2.1.1. Let G be a p-adic Lie group of dimension d. By Theorem 27.1 of [Reference SchneiderSch11], there exists a compact open subgroup $G_0$ of G equipped with an integer-valued, saturated p-valuation. As a consequence, there exist $g_1,\cdots ,g_d\in G_0$ (an ordered basis) such that the map $c:\mathbb {Z}_p^d\to G_0$ defined by $(x_1,\cdots ,x_d)\mapsto g_1^{x_1}\cdots g_d^{x_d}$ is a homeomorphism. Let $G_n=G_0^{p^n}=\{g^{p^n},g\in G_0\}$ for $n\geq 1$ . This is in fact a subgroup, and $c^{-1}(G_n)=(p^n\mathbb {Z}_p)^d$ . So $\{G_n\}_n$ forms a fundamental system of open neighbourhoods of the identity element $1$ in G. We will fix such a c in the following discussion, but all of the definitions we make below will not depend on the choice of c. See Section 23 and Section 26 of [Reference SchneiderSch11] for more details.

If B is a $\mathbb {Q}_p$ -Banach space with norm $\|\cdot \|$ , we denote by $\mathscr {C}(G_n,B)$ the space of B-valued continuous functions on $G_n$ and by $\mathscr {C}^{\mathrm {an}}(G_n,B)$ the subspace of B-valued analytic functions. More precisely, $f:G_n\to B$ is analytic if its pullback under c can be written as

$$ \begin{align*}\mathbf{x}=(x_1,\cdots,x_d)\in (p^n\mathbb{Z}_p)^d\mapsto \sum_{\mathbf{k}=(k_1,\cdots,k_d)\in \mathbb{N}^d} b_{\mathbf{k}} x_1^{k_1}\cdots x_d^{k_d}\end{align*} $$

for some $b_{\mathbf {k}}\in B$ such that $p^{n|\mathbf {k}|}b_{\mathbf {k}}\to 0$ as $|\mathbf {k}|\to \infty $ . We define $\|f\|_{G_n}=\sup _{\mathbf {k}\in \mathbb {N}^d}\|p^{n|\mathbf {k}|}b_{\mathbf {k}}\|$ and $\mathscr {C}^{\mathrm {an}}(G_n,B)$ is a $\mathbb {Q}_p$ -Banach space under this norm. There is a natural norm-preserving isomorphism $\mathscr {C}^{\mathrm {an}}(G_n,\mathbb {Q}_p) \widehat \otimes _{\mathbb {Q}_p} B\cong \mathscr {C}^{\mathrm {an}}(G_n,B)$ induced by the $\mathbb {Q}_p$ -linear structure of B. When $B=\mathbb {Q}_p$ , the usual point-wise multiplication of two functions defines a natural $\mathbb {Q}_p$ -algebra structure on $\mathscr {C}^{\mathrm {an}}(G_n,\mathbb {Q}_p)$ .

To see this definition is independent of the choice of $g_1,\cdots ,g_d$ , we introduce another set of coordinates of $G_n$ given by the the exponential map. These coordinates are called coordinates of the first kind in Section 34 of [Reference SchneiderSch11]. More precisely, let $\operatorname {\mathrm {Lie}}(G)$ be the Lie algebra of G and consider the logarithm map

$$ \begin{align*}\log:G_0\to\operatorname{\mathrm{Lie}}(G),\end{align*} $$

which is a homeomorphism onto its image. In fact, its image $\log (G_0)$ is a free $\mathbb {Z}_p$ -module with a basis $\log (g_1),\cdots ,\log (g_d)$ . Hence,

$$ \begin{align*}e:\mathbb{Z}_p^{d}\to G_0,\, (y_1,\cdots,y_d)\mapsto \exp(y_1\log(g_1)+\cdots+y_d\log(g_d))\end{align*} $$

defines a homeomorphism. Here $\exp $ is the inverse of $\log $ (exponential map). Under this parametrisation, $G_n$ is identified with the image of $(p^n\mathbb {Z}_p)^d$ . As before, a function on $G_n$ is called analytic if its pullback under e can be written as a power series in $y_i$ with certain growth conditions on its coefficients. This definition of analyticity agrees with the previous definition as the transition functions between the coordinates $(x_1,\cdots ,x_d)$ and $(y_1,\cdots ,y_d)$ are given by convergent power series with coefficients in $\mathbb {Z}_p$ ; cf. Proposition 34.1 of [Reference SchneiderSch11].

Different choices of ordered bases $g_1,\cdots ,g_d$ give rise to different bases of $\log (G_0)$ over $\mathbb {Z}_p$ (as a $\mathbb {Z}_p$ -module). It is easy to see now that $\mathscr {C}^{\mathrm {an}}(G_n,B)$ and $\|\cdot \|_{G_n}$ are independent of choice of an ordered basis. As a corollary, $\mathscr {C}^{\mathrm {an}}(G_n,B)$ and $\|\cdot \|_{G_n}$ are invariant under both the left and right translation actions of $G_0$ . We sketch a proof for the left translation by $G_0$ here. The case for right translation and general n will be exactly the same. Let x be a nontrivial element of $G_0$ . It suffices to prove the claim under the assumption that x is not a pth power. Choose an ordered basis $g_1,\cdots ,g_d$ with $g_1=x$ . This is possible by our assumption on $G_0$ ; see Section 26 of [Reference SchneiderSch11]. To compute the left translation action of x, we are essentially reduced to the case $G_0=\mathbb {Z}_p$ , which can be verified directly. In fact, this argument shows a bit more.

Proof. One can reduce to the trivial case $G_n=\mathbb {Z}_p$ by the same argument.

Proof. This is essentially proved in the proof of Theorem 6.1 of [Reference Berger and ColmezBC16]. We recall their argument here. The rough idea is to reduce to the case $G=\mathrm {GL}_N(\mathbb {Z}_p)$ , in which case the algebraic functions on $\mathrm {GL}_N$ are dense in the space of analytic functions.

For n large enough, we may find an embedding $G_n$ into $1+p^2 M_N(\mathbb {Z}_p)\subset \mathrm {GL}_N(\mathbb {Z}_p)$ for some N, where $M_N(\mathbb {Z}_p)$ denotes the space of $N\times N$ matrices over $\mathbb {Z}_p$ . We sketch a proof of this well-known result here because it seems to lack a suitable reference. By Ado’s theorem (Section 7, , Theorem 3 of [Reference BourbakiBou60]), there is a faithful representation of $\operatorname {\mathrm {Lie}}(G)$ over $\mathbb {Q}_p^{\oplus N}$ for some N. Choose n large enough so that $\log (G_n)$ maps to $p^2 M_N(\mathbb {Z}_p)$ . Then this Lie algebra morphism can be integrated to a group homomorphism $G_n\to 1+p^2 M_N(\mathbb {Z}_p)$ , which is necessarily a closed embedding. This can be deduced from results in Section 31, notably Theorem 31.5 of [Reference SchneiderSch11].

Fix such an embedding. $G_n$ is now viewed as a subset of $M_N(\mathbb {Q}_p)$ . For $k\in N$ , let $V_k$ be the space of $\mathbb {Q}_p$ -valued functions on $G_n$ that are restrictions of polynomials of degree $\leq k$ on $M_N(\mathbb {Q}_p)$ . Here we identify $M_N(\mathbb {Q}_p)$ with an affine space of dimension $N^2$ (over $\mathbb {Q}_p$ ). We claim that the union of all $V_k$ is dense in $\mathscr {C}^{\mathrm {an}}(G_n,\mathbb {Q}_p)$ . To see this, note that the space of polynomial functions on $M_N(\mathbb {Q}_p)$ is dense in $\mathscr {C}^{\mathrm {an}}(1+p^2M_N(\mathbb {Z}_p),\mathbb {Q}_p)$ when restricted to $1+p^2M_N(\mathbb {Z}_p)$ . Hence, its pullback under the exponential map is dense in $\mathscr {C}^{\mathrm {an}}(p^2M_N(\mathbb {Z}_p),\mathbb {Q}_p)$ . As the image of $\mathscr {C}^{\mathrm {an}}(p^2M_N(\mathbb {Z}_p),\mathbb {Q}_p)\to \mathscr {C}^{\mathrm {an}}(\log (G_n),\mathbb {Q}_p)$ is dense, this implies our claim.

It is clear from the construction that all $V_k$ are stable under left and right translation actions and closed under multiplications.

2.1.4. Now if B is a $\mathbb {Q}_p$ -Banach space equipped with a continuous action of $G_n$ , then we denote by $B^{G_n-\mathrm {an}}\subset B$ the subset of $G_n$ -analytic vectors. One can define this as follows: there is a left action on $\mathscr {C}^{\mathrm {an}}(G_n,B)$ by $(g\cdot f)(h)=g(f(g^{-1}h)),g,h\in G_n,f\in \mathscr {C}^{\mathrm {an}}(G_n,B)$ . We define $B^{G_n-\mathrm {an}}$ as the image of $\mathscr {C}^{\mathrm {an}}(G_n,B)^{G_n}\to B$ under the evaluation map at the identity element in $G_n$ . Equivalently, $v\in B^{G_n-\mathrm {an}}$ if $f_v:G_n\to B,~f_v(g)=g\cdot v$ lies in $\mathscr {C}^{\mathrm {an}}(G_n,B)$ . It is clear that $\|\cdot \|_{G_n}$ induces a norm on $B^{G_n-\mathrm {an}}$ , which we denote by the same notation. Now $G_n$ acts on $B^{G_n-\mathrm {an}}=\mathscr {C}^{\mathrm {an}}(G_n,B)^{G_n}$ by right translation. One checks easily that this action is unitary and equivariant with respect to the inclusion $B^{G_n-\mathrm {an}}\subset B$ .

We say B is an analytic representation of $G_n$ if $B=B^{G_n-\mathrm {an}}$ .

One has the following lemma (cf. Lemme 2.4. of [Reference Berger and ColmezBC16]).

Proof. For the first part, since all of the homomorphisms are strict, $H^i(M^{\bullet ,o})$ can be killed by a bounded power of p, where $M^{\bullet ,o}$ denotes the open unit ball of $M^{\bullet }$ . Therefore, $M^{\bullet }\widehat {\otimes }_{\mathbb {Q}_p}\mathscr {C}^{\mathrm {an}}(G_n,\mathbb {Q}_p)$ remains exact for any n and we obtain the desired long exact sequence by taking $G_n$ -cohomology and passing to the direct limit over n.

The second and third parts follow from the first part by a simple induction argument. For example, one can prove that the kernel and image of $M^q\to M^{q+1}$ are $\mathfrak {LA}$ -acyclic by induction on q. This certainly implies that $(H^q(M^{\bullet }))^{\mathrm {la}}=H^q((M^{\bullet })^{\mathrm {la}})$ . We leave the third part as an exercise.

Proof. Since B is admissible, there is a $G_0$ -equivariant embedding $B \hookrightarrow \mathscr {C}(G_0,\mathbb {Q}_p)^{\oplus d}$ for some d. Then the quotient C is an admissible $\mathbb {Q}_p$ -Banach representation of $G_0$ . Note that $ \mathscr {C}(G_0,\mathbb {Q}_p)\cong \mathscr {C}(G_n,\mathbb {Q}_p)^{\oplus d_n}$ as a representation of $G_n$ for some $d_n$ . It follows from Shapiro’s lemma for continuous cohomology that $H_{\mathrm {cont}}^1(G_n,\mathscr {C}(G_0,\mathbb {Q}_p)\widehat {\otimes }_{\mathbb {Q}_p}\mathscr {C}^{\mathrm {an}}(G_n,\mathbb {Q}_p))=0$ . Hence,

$$ \begin{align*}(\mathscr{C}(G_0,\mathbb{Q}_p)^{\oplus d})^{G_n-\mathrm{an}}\to C^{G_n-\mathrm{an}} \to H^1_{\mathrm{cont}}(G_n,B\widehat{\otimes}_{\mathbb{Q}_p}\mathscr{C}^{\mathrm{an}}(G_n,\mathbb{Q}_p))\to 0.\end{align*} $$

By Corollaire IV.14. of [Reference Colmez and DospinescuCD14], we have an exact sequence

$$ \begin{align*}0\to B^{(m)}\to (\mathscr{C}(G_0,\mathbb{Q}_p)^{\oplus d})^{(m)}\to C^{(m)}\to 0.\end{align*} $$

See [Reference Colmez and DospinescuCD14, §IV] for the notation here. One can check $C^{G_n-\mathrm {an}}\subset C^{(n+2)}\subset C^{G_{n+1}-\mathrm {an}}$ . This means that the image of $(\mathscr {C}(G_0,\mathbb {Q}_p)^{\oplus d})^{G_{n+1}-\mathrm {an}}$ in $C^{G_{n+1}-\mathrm {an}}$ contains $C^{G_n-\mathrm {an}}$ . Hence, $H^1_{\mathrm {cont}}(G_n,B\widehat {\otimes }_{\mathbb {Q}_p}\mathscr {C}^{\mathrm {an}}(G_n,\mathbb {Q}_p)) \to H^1_{\mathrm {cont}}(G_{n+1},B\widehat {\otimes }_{\mathbb {Q}_p}\mathscr {C}^{\mathrm {an}}(G_{n+1},\mathbb {Q}_p))$ has zero image. A simple induction on cohomology degree $i\geq 1$ proves the theorem.

Proof. Let $B^o$ be the unit open ball of B. Take n large enough so that $B^o$ is $G_n$ -stable. We claim that $H^{\bullet }_{\mathrm {cont}}(G_n,B^o)$ is a finitely generated $\mathbb {Z}_p$ -module and hence has bounded p-torsion. Indeed, since B is admissible, there exists a $\mathbb {Z}_p[G_n]$ -equivariant injection $B^o \hookrightarrow \mathscr {C}(G_n,\mathbb {Z}_p)^{\oplus d}$ for some d, whose quotient is p-torsion free. An induction argument on i implies the finiteness here. It is well-known that $H^{\bullet }_{\mathrm {cont}}(G_n,B)$ can be computed by a cochain complex $(\mathscr {C}(G_n,B)^{\oplus i},d^i)$ . This is a strict complex because $(\mathscr {C}(G_n,B^o)^{\oplus i},d^i)$ computes $H^{\bullet }_{\mathrm {cont}}(G_n,B^o)$ . Since $\mathscr {C}(G_n,B\widehat {\otimes }_{\mathbb {Q}_p}M)\cong \mathscr {C}(G_n,B)\widehat {\otimes }_{\mathbb {Q}_p}M$ , we know that $(\mathscr {C}(G_n,B)^{\oplus i}\widehat {\otimes }_{\mathbb {Q}_p}M,d^i\otimes \mathbf {1})$ computes $H^{\bullet }_{\mathrm {cont}}(G_n,B\widehat {\otimes }_{\mathbb {Q}_p}M)$ . Therefore,

$$ \begin{align*} H^{\bullet}_{\mathrm{cont}}(G_n,B\widehat{\otimes}_{\mathbb{Q}_p}M)\cong H^{\bullet}_{\mathrm{cont}}(G_n,B)\widehat{\otimes}_{\mathbb{Q}_p}M.\end{align*} $$

Our claim now follows from the previous theorem.

3.1.1. Fix a complete algebraically closed non-Archimedean field C of characteristic zero. Recall that a non-Archimedean field is a topological field whose topology is induced by a non-Archimedean norm $|\cdot |:C\to \mathbb {R}_{\geq 0}$ . It is naturally an extension of $\mathbb {Q}_p$ for some p and we assume its norm agrees with the usual p-adic norm $|\cdot |_p=p^{-\mathrm {val}_p(\cdot )}$ on $\mathbb {Q}_p$ . Let $\mathcal {O}_C$ be the ring of integers of C. Our setup is as follows:

• • $X=\operatorname {\mathrm {Spa}}(A,A^+)$ : a 1-dimensional smooth affinoid adic space over $\operatorname {\mathrm {Spa}}(C,\mathcal {O}_C)$ ;

• • G: a finite-dimensional compact p-adic Lie group;

• • $\widetilde X=\operatorname {\mathrm {Spa}}(B,B^+)$ : an affinoid perfectoid algebra over $\operatorname {\mathrm {Spa}}(C,\mathcal {O}_C)$ , which is a ‘log G-Galois pro-étale perfectoid covering’ of X. More precisely, this means that there is a finite set S of classical points in X and $\widetilde X\sim \varprojlim _{i\in I} X_i$ in the sense of Definition 7.14 of [Reference ScholzeSch12] for some index set I, where each $X_i=\operatorname {\mathrm {Spa}}(B_i,B_i^+)$ is a finite Galois covering of X unramified outside of S, and $B^{+}$ is the p-adic completion of $\varinjlim _i B_i^{+}$ . Moreover, the inverse limit of the Galois group of $X_i$ over X is identified with G. When S is nonempty, we further assume for each point s in S, the ramification-index $e_i$ of $X_i\to X$ at s is a p-power for any i and $\{e_i\}_{i\in I}$ is unbounded;

• • assume X is small in the sense that S contains at most one element and there is an étale map $X\to \mathbb {T}^1=\operatorname {\mathrm {Spa}}(C\langle T^{\pm 1}\rangle , \mathcal {O}_C\langle T^{\pm 1}\rangle )$ (respectively $X\to \mathbb {B}^1=\operatorname {\mathrm {Spa}}(C\langle T\rangle , \mathcal {O}_C\langle T\rangle )$ mapping S to $0$ ) when S is empty (respectively otherwise) which factors as a composite of rational embeddings and finite étale maps.

Note that G acts continuously on the Banach algebra B and we can consider the locally analytic vectors $B^{\mathrm {la}}\subset B$ . Let $\operatorname {\mathrm {Lie}}(G)$ be the Lie algebra of G. It acts on $B^{\mathrm {la}}$ , and this action can be extended B-linearly to a map $B\otimes _{\mathbb {Q}_p}\operatorname {\mathrm {Lie}}(G) \times B^{\mathrm {la}} \to B$ .

3.2.1. Fix an étale map $f_1:X\to Y$ which factors as a composite of rational embeddings and finite étale maps, where Y is either $\mathbb {T}^1=\operatorname {\mathrm {Spa}}(C\langle T^{\pm 1}\rangle , \mathcal {O}_C\langle T^{\pm 1}\rangle )$ or $\mathbb {B}^1=\operatorname {\mathrm {Spa}}(C\langle T\rangle , \mathcal {O}_C\langle T\rangle )$ . In the latter case, we assume the image of S in Y is $0$ . We will fix such a choice of $f_1:X\to Y$ from now on.

For any $n\in \mathbb {N}$ , when $Y=\mathbb {T}^1$ (respectively $\mathbb {B}^1$ ), let

$$ \begin{align*}Y_n=\operatorname{\mathrm{Spa}}(R_n,R_n^+):=\operatorname{\mathrm{Spa}}(C\langle T^{\pm \frac{1}{p^{n}}}\rangle, \mathcal{O}_C\langle T^{\pm\frac{1}{p^{n}}}\rangle)~ (\mbox{respectively } \operatorname{\mathrm{Spa}}(C\langle T^{ \frac{1}{p^{n}}}\rangle, \mathcal{O}_C\langle T^{\frac{1}{p^{n}}}\rangle)),\end{align*} $$

$$ \begin{align*} Y_{\infty}=\operatorname{\mathrm{Spa}}(R,R^+):=\operatorname{\mathrm{Spa}}(C\langle T^{\pm \frac{1}{p^{\infty}}}\rangle, \mathcal{O}_C\langle T^{\pm\frac{1}{p^{\infty}}}\rangle)~ (\mbox{respectively } \operatorname{\mathrm{Spa}}(C\langle T^{ \frac{1}{p^{\infty}}}\rangle, \mathcal{O}_C\langle T^{\frac{1}{p^{\infty}}}\rangle)).\end{align*} $$

This is the example considered in Example 3.1.6. Note that $Y_{\infty }\sim \varprojlim _{i\in \mathbb {N}} Y_i$ is a Galois covering of Y and $R^+$ is the p-adic completion of $\varinjlim R^+_n$ . We denote the Galois group by $\Gamma $ , which can be identified with $\mathbb {Z}_p$ noncanonically. For $n\geq m$ , there is the usual trace map

$$ \begin{align*}\mathrm{tr}_{Y,n,m}:R_n^+\to R_m^+.\end{align*} $$

Concretely, it sends $T^{\frac {l}{p^k}},(l,p)=1$ to $p^{n-m}T^{\frac {l}{p^k}}$ if $k\leq m$ and $0$ otherwise. Clearly, $\frac {1}{p^{n-m}}\mathrm {tr}_{Y,n,m}$ are compatible when n varies and extends to a $R_m^+$ -linear map

$$ \begin{align*}\overline{\mathrm{tr}}_{Y,m}^+:R^+\to R^+_m,\end{align*} $$

which is a left inverse of the inclusion $R^+_m\to R^+$ . It commutes with the action of $\Gamma $ as taking traces commutes with the Galois action. Moreover, for any $x\in R^+$ ,

$$ \begin{align*}\lim_{m\to +\infty}\overline{\mathrm{tr}}_{Y,m}^+(x)=x.\end{align*} $$

After inverting p, we get a $R_m$ -linear map $\overline {\mathrm {tr}}_{Y,m} :R\to R_m$ (Tate’s normalised trace). Let $\gamma $ be any topological generator of $p^m\Gamma $ . It is easy to see that $\gamma -1$ is invertible on $\ker (\overline {\mathrm {tr}}_{Y,m})$ . Moreover, the norm $\|(\gamma -1)^{-1}\|$ on $\ker (\overline {\mathrm {tr}}_{Y,m})$ equals $|(\zeta _{p^{m+1}}-1)^{-1}|_p=p^{\frac {1}{p^m(p-1)}}$ , which converges to $1$ as $m\to \infty $ .

3.2.2. The material here should be a consequence of some general results of Diao–Lan–Liu–Zhu on log affinoid perfectoid spaces. See 5.3 of [Reference Diao, Lan, Liu and ZhuDLLZ19]. For our later applications, we give a more explicit presentation.

Now we base change everything along the map $X\to Y$ :

$$ \begin{align*}X_n=\operatorname{\mathrm{Spa}}(A_n,A_n^+):=X\times_Y Y_n,\end{align*} $$

$$ \begin{align*}X_{\infty}=\operatorname{\mathrm{Spa}}(A_{\infty},A_{\infty}^+):=X\times_Y Y_{\infty},\end{align*} $$

where all of the fibre products are taken in the category of adic spaces over C. We remark that $X\to Y$ is locally of finite type in the sense of [Reference HuberHub96, Definition 1.2.1] and the fibre product exists by [Reference HuberHub96, Proposition 1.2.2]. All of the fibre products we consider below will exist for the same reason.

Recall that C is equipped with an non-Archimedean norm $|\cdot |$ . For any $|p|^{c}\in |C^{\times }|\subset \mathbb {R}^{\times }_{\geq 0}$ , we fix a choice of element in C, formally written as $p^c$ , with norm $|p|^{c}$ .

Proof. As $X\to Y$ can be written as a composite of rational embeddings and finite étale maps, we may apply Lemma 4.5 of [Reference ScholzeSch13a] here. Strictly speaking, Lemma 4.5 assumes $Y_n$ is étale over Y, but the same argument works here using that R is a perfectoid C-algebra and $R^+$ is the p-adic completion of $\varinjlim R^+_n$ .

Proof. For $m\geq n$ , as $\overline {\mathrm {tr}}_{X,m}$ is continuous and $\Gamma $ -equivariant, it can be restricted to the $p^n\Gamma $ -analytic vectors

$$ \begin{align*}(A_{\infty})^{p^n\Gamma-\mathrm{an}}\xrightarrow{\overline{\mathrm{tr}}_{X,m}} (A_m)^{p^n\Gamma-\mathrm{an}}=(A_m)^{p^n\Gamma}=A_n.\end{align*} $$

Note that the action of $p^n\Gamma $ on $A_m$ is trivial on $p^m\Gamma $ ; hence, the analyticity implies the first equality. Now the lemma follows from $\lim _{m\to +\infty }\overline {\mathrm {tr}}_{X,m}(x)=x$ .

3.2.6. Next we base change the tower $\{X_n\}_n$ to $\widetilde {X}$ . Suppose $G_0$ is an open subgroup of G. Let

$$ \begin{align*}X_{G_0}:=\operatorname{\mathrm{Spa}}(B^{G_0},(B^+)^{G_0})\end{align*} $$

be the corresponding covering of X. We can take the fibre product in the category of adic spaces over C

$$ \begin{align*}X^{\prime}_{G_0,n}:=X_{G_0}\times_X X_n=X_{G_0}\times_Y Y_n,\end{align*} $$

and take its normalisation

$$ \begin{align*}X_{G_0,n}=\operatorname{\mathrm{Spa}}(B_{G_0,n},B_{G_0,n}^+).\end{align*} $$

Note that by Abhyankar’s lemma for rigid analytic spaces (cf. Lemma 4.2.2, 4.2.3 of [Reference Diao, Lan, Liu and ZhuDLLZ19], which is based on earlier work of Lütkebohmert), $X_{G_0,n}\to Y_n$ is unramified when n is sufficiently large; hence, $X_{G_0,n}\to X_n$ is finite étale and

$$ \begin{align*}X_{G_0,n}=X_{G_0,m}\times_{X_m} X_n= X_{G_0,m}\times_{Y_m} Y_n\end{align*} $$

for sufficiently large $m<n$ . This implies that when n is sufficiently large,

$$ \begin{align*}X_{G_0,\infty}=\operatorname{\mathrm{Spa}}(B_{G_0,\infty},B_{G_0,\infty}^+):=X_{G_0,n}\times_{X_n} X_{\infty}=X_{G_0,n}\times_{Y_n} Y_{\infty}\end{align*} $$

is independent of n. Since $X_{G_0,n}\to Y_n$ can be written as a composite of rational embeddings and finite étale maps, Lemma 3.2.3 still holds in this setting. For example, $B_{G_0,\infty }^+$ is the p-adic completion of $\varinjlim _n B_{G_0,n}^+$ . For sufficiently large n, there exist Tate’s normalised traces $\overline {\mathrm {tr}}_{X_{G_0},n}:B_{G_0,\infty }\to B_{G_0,n}$ with same properties as in the case $\{X_n\}_n$ . In particular, $(B_{G_0,\infty })^{p^n\Gamma -\mathrm {an}}=B_{G_0,n}$ for n large enough.

Suppose $G^{\prime }_0$ is another open subgroup of G containing $G_0$ . We have compatible maps $X_{G_0,n}\to X_{G^{\prime }_0,n}$ which are finite étale when n is sufficiently large. Hence, $X_{G_0,\infty }\to X_{G^{\prime }_0,\infty }$ is a finite étale covering of affinoid perfectoid spaces. Let $B_{\infty }^+$ be the p-adic completion of $\varinjlim _{G_0}B_{G_0,\infty }^+$ over all open subgroups $G_0$ of G and $B_{\infty }=B_{\infty }^+[\frac {1}{p}]$ equipped with the norm induced from $B_{G_0,\infty }^+$ . Then $(B_{\infty }, B_{\infty }^+)$ is again an affinoid perfectoid $(C,\mathcal {O}_C)$ -algebra as it is the completion of a direct limit of perfectoid affinoid $(C,\mathcal {O}_C)$ -algebras. We denote $\operatorname {\mathrm {Spa}}(B_{\infty }, B_{\infty }^+)$ by $\widetilde X_{\infty }$ . Note that $\widetilde {X}_{\infty }$ also agrees with the fibre product of $\tilde {X}=\varprojlim X_{G_0}$ and $X_{\infty }=\varprojlim X_n$ in the pro-Kummer étale site of X equipped with the natural log structure defined by S; cf. [Reference Diao, Lan, Liu and ZhuDLLZ18].

Clearly, $G\times \Gamma $ acts on $B_{\infty }^+$ and $B_{\infty }$ . Then it follows from Faltings’s almost purity theorem (cf. [Reference ScholzeSch12, Theorem 7.9 (iii)]) that $(B_{\infty }^+/(p))^{G_0}$ almost equals $B^{+}_{G_0,\infty }/(p)$ . Hence, $(B_{\infty })^{G_0}=B_{G_0,\infty }$ and $(B_{\infty }^+)^{G_0}$ almost equals $B_{G_0,\infty }^+$ .

Proof. To see (TS1), for any open subgroups $H_1\subset H_2$ of G, we know that $(B_{\infty })^{H_2}=B_{H_2,\infty }\to (B_{\infty })^{H_1}=B_{H_1,\infty }$ is a finite étale map between perfectoid algebras; hence, by Faltings’s almost purity theorem, $B_{H_2,\infty }^+\to B_{H_1,\infty }^+$ is almost finite étale. All of the claims in (TS2), (TS3) are essentially verified in 3.2.2 and the above discussion.

Proof. Recall that $\widetilde {X}=\operatorname {\mathrm {Spa}}(B,B^+)$ is perfectoid and $B^+$ is the p-adic completion of the direct limit $\varinjlim _{G_0} B^+_{G_0,0}$ over all open subgroups $G_0$ of G. Let n be a positive integer. Note that by our assumption on the ramification-index in Section 3.1.1 and Abhyankar’s lemma (cf. Lemma 4.2.2, 4.2.3 of [Reference Diao, Lan, Liu and ZhuDLLZ19]), $X_{G_0,n}\to X_{G_0,0}$ is finite étale for sufficiently small $G_0$ . Hence, for a sufficiently small subgroup $G_0$ , the fibre product

$$ \begin{align*}\widetilde{X}_n=\operatorname{\mathrm{Spa}}(B_n,B_n^+):=\widetilde{X}\times_{X_{G_0,0}} X_{G_0,n}\end{align*} $$

is independent of the choice of $G_0$ and $\widetilde {X}_n\to \widetilde {X}$ is a finite étale map between affinoid perfectoid spaces. By the same argument as in the proof of Lemma 3.2.3, $B^+_n$ is the p-adic completion of $\varinjlim _{G_0} B^+_{G_0,n}$ . Hence, it follows from our discussion in Section 3.2.6 that $B_{\infty }^+$ is the p-adic completion of $\varinjlim _n B^+_n$ . By Faltings’s almost purity theorem, we see that $(B^+_{\infty })^{\Gamma }$ is almost $B^+$ and $(B_{\infty })^{\Gamma }=B$ .

3.2.10. Now suppose V is a finite-dimensional continuous representation of G over $\mathbb {Q}_p$ . We are going to construct a morphism: $\operatorname {\mathrm {Lie}}(\Gamma )\to \operatorname {\mathrm {End}}_{B_{\infty }}(B_{\infty }\otimes _{\mathbb {Q}_p}V)$ ; that is, a Lie algebra representation of $\operatorname {\mathrm {Lie}}(\Gamma )$ on $B_{\infty }\otimes _{\mathbb {Q}_p}V$ .

As G is compact, there exists a G-stable $\mathbb {Z}_p$ -lattice $T\subset V$ . Let $(B_{\infty })^{\circ }\subset B_{\infty }$ be the subset of power-bounded elements (equivalently elements with spectral norm at most $1$ ). Note that $(B_{\infty })^{\circ }\otimes _{\mathbb {Z}_p}T$ carries a diagonal action of G and an action of $\Gamma $ on the first factor. This defines an action of $G\times \Gamma $ on $(B_{\infty })^{\circ }\otimes _{\mathbb {Z}_p}T$ .

Proof. This follows from Proposition 3.3.1 of [Reference Berger and ColmezBC08] by choosing $c_3=c$ and $c_1,c_2$ sufficiently small such that $c_1+2c_2+2c_3< 1$ . Note that in our setup, we may choose the constants $c_1,c_2,c_3$ in the Tate–Sen conditions to be arbitrarily small. The first three parts are the same as [Reference Berger and ColmezBC08, 3.3.1]. The last part is a consequence of the third one.

Proof. For any $k\geq m$ , the normalised trace $\overline {\mathrm {tr}}_{X_{G_0},k}:B_{G_0,\infty }\to B_{G_0,k}$ induces a map

$$ \begin{align*}(B_{\infty}\otimes_{\mathbb{Q}_p} V)^{G_0,p^m\Gamma-\mathrm{an}}=(B_{G_0,\infty}\otimes_{B_{G_0,n}^{+}} D_{G_0,n}^{+}(T))^{p^m\Gamma-\mathrm{an}}\to (B_{G_0,k}\otimes_{B_{G_0,n}^{+}} D_{G_0,n}^{+}(T))^{p^m\Gamma-\mathrm{an}}.\end{align*} $$

As the action of $p^m\Gamma $ on $B_{G_0,k}$ is trivial on $p^k\Gamma $ , we have a natural decomposition

$$ \begin{align*}B_{G_0,k}=\bigoplus_{\chi: p^m\Gamma/p^k\Gamma\to C^{\times}}B_{G_0,k}[\chi]\end{align*} $$

into the direct sum of $\chi $ -isotypic components, where $\chi $ runs through all characters of $p^m\Gamma /p^k\Gamma $ . Note that $\chi $ is an analytic function on $p^m\Gamma $ only when $\chi $ is trivial. Hence,

$$ \begin{align*}(B_{G_0,k}\otimes_{B_{G_0,n}^{+}} D_{G_0,n}^{+}(T))^{p^m\Gamma-\mathrm{an}}=(B_{G_0,k})^{p^m\Gamma}\otimes_{B_{G_0,n}^{+}} D_{G_0,n}^{+}(T)=B_{G_0,m}\otimes_{B_{G_0,n}^{+}} D_{G_0,n}^{+}(T).\end{align*} $$

The rest of the proof is the same as the one of Lemma 3.2.5.

Proof. It is easy to check all these properties on the $\Gamma $ -locally analytic vectors in $(B_{\infty }\otimes _{\mathbb {Q}_p}V)^G$ . We omit the details here.

3.3.1. We will first show that the action in Proposition 3.2.13 factors through $B\otimes \operatorname {\mathrm {Lie}}(G)$ . As a byproduct, this will imply Theorem 3.1.2. The strategy is to apply Proposition 3.2.13 to $V=\mathscr {C}^{\mathrm {an}}(G,\mathbb {Q}_p)$ , the space of analytic functions on G. However, as this is an infinite-dimensional vector space over $\mathbb {Q}_p$ , it requires a limiting argument plus some extra work.

Let $G_0\subset G$ be a compact open subgroup equipped with an integer-valued, saturated p-valuation as in Section 2.1.1. By Proposition 2.1.3, we may replace $G_0$ by a smaller subgroup and assume that there is a dense sub-algebra $\varinjlim _{k\in \mathbb {N}} V_k\subset \mathscr {C}^{\mathrm {an}}(G_0,\mathbb {Q}_p)$ , where each $V_k$ is a finite-dimensional subspace of $\mathscr {C}^{\mathrm {an}}(G_0,\mathbb {Q}_p)$ stable under both the left and right translation actions of $G_0$ . We will always view $V_k$ as a representation of $G_0$ using the left translation action.

Let $\mathscr {C}^{\mathrm {an}}(G_0,\mathbb {Q}_p)^o$ be the unit ball of $\mathscr {C}^{\mathrm {an}}(G_0,\mathbb {Q}_p)$ with respect to the norm $\|\cdot \|_{G_0}$ (see Section 2.1.1 for the notation here). Then

$$ \begin{align*}V_k^o:=V_k\cap \mathscr{C}^{\mathrm{an}}(G_0,\mathbb{Q}_p)^o\end{align*} $$

is a $G_0$ -stable lattice. Moreover, it follows from Lemma 2.1.2 that $V_k^o/p$ is fixed by an open subgroup $G_1$ of $G_0$ for all k. Now we apply Proposition 3.2.11 to $V_k$ , with $G=G_0$ : fix a constant c as in the proposition and $n\geq n(G_1)$ . There is $D^+_{G_1,n}(V_k^o)\subset (B_{\infty })^o\otimes V_k^o$ such that

$$ \begin{align*}(B_{\infty})^{\circ}\otimes_{B_{G_1,n}^{+}} D_{G_1,n}^{+}(V_k^o)=(B_{\infty})^{\circ}\otimes_{\mathbb{Z}_p}V_k^o.\end{align*} $$

By the uniqueness of $D_{G_1,n}^{+}(V_k^o)$ , these $\{D_{G_1,n}^{+}(V_k^o)\}_k$ form a direct system. Hence, we may take the direct limit of this equality over k:

$$ \begin{align*}(B_{\infty})^{\circ}\otimes_{B_{G_1,n}^{+}} \varinjlim_k D_{G_1,n}^{+}(V_k^o)=(B_{\infty})^{\circ}\otimes_{\mathbb{Z}_p}\varinjlim_k V_k^o.\end{align*} $$

By taking the p-adic completion and inverting p, we obtain

$$ \begin{align*}B_{\infty}\widehat{\otimes}_{B_{G_1,n}} D_{G_1,n}=B_{\infty}\widehat{\otimes}_{\mathbb{Q}_p}\mathscr{C}^{\mathrm{an}}(G_0,\mathbb{Q}_p).\end{align*} $$

Here $D^+_{G_1,n}$ is the p-adic completion of $\varinjlim _k D_{G_1,n}^{+}(V_k^o)$ and $D_{G_1,n}=D^+_{G_1,n}\otimes _{\mathbb {Z}_p}\mathbb {Q}_p$ equipped with the p-adic topology. The right-hand side becomes $\mathscr {C}^{\mathrm {an}}(G_0,\mathbb {Q}_p)$ as $\varinjlim _k V_k$ is dense inside of it.

Now we can construct an action of $\operatorname {\mathrm {Lie}}(\Gamma )$ as before: there is an integer $m\geq 0$ such that $(\gamma -1)^mD^+_{G_1,n}(V_k^o)\subset pD^+_{G_1,n}(V_k^o)$ for all $k\geq 0$ and $\gamma \in \Gamma $ . Hence,

$$ \begin{align*}(\gamma-1)^mD^+_{G_1,n}\subset pD^+_{G_1,n}\end{align*} $$

for any $\gamma \in \Gamma $ . By Example 2.1.9, the action of $\Gamma $ on $D_{G_1,n}$ is locally analytic. Its Lie algebra action can be extended $B_{\infty }$ -linearly to an action on $B_{\infty }\widehat {\otimes }_{B_{G_1,n}} D_{G_1,n}=B_{\infty }\widehat {\otimes }_{\mathbb {Q}_p}\mathscr {C}^{\mathrm {an}}(G_0,\mathbb {Q}_p)$ :

$$ \begin{align*}\phi_{G_0}:\operatorname{\mathrm{Lie}}(\Gamma)\to\operatorname{\mathrm{End}}_{B_{\infty}}(B_{\infty}\widehat{\otimes}_{\mathbb{Q}_p}\mathscr{C}^{\mathrm{an}}(G_0,\mathbb{Q}_p)).\end{align*} $$

Again this action uniquely extends the natural action of $\operatorname {\mathrm {Lie}}(\Gamma )$ on the $\Gamma $ -locally analytic vectors in $(B_{\infty }\widehat {\otimes }_{\mathbb {Q}_p}\mathscr {C}^{\mathrm {an}}(G_0,\mathbb {Q}_p))^{G_0}$ , where $G_0$ acts diagonally on $B_{\infty }$ and $\mathscr {C}^{\mathrm {an}}(G_0,\mathbb {Q}_p)$ . In fact, let $D_{G_0,n}^{+}=(D_{G_1,n}^{+})^{G_0}$ and $D_{G_0,n}=(D_{G_1,n})^{G_0}=D_{G_0,n}^{+}\otimes _{\mathbb {Z}_p}{\mathbb {Q}_p}$ . Then by arguments similar to Lemma 3.2.12 and the paragraph below it, we have the following lemma.

Proof. Clearly, $\phi _{G_0}$ factors through $B_{\infty }\otimes _{\mathbb {Q}_p}\operatorname {\mathrm {Lie}}(G_0)$ . As it commutes with $\Gamma $ , it also factors through $(B_{\infty }\otimes _{\mathbb {Q}_p}\operatorname {\mathrm {Lie}}(G_0))^{\Gamma }=B\otimes _{\mathbb {Q}_p}\operatorname {\mathrm {Lie}}(G_0)$ by Lemma 3.2.9.

Proof. Recall that (see Section 2.1.1)

$$ \begin{align*}B^{G_0-\mathrm{an}}=(B\widehat{\otimes}_{\mathbb{Q}_p}\mathscr{C}^{\mathrm{an}}(G_0,\mathbb{Q}_p))^{G_0}=(B_{\infty}\widehat{\otimes}_{\mathbb{Q}_p}\mathscr{C}^{\mathrm{an}}(G_0,\mathbb{Q}_p))^{G_0\times\Gamma},\end{align*} $$

where the second equality follows from Lemma 3.2.9. Hence, by our construction of $\phi _{G_0}$ , the action of $\operatorname {\mathrm {Lie}}(\Gamma )$ is trivial on $B^{G_0-\mathrm {an}}$ . An easy computation shows that this action is nothing but $\phi _{\widetilde {X}}$ . Note that this argument works for all sufficiently small $G_0$ , which clearly implies the claim in the corollary.

Proof. Let $G_0$ be a sufficiently small open subgroup of G as in the beginning of Section 3.3.1. Moreover, we may assume V is a $G_0$ -analytic representation. Let $V^*$ be the dual vector space of V. Then taking the matrix coefficients of V induces a map

$$ \begin{align*}m_V:V\otimes_{\mathbb{Q}_p}V^*\to\mathscr{C}^{\mathrm{an}}(G_0,\mathbb{Q}_p),\end{align*} $$

$$ \begin{align*}m_V(v\otimes l)(g)=l(g^{-1}\cdot v),~v\in V,l\in V^*,g\in G_0.\end{align*} $$

This map is $G_0$ -equivariant where $G_0$ acts only on the first factor of $V\otimes V^*$ and acts via the left translation on $\mathscr {C}^{\mathrm {an}}(G_0,\mathbb {Q}_p)$ . The induced map

$$ \begin{align*}\mathbf{1}_{B_{\infty}}\otimes m_V:B_{\infty}\otimes_{\mathbb{Q}_p}V\otimes_{\mathbb{Q}_p}V^*\to B_{\infty}\widehat{\otimes}_{\mathbb{Q}_p}\mathscr{C}^{\mathrm{an}}(G_0,\mathbb{Q}_p)\end{align*} $$

intertwines $\phi _{V\otimes V^*}=\phi _V\otimes \mathbf {1}_{V^*}$ and $\phi _{G_0}=\phi _{\widetilde {X}}$ . One can check this on the $G_0$ -fixed, $\Gamma $ -locally analytic vectors. Trivially, this map also intertwines $\phi _{\widetilde {X}}\otimes \mathbf {1}_{V^*}$ and $\phi _{\widetilde {X}}$ . Now for any nonzero $v\in V$ , there exists $l\in V^*$ such that $m_V(v\otimes l)\neq 0$ . From this, it is easy to see that $\phi _V$ has to factor through $\phi _{\widetilde {X}}$ .

Proof. It is enough to check that the formulation of $\phi _V$ in Proposition 3.2.13 is functorial in a similar sense. But this basically follows from the construction. We omit the details here.

Proof. We will freely use notation introduced in Section 3.3.1. Let $G_0$ be a compact open subgroup of G considered there. By Lemma 3.3.2, we have an isomorphism for sufficiently large n

$$ \begin{align*}B_{\infty}\widehat{\otimes}_{B_{G_0,n}} (B_{\infty})^{G_0-\mathrm{an},p^n\Gamma-\mathrm{an}}=B_{\infty}\widehat{\otimes}_{\mathbb{Q}_p}\mathscr{C}^{\mathrm{an}}(G_0,\mathbb{Q}_p).\end{align*} $$

It follows from Section 2.1.4 that this is equivariant with respect to the following actions of $G_0$ : the natural action on $(B_{\infty })^{G_0-\mathrm {an},p^n\Gamma -\mathrm {an}}$ , the right translation action on $\mathscr {C}^{\mathrm {an}}(G_0,\mathbb {Q}_p)$ and trivial actions on both  $B_{\infty }$ . Using this action of $G_0$ , we get $B_{\infty }$ -linear actions of $B_{\infty }\otimes _{\mathbb {Q}_p}\operatorname {\mathrm {Lie}}(G)$ on both sides. In particular, D annihilates both sides. However, if $D\neq 0$ , we can always find a $B_{\infty }$ -valued analytic function on $G_0$ which is not annihilated by D. Contradiction. This proves our claim.

Proof. We note that by the functorial property Corollary 3.3.9, $\phi _{\widetilde {X}_{\infty }}\equiv \phi _{{X}_{\infty }}\mod B_{\infty }\otimes _{\mathbb {Q}_p}\operatorname {\mathrm {Lie}}(G)$ ; hence, the composite of $\phi _{\widetilde {X}_{\infty }}$ and the projection map

$$ \begin{align*}\operatorname{\mathrm{Lie}}(\Gamma)\xrightarrow{\phi_{\widetilde{X}_{\infty}}}B_{\infty}\otimes_{\mathbb{Q}_p}\operatorname{\mathrm{Lie}}(G\times\Gamma)\to B_{\infty}\otimes_{\mathbb{Q}_p}\operatorname{\mathrm{Lie}}(\Gamma)\end{align*} $$

is simply the identity map. In particular, we can find $b\in B_{\infty }$ and $d\in \phi _{\widetilde {X}\infty }(\operatorname {\mathrm {Lie}}(\Gamma ))$ such that $D-bd\in B_{\infty }\otimes _{\mathbb {Q}_p}\operatorname {\mathrm {Lie}}(G)$ . The rest follows from Corollary 3.3.5 and Proposition 3.3.10.

3.4.1. In the previous section, we proved the existence of $\theta $ in Theorem 3.1.2. Note that the construction depends on a choice of an étale map $f_1:X\to Y=\mathbb {T}^1$ or $\mathbb {B}^1$ in Section 3.2.1. In this section, we will show that in fact $\theta $ is well-defined up to $A^{\times }$ . People who are interested in global applications can skip reading this part as this part will not play any role later.

3.4.2. Consider another étale map $f_2:X\to Y$ which factors as a composite of rational embeddings and finite étale maps. Let

$$ \begin{align*}\widetilde{X}':=X\times_{f_2,Y}Y_{\infty}\end{align*} $$

be the pullback of the profinite covering $Y_{\infty }$ along $f_2$ . See Section 3.2.1 for the definition of $Y_{\infty }$ . In the notation below, we put a superscript $'$ for everything constructed using $f_2$ instead of $f_1$ . For example, $\widetilde X'=\operatorname {\mathrm {Spa}}(B',B^{\prime +})$ is an affinoid ‘log $G'$ -Galois pro-étale perfectoid covering’ of X with $G'=\Gamma $ . We would like to compute $\phi _{\widetilde {X}'}:\operatorname {\mathrm {Lie}}(\Gamma )\to B'\otimes _{\mathbb {Q}_p}\operatorname {\mathrm {Lie}}(G')=B'\otimes _{\mathbb {Q}_p}\operatorname {\mathrm {Lie}}(\Gamma )$ first. In general, we will use the functorial property to reduce to this computation.

We will freely use the notation introduced in the previous subsections. In particular, $\widetilde {X}_{\infty }'=\operatorname {\mathrm {Spa}}(B_{\infty }',B_{\infty }^{\prime +})$ is a $G'\times \Gamma $ -covering of X. It is also a $G'$ -covering of $X_{\infty }=\operatorname {\mathrm {Spa}}(A_{\infty },A_{\infty }^+)$ and a $\Gamma $ -covering of $\widetilde {X}'=\operatorname {\mathrm {Spa}}(B',B^{\prime +})$ . Note that by Abhyankar’s lemma, both coverings are profinite étale of perfectoid algebras. By the Hochschild–Serre spectral sequence, we have

$$ \begin{align*}H^1_{\mathrm{cont}}(G'\times\Gamma,B_{\infty}')\cong H^1_{\mathrm{cont}}(\Gamma,(B_{\infty}')^{G'\times{\{\mathbf{1}\}}})=H^1_{\mathrm{cont}}(\Gamma,A_{\infty})\cong H^1_{\mathrm{cont}}(\Gamma,A).\end{align*} $$

The first isomorphism follows from $H^1_{\mathrm {cont}}(G'\times \{\mathbf {1}\},B_{\infty }')=0$ , which is a consequence of the almost purity theorem. The last isomorphism follows from the existence and properties of the Tate normalised trace $\overline {\mathrm {tr}}_{X,0}:A_{\infty }\to A$ . We denote by $\tau _1$ the composite isomorphism $H^1_{\mathrm {cont}}(G'\times \Gamma ,B_{\infty }')\xrightarrow {\sim } H^1_{\mathrm {cont}}(\Gamma ,A)$ .

Symmetrically, we can get another isomorphism $\tau _2:H^1_{\mathrm {cont}}(G'\times \Gamma ,B_{\infty }')\xrightarrow {\sim }H^1_{\mathrm {cont}}(\Gamma ,A)$ by

$$ \begin{align*}H^1_{\mathrm{cont}}(G'\times\Gamma,B_{\infty}')\cong H^1_{\mathrm{cont}}(G',(B_{\infty}')^{{\{\mathbf{1}\}}\times\Gamma})=H^1_{\mathrm{cont}}(G',B')=H^1_{\mathrm{cont}}(\Gamma,B')\cong H^1_{\mathrm{cont}}(\Gamma,A).\end{align*} $$

As $H^1_{\mathrm {cont}}(\Gamma ,A)$ is a free A-module of rank 1, the composite $\tau _1\circ \tau _2^{-1}\in \operatorname {\mathrm {End}}_A(H^1_{\mathrm {cont}}(\Gamma ,A))=A$ is given by an element $a\in A^{\times }$ .

Proof. Fix a topological generator $\gamma $ of $\Gamma $ . Unravelling all of the isomorphisms above, we can find an element $b\in B_{\infty }'$ such that

(3.4.1) $$ \begin{align} (\gamma,\mathbf{1})\cdot b-b&= -1 \end{align} $$

(3.4.2) $$ \begin{align} (\mathbf{1},\gamma)\cdot b-b&= a. \end{align} $$

Here $(\gamma ,\mathbf {1}), (\mathbf {1},\gamma )$ are elements in $G'\times \Gamma =\Gamma \times \Gamma $ . In particular, both actions of $\Gamma $ on b are analytic.

As pointed out in Remark 3.3.7, in order to compute $\phi _{\widetilde {X}'}$ , we choose a faithful representation of $G'=\Gamma $ on $V=\mathbb {Q}_p^{\oplus 2}$ :

$$ \begin{align*}\Gamma\to\mathrm{GL}_2(\mathbb{Q}_p),\, \gamma\mapsto \begin{pmatrix} 1 & 1 \\ 0 &1 \end{pmatrix}.\end{align*} $$

Then $B_{\infty }'\otimes _{\mathbb {Q}_p} V=(B_{\infty }')^{\oplus 2}$ has a $G'$ -fixed basis over $B_{\infty }'$ :

$$ \begin{align*}e_1=(1,0),e_2=(b,1).\end{align*} $$

Moreover, it is easy to see that the action of $\Gamma $ on this basis is (locally) analytic. Therefore, $\phi _V$ is obtained by $B_{\infty }'$ -linearly extending the action of $\operatorname {\mathrm {Lie}}(\Gamma )$ on $e_1,e_2$ . Let $t\in \operatorname {\mathrm {Lie}}(\Gamma )$ be the logarithm of $\gamma $ . Then $t\cdot e_1=0$ and it follows from Equation (3.4.2) that $t\cdot e_2=ae_1$ . Hence,

$$ \begin{align*}\phi_V:\operatorname{\mathrm{Lie}}(\Gamma)\to B_{\infty}'\otimes_{\mathbb{Q}_p}\mathfrak{gl}_2(\mathbb{Q}_p)\end{align*} $$

$$ \begin{align*}t\mapsto \begin{pmatrix} 0 & a \\ 0 &0 \end{pmatrix}.\end{align*} $$

Comparing this with the definition of V, we see that $\phi _{\widetilde {X}'}$ is just multiplication by a.

3.4.4. Now we are ready to prove $\phi _{\widetilde {X}}$ is well-defined up to $A^{\times }$ . We will keep using the notation introduced in Section 3.4.2. Suppose $\widetilde {X}$ is ‘log G-Galois pro-étale perfectoid covering’ of X. Now using $f_2:X\to Y$ instead of $f_1:X\to Y$ in the setup Section 3.2.1 and redoing everything before, then rather than $\phi _{\widetilde {X}}$ , we get

$$ \begin{align*}\phi^{\prime}_{\widetilde{X}}:\operatorname{\mathrm{Lie}}(\Gamma)\to B\otimes_{\mathbb{Q}_p}\operatorname{\mathrm{Lie}}(G).\end{align*} $$

Proof. First note that the special case $\widetilde {X}=X_{\infty }=\operatorname {\mathrm {Spa}}(A_{\infty },A_{\infty }^+)$ is essentially proved by the same computation as above (after switching the role of $f_1$ and $f_2$ ). In general, let V be a finite-dimensional continuous representation of G over $\mathbb {Q}_p$ . It is enough to show that $a^{-1}\phi _V$ on $B_{\infty }\otimes _{\mathbb {Q}_p} V$ agrees with the action of $\operatorname {\mathrm {Lie}}(\Gamma )$ induced from $\phi ^{\prime }_{\widetilde {X}}$ . Consider $\widetilde {X}_{\infty }:=\operatorname {\mathrm {Spa}}(B_{\infty },B_{\infty }^+)$ . This is a ‘log $G\times \Gamma $ -Galois pro-étale perfectoid covering’ of X. Then by the functorial property Corollary 3.3.9,

$$ \begin{align*}\phi^{\prime}_{\widetilde{X}}\equiv \phi^{\prime}_{\widetilde{X}_{\infty}} \mod B_{\infty}\otimes_{\mathbb{Q}_p}\operatorname{\mathrm{Lie}}(\Gamma).\end{align*} $$

Hence, the actions of $\operatorname {\mathrm {Lie}}(\Gamma )$ on $B_{\infty }\otimes _{\mathbb {Q}_p} V$ induced from $\phi ^{\prime }_{\widetilde {X}}$ and $\phi ^{\prime }_{\widetilde {X}_{\infty }}$ are the same because $\Gamma $ acts trivially on V. So it suffices to compare $a^{-1}\phi _V$ and $\phi ^{\prime }_{\widetilde {X}_{\infty }}$ .

Since both actions are $B_{\infty }$ -linear, it follows from Proposition 3.2.11 that we only need to compare two actions on the $\Gamma $ -locally analytic vectors in $(B_{\infty }\otimes _{\mathbb {Q}_p} V)^{G}$ . Now on this G-fixed subspace, $\phi _V$ acts by the natural Lie algebra action of $\operatorname {\mathrm {Lie}}(\Gamma )$ , while by Remark 3.3.8 and the functorial property Corollary 3.3.9,

$$ \begin{align*}\phi^{\prime}_{\widetilde{X}_{\infty}}\equiv\phi^{\prime}_{X_{\infty}}\mod B_{\infty}\otimes_{\mathbb{Q}_p}\operatorname{\mathrm{Lie}}(G),\end{align*} $$

the action on $(B_{\infty }\otimes _{\mathbb {Q}_p} V)^{G,\Gamma -\mathrm {la}}$ induced by $\phi ^{\prime }_{\widetilde {X}_{\infty }}$ is just $\phi ^{\prime }_{X_{\infty }}$ . Hence, the desired equality is a direct consequence of the special case $\widetilde {X}=X_{\infty }$ .

3.5.1. The goal of this subsection is to give a sufficient condition for $\theta $ to be nonzero. We will continue using the notation introduced before. In particular, there is a fixed étale map $f_1:X\to Y$ as in Section 3.2.1. First, let’s recall Faltings’s extension in the rigid analytic variety setting; cf. Corollary 6.14 of [Reference ScholzeSch13a], Corollary 2.4.5 of [Reference Diao, Lan, Liu and ZhuDLLZ18]. However, as both references assume X is defined over a discretely valued complete non-Archimedean extension of $\mathbb {Q}_p$ , we give a rather ad hoc definition here which will be sufficient for our purpose.

Fix a generator $\gamma $ of $\Gamma $ from now on and consider the following unipotent representation of $\Gamma $ on $V=\mathbb {Q}_p^{\oplus 2}$ :

$$ \begin{align*}\Gamma\to\mathrm{GL}_2(\mathbb{Q}_p):~\gamma\mapsto \begin{pmatrix} 1 & 1 \\ 0 &1 \end{pmatrix}.\end{align*} $$

Clearly, V is an extension of the trivial representation by itself:

$$ \begin{align*}0\to\mathbb{Q}_p\to V\to \mathbb{Q}_p\to0.\end{align*} $$

Tensor this exact sequence with $B_{\infty }$ over $\mathbb {Q}_p$ and take $\Gamma $ -invariants with respect to the diagonal actions: