]

Riou’s \(r\)-twisted gysin class for \(g\) is an isomorphism: \[Cl_g^{r}\colon \mu_2^{\otimes r} \xrightarrow{\;\sim\;} Rg^!\mu_2^{\otimes r+d}[2d].\] Applying \(\widehat{R}\Gamma_{c}(Z,-)\) and \(R\Gamma_{\text{\'{e}t}}(Z,-)\), and passing to cohomology gives the Thom isomorphisms: \[\widehat\phi_{g,c}^{r}\colon \widehat{H}_{c}^q(Z;\mu_2^{\otimes r}) \xrightarrow{\sim} \widehat{H}_{c,Z}^{q+2d}(T;\mu_2^{\otimes r+d}),\qquad \phi_g^{r}\colon H_{\mathrm{\acute et}}^q(Z;\mu_2^{\otimes r}) \xrightarrow{\sim} H_Z^{q+2d}(T;\mu_2^{\otimes r+d}).\]

We denote the corresponding ordinary Thom class by \[s_{g,Z}:=\phi_g^0(1)\in H_Z^{2d}(T;\mu_2^{\otimes d}).\] The point we shall use is that Riou’s purity classes are compatible with tensoring coefficients. Namely, \(Cl_g^r\) is the composite \[\mu_2^{\otimes r} \simeq g^*\mu_2^{\otimes r}\otimes \FF_{2,Z} \xrightarrow{\;\id\otimes Cl_g^0\;} g^*\mu_2^{\otimes r}\otimes Rg^!\mu_2^{\otimes d}[2d] \longrightarrow Rg^!\mu_2^{\otimes r+d}[2d],\] where the last arrow is the natural \(g^*(-)\)-module structure on \(Rg^!(-)\). Thus, under the identifications \[H_Z^*(T;-)=H^*(Z;Rg^!(-)),\qquad \widehat H_{c,Z}^*(T;-)=\widehat H_c^*(Z;Rg^!(-)),\] the maps induced by \(Cl_g^r\) are multiplication by the class represented by \(Cl_g^0\), namely \(s_{g,Z}\). Hence, for all \(y\in H_{\acute et}^q(Z;\mu_2^{\otimes r})\) and \(x\in \widehat H_c^q(Z;\mu_2^{\otimes r})\), \[\phi_g^r(y)=y\cdot s_{g,Z},\qquad \widehat\phi_{g,c}^r(x)=x\cdot s_{g,Z},\] where the second product is the mixed product between modified compact-support cohomology and ordinary cohomology with support in \(Z\).

This allows us to reduce the regular closed-immersion case to Benoist’s computation. More precisely, Benoist proves that for a regular closed immersion \(g\), one has⁵ \[\Sq_Z\bigl(\phi_g^{r}(y)\bigr) = \phi_g^{r}\bigl(Sq(y)\cdot w^{\mathrm{\acute et}}(N_g)\bigr)\] in \(H_Z^*(T;\mu_2^{\otimes *})\); see [8]. Taking \(r=0\), \(y=1\), we obtain \[\Sq_Z(s_{g,Z})=w^{\mathrm{\acute et}}(N_g)\cdot s_{g,Z}.\] Now let \[x\in \widehat H_c^q(Z;\mu_2^{\otimes r}).\] Using Cartan, we compute \[\begin{align} \widehat{\Sq}_{c,Z}\bigl(\widehat\phi_{g,c}^{r}(x)\bigr) &= \widehat{\Sq}_{c,Z}(x\cdot s_{g,Z})\\ &= \widehat{\Sq}_c(x)\cdot \Sq_Z(s_{g,Z})\\ &= \widehat{\Sq}_c(x)\cdot w^{\mathrm{\acute et}}(N_g)\cdot s_{g,Z}\\ &= \widehat\phi_{g,c}^{r}\bigl(\widehat{\Sq}_c(x)\cdot w^{\mathrm{\acute et}}(N_g)\bigr). \end{align}\] Finally, by construction \(\widehat g_{*,c}\) is the composite of \(\widehat\phi_{g,c}^{r}\) with the forget-support morphism \[\widehat H_{c,Z}^*(T;\mu_2^{\otimes *}) \longrightarrow \widehat H_c^*(T;\mu_2^{\otimes *}),\] so forgetting support yields \[\widehat{\Sq}_c(\widehat g_{*,c}x) = \widehat g_{*,c}\bigl(\widehat{\Sq}_c(x)\cdot w^{\mathrm{\acute et}}(N_g)\bigr).\] This proves the formula for regular closed immersions.

Case 2: projective bundles. Let \[p\colon P:=\mathbb{P}^n_X\longrightarrow X\] be the projection. Since \(p\) is smooth of relative dimension \(n\), its virtual normal bundle is \[\nu_p=-[T_{P/X}]\in K_0(P).\] We write \[\lambda:=c_1^{\mathrm{\acute et}}\!\bigl(\mathcal{O}_P(1)\bigr)\in H_{\mathrm{\acute et}}^2(P;\mu_2).\]

For every integers \(q,r\), Riou’s projective bundle formula gives an isomorphism in \(D(X_{\mathrm{\acute et}},\mathbf{Z}/2)\) \[\bigoplus_{m=0}^n \mu_2^{\otimes (r-m)}[-2m] \xrightarrow{\;\sim\;} Rp_*\mu_2^{\otimes r},\] whose \(m\)-th component is cup product with \(\lambda^m\) [23]. Applying \[\widehat R\Gamma_c(X,-)\] yields an isomorphism \[\widehat\Phi_{p,c}^{q,r}\colon \bigoplus_{m=0}^n \widehat H_c^{q-2m}(X;\mu_2^{\otimes (r-m)}) \xrightarrow{\;\sim\;} \widehat H_c^{q}(P;\mu_2^{\otimes r}),\] given by \[(y_0,\dots,y_n)\longmapsto \sum_{m=0}^n \widehat p_c^*(y_m)\cdot \lambda^m,\] where \(\widehat p_c^*\) denotes the pullback on modified compact-support cohomology induced by \(p\). Thus it is enough to prove the desired identity for a class of the form \[x=\widehat p_c^*(y)\cdot \lambda^m, \qquad 0\le m\le n,\] with \[y\in \widehat H_c^{q-2m}(X;\mu_2^{\otimes (r-m)}).\]

We first compute \(\widehat p_{*,c}(x)\). Let \[p_*\colon H_{\mathrm{\acute et}}^a(P;\mu_2^{\otimes b}) \longrightarrow H_{\mathrm{\acute et}}^{a-2n}(X;\mu_2^{\otimes (b-n)})\] denote the ordinary Gysin pushforward. Because \(p\) is smooth, Riou’s Gysin map is induced by the trace map [23]. Under the projective bundle formula, this pushforward extracts the \(\lambda^n\)-coefficient; equivalently, \[p_*(\lambda^m)=0 \quad (0\le m<n), \qquad p_*(\lambda^n)=1\in H_{\mathrm{\acute et}}^0(X;\mathbf{Z}/2),\] the normalization of the top class being fixed by [23]. Hence, by the modified compact-support projection formula, \[\widehat p_{*,c}(x) = \widehat p_{*,c}\bigl(\widehat p_c^*(y)\cdot \lambda^m\bigr) = y\cdot p_*(\lambda^m) = \begin{cases} 0,& 0\le m<n,\\ y,& m=n. \end{cases}\] Therefore \[\label{eq:lhs-benoist-modified-projective-case} \widehat{\Sq}_c\bigl(\widehat p_{*,c}(x)\bigr) = \begin{cases} 0,& 0\le m<n,\\ \widehat{\Sq}_c(y),& m=n. \end{cases}\tag{5}\]

We now compute the right-hand side. By naturality of \(\widehat{\Sq}_c\) with respect to proper-pullback and by the mixed Cartan formula, one has \[\widehat{\Sq}_c(x) = \widehat{\Sq}_c\bigl(\widehat p_c^*(y)\cdot \lambda^m\bigr) = \widehat p_c^*\bigl(\widehat{\Sq}_c(y)\bigr)\cdot \Sq(\lambda^m),\] where \(\Sq(\lambda^m)\) is the ordinary (total) Steenrod square of the ordinary class \(\lambda^m\in H_{\mathrm{\acute et}}^{2m}(P;\mu_2^{\otimes m})\). Multiplying by the ordinary class \(w^{\mathrm{\acute et}}(\nu_p)\) and applying the modified compact-support projection formula again gives \[\widehat p_{*,c}\bigl(\widehat{\Sq}_c(x)\cdot w^{\mathrm{\acute et}}(\nu_p)\bigr) = \widehat{\Sq}_c(y)\cdot p_*\bigl(\Sq(\lambda^m)\cdot w^{\mathrm{\acute et}}(\nu_p)\bigr).\]

At this point we reduce to Benoist’s ordinary theorem. Since \(X\) is regular Noetherian and admits an ample invertible sheaf, the same holds for \(P=\mathbb{P}^n_X\). Thus Benoist’s ordinary relative Wu formula, recalled above as Theorem 12 and originally proved in [8], applies to the proper morphism \(p\colon P\to X\) and to the ordinary class \[\lambda^m\in H_{\mathrm{\acute et}}^{2m}(P;\mu_2^{\otimes m}).\] It yields \[p_*\bigl(\Sq(\lambda^m)\cdot w^{\mathrm{\acute et}}(\nu_p)\bigr) = \Sq\bigl(p_*(\lambda^m)\bigr).\] Using the above computation of \(p_*(\lambda^m)\), and the identity \(\Sq(1)=1\), we obtain \[p_*\bigl(\Sq(\lambda^m)\cdot w^{\mathrm{\acute et}}(\nu_p)\bigr) = \begin{cases} 0,& 0\le m<n,\\ 1,& m=n. \end{cases}\] Consequently, \[\widehat p_{*,c}\bigl(\widehat{\Sq}_c(x)\cdot w^{\mathrm{\acute et}}(\nu_p)\bigr) = \begin{cases} 0,& 0\le m<n,\\ \widehat{\Sq}_c(y),& m=n. \end{cases}\] Comparing with 5 , we conclude that \[\widehat{\Sq}_c(\widehat p_{*,c}x) = \widehat p_{*,c}\bigl(\widehat{\Sq}_c(x)\cdot w^{\mathrm{\acute et}}(\nu_p)\bigr)\] for all \(x\in \widehat H_c^*(P;\mu_2^{\otimes *})\). This proves the projective bundle case, and completes the proof of the theorem. ◻

Proof. Set \(D(F):=\underline{\Hom}(F;\mathbf{G}_m)\). Since \(2\) is invertible on \(B\), we have canonical identifications \[D(\mu_\ell)\simeq \mathbf{Z}/\ell, \qquad D(\mathbf{Z}/\ell)\simeq \mu_\ell,\] and the short exact sequence \[0\longrightarrow \mu_2\longrightarrow \mu_4\longrightarrow \mu_2\longrightarrow 0\] is Cartier dual to \[0\longrightarrow \mathbf{Z}/2\longrightarrow \mathbf{Z}/4\longrightarrow \mathbf{Z}/2\longrightarrow 0.\]

By Artin–Verdier duality, for every constructible sheaf \(F\) on \(B\) there is a functorial perfect pairing \[H^i(B;D(F))\times \widehat H_c^{3-i}(B;F)\longrightarrow \mathbf{Q}/\mathbf{Z}.\] Applying this to the above dual pair of short exact sequences, we obtain a diagram, which is commutative up to a sign [27]; see also [25] \[\begin{tikzcd} H^0(B;\mu_2) \arrow[r,"\delta_1"] \arrow[d,"\sim"'] & H^1(B;\mu_2)\arrow[d,"\sim"]\\ \widehat H_c^3(B;\mathbf{Z}/2)^\vee \arrow[r,"(\widehat\delta_0)^\vee"'] & \widehat H_c^2(B;\mathbf{Z}/2)^\vee. \end{tikzcd}\] Since we are working modulo \(2\), there is no sign ambiguity, hence the diagram commutes. ◻

Proof. By Artin–Verdier duality for arithmetic curves, the trace pairings \[H^i(B;\mathbf{Z}/2)\times \widehat H_c^{3-i}(B;\mu_2)\longrightarrow \mathbf{Z}/2\] are perfect for every \(i\); see [25], see also Theorem 8. We need to be a little careful due to the fact that the Steenrod algebra on modified compact support cohomology is the extended Steenrod algebra, hence supported in negative indices as well. It turns out however that these negative indexed Steenrod squares shall not play a role.

Indeed, let \(x\in H_c^{3-i}(B;\mu_2)\) be arbitrary. Then \[\int_B \widehat{\Sq}_c(x) = \int_B \sum_j\widehat{\Sq}^j_c(x) = \int_B \widehat{\Sq}^{i}_c(x),\] since the trace pairing only detects classes of degree \(3\). By the instability relations, \(\widehat{\Sq}^{i}_c(x) = 0\) if \(i > 3-i\iff i \ge 2\). Hence, we may assume \(i \le 1\) so that \(|x|\ge 2\). Since \(\widehat H_c^{k}(B;\mu_2) \cong H_c^{k}(B;\mu_2)\) for \(k\ge 2\) (see Proposition [prop:sq0-sq1-boundary-operations]), and since the Steenrod algebra on \(H_c^{k}(B;\mu_2)\) is non-negatively graded, \[|x| > 3 \implies \widehat{\Sq}_c(x) = 0.\] Hence \[\int_B \widehat{\Sq}_c(x) \neq 0\implies 2\le |x|\le 3,\] and consequentially, we have \[v_{B,i}=0\qquad (i<0\quad \text{or}\quad i>2).\] Now let \(x\in \widehat H_c^3(B;\mu_2)\). Since \(\widehat{\Sq}_c^0=\id\) on \(x\) of degree at least \(2\), we conclude that \(v_{B,0}=1\).

Finally, let \(x\in \widehat H_c^2(B;\mu_2)\). By Proposition [prop:sq0-sq1-boundary-operations] and Lemma 3, \[\int_B \widehat{\Sq}_c^1(x) = \langle 1,\widehat{\delta}_0(x)\rangle = \langle \delta_1(1_{\mu}),x\rangle = \langle \beta_B,x\rangle = \int_B \beta_B\cup x,\] where \(1\in H^0(B;\mathbf{Z}/2)\) corresponds to \(1_{\mu}\in H^0(B;\mu_2)\) under the canonical identification \(\mathbf{Z}/2\simeq \mu_2\). It follows that \(v_{B,1}=\beta_B\), and combining this with the vanishing of \(v_{B,i}\) for \(i\ge 2\) and the identity \(v_{B,0}=1\), we conclude that \[v_B=1+\beta_B.\] ◻

Proof. Consider the commutative diagram of short exact sequences of étale sheaves on \(T\): \[\begin{array}{ccccccccc} 0 & \longrightarrow & \mu_2 & \longrightarrow & \mu_4 & \xrightarrow{(\cdot)^2} & \mu_2 & \longrightarrow & 0 \\ && \Vert && \downarrow && \downarrow \\ 0 & \longrightarrow & \mu_2 & \longrightarrow & \mathbf{G}_m & \xrightarrow{(\cdot)^2} & \mathbf{G}_m & \longrightarrow & 0. \end{array}\] Applying cohomology and using the naturality of connecting morphisms for the section \[1_\mu=-1\in H^0(T;\mu_2)\subseteq H^0(T;\mathbf{G}_m)\] shows that the Bockstein class \(\beta_T\) agrees with the Kummer class of \(-1\). ◻

Proof. This is immediate from Proposition [prop:absolute-arithmetic-wu-formula] and Lemma 4. ◻

Proof. By Proposition [prop:beta-square-generic-point], \((\beta_B^2)|_\eta\) is the Brauer class of \(({-1},{-1})\), so it vanishes if and only if \(({-1},{-1})\) is split. By [29], a quaternion algebra \((a,b)\) is split if and only if \(b\) is a norm from \(K(\sqrt a)\). Applying this with \(a=b=-1\) gives the last equivalence. ◻

Proof. Let \(\Omega_R\) be the set of real places of \(K\), and put \(r=|\Omega_R|\). Since \(2\) is invertible on \(B\), every place of \(K\) above \(2\) lies in \(S\); in particular \(S\neq \varnothing\). Hence \(B\) is exactly the open subscheme \(U=X\setminus S\) covered by [28], and that theorem gives \[H^i_{\acute et}(B;\mathbf{Z}/2)\cong (\mathbf{Z}/2)^r \qquad\text{for all }i\ge 3.\] If \(r=0\), then \(H^3_{\acute et}(B;\mathbf{Z}/2)=0\), so \(\beta_B^k=0\).

Assume now that \(r>0\). By the discussion in [28], for every \(i\ge 3\) the restriction map \[\rho_i\colon H^i_{\acute et}(B;\mathbf{Z}/2) \xrightarrow{\;\sim\;} \prod_{\nu\in \Omega_R} H^i_{\acute et}(K_\nu;\mathbf{Z}/2)\] is an isomorphism, and cup products in total degree \(\ge 3\) are computed after restriction to the real places.

For each real place \(\nu\), the restriction of \(\beta_B\) to \(H^1_{\acute et}(K_\nu;\mathbf{Z}/2)=H^1_{\acute et}(\mathbf{R};\mathbf{Z}/2)\) is the Kummer class of \(-1\in \mathbf{R}^\times\), hence is the unique nonzero class \(u_\nu\). Since \(\Gal(\mathbf{C}/\mathbf{R})\cong \mathbf{Z}/2\), the standard computation of the cohomology ring of \(\mathbf{Z}/2\) gives \[H^\ast_{\acute et}(\mathbf{R};\mathbf{Z}/2)\cong \mathbb{F}_2[u_\nu], \qquad \deg(u_\nu)=1,\] so \(u_\nu^k\neq 0\) in \(H^k_{\acute et}(\mathbf{R};\mathbf{Z}/2)\). Therefore \[\rho_k(\beta_B^k) = \bigl(u_\nu^k\bigr)_{\nu\in \Omega_R} \neq 0.\] Since \(\rho_k\) is injective, it follows that \(\beta_B^k\neq 0\). ◻

Proof. This is immediate from the corresponding properties of \(w^{\mathrm{\acute et}}\) and the fact that \(\Sq^{-1}\) is a ring automorphism. ◻

Proof. Write \[Q(2z)=\sum_{m\ge 0} a_m z^m, \qquad a_m=\frac{2^m B_m}{m!}.\] We have \(a_0=1\) and \(a_1=2B_1=-1\). If \(m>1\) is odd, then \(B_m=0\), so \(a_m=0\). It therefore remains to consider the coefficients \(a_{2n}\) with \(n\ge 1\).

By the von Staudt–Clausen theorem (see, for example, [30]), one has \(v_2(B_{2n})=-1\). Hence \[v_2(a_{2n})=2n-1-v_2((2n)!).\] Let \(s_2(N)\) denote the sum of the binary digits of \(N\). By Legendre’s formula, \[v_2((2n)!)=2n-s_2(2n)=2n-s_2(n),\] and therefore \[v_2(a_{2n})=s_2(n)-1\ge 0.\] This proves that \(Q(2z)\) is \(2\)-integral.

Moreover, \(a_{2n}\) is odd if and only if \(s_2(n)=1\), that is, if and only if \(n\) is a power of \(2\), it follows that the reduction modulo \(2\) of \(Q(2z)\) is \[1+\sum_{r\ge 0} z^{2^r},\] as claimed. ◻

Proof. All formal power series below are evaluated degreewise in the completed graded ring \(\prod_{m\ge 0} H^m_{\mathrm{\acute et}}(X;\mathbf{Z}/2)\). Write \[\beta:=\beta_X,\qquad x:=\bar x,\qquad A:=T(\beta).\] Since \(T(z)\) has constant term \(1\), the class \(A\) is invertible.

First note that, by ?? , one has the formal identity \[T(z+z^2) = 1+\sum_{r\ge 0}(z+z^2)^{2^r} = 1+\sum_{r\ge 0}\bigl(z^{2^r}+z^{2^{r+1}}\bigr) = 1+z.\] Since \(\deg(\beta)=1\), one has \(\Sq(\beta)=\beta+\beta^2\), hence \[\Sq(A)=T(\Sq(\beta))=T(\beta+\beta^2)=1+\beta.\]

Next, because \(\deg(x)=2\), the instability relations give \[\Sq(x)=x+\Sq^1(x)+x^2.\] By Lemma 2, \[\Sq^1(a)=\delta_1(a)+\beta\cdot a,\] where \(\delta_1\) is the Bockstein attached to the exact sequence \[0\longrightarrow \mu_2\longrightarrow \mu_4\longrightarrow \mu_2\longrightarrow 0.\] Now \(x\) is the reduction of \(c_1(L)\in H^2_{\mathrm{\acute et}}(X;\mathbf{Z}_2(1))\), so its reduction modulo \(4\) lifts to \(H^2_{\mathrm{\acute et}}(X;\mathbf{Z}/4(1)) \cong H^2_{\mathrm{\acute et}}(X;\mu_4)\). Therefore \(\delta_1(x)=0\), and so \[\Sq(x)=x+\beta x+x^2=x(1+\beta+x).\]

Since \[2td(L)=\Sq^{-1}\bigl(w^{\mathrm{\acute et}}(L)\bigr),\] it is enough to prove that \[\Sq\bigl(\Theta_\beta(x)\bigr)=w^{\mathrm{\acute et}}(L).\] By Theorem 7, \[w^{\mathrm{\acute et}}(L)=1+\beta+x.\]

Now \[\begin{align} \Sq\bigl(\Theta_\beta(x)\bigr) &= \Sq(A)\cdot \Sq\!\left(T\!\bigl(A^{-2}x\bigr)\right) \\ &= (1+\beta)\cdot T\!\left(\Sq(A)^{-2}\Sq(x)\right) \\ &= (1+\beta)\cdot T\!\left(\frac{x(1+\beta+x)}{(1+\beta)^2}\right). \end{align}\] Since \(1+\beta\) is invertible in \(\widehat H^*_{\mathrm{\acute et}}(X;\mathbf{Z}/2)\), we may set \[u:=\frac{x}{1+\beta}.\] Then \[\frac{x(1+\beta+x)}{(1+\beta)^2}=u+u^2.\] Using the identity \(T(u+u^2)=1+u\), we obtain \[\Sq\bigl(\Theta_\beta(x)\bigr) = (1+\beta)\bigl(1+u\bigr) = 1+\beta+x.\] This is exactly \(w^{\mathrm{\acute et}}(L)\), and the result follows. ◻

Proof. By Proposition [prop:2td-basic] and Grothendieck’s splitting principle (see Remark [rem:etale-sw-splitting]), \[\pi^*2td(\xi)=2td(\pi^*\xi)=\prod_{i=1}^r 2td(L_i).\] Lemma 6 identifies each factor with \(\Theta_{\pi^*\beta_X}(x_i)\), which proves ?? . ◻

Proof. Since \(P_r(\beta_X, \overline{x}_1,\dots,\overline{x}_r) := \prod_{i=1}^r\Theta_{\beta_X}(\overline{x}_i)\) is a unique polynomial, whose \(m\)’th homogeneous piece computes the degree \(m\)’th \(2td\) class of any bundle \(\xi\) upon the substitution \(\overline{x}_i \mapsto \overline{x}_i(\xi)\), where the \(\overline{x}_i(\xi)\) denote the Chern roots of \(\xi\), and since \(P_r\) is invariant under permutations of the \(\overline{x}_i\), it follows that \(P_r\) may be expanded uniquely in terms of homogenous symmetric functions in the Chern roots (with coefficients in \(\FF_2\)), and hence may be rewritten as a polynomial in the mod-\(2\) reductions of the Chern classes (with coefficients in \(\FF_2\)). This proves the first part of the corollary.

For the second part, recall that by 7 , \[v_X = f^*v_B\cdot \Sq^{-1}(w^{\text{\'{e}t}}(\tau_f)) = Q_B(\beta_X)\cdot P_d(\beta_X, \overline{x}_1(\tau_f),\dots,\overline{x}_d(\tau_f)),\] where \[Q_B(\beta) = \begin{cases} 1+\beta_B& B = \Spec\,\mathcal{O}_{K,S}\\ 1& B = \Spec\,\mathbb{F}_{q}, \end{cases}\] and take the \(m\)’th homogeneous part of \(Q_B(\beta)\cdot P_d(\beta, \overline{x}_1,\dots,\overline{x}_d)\) for \(P^{\mathrm{abs}}_{m,d}\). ◻

Proof. \(X\) satisfies Artin–Verdier duality in total degree \[N_X=3.\]

Let \(\tau_f\in K^0(X)\) denote the virtual relative tangent class of \(f\). By the absolute formula, \[v_X=(1+\beta_X)\cdot 2td(\tau_f).\]

We first identify the degree-\(2\) part of this class. For a line bundle \(M\) on \(X\), Lemma 6 gives \[2td(M)=1+\beta_X+\beta_X^2 + \overline{c}_1(M)+\text{(terms of degree \ge 3)}.\] Using multiplicativity and the splitting principle, it follows that for every \(\xi\in K^0(X)\), \[2td(\xi)= 1+\rk(\xi)\beta_X+\Bigl(\binom{\rk(\xi)+1}{2}\beta_X^2+\bar c_1(\xi)\Bigr)+\text{(terms of degree \ge 3)}.\] Since \(\rk(\tau_f)=0\), we obtain \[v_{X,2} = \bigl((1+\beta_X)\cdot 2td(\tau_f)\bigr)_2 = 2td(\tau_f)_2 = \overline{c}_1(\tau_f).\]

Now let \(\mathfrak D_{X/B}\subset \mathcal{O}_X\) be the different ideal, and write \[\mathfrak D_{X/B}=\mathcal{O}_X(-R)\] for the associated effective divisor \(R\) on \(X\). Thus \(R\) is the different divisor of \(f\).

By [11], the different ideal sheaf is the inverse of the relative dualizing sheaf, i.e. \[\omega_{X/B}\cong \mathcal{O}_X(R).\] On the other hand, \[\det(\tau_f)\cong \omega_{X/B}^{-1},\] hence \[\overline{c}_1(\tau_f) = \overline{c}_1(\det \tau_f) = \overline{c}_1(\omega_{X/B}^{-1}) = \overline{c}_1(\mathcal{O}_X(R)),\] the last equality because signs disappear modulo \(2\). Therefore \[v_{X,2}=\overline{c}_1(\mathcal{O}_X(R)).\]

Finally, since \(N_X=3\), the instability relations imply \[v_{X,2}=0.\] Consequently, \[\overline{c}_1(\mathcal{O}_X(R))=0 \qquad\text{in }H^2_{\acute et}(X;\mathbf{Z}/2).\] By the Kummer exact sequence, this is equivalent to the existence of a line bundle \(M\) on \(X\) such that \[\mathcal{O}_X(R)\cong M^{\otimes 2}.\] Equivalently, the class of the different divisor \(R\) is even in \(\Pic(X)\). For the Dedekind scheme \(X\), this is the usual statement that the ideal class of the different is a square. ◻

Proof. Since \(X\) satisfies Artin–Verdier duality in total degree \(N_X=3\), Theorem 14 gives \[v_{X,2}=0.\]

The proof is the same as that of Theorem 16, one minor difference is that now the virtual relative tangent bundle \(\tau_f\) is an honest vector bundle, which we denote by \(T_X\). The only new point is that \(\rk(T_X)=1\), so the degree-\(2\) term is \[v_{X,2}= \overline{c}_1(T_X).\] Hence \[0=v_{X,2}=\overline{c}_1(T_X)=\overline{c}_1(\omega_X),\] because \(T_X\cong \omega_X^{-1}\) and signs disappear modulo \(2\). As \(2\in k^\times\), the Kummer exact sequence shows that \(\omega_X\in 2\Pic(X)\), equivalently that \(\omega_X\cong \vartheta^{\otimes 2}\) for some line bundle \(\vartheta\) on \(X\). ◻

Proof. Theorem 17 shows that both \(X\) and \(Y\) have theta-characteristics. A repeat of the argument in the proof of Theorem 16 but in the category of varieties over finite fields in odd characteristics shows that (Riemann–Hurwitz) \[\omega_X \cong f^*\omega_Y\otimes\mathcal{O}_X(R_f),\] and that the divisor class of the line bundle \(\mathcal{O}_X(R_f)\) admits a square root (generalized Hecke). The odd tame ramification assumption implies (see [11]) that \[R_f = \sum_{x\in X}(e_x-1)\cdot [x] = 2\cdot \sum_{x\in X}\frac{e_x-1}{2}\cdot [x] =: 2D_f \in \mathrm{Div}(X)\] in the divisor group, hence \(R_f\) admits a square root not only in \(\Pic(X)\), but in \(\mathrm{Div}(X)\) itself. This \(D_f\) gives us a canonical square root of the different/ramification divisor \(R_f\), which allows us to produce a canonical lift to any theta characteristic on \(Y\).

Indeed, let \(\vartheta_Y\) be a theta characteristic on \(Y\), and set \(\vartheta_X := f^*\vartheta_Y\otimes \mathcal{O}_X(D_f)\). Clearly, \[\vartheta_X^{\otimes 2} := (f^*\vartheta_Y\otimes \mathcal{O}_X(D_f))^{\otimes 2} = f^*\vartheta_Y^{\otimes 2}\otimes \mathcal{O}_X(2D_f) = f^*\omega_Y\otimes \mathcal{O}_X(R_f),\] which is a canonical bundle representative on \(X\) by Riemann–Hurwitz, making \(\vartheta_X\) into a theta characteristic on \(X\). ◻

Proof. Write \(\widetilde{L}:=f^{-1}(L)\) and \(U:=M\setminus \widetilde{L}\). For each connected component \(\widetilde{K}\subset \widetilde{L}\), let \(e_{\widetilde{K}}\geq 1\) denote the ramification index of \(f\) along \(\widetilde{K}\). Since \(f\) is a smooth branched cover in the sense of Definition 29, the ramification index is locally constant on the branch locus, hence constant on each connected component \(\widetilde{K}\).

All cohomology groups below are taken with coefficients in \(\mathbf{Z}/2\), unless explicitly indicated otherwise.

Let \[\tau_f:=[T_M]-f^*[T_N]\in KO^0(M)\] be the virtual relative tangent bundle. By definition, \[w(\tau_f)=w(T_M)\,w(f^*T_N)^{-1}.\]

We first prove that \(w_2(\tau_f)=0\). By the instability relations, we find that \(v_{X,2} = 0\). The Wu formula \[w(T_X)=\Sq(v_X)\] therefore yields \[w_1(T_X)=v_{X,1}, \qquad w_2(T_X)=v_{X,2}+\Sq^1(v_{X,1})=\Sq^1(v_{X,1})=v_{X,1}^2=w_1(T_X)^2.\] Thus, for every closed smooth \(3\)-manifold \(X\), \[\label{eq:w2-w1-square} w_2(T_X)=w_1(T_X)^2.\tag{8}\]

Next we claim that \[w_1(T_M)=f^*w_1(T_N)\qquad\text{in }H^1(M).\] Indeed, on \(U=M\setminus \widetilde{L}\) the differential is an isomorphism \[df\colon T_M|_U \xrightarrow{\sim} f^*T_N|_U,\] so the two classes agree after restriction to \(U\).

Let \(\nu(\widetilde{L})\) be a tubular neighbourhood of \(\widetilde{L}\) in \(M\). By excision and the Thom isomorphism for the normal \(2\)-plane bundle of \(\widetilde{L}\subset M\), \[H^1(M;U) \cong H^1\!\bigl(\nu(\widetilde{L});\nu(\widetilde{L})\setminus \widetilde{L}\bigr) \cong H^{-1}(\widetilde{L}) =0.\] Hence the restriction map \[H^1(M)\longrightarrow H^1(U)\] is injective, and the claim follows. In particular, \[w_1(\tau_f)=w_1(T_M)+f^*w_1(T_N)=0.\]

Now set \[x:=f^*w_1(T_N),\qquad y:=f^*w_2(T_N).\] Up to degree \(2\), one has \[w(f^*T_N)=1+x+y, \qquad w(f^*T_N)^{-1}=1+x+x^2+y,\] since \[(1+x+y)(1+x+x^2+y)=1\] modulo terms of degree \(\ge 3\). Therefore \[w_2(\tau_f)=w_2(T_M)+w_1(T_M)x+x^2+y.\] Using \(w_1(T_M)=x\) and 8 for \(X=M\) and \(X=N\), we obtain \[w_2(\tau_f)=x^2+x^2+x^2+x^2=0.\]

We now compute the same class \(w_2(\tau_f)\) by localizing it along the ramification link \(\widetilde{L}\).

Since \(df\) is an isomorphism over \(U\), it gives a stable trivialization of \(\tau_f|_U\). Let \[w_2^{\mathrm{rel}}(\tau_f,df)\in H^2(M;U)\] denote the corresponding relative second Stiefel–Whitney class. By construction, its image under the natural map \[H^2(M;U)\longrightarrow H^2(M)\] is the absolute class \(w_2(\tau_f)\).

Choose pairwise disjoint tubular neighbourhoods \(T_{\widetilde{K}}\) of the connected components \(\widetilde{K}\subset \widetilde{L}\). Excision gives \[H^2(M;U) \cong \bigoplus_{\widetilde{K}\subset \widetilde{L}} H^2(T_{\widetilde{K}};T_{\widetilde{K}}\setminus \widetilde{K}).\] For each \(\widetilde{K}\), let \[u_{\widetilde{K}}\in H^2(T_{\widetilde{K}};T_{\widetilde{K}}\setminus \widetilde{K})\] be the Thom class of the normal bundle of \(\widetilde{K}\subset M\). Since \(\widetilde{K}\) is connected, the Thom isomorphism identifies \[H^2(T_{\widetilde{K}};T_{\widetilde{K}}\setminus \widetilde{K}) \cong H^0(\widetilde{K})\cong \mathbf{Z}/2,\] with generator \(u_{\widetilde{K}}\). Hence there are unique coefficients \(a_{\widetilde{K}}\in \mathbf{Z}/2\) such that \[w_2^{\mathrm{rel}}(\tau_f,df) = \sum_{\widetilde{K}\subset \widetilde{L}} a_{\widetilde{K}}\,u_{\widetilde{K}}.\] Under the map \(H^2(M;U)\to H^2(M)\), the class \(u_{\widetilde{K}}\) maps to \(\mathrm{PD}([\widetilde{K}])\). Consequently, \[\label{eq:w2-sum-ak} w_2(\tau_f) = \sum_{\widetilde{K}\subset \widetilde{L}} a_{\widetilde{K}}\,\mathrm{PD}([\widetilde{K}]).\tag{9}\]

It remains to determine \(a_{\widetilde{K}}\) for a fixed component \(\widetilde{K}\subset \widetilde{L}\).

Fix a point \(p\in \widetilde{K}\), and let \[\iota_p\colon (\Delta,\partial\Delta)\hookrightarrow (T_{\widetilde{K}},T_{\widetilde{K}}\setminus \widetilde{K})\] be the inclusion of a meridional disk, i.e. of a fiber disk of the normal bundle over \(p\). By the defining property of the Thom class, \[\iota_p^*(u_{\widetilde{K}})\] is the generator of \[H^2(\Delta;\partial\Delta)\cong \mathbf{Z}/2.\] Since both source and target are \(1\)-dimensional over \(\mathbf{Z}/2\), the map \[\iota_p^*\colon H^2(T_{\widetilde{K}};T_{\widetilde{K}}\setminus \widetilde{K}) \longrightarrow H^2(\Delta;\partial\Delta)\] is an isomorphism. Therefore \(a_{\widetilde{K}}\) is exactly the image of \(w_2^{\mathrm{rel}}(\tau_f,df)\) in \(H^2(\Delta;\partial\Delta)\).

By Definition 29, after shrinking neighbourhoods of \(p\) and \(f(p)\), there exist local coordinates \[(-\varepsilon,\varepsilon)\times D \subset \mathbf{R}\times \mathbf{C}\] centered at \(p\) and \(f(p)\) in which \(\widetilde{K}\) and \(L\) are given by \((-\varepsilon,\varepsilon)\times\{0\}\), and \(f\) is given by \[f(u,z)=(u,z^{e_{\widetilde{K}}}).\] Take \(\Delta=\{0\}\times D\). Over \(\Delta\), the tangent bundles split as \[T_M|_\Delta \cong \varepsilon^1\oplus T_\Delta, \qquad f^*T_N|_\Delta \cong \varepsilon^1\oplus (f|_\Delta)^*T_\Delta,\] where the first summand is the tangent line in the \(u\)-direction. On \(\Delta^\times:=\Delta\setminus\{0\}\), the isomorphism \(df\) is the identity on the first summand and the differential of \(z\mapsto z^{e_{\widetilde{K}}}\) on the second. By Whitney additivity for relative Stiefel–Whitney classes, adjoining the common trivial line \(\varepsilon^1\) does not affect the degree-\(2\) class. Thus the restriction of \(w_2^{\mathrm{rel}}(\tau_f,df)\) to \(H^2(\Delta;\partial\Delta)\) is the relative second Stiefel–Whitney class of the rank-\(2\) datum \[\bigl(T_\Delta,(f|_\Delta)^*T_\Delta,df|_{\partial\Delta}\bigr).\]

Collapsing \(\partial\Delta\) to a point identifies \(H^2(\Delta;\partial\Delta)\) with \(H^2(S^2)\). Under this identification, the preceding relative class becomes the ordinary \(w_2\) of the oriented real \(2\)-plane bundle \(\xi_{\widetilde{K}}\to S^2\) obtained by clutching the two trivial bundles \(T_\Delta\) and \((f|_\Delta)^*T_\Delta\) along \(\partial\Delta=S^1\) via the boundary isomorphism \(df\).

Using the complex coordinate \(z\) to orient and trivialize both bundles, the clutching map is \[S^1\longrightarrow \mathbf{C}^\times, \qquad \zeta\longmapsto e_{\widetilde{K}}\zeta^{\,e_{\widetilde{K}}-1},\] because \[d(z^{e_{\widetilde{K}}})=e_{\widetilde{K}}z^{e_{\widetilde{K}}-1}dz.\] Since multiplication by the positive real number \(e_{\widetilde{K}}\) is homotopic to the identity in \(\mathbf{C}^\times\), this loop is homotopic to \[\zeta\longmapsto \zeta^{\,e_{\widetilde{K}}-1}.\] Hence \(\xi_{\widetilde{K}}\) is the underlying real bundle of the complex line bundle on \(S^2\cong \mathbf{CP}^1\) with clutching function \(\zeta\mapsto \zeta^{\,e_{\widetilde{K}}-1}\). Its first Chern class is therefore \((e_{\widetilde{K}}-1)\) times the positive generator of \(H^2(S^2;\mathbf{Z})\), and so \[w_2(\xi_{\widetilde{K}}) \equiv e_{\widetilde{K}}-1 \pmod 2\] in \(H^2(S^2)\cong \mathbf{Z}/2\). It follows that \[a_{\widetilde{K}}=e_{\widetilde{K}}-1\in \mathbf{Z}/2.\]

Substituting this into 9 , we obtain \[w_2(\tau_f) = \sum_{\widetilde{K}\subset \widetilde{L}} (e_{\widetilde{K}}-1)\,\mathrm{PD}([\widetilde{K}]) = \mathrm{PD}\!\left( \sum_{\widetilde{K}\subset \widetilde{L}} (e_{\widetilde{K}}-1)[\widetilde{K}] \right) = \mathrm{PD}(D_f).\] Since we already proved that \(w_2(\tau_f)=0\), we conclude that \[\mathrm{PD}(D_f)=0.\] By Poincaré duality over \(\mathbf{Z}/2\), this implies \[D_f=0\qquad\text{in }H_1(M;\mathbf{Z}/2).\] ◻

Proof. For each \(i\), set \[A^i:=H^i(X;\mathbf{Z}_2(d)).\] By the finiteness theorem for smooth proper étale cohomology over finite fields, each \(A^i\) is a finitely generated \(\mathbf{Z}_2\)-module. Write \[A^i\simeq \mathbf{Z}_2^{b_i}\oplus T^i,\] where \(T^i\) is finite \(2\)-primary, and put \[\tau_i:=\dim_{\mathbb{F}_2}(T^i/2T^i).\] The long exact sequence in cohomology associated to the coefficient short exact sequence \[0\to \mathbf{Z}_2(d)\xrightarrow{\cdot 2} \mathbf{Z}_2(d)\longrightarrow \mathbb{F}_2\to 0,\] gives a natural short exact sequence \[0 \longrightarrow A^i\otimes_{\mathbf{Z}_2}\mathbb{F}_2 \longrightarrow H^i(X;\mathbb{F}_2) \longrightarrow A^{i+1}[2] \longrightarrow 0.\] Hence \[\dim_{\mathbb{F}_2}H^i(X;\mathbb{F}_2) = b_i+\tau_i+\tau_{i+1}.\] On the other hand, \[\dim_{\mathbf{Q}_2}H^i(X;\mathbf{Q}_2(d))=b_i.\] Therefore \[\begin{align} \Delta_2(X) &= \sum_{i=0}^{2d}(\tau_i+\tau_{i+1}) \pmod 2 \\ &= \tau_0+\tau_{2d+1} \pmod 2. \end{align}\] The group \[A^0=H^0(X;\mathbf{Z}_2(d))\] is torsion-free, possibly zero, so \(\tau_0=0\). Thus \[\Delta_2(X) = \tau_{2d+1} = \dim_{\mathbb{F}_2} \bigl(H^{2d+1}(X;\mathbf{Z}_2(d))_{\mathrm{tors}}/2\bigr) = \epsilon_2(X).\] ◻

Proof. For \(e\ge 1\), set \[V_e := T[2^e]\Big/\bigl(T[2^{e-1}]+2T[2^{e+1}]\bigr).\] A direct check from the elementary divisor decomposition shows that \[\dim_{\mathbb{F}_2}V_e=m_e.\] The form induces a bilinear pairing \[\lambda_e:V_e\times V_e \longrightarrow \tfrac12\mathbf{Z}_2/\mathbf{Z}_2\simeq \mathbb{F}_2\] by the formula \[\lambda_e(\overline{x},\overline{y}) := 2^{e-1}\lambda(x,y).\] This is well-defined. If \(x\) is changed by an element of \(T[2^{e-1}]\), then \(2^{e-1}\lambda(x,y)=0\). If \(x\) is changed by an element \(2z\), with \(z\in T[2^{e+1}]\), then \[2^{e-1}\lambda(2z,y) = 2^e\lambda(z,y) = \lambda(z,2^ey) = 0,\] because \(y\in T[2^e]\). The same argument applies in the second variable.

The form \(\lambda_e\) is nonsingular. Suppose that \(\overline{x}\in V_e\) pairs trivially with every element of \(V_e\). Then \[2^{e-1}\lambda(x,y)=0 \qquad \text{for all } y\in T[2^e],\] or equivalently \[\lambda(2^{e-1}x,y)=0 \qquad \text{for all } y\in T[2^e].\] For a nonsingular form, the annihilator of \(T[2^e]\) is \(2^eT\). Thus \(2^{e-1}x\in 2^eT\). Choose \(z\in T\) such that \[2^{e-1}x=2^ez.\] Since \(x\in T[2^e]\), multiplying by \(2\) gives \(z\in T[2^{e+1}]\). Hence \[x-2z\in T[2^{e-1}],\] and therefore \[x\in T[2^{e-1}]+2T[2^{e+1}].\] Thus \(\overline{x}=0\), proving nonsingularity.

Assume first only that \(\lambda\) is skew-symmetric. Then \[2\lambda(x,x)=0 \qquad \text{for all }x\in T.\] If \(e\ge 2\), it follows that \[\lambda_e(\overline{x},\overline{x}) = 2^{e-1}\lambda(x,x) = 2^{e-2}\bigl(2\lambda(x,x)\bigr) = 0.\] Thus \(\lambda_e\) is nonsingular and alternating over \(\mathbb{F}_2\), so \(V_e\) has even dimension. Hence \(m_e\) is even for every \(e\ge 2\).

If \(\lambda\) is alternating, then \(\lambda(x,x)=0\) for all \(x\), and the same argument applies also for \(e=1\). Since \[\dim_{\mathbb{F}_2}(T/2T)=\sum_{e\ge 1}m_e,\] the assertion follows. ◻

Proof. By Poincaré duality, the adjoint of \(B_X^{(d)}\) is precisely \[\delta_d: H^{2d}(X;\mathbb{F}_2) \longrightarrow H^{2d+1}(X;\mathbb{F}_2).\] Therefore \[\operatorname{rank}B_X^{(d)} = \operatorname{rank}\delta_d.\]

Write \[H^{2d+1}(X;\mathbf{Z}_2(d))_{\mathrm{tors}} \simeq \bigoplus_{e\ge 1} (\mathbf{Z}/2^e\mathbf{Z})^{m_e}.\] We first compute the rank of \(\delta_d\). Let \[A^i:=H^i(X;\mathbf{Z}_2(d)).\] For \(n=2,4\), the universal coefficient sequence gives \[0 \longrightarrow A^{2d}/n \longrightarrow H^{2d}(X;\mathbf{Z}/n(d)) \longrightarrow A^{2d+1}[n] \longrightarrow 0.\] The reduction map from \(n=4\) to \(n=2\) gives a morphism between these short exact sequences, and the left vertical map \[A^{2d}/4\longrightarrow A^{2d}/2\] is surjective. Hence the cokernel of \[H^{2d}(X;\mathbf{Z}/4(d)) \longrightarrow H^{2d}(X;\mathbf{Z}/2(d))\] is naturally identified with the cokernel of \[A^{2d+1}[4]\longrightarrow A^{2d+1}[2].\] The image of the Bockstein \(\delta_d\) is exactly this cokernel.

It remains to compute this cokernel on elementary divisors. A free \(\mathbf{Z}_2\)-summand contributes nothing. For a summand \(\mathbf{Z}/2^e\mathbf{Z}\), the target \(A^{2d+1}[2]\) is one-dimensional over \(\mathbb{F}_2\). If \(e=1\), then the induced map \[A^{2d+1}[4]\longrightarrow A^{2d+1}[2]\] is zero; if \(e\ge 2\), it is surjective. Thus \[\operatorname{rank}_{\mathbb{F}_2}\delta_d=m_1.\]

On the other hand, \[\epsilon_2(X) = \dim_{\mathbb{F}_2}(\mathcal{T}_X/2\mathcal{T}_X) \equiv \sum_{e\ge 1}m_e \pmod 2.\] The Artin–Tate linking form \(\lambda_X\) is nonsingular and skew-symmetric. By Lemma 7, \(m_e\) is even for every \(e\ge 2\). Hence \[\epsilon_2(X) \equiv m_1 = \operatorname{rank}_{\mathbb{F}_2}B_X^{(d)} \pmod 2.\] ◻

Proof. Quotienting by the radical does not change the rank. The image of \(c\) is again characteristic for the induced nondegenerate symmetric form. Thus we may assume that \(B\) is nondegenerate.

If \(B(c,c)=1\), then the line \(\mathbb{F}_2c\) is nondegenerate, and its orthogonal complement has characteristic vector \(0\). Hence the orthogonal complement is alternating and has even dimension. Therefore \(\operatorname{rank}(B)\) is odd.

If \(B(c,c)=0\), then either \(c=0\), in which case \(B\) is alternating and therefore has even rank, or \(c\neq 0\). In the latter case choose \(x\) with \(B(c,x)=1\). The span of \(c\) and \(x\) is a hyperbolic plane, and its orthogonal complement has characteristic vector \(0\). Again the total rank is even. Thus in all cases \[\operatorname{rank}(B)\equiv B(c,c)\pmod 2.\] ◻

Proof. By Proposition [prop:function-field-defect-middle-torsion], \[\Delta_2(X)=\epsilon_2(X).\] By Proposition [prop:function-field-derham-milnor-parity], \[\epsilon_2(X) \equiv \operatorname{rank}_{\mathbb{F}_2}B_X^{(d)} \pmod 2.\] By Proposition [prop:function-field-middle-wu-characteristic], \[\operatorname{rank}_{\mathbb{F}_2}B_X^{(d)} \equiv B_X^{(d)}(v_{2d}(X),v_{2d}(X)) \pmod 2.\] By definition of \(B_X^{(d)}\), this last expression is \[\int_X v_{2d}(X)\,\delta_d(v_{2d}(X)).\] This completes the proof. ◻

Proof. The Leray spectral sequence \[E_2^{a,b}=H^a\bigl(B;R^bf_*\Lambda\bigr) \Longrightarrow H^{a+b}(X;\Lambda)\] yields the low-degree exact sequence \[0\to H^1(B;\Lambda) \xrightarrow{f^*} H^1(X;\Lambda) \xrightarrow{\epsilon} H^0\bigl(B;R^1f_*\Lambda\bigr) \xrightarrow{d_2^{0,1}} H^2(B;\Lambda),\] because \(f_*\Lambda=\Lambda\). By proper and smooth base change, \(R^1f_*\Lambda\) is a finite locally constant sheaf on \(B\), and for every geometric point \(\bar s\to B\) there is a canonical identification \[\bigl(R^1f_*\Lambda\bigr)_{\bar s}\simeq H^1(X_{\bar s};\Lambda).\] Under this identification, the stalk of \(\epsilon(\Delta_1)\) at \(\bar s\) is the image of \(i_s^*(\Delta_1)\) in \(H^1(X_{\bar s};\Lambda)\). By 11 , this stalk is zero for every closed point \(s\), hence \(\epsilon(\Delta_1)=0\). Therefore there exists a class \(\chi\in H^1(B;\Lambda)\) such that \[\Delta_1=f^*\chi.\] Suppose that \(\chi\neq 0\). Since \[H^1(B;\Lambda)\cong \operatorname{Hom}_{\mathrm{cont}}\bigl(\pi_1^{\text{\'{e}t}}(B);\Lambda\bigr),\] Chebotarev’s density theorem yields a closed point \(s\in B\) such that \(\chi(\Frob_s)\neq 0\). On the other hand, for the geometrically connected fibre \(X_s/k(s)\), the exact sequence of étale fundamental groups \[1\to \pi_1^{\text{\'{e}t}}(X_{\bar s})\to \pi_1^{\text{\'{e}t}}(X_s)\to G_{k(s)}\to 1\] implies, by the associated low-degree sequence in cohomology, that the pullback map \[H^1\bigl(k(s);\Lambda\bigr)\longrightarrow H^1(X_s;\Lambda)\] is injective. Hence \[i_s^*(\Delta_1)=f_s^*\bigl(\chi|_{k(s)}\bigr)\neq 0,\] contrary to 11 . Thus \(\chi=0\), and therefore \(\Delta_1=0\). ◻

Proof. Fix \(\alpha\in \Obs_2(X/B)\). By proper and smooth base change, \(R^2f_*\Lambda\) is a finite locally constant sheaf, and the edge map \[H^2(X;\Lambda)\longrightarrow H^0\bigl(B;R^2f_*\Lambda\bigr)\] has the property that its stalk at \(\bar s\) is the image of \(i_s^*(\alpha)\) in \(H^2(X_{\bar s};\Lambda)\). Since \(i_s^*(\alpha)=0\) for every closed \(s\), all these stalks vanish, so the image of \(\alpha\) in \(H^0(B;R^2f_*\Lambda)\) is zero. Thus \(\alpha\in F^1H^2(X;\Lambda)\). By 12 , its image \(\rho(\alpha)\) lies in \(\ker(d_2^{1,1})\).

Now fix a closed point \(s\in B\). Because \(k(s)\) is a finite field, it has cohomological dimension \(1\) for torsion coefficients prime to \(\operatorname{char} k(s)\). Hence the Leray spectral sequence for \(X_s\to \Spec k(s)\) yields a short exact sequence \[\label{eq:HS-fibre-degree-two} 0\to H^1\bigl(k(s);H^1(X_{\bar s};\Lambda)\bigr) \to H^2(X_s;\Lambda) \to H^0\bigl(k(s);H^2(X_{\bar s};\Lambda)\bigr) \to 0.\tag{13}\] Using proper base change again, we identify \[H^1(X_{\bar s};\Lambda)\simeq (R^1f_*\Lambda)_{\bar s}, \qquad H^2(X_{\bar s};\Lambda)\simeq \bigl(R^2f_*\Lambda\bigr)_{\bar s}.\] By functoriality of the Leray spectral sequence, the image of \(\operatorname{loc}_s\bigl(\rho(\alpha)\bigr)\) in \(H^2(X_s; \Lambda)\) under the inclusion of 13 is precisely \(i_s^*(\alpha)\), hence vanishes. As this holds for every closed \(s\), we obtain \[\rho(\alpha)\in \ker\!\bigl(d_2^{1,1}\bigr)\cap \Sha^1_{\mathrm{fib}}(B;R^1f_*\Lambda).\] This defines a homomorphism \[\Obs_2(X/B) \longrightarrow \ker\!\bigl(d_2^{1,1}\bigr)\cap \Sha^1_{\mathrm{fib}}(B;R^1f_*\Lambda).\] Its kernel consists exactly of those classes in \(\Obs_2(X/B)\) lying in \(F^2H^2(X;\Lambda)\), which is all of \(F^2H^2(X;\Lambda)\). Indeed, due to the functoriality of Leray with respect to the filtration, we have \(i_s^*: F^2H^2(X;\Lambda)\to F^2H^2(X_s;\Lambda) = 0\). Therefore, the kernel is isomorphic to \(F^2H^2(X;\Lambda) \cong \operatorname{coker}(d_2^{0,1})\). This implies the exactness of the sequence: \[0\to \operatorname{coker}\!\bigl(d_2^{0,1}\colon H^0(B;R^1f_*\Lambda)\to H^2(B;\Lambda)\bigr) \longrightarrow \Obs_2(X/B) \xrightarrow{\rho} \ker\!\bigl(d_2^{1,1}\bigr)\cap \Sha^1_{\mathrm{fib}}(B;R^1f_*\Lambda).\]

It remains to show that the map \(\Obs_2(X/B) \xrightarrow{\rho} \ker\!\bigl(d_2^{1,1}\bigr)\cap \Sha^1_{\mathrm{fib}}(B;R^1f_*\Lambda)\) is surjective. Let \[\beta\in \ker\!\bigl(d_2^{1,1}\bigr)\cap \Sha^1_{\mathrm{fib}}(B;R^1f_*\Lambda).\] By 12 , choose a lift \(\alpha\in F^1H^2(X;\Lambda)\) with \(\rho(\alpha)=\beta\). For every closed point \(s\), the image of \(i_s^*(\alpha)\) in \(H^0(k(s);H^2(X_{\bar s};\Lambda))\) is zero because \(\alpha\in F^1H^2(X;\Lambda)\), and the image of \(i_s^*(\alpha)\) in \(H^1(k(s);H^1(X_{\bar s};\Lambda))\) is \(\operatorname{loc}_s(\beta)=0\). By the exact sequence 13 , this implies \(i_s^*(\alpha)=0\) for every \(s\). Thus \(\alpha\in \Obs_2(X/B)\), and the displayed map is surjective. This proves the proposition. ◻