Proof. The preservation of cones is obvious. By Lemma 3.2 loc.cit.the functor \(F\) maps complexes concentrated in one degree to \(\mathcal{O}_E\)-complexes. By an immediate induction it maps bounded complexes to objects of \(D^b(K, \mathcal{O}_E)\).

We are left to check that \(F\) maps exact complexes to negligible complexes. By Lemma 1.4 loc.cit.it is enough to check that the image of an exact complex is essentially zero, which easily follows from the Artin-Rees lemma for \(\mathcal{O}_E\)-modules. ◻

Proof. We check the axioms of [12]. We first check axiom TR1. The first part follows immediately from the definition. The fact that \(X \to X \to 0 \to \Sigma X\) is distinguished for any object \(X\) is obvious. Any morphism \(X \to Y\) in \(\mathcal{D}[\ell^{-1}]\) is of the form \(\ell^{-k} f\) for some integer \(k \geq 0\) and some morphism \(f : X \to Y\) in \(\mathcal{D}\). We have a commutative diagram in \(\mathcal{D}[\ell^{-1}]\) \[\begin{tikzcd} X \arrow[r, "{f}"] \arrow[d, "{1}"] & Y \arrow[d, "{\ell^{-k}}"] \\ X \arrow[r, "{\ell^{-k} f}"] & Y \end{tikzcd}\] where the vertical morphism are isomorphisms, and the existence of a distinguished triangle containing \(\ell^{-k} f\) in \(\mathcal{D}[\ell^{-1}]\) follows from the existence of a distinguished triangle containing \(f\) in \(\mathcal{D}\). The last part of TR1 is tautological.

Axiom TR2 is formal.

To check axiom TR3 we can reduce to the case of a diagram \[\begin{tikzcd} X \arrow[r, "{u}"] \arrow[d, "{\ell^{-a} f}"] & Y \arrow[d, "{\ell^{-b} g}"] \arrow[r, "{v}"] & Z \arrow[r, "{w}"] & \Sigma X \arrow[d, "{\ell^{-a} \Sigma f}"] \\ X' \arrow[r, "{u'}"] & Y' \arrow[r, "{v'}"] & Z' \arrow[r, "{w'}"] & \Sigma X' \end{tikzcd}\] where \(u,v,w,u',v',w',f,g\) are morphisms in \(\mathcal{D}\) and both rows are distinguished triangles in \(\mathcal{D}\). Letting \(k=\max(a,b)\) and using the commutative diagram \[\begin{tikzcd} X \arrow[r, "{u}"] \arrow[d, "{\ell^k}"] & Y \arrow[d, "{\ell^k}"] \arrow[r, "{v}"] & Z \arrow[d, "{\ell^k}"] \arrow[r, "{w}"] & \Sigma X \arrow[d, "{\ell^k}"] \\ X \arrow[r, "{u}"] & Y \arrow[r, "{v}"] & Z \arrow[r, "{w}"] & \Sigma X \end{tikzcd}\] where all vertical morphisms are isomorphisms in \(\mathcal{D}[\ell^{-1}]\), we reduce to the case where \(a=b=0\), and apply TR3 for \(\mathcal{D}\).

In order to check the octahedral axiom TR4 we may reduce, thanks to TR3, to the case where the three initial distinguished triangles are obtained (in a strict sense) from distinguished triangles in \(\mathcal{D}\). ◻

Proof. The fact that this functor is exact is obvious. Essential surjectivity follows from the fact that for a finite free \(\mathcal{O}_E\)-module \(\Lambda\) any compact subset of \(E \otimes_{\mathcal{O}_E} \Lambda\) is contained in some \(\mathfrak{m}_E^{-n} \Lambda\).

Let us show that the functor is conservative. By exactness it is enough to show that an object \(A\) of \(D^b(\operatorname{Rep}_{\mathrm{fg}, \mathrm{cont}}(K, \mathcal{O}_E))[\ell^{-1}]\) is zero if its image is zero. This last condition means that the cohomology groups of \(A\) are killed by \(\ell^k\) for some integer \(k \geq 0\). There exist integers \(a \leq b\) such that \(H^i(A) = 0\) for \(i \not\in [a,b]\), and we proceed by induction on \(b-a\). If \(b-a=0\) then \(A\) is quasi-isomorphic to a complex concentrated in one degree, and its endomorphism \(\ell^k\) is both invertible and zero, so \(A\) is a zero object. If \(b-a>0\) we use the distinguished triangle in \(D^b(\operatorname{Rep}_{\mathrm{fg},\mathrm{cont}}(K,\mathcal{O}_E))\) \[H^a(A)[-a] \to A \to B \xrightarrow{+1}\] where \(H^i(B)=0\) for \(i \not\in [a+1,b]\). By induction hypothesis both \(H^a(A)\) and \(B\) are zero, so \(A\) is zero too.

Let \(A,B\) be objects of \(D^b(\operatorname{Rep}_{\mathrm{fg},\mathrm{cont}}(K, \mathcal{O}_E))\). It is clear that every morphism of complexes \(E \otimes_{\mathcal{O}_E} A \to E \otimes_{\mathcal{O}_E} B\) is of the form \(\ell^{-k} f\) for some morphism of complexes \(f: A \to B\). This fact and conservativity imply fullness.

To check faithfulness it is enough to prove that if \(f: A \to B\) is a morphism between bounded complexes of objects in \(\operatorname{Rep}_{\mathrm{fg},\mathrm{cont}}(K, \mathcal{O}_E)\) mapping to the zero morphism in \(D^b(\operatorname{Rep}_{\mathrm{fg},\mathrm{cont}}(K, E))\) then it already maps to the zero morphism in \(D^b(\operatorname{Rep}_{\mathrm{fg},\mathrm{cont}}(K, \mathcal{O}_E))[\ell^{-1}]\). The condition means that there exists a quasi-isomorphism \(g_E: E \otimes_{\mathcal{O}_E} B \to C_E\) of bounded complexes of objects in \(\operatorname{Rep}_{\mathrm{fg},\mathrm{cont}}(K, E)\) such that the composition \(E \otimes_{\mathcal{O}_E} A \to C_E\) is homotopic to zero. There exists a bounded complex \(C\) of objects in \(\operatorname{Rep}_{\mathrm{fg},\mathrm{cont}}(K, \mathcal{O}_E)\), an isomorphism \(E \otimes_{\mathcal{O}_E} C \simeq C_E\) and a morphism of complexes \(g: B \to C\) inducing \(g_E\) such that the composition \(g \circ f: A \to C\) is homotopic to zero. We have proved above that \(g\) becomes an isomorphism in \(D^b(\operatorname{Rep}_{\mathrm{fg},\mathrm{cont}}(K, \mathcal{O}_E))[\ell^{-1}]\), and so \(f\) becomes the zero morphism in this category. ◻

Proof. If \(K'\) is an open subgroup of \(K\) then \(f^* = \operatorname{res}_{K'}\) admits a left adjoint, namely the functor \(\mathrm{find}_{K'}^K\) of induction with finite support in \(K' \backslash K\). More precisely the unit for this adjunction is given by the obvious identification of an \((\mathcal{O}_E/\mathfrak{m}_E^m)_\bullet\)-module \(A_m\) in \(S_{K'}\) with the subobject of \(\operatorname{res}_{K'} \mathrm{find}_{K'}^K A_m\) consisting of all functions supported on \(K'\). The functor \(\mathrm{find}_{K'}^K\) is obviously exact so the lemma follows from this adjunction in this case.

Now consider the case where \(K\) is profinite. Let \(I_\bullet\) be an injective \((\mathcal{O}_E)_\bullet\)-module in \(S_K^{\mathbb{N}}\). Consider an embedding \(A_\bullet \subset B_\bullet\) of \((\mathcal{O}_E)_\bullet\)-modules in \(S_{K'}^{\mathbb{N}}\) and a morphism \(\varphi: A_\bullet \to \operatorname{res}_{K'} I_\bullet\). We wish to extend \(\varphi\) to \(B_\bullet\). Assume that \(B_\bullet\) is not equal to \(A_\bullet\), i.e.there exists \(n \geq 1\) such that we have \(A_n \subsetneq B_n\). Choose \(x \in B_n \smallsetminus A_n\). For \(m \leq n\) let \(C_m\) be the sub-\(\mathcal{O}_E/\mathfrak{m}_E^m[K']\)-module of \(B_m\) generated by the image of \(x\), and for \(m>n\) let \(C_m=0\). This defines a sub-\((\mathcal{O}_E)_\bullet\)-module of \(B_\bullet\), and to extend \(\varphi\) to \(A_\bullet+C_\bullet\) it is enough to extend \(\varphi|_{A_\bullet \cap C_\bullet}\) to \(C_\bullet\). There exists an open and normal subgroup \(U\) of \(K\) such that \(x\) is fixed by \(U \cap K'\). In particular each \(C_m\) is an \(\mathcal{O}_E/\mathfrak{m}_E^m[K'/(U \cap K')]\)-module. Since \(K'/(U \cap K')\) is finite and \(\mathcal{O}_E\) is noetherian each \(A_m \cap C_m\) is a finitely generated \(\mathcal{O}_E/\mathfrak{m}_E^m[K'/(U \cap K')]\)-module. This property (and \(C_m=0\) for \(m>n\)) imply that up to replacing \(U\) by a smaller normal open subgroup of \(K\) we have \(\varphi(A_m \cap C_m) \subset I_m^U\) for all \(m\). Using the isomorphism \(K'/(U \cap K') \simeq K'U/U\) we may see \(C_\bullet\) as a \((S_{K'U}^\mathbb{N}, (\mathcal{O}_E)_\bullet)\)-module, \(A_\bullet \cap C_\bullet\) as a submodule and \(\varphi|_{A_\bullet \cap C_\bullet}\) as a morphism to \(\operatorname{res}_{K'U} I_\bullet\). Now \(K'U\) is an open subgroup of \(K\) so by the previous case we can extend \(\varphi|_{A_\bullet \cap C_\bullet}\) to \(C_\bullet\). So \(\varphi\) can be extended to give a morphism \(A_\bullet + C_\bullet \to \operatorname{res}_{K'} I_\bullet\). By the usual argument using Zorn’s lemma we conclude that any \(\varphi\) as above can be extended to give a morphism \(B_\bullet \to \operatorname{res}_{K'} I_\bullet\). ◻

Proof. The functors \(C^n(N, -)\) are exact so the preservation of quasi-isomorphisms in the first point follows from [13]. The fact that the functor \(R\Gamma_{\mathrm{cont}}(N, -)\) is exact, i.e.maps cones to cones, is a routine computation with double complexes.

The morphism of functors \(R\Gamma(N,-) \to R\Gamma_{\mathrm{cont}}(N, -)\) is defined as follows. For an object \(A^\bullet\) of \(D^+(S_K^{\mathbb{N}}, (\mathcal{O}_E)_\bullet)\) choose a quasi-isomorphism \(A^\bullet \to I^\bullet\) where \(I^n = 0\) for \(n<<0\) and each \(I^n\) is an injective \((S_K^{\mathbb{N}}, (\mathcal{O}_E)_\bullet)\)-module. This induces a quasi-isomorphism \(R\Gamma_{\mathrm{cont}}(N, A^\bullet) \to R\Gamma_{\mathrm{cont}}(N, I^\bullet)\). We have a natural morphism of double complexes \((I^\bullet)^N \to C^\bullet(N, I^\bullet)\) where \((I^\bullet)^N\) abusively denotes the double complex whose \(0\)-th row is \((I^\bullet)^N\) and whose other rows are zero. By definition we have \(R\Gamma(N, A^\bullet) = (I^\bullet)^N\). The composition in \(D^+(S_H^{\mathbb{N}}, (\mathcal{O}_E)_\bullet)\) \[R\Gamma(N, A^\bullet) = (I^\bullet)^N \longrightarrow R\Gamma_{\mathrm{cont}}(N, I^\bullet) \xleftarrow{\sim} R\Gamma_{\mathrm{cont}}(N, A^\bullet)\] is functorial in \(A^\bullet\), as usual because \(D^+(S_K^{\mathbb{N}}, (\mathcal{O}_E)_\bullet)\) is equivalent to the homotopy category of bounded below chain complexes consisting of injective \((S_K^{\mathbb{N}}, (\mathcal{O}_E)_\bullet)\)-modules.

To check that this morphism of functors is an isomorphism it is enough to check that for any injective \((S_K^{\mathbb{N}}, (\mathcal{O}_E)_\bullet)\)-module \(I\) the complex \[I^N \longrightarrow C^\bullet(N, I)\] is a resolution of the \((S_H^{\mathbb{N}}, (\mathcal{O}_E)_\bullet)\)-module \(I^N\). Thanks to Lemma 7 we may assume \(K=N\). For any normal open subgroup \(U\) of \(N\) the \((S_{N/U}^{\mathbb{N}}, (\mathcal{O}_E)_\bullet)\)-module \(I^U\) is injective. For any integer \(n \geq 0\) and any integer \(i \geq 1\) we have an identification \[C^n(N/U, I_i^U) = \operatorname{Hom}_{(S_{N/U}^{\mathbb{N}}, (\mathcal{O}_E)_\bullet)}(R(i,n), I^U)\] where \(R(i,n)\) is the \((S_{N/U}, (\mathcal{O}_E)_\bullet)\)-module defined by \[R(i,n)_j = \begin{cases} \mathcal{O}_E/\mathfrak{m}_E^j[(N/U)^{n+1}] & \text{ if } 1 \leq j \leq i \\ 0 & \text{ if } j>i \end{cases}\] with obvious transition morphisms. Defining an \((S_{N/U}, (\mathcal{O}_E)_\bullet)\)-module \(R(i)\) similarly: \[R(i)_j = \begin{cases} \mathcal{O}_E/\mathfrak{m}_E^j & \text{ if } 1 \leq j \leq i \\ 0 & \text{ if } j>i, \end{cases}\] it is well-known that the complex \[\dots \to R(i, 2) \to R(i, 1) \to R(i, 0) \to R(i) \to 0\] is exact, and we obtain, by almost the same argument as in the case of cohomology for finite groups, that \(C^\bullet(N/U, I^U)\) is a resolution of \((I^U)^{N/U} = I^N\). We conclude by taking the direct limit over all normal open subgroups \(U\) of \(N\). ◻

Proof. Commutativity of the top square is a special case of 3 , so we are left to prove commutativity of the bottom square.

We may restrict to considering bounded-below complexes \(I^\bullet\) consisting of injective \((S_N^{\mathbb{N}}, (\mathcal{O}_E)_\bullet)\)-modules, in which case we have \(R\Gamma(N, I^\bullet) = (I^\bullet)^N\) and also \[R\Gamma(N', \operatorname{res}_{N'}(I^\bullet)) = (\operatorname{res}_{N'}(I^\bullet))^{N'}\] thanks to Lemma 7. The following diagram of double complexes, where as in the previous proof complexes are considered as double complexes concentrated in one row, is clearly commutative. \[\begin{tikzcd} (I^\bullet)^N \arrow[d] \arrow[r] & (\operatorname{res}_{N'} I^\bullet)^{N'} \arrow[d] \\ C^\bullet_{\mathrm{cont}}(N, I^\bullet) \arrow[r] & C^\bullet_{\mathrm{cont}}(N', \operatorname{res}_{N'}(I^\bullet)) \end{tikzcd}\] Taking total complexes yields the result. ◻

Proof. The first point is easy and left to the reader. This other points are dual to the Eilenberg-Zilber theorem [14], following [15]. Our operator \(\Psi\) is dual to the one defined on p.55 of [14]. The closed formula for \(\Psi\) is a slight reformulation of the dual to the opposite of the “Shih operator” [16]. It is not difficult to check that the opposite of Rubio’s \(\mathrm{SHI}\) satisfies the formula defining Eilenberg-MacLane’s \(\Psi\) by induction. Details are left to the reader. ◻

Proof. Two choices of isomorphisms \(N \simeq \mathbb{Z}_{\ell}^r\) differ by an element of \(\mathrm{GL}_r(\mathbb{Z}_{\ell})\). The lemma follows from Vandermonde’s identity and the expansion \[\label{eq:change95var95pol95Hilb} \binom{\lambda x}{k} = \lambda^k \binom{x}{k}+ \sum_{0 \leq i \leq k-1} a_i(\lambda, k) \binom{x}{i}\tag{15}\] with \(a_i(\lambda, k) \in \mathbb{Z}_{\ell}\), valid for any \(\lambda \in \mathbb{Z}_{\ell}\) and \(k \in \mathbb{Z}_{\geq 0}\). Details are left to the reader. ◻

Proof. To avoid confusion we will denote the action of \(\mathbb{Z}_{\ell}\) on \(A\) as \[\begin{align} \mathbb{Z}_{\ell}\times A & \longrightarrow A \\ (x, a) & \longmapsto \gamma^x \cdot a. \end{align}\] The fact that the morphism in the lemma is well-defined is clear, and injectivity immediately follows from the equivariance condition which reads \[\varphi(x_0, \dots, x_n) = \gamma^{x_0} \cdot \varphi(0, x_1-x_0, \dots, x_n-x_0).\] We are left to check surjectivity. Let \(\psi \in \operatorname{Fil}^d_w \operatorname{Pol}(\mathbb{Z}_{\ell}^n, A)\). The polynomial function \[(x_0, \dots, x_n) \mapsto \psi(x_1-x_0, \dots, x_n-x_0)\] clearly belongs to \(\operatorname{Fil}^d_w \operatorname{Pol}(\mathbb{Z}_{\ell}^{n+1}, A)\), and so there is an expansion \[\psi(x_1-x_0, \dots, x_n-x_0) = \sum_{i \in I} P_i(x_0, \dots, x_n) a_i\] where \(I\) is a finite set, \(P_i \in \operatorname{Pol}(\mathbb{Z}_{\ell}^{n+1}, \mathcal{O}_E)\) has total degree \(\leq d_i\) and \(a_i \in \operatorname{Fil}^{d-wd_i} A\). Fix \(j_0 \in \mathbb{Z}\) for which we have \(\operatorname{Fil}^{j_0} A = 0\). For \(x_0 \in \mathbb{Z}_{\geq 0}\) we have \[\begin{align} \gamma^{x_0} \cdot \psi(x_1-x_0, \dots, x_n-x_0) &= \sum_{i \in I} P_i(x_0, \dots, x_n) \gamma^{x_0} \cdot a_i \\ &= \sum_{i \in I} \sum_{0 \leq k < (d-j_0)/w-d_i} P_i(x_0, \dots, x_n) \binom{x_0}{k} (\gamma-1)^k a_i \end{align}\] where the last equality follows from the usual binomial identity for \[\gamma^{x_0} = ((\gamma-1) + 1)^{x_0} \in \mathrm{End}(A)\] and the fact that \((\gamma-1)^k\) maps \(\operatorname{Fil}^{d-wd_i} A\) to \(\operatorname{Fil}^{d-wd_i-wk} A\). By continuity of the action this equality holds for arbitrary \(x_0 \in \mathbb{Z}_{\ell}\) and it is clear on this expression that \[(x_0, \dots, x_n) \longmapsto \gamma^{x_0} \cdot \psi(x_1-x_0, \dots, x_n-x_0)\] belongs to \(\operatorname{Fil}^d_w \operatorname{Pol}(\mathbb{Z}_{\ell}^{n+1}, A)\). ◻

Proof. To avoid confusion we will denote the action of \(\mathbb{Z}_{\ell}\) on \(A\) as \[\begin{align} \mathbb{Z}_{\ell}\times A & \longrightarrow A \\ (x, a) & \longmapsto \gamma^x \cdot a. \end{align}\] Let \(\sigma_w \operatorname{Pol}(\mathbb{Z}_{\ell}, A)\) be \(\operatorname{Pol}(\mathbb{Z}_{\ell}, A)\) but with shifted filtration: \[\operatorname{Fil}^j \sigma_w \operatorname{Pol}(\mathbb{Z}_{\ell}, A) := \operatorname{Fil}^{j-w}_w \operatorname{Pol}(\mathbb{Z}_{\ell}, A).\] We use the resolution (short exact sequence) \[0 \to A \to \operatorname{Pol}(\mathbb{Z}_{\ell}, A) \to \sigma_w \operatorname{Pol}(\mathbb{Z}_{\ell}, A) \to 0\] where the first map is \(a \mapsto (x \mapsto a)\) and the second map is \(\Delta: f \mapsto (x \mapsto f(x+1) - f(x))\). Note that both morphisms are strict, in fact we have a section \(s\) of \(\Delta\): \[\left( x \mapsto \binom{x}{i} a \right) \mapsto \left( x \mapsto \binom{x}{i+1} a \right)\] which identifies \(\operatorname{Pol}(\mathbb{Z}_{\ell}, A)\) with \(A \oplus \sigma_w \operatorname{Pol}(\mathbb{Z}_{\ell}, A)\) as a filtered \(\mathcal{O}_E\)-module, but \(s\) is not \(\mathbb{Z}_{\ell}\)-equivariant. We get for each integer \(n \geq 0\) a short exact sequence \[0 \to \operatorname{Pol}(\mathbb{Z}_{\ell}^{n+1}, A) \to \operatorname{Pol}(\mathbb{Z}_{\ell}^{n+1}, \operatorname{Pol}(\mathbb{Z}_{\ell}, A)) \to \operatorname{Pol}(\mathbb{Z}_{\ell}^{n+1}, \sigma_w \operatorname{Pol}(\mathbb{Z}_{\ell}, A)) \to 0\] in which both morphisms are strict. By the previous lemma taking \(\operatorname{Fil}^d\) and \(\mathbb{Z}_{\ell}\)-invariants yields a short exact sequence \[\label{eq:short95ex95seq95alg95cocycles} 0 \to C^n_{\operatorname{pol}, \leq d}(\mathbb{Z}_{\ell}, A) \to C^n_{\operatorname{pol}, \leq d}(\mathbb{Z}_{\ell}, \operatorname{Pol}(\mathbb{Z}_{\ell}, A)) \to C^n_{\operatorname{pol}, \leq d}(\mathbb{Z}_{\ell}, \sigma_w \operatorname{Pol}(\mathbb{Z}_{\ell}, A)) \to 0\tag{16}\]

Next we claim that the complex \[C^\bullet_{\operatorname{pol}, \leq d}(\mathbb{Z}_{\ell}, \operatorname{Pol}(\mathbb{Z}_{\ell}, A))\] is exact in positive degree. Using the identification of filtered \(\mathcal{O}_E[N]\)-modules \[\begin{align} \operatorname{Pol}(\mathbb{Z}_{\ell}^{n+1}, \operatorname{Pol}(\mathbb{Z}_{\ell}, A)) & \longrightarrow \operatorname{Pol}(\mathbb{Z}_{\ell}^{n+2}, A) \\ \varphi & \longmapsto \left( (x_0, \dots, x_{n+1}) \mapsto \varphi(x_1, \dots, x_{n+1})(x_0) \right) \end{align}\] we obtain \[\operatorname{Fil}^d_w C^n_{\operatorname{pol}}(\mathbb{Z}_{\ell}, \operatorname{Pol}(\mathbb{Z}_{\ell}, A)) \simeq \operatorname{Fil}^d_w C^{n+1}_{\operatorname{pol}}(\mathbb{Z}_{\ell}, A)\] and applying the previous lemma to the right hand side gives us an isomorphism \[\begin{align} \operatorname{Fil}^d_w C^n_{\operatorname{pol}}(\mathbb{Z}_{\ell}, \operatorname{Pol}(\mathbb{Z}_{\ell}, A)) & \longrightarrow \operatorname{Fil}^d_w \operatorname{Pol}(\mathbb{Z}_{\ell}^{n+1}, A) \\ \varphi & \longmapsto \left( (x_0, \dots, x_n) \mapsto \varphi(x_0, \dots, x_n)(0) \right). \end{align}\] Via this identification the exactness of \(C^\bullet_{\operatorname{pol}, \leq d}(\mathbb{Z}_{\ell}, \operatorname{Pol}(\mathbb{Z}_{\ell}, A))\) in positive degree is obtained as usual using the homotopy corresponding to \[\begin{align} \operatorname{Fil}^d \operatorname{Pol}(\mathbb{Z}_{\ell}^{n+2}, A) & \longrightarrow \operatorname{Fil}^d \operatorname{Pol}(\mathbb{Z}_{\ell}^{n+1}, A) \\ \psi & \longrightarrow \left( (x_0, \dots, x_n) \mapsto \psi(0, x_0, \dots, x_n) \right). \end{align}\] In degree \(0\) we find that the kernel of \(\operatorname{Fil}^d_w C^0_{\operatorname{pol}}(\mathbb{Z}_{\ell}, \operatorname{Pol}(\mathbb{Z}_{\ell}, A)) \to \operatorname{Fil}^d_w C^1_{\operatorname{pol}}(\mathbb{Z}_{\ell}, \operatorname{Pol}(\mathbb{Z}_{\ell}, A))\) is naturally isomorphic to \(\operatorname{Fil}^d A\).

The same arguments are valid with \(\operatorname{Pol}(\mathbb{Z}_{\ell}, A)\) replaced by \(\sigma_w \operatorname{Pol}(\mathbb{Z}_{\ell}, A)\) (e.g.by shifting the filtration on \(A\)). Using the short exact sequences 16 and a familiar double complex argument we obtain that the complexes \(\operatorname{Fil}^d_w C^\bullet_{\operatorname{pol}}(\mathbb{Z}_{\ell}, A)\) and \[\label{eq:FildA95gm195FildmwA} \operatorname{Fil}^d A \xlongrightarrow{\gamma-1} \operatorname{Fil}^{d-w} A.\tag{17}\] are both quasi-isomorphic to the same complex. By making this explicit or by direct computation now that we know that \(\operatorname{Fil}^d_w C^\bullet_{\operatorname{pol}}(\mathbb{Z}_{\ell}, A)\) is exact in degree \(>1\) we find that the complex \(\tau_{\leq 1} \operatorname{Fil}^d_w C^\bullet_{\operatorname{pol}}(\mathbb{Z}_{\ell}, A)\) is isomorphic to 17 via the isomorphisms \[\begin{align} \operatorname{Fil}^d_w C^0_{\operatorname{pol}}(\mathbb{Z}_{\ell}, A) & \longrightarrow \operatorname{Fil}^d A \\ \varphi & \longmapsto \varphi(0) \end{align}\] and \[\begin{align} \operatorname{Fil}^d_w Z^1_{\operatorname{pol}}(\mathbb{Z}_{\ell}, A) & \longrightarrow \operatorname{Fil}^{d-w} A \\ \varphi & \longmapsto \varphi(0, 1). \end{align}\] The last assertion of the lemma follows using the quasi-isomorphism 13 recalled above. ◻

Proof.

1. By Lemma 14 the embedding of double complexes \[\label{eq:dbl95cplxes95FCpolZell} \left[ \operatorname{Fil}^d A^\bullet \xrightarrow{\gamma-1} \operatorname{Fil}^{d-w} A^\bullet \right] \to \operatorname{Fil}^d_w C_{\operatorname{pol}}^\bullet(\mathbb{Z}_{\ell}, A^\bullet)\tag{18}\] where the left hand side has two rows in degrees \(0\) and \(1\), consists of quasi-isomorphisms on columns. Because we assume \(j \geq d \geq d_0+w\) the embedding of double complexes \[\left[ \operatorname{Fil}^d A^\bullet \xrightarrow{\gamma-1} \operatorname{Fil}^{d-w} A^\bullet \right] \to \left[ \operatorname{Fil}^j A^\bullet \xrightarrow{\gamma-1} \operatorname{Fil}^j A^\bullet \right]\] consists of quasi-isomorphisms on all rows. We also know that the embedding of double complexes \[\left[ \operatorname{Fil}^j A^\bullet \xrightarrow{\gamma-1} \operatorname{Fil}^j A^\bullet \right] \to C^\bullet(\mathbb{Z}_{\ell}, \operatorname{Fil}^j A^\bullet)\] consists of quasi-isomorphisms on all columns (this is where we use the assumption that each \(\operatorname{Fil}^j A^n\) is torsion). These three embeddings and the embedding \[\operatorname{Fil}^d_w C^\bullet_{\operatorname{pol}}(\mathbb{Z}_{\ell}, A^\bullet) \to C^\bullet(\mathbb{Z}_{\ell}, \operatorname{Fil}^j A^\bullet)\] sit in an obvious commutative diagram, and the result follows.

2. This also follows from the fact that the embedding of double complexes 18 consists of quasi-isomorphisms on columns.

 ◻

Proof. The fact that [it:central95series95Zell95ri] implies [it:tors95free95nilp] is easy and left to the reader. For the other implication we claim that we can take \((N_i)_i\) to be the upper central series of \(N\). Let \((N_j')_j\) be the lower central series of \(N\). Applying [18] to finite quotients of \(N\) we see that each \(N'_j\) is also topologically finitely generated, and so we have isomorphisms \(N'_j/N'_{j+1} \simeq \mathbb{Z}_{\ell}^{r_j'} \times \mathbb{Z}/\ell^{s'_j}\mathbb{Z}\) for all \(j\). Note that any closed subgroup of \(\mathbb{Z}_{\ell}^{r_j'} \times \mathbb{Z}/\ell^{s'_j}\mathbb{Z}\) is also topologically finitely generated. Intersecting \((N'_j)_j\) with \(N_i\) then shows that each \(N_i\) is also topologically finitely generated, and so we have isomorphisms \(N_i/N_{i+1} \simeq \mathbb{Z}_{\ell}^{r_i} \times \mathbb{Z}/\ell^{s_i}\mathbb{Z}\). We are left to show that all \(s_i\) vanish, i.e.that \(N_i/N_{i+1}\) is \(\ell\)-torsion-free. This can be proved by descending induction on \(i \in \{0, \dots, m\}\), using the well-known fact that for any \(i \in \{0, \dots, m-1\}\), \(x \in N_i\) and \(y \in N\) we have \([x^\ell,y] = [x,y]^\ell\) in \(N_{i+1}/N_{i+2}\). ◻

Proof. Recall from [20] that unipotent algebraic groups over a field of characteristic zero are equivalent to nilpotent finite-dimensional Lie algebras via the exponential map and the Campbell-Hausdorff formula, which are polynomial in this case. Let \(\mathbf{B}\) be the open affine subscheme of \(\mathbf{N}_0^{r_0} \times_{\mathbb{Q}_{\ell}} \dots \times_{\mathbb{Q}_{\ell}} \mathbf{N}_m^{r_m}\) parametrizing families \(\underline{g} = (g_{i,j})_{0 \leq i \leq m, 1 \leq j \leq r_i}\) such that for any \(0 \leq i \leq m\) the image of \((g_{i,j})_j\) in the vector group \(\mathbf{N}_i/\mathbf{N}_{i+1}\) is a basis. Then the morphism of schemes over \(\mathbb{Q}_{\ell}\) \[\begin{align} \Theta: \mathbf{B}\times_{\mathbb{Q}_{\ell}} \mathbb{A}_{\mathbb{Q}_{\ell}}^{\sum_i r_i} & \longrightarrow \mathbf{B}\times_{\mathbb{Q}_{\ell}} \mathbf{N}\\ (\underline{g}, \underline{x}) & \longmapsto (\underline{g}, \psi_{\underline{g}}(\underline{x})), \end{align}\] where \(\psi_{\underline{g}}\) is defined using the same formula 19 , is easily seen to be an isomorphism using the exponential map. Choose an arbitrary isomorphism \(\mathbf{N}\simeq \mathbb{A}_{\mathbb{Q}_{\ell}}^{\sum_i r_i}\) of schemes over \(\mathbb{Q}_{\ell}\) (e.g.one given by a point in \(\mathbf{B}(\mathbb{Q}_{\ell})\)), identifying \(\Theta\) with an automorphism of \(\mathbb{A}_{\mathbf{B}}^{\sum_i r_i}\) over \(\mathbf{B}\). Then the degrees of the specialization of \(\Theta\) and \(\Theta^{-1}\) at any point of \(\mathbf{B}\) are uniformly bounded, implying the first bound in Assumption 17.

The second bound is obtained using a similar argument for the composition of \[\begin{align} \mathbf{B}\times_{\mathbb{Q}_{\ell}} \mathbb{A}_{\mathbb{Q}_{\ell}}^{2\sum_i r_i} & \longrightarrow \mathbf{B}\times_{\mathbb{Q}_{\ell}} \mathbf{N}\\ (\underline{g}, \underline{y}, \underline{x}) & \longmapsto (\underline{g}, \psi_{\underline{g}}(\underline{y}) \psi_{\underline{g}}(\underline{x})) \end{align}\] with \(\Theta^{-1}\). ◻

Proof.

1. For \(h \in H\) we know that \(\psi_{\underline{g}}^{-1}(h^{-1} \psi_{\underline{g}}(\underline{x}_m, \dots, \underline{x}_0) h)\) is given as in 20 by a family \((\alpha_i)_i\) with \(\alpha_i \in \mathrm{GL}_{r_i}(\mathbb{Z}_{\ell})\) and a family of polynomials \((\beta_i)_i\) of degree at most \(d\) where \(d \in \mathbb{Z}_{\geq 0}\) is as in Definition 20. We have \(w_i \geq dw_j\) whenever \(i > j\) and the fact that \(H\) preserves each \(\operatorname{Fil}^j_{\underline{w}} \operatorname{Pol}(N^n, A)\) easily follows.

2. Under the additional assumption we also have \(\alpha_i = 1\) for all \(i\) in 20 . Using the inequalities \(w_i \geq dw_j + w_0\) whenever \(i>j\), as well as the inequality \(w_A \geq w_0\), we conclude that \(H\) acts trivially on each \(\operatorname{Fil}^j_{\underline{w}} \operatorname{Pol}(N^n, A) / \operatorname{Fil}^{j-w_0}_{\underline{w}} \operatorname{Pol}(N^n, A)\).

 ◻

Proof. We proceed by induction on \(\sum_i r_i \in \mathbb{Z}_{\geq 0}\), the initial case being trivial. So assume that we have \(\sum_i r_i > 0\). We may assume \(r_0>0\) and realize \(N\) as \(N' \rtimes \mathbb{Z}_{\ell}\). By induction hypothesis we have isomorphisms \[\begin{align} \operatorname{Fil}^d_{\underline{w}'} C^n_{\operatorname{pol}}(N', A) & \longrightarrow \operatorname{Fil}^d_{\underline{w}'} \operatorname{Pol}((N')^n, A) \\ \varphi & \longmapsto ((x_1, \dots, x_n) \mapsto \varphi(1,x_1,\dots,x_n)). \end{align}\] These isomorphisms are \(\mathbb{Z}_{\ell}\)-equivariant (for the usual action given by formula 5 on both sides), and we see them as an isomorphism of filtered \(\mathcal{O}_E[\mathbb{Z}_{\ell}]\)-modules \[C^n_{\operatorname{pol}}(N', A) \longrightarrow \operatorname{Pol}((N')^n, A).\] By the second part of 21 we know that the action of \(\mathbb{Z}_{\ell}\) on each \(\operatorname{Fil}^d_{\underline{w}'} / \operatorname{Fil}^{d-w_0}_{\underline{w}'}\) (on either side) is trivial. We also have identifications \[\begin{align} \beta_n: \operatorname{Pol}(N^n, A) & \longrightarrow \operatorname{Pol}(\mathbb{Z}_{\ell}^n, \operatorname{Pol}((N')^n, A)) \\ \varphi & \longmapsto ((y_1, \dots, y_n) \mapsto ((x_1, \dots, x_n) \mapsto \varphi(x_1 y_1, \dots, x_n y_n))) \end{align}\] which clearly identify the filtration \(\operatorname{Fil}^\bullet_{\underline{w}}\) on the left-hand side with \(\operatorname{Fil}^\bullet_{w_0}\) on the right-hand side (here \(\operatorname{Pol}((N')^n, A)\) is endowed with the filtration \(\operatorname{Fil}^\bullet_{\underline{w}'}\)). These maps are also \(N\)-equivariant (see Section 5) and so we also have isomorphisms of filtered \(\mathcal{O}_E\)-modules \[\begin{align} \beta_{n+1}^N: C^n_{\operatorname{pol}}(N, A) & \longrightarrow C^n_{\operatorname{pol}}(\mathbb{Z}_{\ell}, C^n_{\operatorname{pol}}(N', A)) \\ \varphi & \longmapsto ((y_0, \dots, y_n) \mapsto ((x_0, \dots, x_n) \mapsto \varphi(x_0 y_0, \dots, x_n y_n))). \end{align}\] We conclude the induction step using the commutative diagram of filtered \(\mathcal{O}_E\)-modules \[\begin{tikzcd} C^n_{\operatorname{pol}}(N, A) \arrow[r, "{\beta_{n+1}^N}"] \arrow[dd, "{\alpha_N}"] & C^n_{\operatorname{pol}}(\mathbb{Z}_{\ell}, C^n_{\operatorname{pol}}(N', A)) \arrow[d, "{C^n_{\operatorname{pol}}(\mathbb{Z}_{\ell}, \alpha_{N'})}"] \\ & \arrow[d, "{\alpha_{\mathbb{Z}_{\ell}}}"] C^n_{\operatorname{pol}}(\mathbb{Z}_{\ell}, \operatorname{Pol}((N')^n, A)) \\ \operatorname{Pol}(N^n, A) \arrow[r, "{\beta_n}"] & \operatorname{Pol}(\mathbb{Z}_{\ell}^n, \operatorname{Pol}((N')^n, A)) \end{tikzcd}\] The maps \(\beta_n\) and \(\beta_{n+1}^N\) are isomorphisms of filtered \(\mathcal{O}_E\)-modules, as are the maps \(C^n_{\operatorname{pol}}(\mathbb{Z}_{\ell}, \alpha_{N'})\) (by induction hypothesis) and \(\alpha_{\mathbb{Z}_{\ell}}\) (thanks to Lemma 13), so \(\alpha_N\) is also an isomorphism of filtered \(\mathcal{O}_E\)-modules. ◻

Proof. We will prove both points by induction on \(\sum_i r_i\). We start with the first point, in particular we assume that each \(\operatorname{Fil}^j A^n\) is torsion. We may reduce to the case where \(r_0>0\). (Note: this is the only place in the induction where we use \(w_A \geq w_m\) rather than the weaker inequality \(w_A \geq w_0\).) Let \(N'\) be the preimage of \(\mathbb{Z}_{\ell}^{r_0-1} \times \{0\}\) in \(N\) and identify \(N\) with \(N' \rtimes \mathbb{Z}_{\ell}\). By induction hypothesis for any \(d \geq d_A w_A + \sum_i r_i w_i - w_0\) and any \(j \geq \lfloor d/w_A \rfloor\) the embedding of complexes \[\operatorname{Tot}^\bullet \left( \operatorname{Fil}^d_{\underline{w}'} C^\bullet_{\operatorname{pol}}(N', A^\bullet) \right) \rightarrow \operatorname{Tot}^\bullet \left( C^\bullet(N', \operatorname{Fil}^j A^\bullet) \right)\] is a quasi-isomorphism. Because for complexes of smooth \(\mathcal{O}_E[\mathbb{Z}_{\ell}]\)-modules the functor \(\operatorname{Tot}^\bullet(C^\bullet(\mathbb{Z}_{\ell}, -))\) respect quasi-isomorphisms⁵, and by associativity of “taking total complexes”, it follows that for such integers \(d\) and \(j\) the embedding \[\operatorname{Tot}^\bullet \left( C^\bullet(\mathbb{Z}_{\ell}, \operatorname{Fil}^d_{\underline{w}'} C^\bullet_{\operatorname{pol}}(N', A^\bullet)) \right) \rightarrow \operatorname{Tot}^\bullet \left( C^\bullet(\mathbb{Z}_{\ell}, C^\bullet(N', \operatorname{Fil}^j A^\bullet)) \right)\] is also a quasi-isomorphism. Note that the induction hypothesis also implies that the embedding \[\operatorname{Tot}^\bullet \left( \operatorname{Fil}^d_{\underline{w}'} C^\bullet_{\operatorname{pol}}(N', A^\bullet) \right) \longrightarrow \operatorname{Tot}^\bullet \left( \operatorname{Fil}^{d+1}_{\underline{w}'} C^\bullet_{\operatorname{pol}}(N', A^\bullet) \right)\] is a quasi-isomorphism for \(d \geq d_A w_A + \sum_i r_i w_i - w_0\). Thanks to Lemma 21 we know that the action 5 of \(\mathbb{Z}_{\ell}\) on \(C^\bullet_{\operatorname{pol}}(N', A)\) preserves the filtration \(\operatorname{Fil}^j_{\underline{w}'}\) and that it acts trivially on each \(\operatorname{Fil}^j_{\underline{w}'} / \operatorname{Fil}^{j-w_0}_{\underline{w}'}\). By the first part of Lemma 15 the embedding of complexes \[\operatorname{Tot}^\bullet \left( \operatorname{Fil}^d_{w_0} C^\bullet_{\operatorname{pol}}(\mathbb{Z}_{\ell}, C^\bullet_{\operatorname{pol}}(N', A^\bullet)) \right) \rightarrow \operatorname{Tot}^\bullet \left( C^\bullet(\mathbb{Z}_{\ell}, \operatorname{Fil}^j_{\underline{w}'} C^\bullet_{\operatorname{pol}}(N', A^\bullet)) \right)\] is a quasi-isomorphism whenever \(d \geq d_A w_A + \sum_i r_i w_i\) and \(j \geq d\).

By Theorem 10 the maps \(\operatorname{AW}\) and \(\operatorname{EML}\) induce quasi-isomorphisms \[\operatorname{Tot}^\bullet( C^\bullet(\mathbb{Z}_{\ell}, C^\bullet(N', A^n))) \leftrightarrow C^\bullet(N, A^n).\] We claim that for every \(n \in \mathbb{Z}\) the maps \(\operatorname{AW}\) and \(\operatorname{EML}\) also induce well defined maps \[\label{eq:EZ95Fil} \operatorname{Tot}^\bullet( \operatorname{Fil}^d_{w_0} C^\bullet_{\operatorname{pol}}(\mathbb{Z}_{\ell}, C^\bullet_{\operatorname{pol}}(N', A^n))) \leftrightarrow \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, A^n)\tag{21}\] which are inverse of each other modulo homotopies, in particular they are quasi-isomorphisms. Here \(C^\bullet_{\operatorname{pol}}(N', A^n)\) is endowed with the filtration \(\operatorname{Fil}^\bullet_{\underline{w}'}\). To prove this claim we simply observe that it is clear on the defining formulas 7 , 10 , 11 and 12 that the maps \(\theta^*\), \(\operatorname{AW}\), \(\operatorname{EML}\) and \(\Psi\) preserve the filtrations, so that the argument in Theorem 10 is still valid for these subcomplexes.

We will conclude using the following commutative diagram where \(d\) is any integer and \(j \geq \lfloor d/w_A \rfloor\). \[\begin{tikzcd} \operatorname{Tot}^\bullet(\operatorname{Fil}^d_{w_0} C^\bullet_{\operatorname{pol}}(\mathbb{Z}_{\ell}, C^\bullet_{\operatorname{pol}}(N', A^\bullet))) \arrow[r] \arrow[d, "{\operatorname{AW}}"] & \operatorname{Tot}^\bullet(C^\bullet(\mathbb{Z}_{\ell}, \operatorname{Fil}^d_{\underline{w}'} C^\bullet_{\operatorname{pol}}(N', A^\bullet))) \arrow[d, "{\mathrm{IH}}"] \\ \operatorname{Tot}^\bullet(\operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, A^\bullet)) \arrow[d, "{\iota}"] & \operatorname{Tot}^\bullet(C^\bullet(\mathbb{Z}_{\ell}, \operatorname{Tot}^\bullet(C^\bullet(N', \operatorname{Fil}^j A^\bullet)))) \arrow[d, equal] \\ \operatorname{Tot}^\bullet(C^\bullet(N, \operatorname{Fil}^j A^\bullet)) & \arrow[l, "{\operatorname{AW}}"] \operatorname{Tot}^\bullet(C^\bullet(\mathbb{Z}_{\ell}, C^\bullet(N', A^\bullet))) \end{tikzcd}\] For \(d \geq d_A w_A + \sum_i r_i w_i\) we have seen above that the top horizontal map is a quasi-isomorphism thanks to Lemma 15, that the map labelled \(\mathrm{IH}\) is a quasi-isomorphism thanks to the induction hypothesis, and that both maps labelled \(\operatorname{AW}\) are quasi-isomorphisms. It follows that the embedding labelled \(\iota\) is also a quasi-isomorphism.

The second point is proved by induction in a similar but simpler fashion using 21 and the second part of Lemma 15. ◻

Proof. For \(n \in \mathbb{Z}\) and an integer \(i \geq 1\) we endow \(A^n / \mathfrak{m}_E^i A^n\) with the image of the given filtration of \(A^n\). By the first part of Theorem 23 we have for every \(i \geq 1\) a quasi-isomorphism \[\operatorname{Tot}^\bullet \left( \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, A^\bullet/\mathfrak{m}_E^i) \right) \to C^\bullet(N, A^\bullet/\mathfrak{m}_E^i)\] and as \(i\) varies these morphisms are clearly compatible. In particular we have an isomorphism in \(D^+(H, \mathcal{O}_E)\) \[\operatorname{Tot}^\bullet \left( \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, F(A^\bullet)) \right) \to R\Gamma_{\mathrm{cont}}(N, F(A^\bullet)).\] Thanks to the second part of Theorem 23 we know that the complex \(\operatorname{Tot}^\bullet \left( \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, A^\bullet) \right)\) has non-vanishing cohomology in only finitely many degrees, in particular we may apply to it the functor \(F\) of Proposition 2. More precisely if we have \(A^j = 0\) for all \(j>r_A\) then we have \[H^j \left( \operatorname{Tot}^\bullet \left( \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, A^\bullet) \right) \right) =0\] for all \(j > r_A + \sum_i r_i\), and letting \(r = 1 + r_A + \sum_i r_i\) we have a canonical isomorphism \[F \left( \operatorname{Tot}^\bullet \left( \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, A^\bullet) \right) \right) \simeq F \left( \tau_{\leq r} \operatorname{Tot}^\bullet \left( \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, A^\bullet) \right) \right).\] (The integer \(r\) is one more than necessary here but this will be useful below.) We have a morphism in \(D^+(H, \mathcal{O}_E)\) \[\label{eq:F95FilCpol95to95FilCpol95F} F\left( \operatorname{Tot}^\bullet \left( \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, A^\bullet) \right) \right) \to \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, F(A^\bullet))\tag{22}\] obtained as the composition of the natural morphism of complexes \[\label{eq:F95FilCpol95to95FilCpol95F95tau} F\left( \tau_{\leq r} \operatorname{Tot}^\bullet \left( \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, A^\bullet) \right) \right) \to \tau_{\leq r} \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, F(A^\bullet))\tag{23}\] with the quasi-isomorphism \[\tau_{\leq r} \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, F(A^\bullet)) \to \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, F(A^\bullet))\] (again using the second part of Theorem 23, this time for \(A^\bullet \otimes_{\mathcal{O}_E} \mathcal{O}_E/\mathfrak{m}_E^i\)). Let us show that 22 is an isomorphism in \(D^+(H, \mathcal{O}_E)\). Thanks to Lemma 22 we may identify the \(\mathcal{O}_E\)-module \(\operatorname{Fil}^d_{\underline{w}} C^n_{\operatorname{pol}}(N,A^m)\) with a direct sum of finitely many \(\operatorname{Fil}^j A^m\). The natural maps of \((\mathcal{O}_E)_\bullet\)-modules \[\left( \operatorname{Fil}^j A^m \otimes_{\mathcal{O}_E} \mathcal{O}_E/\mathfrak{m}_E^i \right)_{i \geq 1} \longrightarrow \left( \left( \operatorname{Fil}^j A^m + \mathfrak{m}_E^i A^m \right) / \mathfrak{m}_E^i A^m \right)_{i \geq 1}\] are clearly surjective, and by the Artin-Rees lemma their kernels are essentially zero. This implies that the morphism of complexes 23 is surjective (in degree \(r\) this uses the fact that the right-hand side has vanishing cohomology, and is the reason for the increment in the definition of \(r\)) and that its kernel is essentially zero. In the derived category this kernel is isomorphic to the cone of 23 up to a shift, and so this cone is essentially zero and thus negligible by [10], i.e.@eq:eq:F95FilCpol95to95FilCpol95F95tau is an isomorphism in \(D^+(H, \mathcal{O}_E)\) and thus so is 22 . ◻

Proof. By the usual double complexes arguments it is enough to consider the case of a single filtered \(\mathcal{O}_E[N]\)-module \((A, \operatorname{Fil}^\bullet A)\). As in the proof of Theorem 23 we endow each \(A/\mathfrak{m}_E^iA\) with the filtration \(((\operatorname{Fil}^j A + \mathfrak{m}_E^i A)/\mathfrak{m}_E^i A)_{j \in \mathbb{Z}}\). From the first part of Theorem 23 we have quasi-isomorphisms \[\operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, A/\mathfrak{m}_E^i A) \longrightarrow C^\bullet(N, A / \mathfrak{m}_E^i A)\] which are compatible as \(i \geq 1\) varies, and we consider them as an embedding of complexes of projective systems of abelian groups. All terms satisfy the Mittag-Leffler condition, in fact all transition morphisms are surjective (thanks to Lemma 22 for the left-hand side), and so applying \(R \varprojlim_i\) we obtain a quasi-isomorphism \[\varprojlim_i \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, A/\mathfrak{m}_E^i A) \longrightarrow C^\bullet_{\mathrm{cont}}(N, A).\] Because each \(\operatorname{Fil}^j A\) is closed in \(A\) the natural map \[\operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, A) \longrightarrow \varprojlim_i \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, A/\mathfrak{m}_E^i A)\] is an isomorphism. ◻

Proof. We proceed by induction on \(m \geq 0\), the case \(m \leq 0\) being trivial. So assume \(m>0\) and consider \(N\) as a central extension of \(N' := N/N_m\) by \(N_m\). We have a continuous section \(s: N/N_m \to N\), for example the section given by Mal’cev coordinates 19 , and so our central extension corresponds to a class in \(H^2_{\mathrm{cont}}(N/N_m, N_m)\). Let \(N'_i = N_i/N_m\) for \(0 \leq i \leq m\), applying the induction hypothesis to \((N', (N'_i)_i)\) we obtain a filtered unipotent group \((\mathbf{N}', (\mathbf{N}'_i)_{0 \leq i \leq m})\). By Lemma 18 there exists \(\underline{w}' = (w_A, w_{m-1}, \dots, w_0)\) such that \((N', (N'_i)_i, \underline{w}')\) is a weighted filtered nilpotent group (Definition 20). By Corollary 25 any class in \(H^2_{\mathrm{cont}}(N/N_m, N_m)\) is represented by a polynomial \(2\)-cocycle, i.e.up to choosing another continuous section \(s\) we may assume that the map \[\begin{align} (N/N_m)^2 & \longmapsto N_m \\ (x,y) & \longmapsto s(x) s(y) s(xy){-1} \end{align}\] is given by a polynomial, and choosing an isomorphism \(N_m \simeq \mathbb{Z}_{\ell}^{r_m}\) we can see this polynomial as a morphism \(c: (\mathbf{N}')^2 \longrightarrow \mathbb{A}_{\mathbb{Q}_{\ell}}^{r_m}\) of schemes over \(\mathbb{Q}_{\ell}\). Using \(c\) we may define a structure of linear algebraic group on the scheme \(\mathbf{N}= \mathbb{A}_{\mathbb{Q}_{\ell}}^{r_m} \times_{\mathbb{Q}_{\ell}} \mathbf{N}'\), define \(\mathbf{N}_m = \mathbb{A}_{\mathbb{Q}_{\ell}}^{r_m} \subset \mathbf{N}\) and \(\mathbf{N}_i = \mathbb{A}_{\mathbb{Q}_{\ell}}^{r_m} \times_{\mathbb{Q}_{\ell}} \mathbf{N}'_i\) for \(0 \leq i < m\) and identify \(N\) with a compact open subgroup of \(\mathbf{N}(\mathbb{Q}_{\ell})\). Details are left to the reader. ◻

Proof. Choose a compact open subgroup \(N\) of \(\mathbf{N}(\mathbb{Q}_{\ell})\) and let \(N_i = N \cap \mathbf{N}_i(\mathbb{Q}_{\ell})\). Choose \(\mathcal{O}_E\)-lattices \(A^n\) in \(V^n\) which are stable under \(N\) and stable under the differentials of \(V^\bullet\), and endow \(A^\bullet\) with the induced filtration. The lemma follows from Corollary 25 and extending scalars from \(\mathcal{O}_E\) to \(E\). ◻

Proof. By the same argument as in the proof of Lemma 5 it is enough to show that the cone of \(r_{K,N,K'}(F(A^\bullet))\) has non-zero cohomology in only finitely many degrees and that all of its cohomology group are killed by some power of \(\ell\). For this we can forget the action of \(K'/N'\), and using Lemma 9 we are left to show this property for the cone of the restriction map on cocycles \[R\Gamma_{\mathrm{cont}}(N, \operatorname{res}_N F(A^\bullet)) \longrightarrow R\Gamma_{\mathrm{cont}}(N', \operatorname{res}_{N'} F(A^\bullet)).\] Let \((N_i)_i\) be the upper central series of \(N\), and choose \(\underline{w}\) as in Definition 20. As in the proof of Corollary 24 we endow each \(A^n/\mathfrak{m}_E^i A^n\) with the filtration induced by that on \(A^n\), and for any large enough integer \(d\) we have thanks to the first part of Theorem 23 a quasi-isomorphism \[\operatorname{Tot}^\bullet(\operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, F(A^\bullet))) \simeq R\Gamma_{\mathrm{cont}}(N, \operatorname{res}_N F(A^\bullet)),\] and similarly with \(N\) replaced by \(N'\) (replacing \(N_i\) by \(N_i' := N_i \cap N'\)). We are left to consider the cone of the natural map of complexes \[\label{eq:res95cocyc95cone} \operatorname{Tot}^\bullet(\operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N, F(A^\bullet))) \longrightarrow \operatorname{Tot}^\bullet(\operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N', F(A^\bullet))).\tag{24}\] By the second part of Theorem 23 we also know that both sides have non-vanishing cohomology in only finitely many degrees.

We claim that for any \(m \geq 0\) and \(n \in \mathbb{Z}\) there exists an integer \(k \geq 0\), which also depends on \(N\), \(N'\), \(\underline{w}\) and on an integer \(j_0>0\) satisfying \(\operatorname{Fil}^{-j_0} A^n = 0\) and \(\operatorname{Fil}^{j_0} A^n = A^n\), such that for any \(i \geq 1\) the kernel and cokernel of the morphism \[\operatorname{Fil}^d_{\underline{w}} C^m_{\operatorname{pol}}(N, A^n/\mathfrak{m}_E^i A^n) \xrightarrow{\operatorname{res}} \operatorname{Fil}^d_{\underline{w}} C^m_{\operatorname{pol}}(N', A^n/\mathfrak{m}_E^i A^n)\] are killed by \(\ell^k\). To prove the claim we use Lemma 22 to consider instead the kernel and cokernel of \[\label{eq:res95Fil95Pol95N95Nprime} \operatorname{Fil}^d_{\underline{w}} \operatorname{Pol}(N^m, A^n/\mathfrak{m}_E^i A^n) \xrightarrow{\operatorname{res}} \operatorname{Fil}^d_{\underline{w}} \operatorname{Pol}((N')^m, A^n/\mathfrak{m}_E^i A^n).\tag{25}\] We also observe that we have an isomorphism of filtered \(\mathcal{O}_E\)-modules \[\operatorname{Pol}(N^m, \mathcal{O}_E) \otimes_{\mathcal{O}_E} A^n/\mathfrak{m}_E^i A^n \simeq \operatorname{Pol}(N^m, A^n/\mathfrak{m}_E^i A^n),\] where \(\mathcal{O}_E\) is endowed with the trivial filtration (\(\operatorname{Fil}^{-1} \mathcal{O}_E = 0\) and \(\operatorname{Fil}^0 \mathcal{O}_E = \mathcal{O}_E\)) and the filtration on \(A^n/\mathfrak{m}_E^i A^n\) is divided by \(w_A\), and similarly for \(N'\). Each restriction map \(\operatorname{Fil}^j_{\underline{w}} \operatorname{Pol}(N^m, \mathcal{O}_E) \to \operatorname{Fil}^j_{\underline{w}} \operatorname{Pol}((N')^m, \mathcal{O}_E)\) is an embedding with finite cokernel because both sides are lattices inside \(\operatorname{Fil}^j_{\underline{w}} \operatorname{Pol}(\mathbf{N}(\mathbb{Q}_{\ell})^m, E)\). It follows that the cokernel of 25 is killed by a power of \(\ell\) which does not depend on \(i\) (note that only the indices \(j\) satisfying \(0 \leq j \leq d + (j_0-1) w_A\) are relevant). The fact that the kernel of 25 is killed by some power of \(\ell\) is slightly more subtle, it follows by induction from the same property for the maps \[\label{eq:res95gr95Pol95N95Nprime} \operatorname{gr}^j_{\underline{w}} \operatorname{Pol}(N^m, A^n/\mathfrak{m}_E^i A^n) \xrightarrow{\operatorname{res}} \operatorname{gr}^j_{\underline{w}} \operatorname{Pol}((N')^m, A^n/\mathfrak{m}_E^i A^n)\tag{26}\] for \(w_A(1-j_0) \leq j \leq d\). We have \[\operatorname{gr}^j_{\underline{w}} \operatorname{Pol}(N^m, A^n/\mathfrak{m}_E^i A^n) \simeq \bigoplus_{\substack{j_P,j_A \in \mathbb{Z}\\ j_P + w_A j_A = j}} \operatorname{gr}^{j_P}_{\underline{w}} \operatorname{Pol}(N^m, \mathcal{O}_E) \otimes_{\mathcal{O}_E} \operatorname{gr}^{j_A} A^n/\mathfrak{m}_E^i A^n\] and similarly for \(N'\). Using the beginning of the long exact sequence for \(\mathrm{Tor}\) groups we deduce that if a power of \(\ell\) kills \[\operatorname{gr}^{j_P}_{\underline{w}} \operatorname{Pol}((N')^m, \mathcal{O}_E) / \operatorname{gr}^{j_P}_{\underline{w}} \operatorname{Pol}(N^m, \mathcal{O}_E)\] for all \(0 \leq j_P \leq j+(j_0-1)w_A\) then it also kills the kernel of 26 . This concludes the proof of the claim. It is easy to deduce that each cohomology group of the cone of 24 is killed by some power of \(\ell\). ◻

Proof. Let \(s: \mathbf{H}\to \mathbf{N}\rtimes \mathbf{H}\) be the obvious section. We introduce the compact open subgroup \(K' = K s(H_K)\) of \((\mathbf{N}\rtimes \mathbf{H})(\mathbb{Q}_{\ell})\). It is open because it contains \(K\), and compact because it is generated by its open subgroup \(K\) and the subset \(X := \{k s(k)^{-1} | k \in K \}\) of \(\mathbf{N}(\mathbb{Q}_{\ell})\), and \(X\) is compact and so there exists a compact subgroup of \(\mathbf{N}(\mathbb{Q}_{\ell})\) which contains \(X \cup N_K\) and is stable under the conjugation action of \(K\). Denoting \(N_{K'} = \mathbf{N}(\mathbb{Q}_{\ell}) \cap K'\), we have \(K' = N_{K'} \rtimes s(H_K)\), in particular \(H_{K'} := K'/N_{K'}\) is equal to \(H_K\).

Let \(A^\bullet\) be a model of \(V^\bullet\), i.e.an object of \(D^b(\operatorname{Rep}_{\mathrm{fg},\mathrm{cont}}(K', \mathcal{O}_E))\) with an isomorphism \(E \otimes_{\mathcal{O}_E} A^\bullet \simeq V^\bullet\) in \(D^b(\operatorname{Rep}_{\mathrm{fg},\mathrm{cont}}(K', E))\). We may and do assume that \(A^\bullet\) is simply a sub-complex of \(V^\bullet\) considered as a complex of \(\mathcal{O}_E[K']\)-modules. Define \(\operatorname{Fil}^j A^n = A^n \cap \operatorname{Fil}^j V^n\) for all \(n,j \in \mathbb{Z}\). By Lemma 8 and Corollary 24 we have an isomorphism in \(D^+(H_K, \mathcal{O}_E)\) \[R\Gamma(N_{K'}, F(A^\bullet)) \simeq F \left( \operatorname{Tot}^\bullet(\operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N_{K'}, A^\bullet)) \right).\] In particular this becomes an isomorphism in \(D^+(H_K, E)\). It is clear that the image of \[\operatorname{Tot}^\bullet(\operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(N_{K'}, A^\bullet))\] in \(D^b(\operatorname{Rep}_{\mathrm{fg},\mathrm{cont}}(H_K, E))\) is equal to \[\operatorname{Tot}^\bullet(\operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(\mathbf{N}(\mathbb{Q}_{\ell}), V^\bullet)).\] Finally by Lemma 29 we have an isomorphism in \(D^+(H_K, E)\) \[R\Gamma(N_{K'}, F(A^\bullet)) \simeq R\Gamma(N_K, \operatorname{res}_K F(A^\bullet)).\] ◻

Proof. We have \(\mathfrak{n}= \mathfrak{n}_1 \oplus \mathfrak{n}_2\) with \([\mathfrak{n}_k, \mathfrak{n}_k] \subset \mathfrak{n}_k\) and \([\mathfrak{n}_1, \mathfrak{n}_2] \subset \mathfrak{n}_1\). The action of \(\mathfrak{n}_2\) on \(C^\bullet(\mathfrak{n}_1, V)\) is easily computed from 27 : \[(Y \cdot f)(X_1, \dots, X_n) = Y \cdot f(X_1, \dots, X_n) + \sum_{i=1}^n f(X_1, \dots, [X_i,Y], \dots, X_n).\] We have isomorphisms \[\begin{align} \bigoplus_{i+j=n} \bigwedge^j \mathfrak{n}_1 \otimes \bigwedge^i \mathfrak{n}_2 & \longrightarrow \bigwedge^n \mathfrak{n}\\ (X_1 \wedge \dots \wedge X_j) \otimes (Y_1 \wedge \dots \wedge Y_i) & \longmapsto X_1 \wedge \dots \wedge X_j \wedge Y_1 \wedge \dots \wedge Y_i \end{align}\] and so the map in the lemma is an isomorphism of rational representations of \(\mathbf{H}\). To check that it is a map of complexes, if \(f \in C^{n-1}(\mathfrak{n}, V)\) corresponds to \((f_{i,j})_{i+j=n-1}\) then for all \(X_1, \dots, X_j \in \mathfrak{n}_1\) and \(Y_1, \dots, Y_i \in \mathfrak{n}_2\) we have \[f(X_1, \dots, X_j, Y_1, \dots, Y_i) = f_{i,j}(Y_1, \dots, Y_i)(X_1, \dots, X_j)\] and we compute \[\begin{align} & (d_{\mathfrak{n}} f)(X_1, \dots, X_j, Y_1, \dots, Y_i) \\ =& \sum_{a=1}^j (-1)^{a+1} X_a \cdot f_{i,j-1}(Y_1, \dots, Y_i)(X_1, \dots, \widehat{X_a}, \dots, X_j) \\ &+ \sum_{a=1}^i (-1)^{j+a+1} Y_a \cdot f_{i-1,j}(Y_1, \dots, \widehat{Y_a}, \dots, Y_i)(X_1, \dots, X_j) \\ &+ \sum_{1 \leq a < b \leq j} (-1)^{a+b} f_{i,j-1}(Y_1, \dots, Y_i)([X_a,X_b], X_1, \dots, \widehat{X_a}, \dots, \widehat{X_b}, \dots, X_j) \\ &+ \sum_{1 \leq a < b \leq i} (-1)^{a+b+j} f_{i-1,j}([Y_a,Y_b], Y_1, \dots, \widehat{Y_a}, \dots, \widehat{Y_b}, \dots, Y_i)(X_1, \dots, X_j) \\ &+ \sum_{1 \leq a \leq j} \sum_{1 \leq b \leq i} (-1)^{a+b+j} f_{i-1,j}(Y_1, \dots, \widehat{Y_b}, \dots, Y_i)([X_a,Y_b], X_1, \dots, \widehat{X_a}, \dots, X_j) \\ =& d_{\mathfrak{n}_1} (f_{i,j-1}(Y_1, \dots, Y_i))(X_1, \dots, X_j) + (-1)^j (d_{\mathfrak{n}_2} f_{i-1,j})(Y_1, \dots, Y_i)(X_1, \dots, X_j). \end{align}\] The second part of the lemma follows using standard arguments with total complexes. ◻

Proof. Neither the tensor product with \(V\) nor the invariance condition under \(\mathbf{N}\) play any role in the proof and so we omit both. Recall that \(\Theta_{\mathbf{N}}\) is defined as the composition of 28 , denoted \(\alpha\) in this proof, and 29 , denoted \(\beta\) in this proof. (This composition is restricted to \(\mathbf{N}\)-invariant forms, but the distinction is irrelevant here.) By definition the subspace of normalized cochains is included in the kernel of \[\begin{align} \pi_i: R(\mathbf{N}^{n+1}) & \longmapsto R(\mathbf{N}^n) \\ f_0 \otimes \dots \otimes f_n & \longmapsto f_i f_{i+1} \otimes f_0 \otimes \dots \otimes f_{i-1} \otimes f_{i+2} \otimes \dots \otimes f_n \end{align}\] for any \(0 \leq i \leq n-1\). With the convention \(\tau_0 = \mathrm{id}\) we compute \[(-1)^i d \circ \alpha \circ \pi_i = \alpha \circ \tau_i + \alpha \circ \tau_{i+1}.\] It follows that on normalized cochains we have \[\alpha = -\alpha \circ \tau_1 = \dots = (-1)^j \alpha \circ \tau_j.\] Applying \(\beta\) yields the desired formula. ◻

Proof. It is enough to show that the restriction of both compositions to \[C^i_{\operatorname{pol},\mathrm{norm}}(\mathbf{N}_2, C^j_{\operatorname{pol},\mathrm{norm}}(\mathbf{N}_1, V))\] are equal. Recall from Theorem 10 that \(\operatorname{AW}\) maps this space to \(C^{i+j}_{\operatorname{pol},\mathrm{norm}}(\mathbf{N}_1 \rtimes \mathbf{N}_2, V)\). For explicit computations it is convenient to denote \[c = f_0 \otimes \dots \otimes f_i \otimes (g_0 \otimes \dots \otimes g_j \otimes v) \in R(\mathbf{N}_2^{i+1}) \otimes_E (R(\mathbf{N}_1^{j+1}) \otimes_E V).\] By definition we have \[\operatorname{AW}(c) = (g_0 \otimes 1) \otimes \dots \otimes (g_{j-1} \otimes 1) \otimes (g_j \otimes f_0) \otimes (1 \otimes f_1) \otimes \dots \otimes (1 \otimes f_i) \otimes v\] in \(R((\mathbf{N}_1 \rtimes \mathbf{N}_2)^{n+1}) \otimes_E V\). Applying \(\Theta_{\mathbf{N}_1 \rtimes \mathbf{N}_2} \circ \tau_j\) and evaluating at \((X_1, \dots, X_n) \in (\mathfrak{n}_1 \oplus \mathfrak{n}_2)^n\) we obtain \[(g_j \otimes f_0) \sum_{\sigma \in S_n} \epsilon(\sigma) d(g_0 \otimes 1)(X_{\sigma(1)}) \dots d(g_{j-1} \otimes 1)(X_{\sigma(j)}) d(1 \otimes f_1)(X_{\sigma(j+1)}) \dots d(1 \otimes f_i)(X_{\sigma(n)}) v.\] If we impose that \(X_k\) belongs to \(\mathfrak{n}_1\) for \(k \leq j'\) and to \(\mathfrak{n}_2\) for \(k>j'\), then the term corresponding to \(\sigma\) vanishes unless we have \(j'=j\) and \(\sigma \in S_j \times S_i\). More precisely we obtain \[\begin{align} & \left( \Theta_{\mathbf{N}_1 \rtimes \mathbf{N}_2} \tau_j \operatorname{AW}(c) \right) (X_1, \dots, X_{j'}, Y_1, \dots, Y_{n-j'}) \\ =& \begin{cases} \left( \Theta_{\mathbf{N}_1} \tau_j \left( \Theta_{\mathbf{N}_2}(c)(Y_1, \dots, Y_i) \right) \right) (X_1, \dots, X_j) & \text{ if } j'=j, \\ 0 & \text{ otherwise.} \end{cases} \end{align}\] whenever \(X_1, \dots, X_{j'} \in \mathfrak{n}_1\) and \(Y_1, \dots, Y_{n-j'} \in \mathfrak{n}_2\). By Lemma 32 this implies \[\Theta_{\mathbf{N}_1 \rtimes \mathbf{N}_2}(\operatorname{AW}(c)) = \Theta_{\mathbf{N}_2} \Theta_{\mathbf{N}_1} (c)\] for any \(c \in C^i_{\mathrm{alg},\mathrm{norm}}(\mathbf{N}_2, C^j_{\mathrm{alg},\mathrm{norm}}(\mathbf{N}_1, V))\) ◻

Proof. Endow \(\mathbf{N}\) with a filtration \((\mathbf{N}_i)_{0 \leq i \leq m+1}\) (e.g.the upper central series) and weights \(\underline{w} = (w_A, w_m, \dots, w_0)\) making it a weighted filtered unipotent algebraic group in the sense of Section 8. Denote \(N_i = N \cap \mathbf{N}_i(\mathbb{Q}_{\ell})\). Define a filtration \(\operatorname{Fil}^\bullet V^\bullet\) on \(V^\bullet\) by \(\operatorname{Fil}^j V^\bullet = 0\) for \(j<0\) and \(\operatorname{Fil}^j V^n = (V^n/\operatorname{Fil}^{j-1} V^n)^{\mathbf{N}_E}\) for \(j \geq 0\). By Proposition 30 we have an isomorphism in \(D^+(H_K, E)\) \[\label{eq:step195pf95vanEst} R\Gamma(N_K, F(V^\bullet)) \simeq F \left( \operatorname{Tot}^\bullet(\operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(\mathbf{N}(\mathbb{Q}_{\ell}), V^\bullet)) \right)\tag{31}\] for any large enough integer \(d\). For large enough \(d\) we also have a quasi-isomorphism \[\operatorname{Tot}^\bullet \left( \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(\mathbf{N}(\mathbb{Q}_{\ell}), V^\bullet) \right) \subset \operatorname{Tot}^\bullet \left( C^\bullet_{\operatorname{pol}}(\mathbf{N}(\mathbb{Q}_{\ell}), V^\bullet) \right)\] thanks to Lemma 28. We also have a quasi-isomorphism 30 \[\operatorname{Tot}^\bullet \left( C^\bullet_{\operatorname{pol}}(\mathbf{N}(\mathbb{Q}_{\ell}), V^\bullet) \right) \xrightarrow{\sim} R\Gamma_{\operatorname{Lie}}(\mathfrak{n}, V^\bullet).\] Composing the two gives an isomorphism in \(D^b(\operatorname{Rep}_{\mathrm{fg},\mathrm{cont}}(H_K, E))\) \[\operatorname{Tot}^\bullet \left( \operatorname{Fil}^d_{\underline{w}} C^\bullet_{\operatorname{pol}}(\mathbf{N}(\mathbb{Q}_{\ell}), V^\bullet) \right) \to R\Gamma_{\operatorname{Lie}}(\mathfrak{n}, V^\bullet).\] Applying the functor \(F\) and composing with 31 gives \(\nu_{\mathbf{N}, \mathbf{H}, K}(V^\bullet)\). It is easy to check that this morphism does not depend on the choices made (filtration, weights, \(d\), model over \(\mathcal{O}_E\) …), by cofinality arguments (e.g.\(\varinjlim_d \operatorname{Fil}^d_{\underline{w}} C^n_{\operatorname{pol}}(\mathbf{N}(\mathbb{Q}_{\ell}), V^m) = C^n_{\operatorname{pol}}(\mathbf{N}(\mathbb{Q}_{\ell}), V^m)\)). Using similar arguments one can check that \(\nu_{\mathbf{N},\mathbf{H},K}(V^\bullet)\) is functorial in \(V^\bullet\). The details of these relatively formal arguments are omitted here.

The proof of the first compatibility is very formal.

The second compatibility property, where \(K'\) is an open subgroup of \(K\), follows from Lemma 29, also using arguments as in the proof of this lemma.

The second compatibility property follows from Proposition 33. ◻

Proof. Denote by \(\mathbb{A}_f^{(\ell)}\) the restricted product of \(\mathbb{Q}_p\) for primes \(p \neq \ell\), so that we have \(\mathbb{A}_f = \mathbb{Q}_{\ell}\times \mathbb{A}_f^{(\ell)}\). Denote \(N_{K,\ell} = N_K \cap \mathbf{N}(\mathbb{Q}_{\ell})\) and \(N_K^{(\ell)} = N_K \cap \mathbf{N}(\mathbb{A}_f^{(\ell)})\). Denote \(K^{(\ell)} = K \cap (\mathbf{N}\rtimes \mathbf{H})(\mathbb{A}_f^{(\ell)}\) and \(K_\ell = K/K^{(\ell)}\), considered as a compact open subgroup of \((\mathbf{N}\rtimes \mathbf{H})(\mathbb{Q}_{\ell})\). We have \(N_K = N_{K,\ell} \times N_K^{(\ell)}\) and \(N_K^{(\ell)}\) is an inverse limit of finite groups having cardinality coprime to \(\ell\). It follows that the functor of \(N_K^{(\ell)}\)-invariants, from \((S_K^{\mathbb{N}}, (\mathcal{O}_E)_\bullet)\)-modules to \((S_{K/N_K^{(\ell)}}^{\mathbb{N}}, (\mathcal{O}_E)_\bullet)\)-modules, is exact. We get an isomorphism of functors \[\begin{tikzcd}[row sep=4em] D^+(K_\ell, E) \arrow[rd, "{\inf}" below left] \arrow[rr, "{\inf}" above] & \arrow[d, Rightarrow, shorten=6mm]& D^+(K/N_K^{(\ell)}, E) \\ & D^+(K, E) \arrow[ru, "{R\Gamma(N_K^{(\ell)}, -)}" below right] & \end{tikzcd}\] where each map labelled \(\inf\) is an inflation map. We also have a morphism of functors \[\begin{tikzcd}[column sep=4em, row sep=4em] D^+(K_\ell, E) \arrow[r, "{R\Gamma(N_{K,\ell}, -)}"] \arrow[d, "{\inf}"] & D^+(K_\ell/N_{K,\ell}, E) \arrow[d, "{\inf}"] \arrow[Rightarrow, ld, shorten=8mm] \\ D^+(K/N_K^{(\ell)}, E) \arrow[r, "{R\Gamma(N_{K,\ell}, -)}"] & D^+(H_K, E) \end{tikzcd}\] which is easily seen to be an isomorphism using the isomorphisms of functors \(r_{K_\ell, N_{K,\ell}, N_{K,\ell}}\) and \(r_{K/N_K^{(\ell)}, N_{K,\ell}, N_{K,\ell}}\) defined in 2 . Combining these two isomorphisms of functors with the isomorphism of functors \[R\Gamma(N_K, -) \simeq R\Gamma(N_{K,\ell}, R\Gamma(N_K^{(\ell)}, -))\] reduces the corollary to Theorem 34. ◻