${\mathbb{P}}^{d}_{K}\setminus{\mathbb{P}}^{r}_{K}(\varepsilon)^{-}=\bigcup_{i={r}+1}^{d}U_{i,\varepsilon}$
(2.12)

by the Weierstraß domains

$\displaystyle U_{i,\varepsilon}=D_{+}(X_{i})_{\varepsilon}$
$\displaystyle\vcentcolon=\left\\{[z_{0}{\,:\,}\ldots{\,:\,}z_{d}]\in{\mathbb{P}}^{d}_{K}\,\middle|\,\forall
j=0,\ldots,d:\varepsilon\,{\lvert{z_{j}}\rvert}\leq{\lvert{z_{i}}\rvert}\right\\}$
$\displaystyle\phantom{\vcentcolon}=\left\\{[z_{0}{\,:\,}\ldots{\,:\,}z_{d}]\in{\mathbb{P}}^{d}_{K}\,\middle|\,\forall
j=0,\ldots,d:\left\lvert\lambda\frac{z_{j}^{m}}{z_{i}^{m}}\right\rvert\leq
1\right\\},$

cf. [7, 6.1.5 Thm. 4]. We will denote this covering (2.12) by
${\mathcal{U}}_{\varepsilon}$ in the following. Note that $U_{i,\varepsilon}$
is contained in the standard open affine subset $D_{+}(X_{i})$ of the
projective space.

Moreover, for $0\leq\varepsilon<1$, $\varepsilon\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$, let

$U_{i,\varepsilon}^{-}=D_{+}(X_{i})_{\varepsilon}^{-}\vcentcolon=\left\\{[z_{0}{\,:\,}\ldots{\,:\,}z_{d}]\in{\mathbb{P}}^{d}_{K}\,\middle|\,\forall
j=0,\ldots,d:\varepsilon\,{\lvert{z_{j}}\rvert}<{\lvert{z_{i}}\rvert}\right\\}.$

Then $U_{i,\varepsilon}^{-}$ becomes an open admissible subdomain of
${\mathbb{P}}^{d}_{K}$ which is quasi-Stein via the admissible covering

$U_{i,\varepsilon}^{-}=\bigcup_{{\lvert{{\mathcal{O}}_{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}}\rvert}\ni\varepsilon_{m}\searrow\varepsilon}U_{i,\varepsilon_{m}},$

for any strictly decreasing sequence of
$(\varepsilon_{m})_{m\in{\mathbb{N}}}\subset{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}^{\times}}\rvert}$, with $\varepsilon_{m}>\varepsilon$,
$\varepsilon_{m}\rightarrow\varepsilon$. The condition on the associated
homomorphisms of affinoid algebras to have dense image is fulfilled for every
inclusion of Weierstraß domains [7, 7.3.4 Prop. 2]. In the extreme case of
$\varepsilon=0$, we have $U_{i,0}^{-}=U_{i}^{\mathrm{rig}}$, cf. [7, 9.3.4
Example 2]. Here $U_{i}^{\mathrm{rig}}$ denotes the rigid analytification of
the open affine subscheme
$U_{i}\vcentcolon=D_{+}(X_{i})\subset{\mathbb{P}}_{K}^{d}$.

We let ${\mathcal{U}}_{\varepsilon}^{-}$ denote the admissible covering

${\mathbb{P}}^{d}_{K}\setminus{\mathbb{P}}^{r}_{K}(\varepsilon)=\bigcup_{i={r}+1}^{d}U_{i,\varepsilon}^{-}.$
(2.13)

In the extreme case $\varepsilon=0$, we also write
$({\mathbb{P}}^{r}_{K})^{\mathrm{rig}}\subset{\mathbb{P}}_{K}^{d}$ when we
want to signify that the complement of
${\mathbb{P}}_{K}^{r}\subset({\mathbb{P}}_{K}^{d})^{\mathrm{rig}}$ is an
admissible open subset

${\mathbb{P}}^{d}_{K}\setminus\big{(}{\mathbb{P}}^{r}_{K}\big{)}^{\mathrm{rig}}=\bigcup_{i={r}+1}^{d}U_{i}^{\mathrm{rig}}=\bigcup_{i={r}+1}^{d}\bigcup_{{\lvert{{\mathcal{O}}_{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}}\rvert}\ni\varepsilon_{m}\searrow 0}U_{i,\varepsilon_{m}}.$

Moreover, we let ${\mathcal{U}}$ denote the standard covering by open affine
subschemes

${\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{r}=\bigcup_{i={r}+1}^{d}U_{i}.$
(2.14)

Finally, we write, for any subset $I\subset\\{0,\ldots,d\\}$,

$U_{I,\varepsilon}\vcentcolon=\bigcap_{i\in I}U_{i,\varepsilon},$ (2.15)

and similarly $U_{I,\varepsilon}^{-}$, $U_{I}$, and $U_{I}^{\mathrm{rig}}$. By
convention we set
$U_{\emptyset}=U_{\emptyset,\varepsilon}=U_{\emptyset,\varepsilon}^{-}\vcentcolon={\mathbb{P}}_{K}^{d}$.

######  Lemma 2.4.1.

Let $0<\varepsilon<1$ with $\varepsilon\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$. Let $I\subset\\{0,\ldots,d\\}$ and denote by
${\mathcal{E}}^{\mathrm{alg}}(U_{I})$ the sections of ${\mathcal{E}}$ on the
algebraic variety $U_{I}$. Then the homomorphism
${\mathcal{E}}^{\mathrm{alg}}(U_{I})\rightarrow{\mathcal{E}}(U_{I,\varepsilon})$
which is induced from $U_{I,\varepsilon}$ being an admissible open subdomain
of $U_{I}^{\mathrm{rig}}$ is injective and its image is dense. In particular,
${\mathcal{E}}(U_{I,\varepsilon})$ is the completion of
${\mathcal{E}}^{\mathrm{alg}}(U_{I})$ when the latter is considered with the
topology coming from the Banach topology of
${\mathcal{E}}(U_{I,\varepsilon})$.

###### Proof.

We may assume that $I=\\{i_{0},\ldots,i_{m}\\}\neq\emptyset$. Then the
isomorphism

$\displaystyle U_{i_{0}}=D_{+}(X_{i_{0}})$
$\displaystyle\overset{\cong}{\longrightarrow}{\mathrm{Spec}}\,K\big{[}T_{(j,i_{0})}\mid
j=0,\ldots,d,j\neq i_{0}\big{]},$ $\displaystyle\frac{X_{j}}{X_{i_{0}}}$
$\displaystyle\longmapsfrom T_{(j,i_{0})},$

of schemes induces an isomorphism
$U_{I}\cong{\mathrm{Spec}}(A/{\mathfrak{a}})$ where

$\displaystyle A$
$\displaystyle=K\big{[}T_{(j,i_{k})}\,\big{|}\,j=0,\ldots,d,k=0,\ldots,m\text{,
with }j\neq i_{k}\big{]},$ $\displaystyle{\mathfrak{a}}$
$\displaystyle=\big{(}T_{(i_{j},i_{k})}T_{(i_{k},i_{j})}-1\,\big{|}\,j,k=0,\ldots,m\text{,
with }j\neq k\big{)}.$

Recall that $U_{I}^{\mathrm{rig}}$ is defined via the admissible covering [7,
9.3.4 Example 2]

$U_{I}^{\mathrm{rig}}=\bigcup_{n\geq 0}U_{I,{\lvert{{\pi}}\rvert}^{n}},$

and we have isomorphisms
$U_{I,{\lvert{{\pi}}\rvert}^{n}}\cong{\mathrm{Sp}}(A_{n}/{\mathfrak{a}}A_{n})$,
for

$\displaystyle
A_{n}=K\big{\langle}{\pi}^{n}T_{(j,i_{k})}\,\big{|}\,j=0,\ldots,d,k=0,\ldots,m\text{,
with }j\neq i_{k}\big{\rangle}.$

We see that $A/{\mathfrak{a}}\hookrightarrow A_{n}/{\mathfrak{a}}A_{n}$ has
dense image as $A$ is dense in $A_{n}$. Furthermore, the Weierstraß subdomain
$U_{I,\varepsilon}\subset U_{I}^{\mathrm{rig}}$ is contained in some
$U_{I,{\lvert{{\pi}}\rvert}^{n}}$ so that
$A/{\mathfrak{a}}\hookrightarrow{\mathcal{O}}(U_{I,\varepsilon})$ is dense as
well [7, 7.3.4 Prop. 2]. This settles the case of the structure sheaf
${\mathcal{E}}={\mathcal{O}}$.

For a general ${\mathrm{\bf G}}$-equivariant vector bundle ${\mathcal{E}}$ on
${\mathbb{P}}_{K}^{d}$ by Lemma 2.4.4, we find a trivialization
${\mathcal{E}}^{\mathrm{alg}}{\\!\,\mid_{U_{i_{0}}}}\cong({\mathcal{O}}^{\mathrm{alg}})^{\oplus
n}{\\!\,\mid_{U_{i_{0}}}}$, for $n\vcentcolon={\rm rk}({\mathcal{E}})$.
Therefore the claim follows from the compatible isomorphisms
${\mathcal{E}}^{\mathrm{alg}}(U_{I})\cong{\mathcal{O}}^{\mathrm{alg}}(U_{I})^{\oplus
n}$ and
${\mathcal{E}}(U_{I,\varepsilon})\cong{\mathcal{O}}(U_{I,\varepsilon})^{\oplus
n}$. ∎

######  Remark 2.4.2.

One can give a concrete description of the Gauss norm
${\lvert{{\,\\_\,}}\rvert}_{\varepsilon}$ on the affinoid algebra
${\mathcal{O}}(U_{I,\varepsilon})$ (and in turn on
${\mathcal{O}}^{\mathrm{alg}}(U_{I})$), cf. [56, Proof of Lemma 1.3.1]:

$\bigg{\lvert}\sum_{{\underline{k}}\in\Lambda_{I}}a_{{\underline{k}}}\,X_{0}^{k_{0}}\cdots
X_{d}^{k_{d}}\bigg{\rvert}_{\varepsilon}=\sup_{{\underline{k}}\in\Lambda_{I}}{\lvert{a_{{\underline{k}}}}\rvert}\bigg{(}\frac{1}{\varepsilon}\bigg{)}^{{\lvert{\max(0,{\underline{k}})}\rvert}}.$
(2.16)

Here we use the notation

$\displaystyle\Lambda_{I}$
$\displaystyle\vcentcolon=\bigg{\\{}{\underline{k}}\in{\mathbb{Z}}^{d+1}\bigg{|}\sum_{i=0}^{d}k_{i}=0\text{
, and }\forall i\in\\{0,\ldots,d\\}\setminus I:k_{i}\geq 0\bigg{\\}},$

and

$\displaystyle\max(0,{\underline{k}})$
$\displaystyle\vcentcolon=\big{(}\max(0,k_{0}),\ldots,\max(0,k_{d})\big{)}.$

We will need the following result about the Čech complex associated to the
covering (2.14)888The statement of Theorem 2.4.3 is found in the proof of [56,
Lemma 1.3.1]. However the justification given there contains some flaws..

######  Theorem 2.4.3.

Let ${\mathcal{E}}$ be a ${\mathrm{\bf G}}$-equivariant vector bundle on
${\mathbb{P}}_{K}^{d}$, and $0<\varepsilon<1$ with
$\varepsilon\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$. For the Čech complex
$C^{\bullet}({\mathcal{U}},{\mathcal{E}})$ associated with the covering (2.14)
which computes the coherent cohomology
$H^{\bullet}({\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{r},{\mathcal{E}})$,
the differentials

$d^{q}\colon C^{q}({\mathcal{U}},{\mathcal{E}})\longrightarrow
C^{q+1}({\mathcal{U}},{\mathcal{E}})$

are strict continuous homomorphisms when each
${\mathcal{E}}(U_{I})\,(={\mathcal{E}}^{\mathrm{alg}}(U_{I}))\subset{\mathcal{E}}(U_{I,\varepsilon})$
is endowed with the topology coming from the Banach topology of
${\mathcal{E}}(U_{I,\varepsilon})$, for $I\subset\\{{r}+1,\ldots,d\\}$.

Let ${\mathrm{\bf T}}\subset{\mathrm{\bf G}}$ denote the split maximal torus
of diagonal matrices. For the group of its characters, we have the usual
identification $X({\mathrm{\bf T}})\cong{\mathbb{Z}}^{d+1}$ by choosing the
characters

$\epsilon_{i}\colon{\mathrm{\bf T}}\longrightarrow{\mathbb{G}}_{m}\,,\quad{\rm
diag}(t_{0},\ldots,t_{d})\longmapsto t_{i}\quad\text{, for $i=0,\ldots,d$,}$

as a ${\mathbb{Z}}$-basis. Recall that for any ${\mathrm{\bf T}}$-module $V$
in the sense of [41, I. §2.7], we have a decomposition into weight spaces [41,
I. §2.11]

$V\cong\bigoplus_{{\underline{\lambda}}\in X({\mathrm{\bf
T}})}V_{\underline{\lambda}}\quad\text{, for
$V_{\underline{\lambda}}\vcentcolon=\big{\\{}v\in V\,\big{|}\,\text{$\forall$
$K$-algebras $R$, $\forall t\in{\mathrm{\bf T}}(R)$}:t.(v\otimes
1)=v\otimes{\underline{\lambda}}(t)\big{\\}}$.}$

In particular this is a decomposition into simultaneous eigenspaces with
respect to the induced action of $T\vcentcolon={\mathrm{\bf T}}(K)$ on $V$. We
say that $v\in V$ is ${\mathrm{\bf T}}$-homogeneous if $v\in
V_{\underline{\lambda}}$, for some ${\underline{\lambda}}\in X({\mathrm{\bf
T}})$.

Note that the open affine subsets $U_{i}\subset{\mathbb{P}}_{K}^{d}$ are
stabilized under the action of ${\mathrm{\bf T}}$, for all $i=0,\ldots,d$.
Therefore, the $K$-vector space of sections ${\mathcal{E}}(U_{i})$ obtains the
structure of a ${\mathrm{\bf T}}$-module which decomposes into weight spaces

${\mathcal{E}}(U_{i})=\bigoplus_{{\underline{\lambda}}\in X({\mathrm{\bf
T}})}{\mathcal{E}}(U_{i})_{\underline{\lambda}}.$ (2.17)

For $x\in{\mathbb{P}}_{K}^{d}$, we let
${\mathcal{E}}(x)\vcentcolon={\mathcal{E}}_{x}/{\mathfrak{m}}_{x}{\mathcal{E}}_{x}$
denote the fibre of ${\mathcal{E}}$ at $x$ where ${\mathfrak{m}}_{x}$ is the
maximal ideal of the local ring ${\mathcal{O}}_{{\mathbb{P}}_{K}^{d},x}$. The
$K$-vector space ${\mathcal{E}}(x)$ is canonically isomorphic to the fibre at
$x$ of the geometric vector bundle associated with ${\mathcal{E}}$.

Note that the ${\mathrm{\bf T}}$-action on ${\mathbb{P}}_{K}^{d}$ has the
fixed points
$x_{i}\vcentcolon=[0{\,:\,}\ldots{\,:\,}0{\,:\,}1{\,:\,}0{\,:\,}\ldots{\,:\,}0]\in
U_{i}$, for $i=0,\ldots,d$. We therefore obtain the structure of a
${\mathrm{\bf T}}$-module on ${\mathcal{E}}_{x_{i}}$. When considering the
structure sheaf ${\mathcal{O}}$ of ${\mathbb{P}}_{K}^{d}$ as a ${\mathrm{\bf
G}}$-equivariant sheaf in the usual way, the maximal ideal
${\mathfrak{m}}_{x_{i}}\subset{\mathcal{O}}_{x_{i}}$ is a ${\mathrm{\bf
T}}$-submodule for the same reason. It follows from

$t.(se)=(t.s)(t.e)\quad\text{, for all $t\in{\mathrm{\bf T}}(R)$,
$s\in{\mathcal{O}}(U_{i}){\,\otimes_{K}\,}R$, and
$e\in{\mathcal{E}}(U_{i}){\,\otimes_{K}\,}R$,}$ (2.18)

that

$t.(se)=(t.s)(t.e)\quad\text{, for all $t\in{\mathrm{\bf T}}(R)$,
$s\in{\mathcal{O}}_{{\mathbb{P}}_{K}^{d},x_{i}}{\,\otimes_{K}\,}R$, and
$e\in{\mathcal{E}}_{x_{i}}{\,\otimes_{K}\,}R$,}$

for all $K$-algebras $R$. Hence
${\mathfrak{m}}_{x_{i}}{\mathcal{E}}_{x_{i}}\subset{\mathcal{E}}_{x_{i}}$ is a
${\mathrm{\bf T}}$-submodule as well, and we obtain the structure of a
${\mathrm{\bf T}}$-module on the quotient ${\mathcal{E}}(x_{i})$.

######  Lemma 2.4.4 (cf. [44, Lemma 4.6]).

Let ${\mathcal{E}}$ be a ${\mathrm{\bf G}}$-equivariant vector bundle on
${\mathbb{P}}_{K}^{d}$ and $i\in\\{0,\ldots,d\\}$. For all open subsets
$U\subset U_{i}$, there are $T$-equivariant isomorphisms of
${\mathcal{O}}(U)$-modules

${\mathcal{E}}(U)\cong{\mathcal{O}}(U){\,\otimes_{K}\,}{\mathcal{E}}(x_{i}),$

which are compatible with respect to the restriction homomorphisms.

###### Proof.

By [41, I.5.16] and [41, II.1.10], the ${\mathrm{\bf G}}$-equivariant vector
bundle ${\mathcal{E}}$ admits a trivialisation on the open affine subset
$U_{i}$. In particular, we can find sections in ${\mathcal{E}}(U_{i})$ which
globally generate ${\mathcal{E}}{\\!\,\mid_{U_{i}}}$. We may take a $K$-basis
$(e_{j})_{j\in J}$, of ${\mathcal{E}}(U_{i})$ to do so. For $U\subset U_{i}$
open, we then define the homomorphism

$\varphi_{U}\colon{\mathcal{E}}(U)\longrightarrow{\mathcal{O}}(U){\,\otimes_{K}\,}{\mathcal{E}}(x_{i})\,,\quad\sum_{j\in
J}s_{j}e_{j}{\\!\,\mid_{U}}\longmapsto\sum_{j\in J}s_{j}\otimes
e_{j}(x_{i})\quad\text{, for $s_{j}\in{\mathcal{O}}(U)$,}$

of ${\mathcal{O}}(U)$-modules. Note that $\varphi_{U}$ is independent of the
choice of the $K$-basis $(e_{j})_{j\in J}$. An inverse to $\varphi_{U}$ is
given by

${\mathcal{O}}(U){\,\otimes_{K}\,}{\mathcal{E}}(x_{i})\longrightarrow{\mathcal{E}}(U)\,,\quad
s\otimes\sum_{j\in J}a_{j}e_{j}(x_{i})\longmapsto\sum_{j\in
J}a_{j}se_{j}{\\!\,\mid_{U}}\quad\text{, for $s\in{\mathcal{O}}(U)$, $a_{j}\in
K$.}$

To show that $\varphi_{U}$ is $T$-equivariant, let $t\in T$, $k\in J$, and
$t.e_{k}=\sum_{j\in J}a_{j}e_{j}$ in ${\mathcal{E}}(U_{i})$, for some
$a_{j}\in K$. It follows from (2.18) that, for all $s\in{\mathcal{O}}(U)$,

$\displaystyle\varphi_{U}\big{(}t.(se_{k}{\\!\,\mid_{U}})\big{)}$
$\displaystyle=\varphi_{U}\bigg{(}\sum_{j\in
J}(t.s)a_{j}e_{j}{\\!\,\mid_{U}}\bigg{)}=t.s\otimes\sum_{j\in
J}a_{j}e_{j}(x_{i})$ $\displaystyle=t.\big{(}s\otimes
e_{k}(x_{i})\big{)}=t.\varphi_{U}\big{(}se_{k}{\\!\,\mid_{U}}\big{)}.$

The compatibility of the $\varphi_{U}$ with the restriction homomorphisms is
immediate. ∎

It will be helpful to fix certain norms on ${\mathcal{E}}(U_{I})$, for non-
empty $I\subset\\{0,\ldots,d\\}$, which realise its locally convex topology.
Recall that

${\mathcal{O}}(U_{I,\varepsilon})=\Bigg{\\{}\sum_{{\underline{k}}\in\Lambda_{I}}a_{\underline{k}}\,X^{\underline{k}}\Bigg{|}{\lvert{a_{\underline{k}}}\rvert}\bigg{(}\frac{1}{\varepsilon}\bigg{)}^{{\lvert{\max(0,{\underline{k}})}\rvert}}\rightarrow
0\text{ as ${\lvert{{\underline{k}}}\rvert}\rightarrow\infty$}\Bigg{\\}},$

with norm given by (2.16). We now choose some $i\in I$, and let
$v_{1},\ldots,v_{n}\in{\mathcal{E}}(x_{i})$ be a $K$-basis of weight vectors
of weights ${\underline{\lambda}}_{1},\ldots,{\underline{\lambda}}_{n}\in
X({\mathrm{\bf T}})$. We endow ${\mathcal{E}}(x_{i})$ with the norm prescribed
by this basis, i.e.

$\bigg{\lvert}\sum_{l=1}^{n}a_{l}\,v_{l}\bigg{\rvert}\vcentcolon=\max_{l=1}^{n}{\lvert{a_{l}}\rvert}\quad\text{,
for $a_{1},\ldots,a_{n}\in K$.}$

We continue to denote the elements of ${\mathcal{E}}(U_{i})$ corresponding to
$1\otimes v_{l}$ under the isomorphism of Lemma 2.4.4 by $v_{l}$.
Consequently, we have an isomorphism of ${\mathcal{O}}(U_{i})$-modules

${\mathcal{O}}(U_{i})^{\oplus
n}\overset{\cong}{\longrightarrow}{\mathcal{E}}(U_{i})\,,\quad(f_{1},\ldots,f_{n})\longmapsto\sum_{l=1}^{n}f_{l}\,v_{l}.$

This induces an isomorphism of ${\mathcal{O}}(U_{I,\varepsilon})$-modules
${\mathcal{E}}(U_{I,\varepsilon})\cong{\mathcal{O}}(U_{I,\varepsilon})^{\oplus
n}$ which endows the former with the norm

$\bigg{\lvert}\sum_{l=1}^{n}f_{l}\,v_{l}\bigg{\lvert}_{\varepsilon}\vcentcolon=\sup_{\begin{subarray}{c}l=1,\ldots,n\\\
{\underline{k}}\in\Lambda_{I}\end{subarray}}{\lvert{a_{l,{\underline{k}}}}\rvert}\bigg{(}\frac{1}{\varepsilon}\bigg{)}^{{\lvert{\max(0,{\underline{k}})}\rvert}}\quad\text{,
for $a_{l,{\underline{k}}}\in K$ with
$f_{l}=\sum_{{\underline{k}}\in\Lambda_{I}}a_{l,{\underline{k}}}\,X^{\underline{k}}\in{\mathcal{O}}(U_{I,\varepsilon})$.}$
(2.19)

We fix this norm on ${\mathcal{E}}(U_{I,\varepsilon})$ and on its subspace
${\mathcal{E}}(U_{I})\,(={\mathcal{E}}^{\mathrm{alg}}(U_{I}))$, and omit the
$\varepsilon$ from the index. Note that for a different choice of $i\in I$ or
the basis of weight vectors of ${\mathcal{E}}(x_{i})$, the above construction
yields an equivalent norm on ${\mathcal{E}}(U_{I,\varepsilon})$ by [7, 3.7.3
Prop. 3].

As the main step towards proving Theorem 2.4.3, we want to show that, for
“large enough” weights ${\underline{\lambda}}\in X({\mathrm{\bf T}})$, the
weight spaces ${\mathcal{E}}(U_{I})_{\underline{\lambda}}$ change in a uniform
way when one varies the “extreme” entries of ${\underline{\lambda}}$. To this
end we define, for $m\in{\mathbb{N}}_{0}$,

${\Delta}_{m}\vcentcolon=\left\\{{\underline{\lambda}}\in X({\mathrm{\bf
T}})\cong{\mathbb{Z}}^{d+1}\,\middle|\,\forall
j=0,\ldots,d:{\lvert{\lambda_{j}}\rvert}\leq m\right\\}.$

Because the ${\mathrm{\bf T}}$-modules ${\mathcal{E}}(x_{i})$ are finite-
dimensional, we find some $M\in{\mathbb{N}}$ such that the weights of all
${\mathcal{E}}(x_{0}),\ldots,{\mathcal{E}}(x_{d})$ are concentrated in
${\Delta}_{M}$, i.e. for all $i\in\\{0,\ldots,d\\}$ and all
${\underline{\lambda}}\in X({\mathrm{\bf T}})$ with
${\mathcal{E}}(x_{i})_{\underline{\lambda}}\neq\\{0\\}$, we have
${\underline{\lambda}}\in{\Delta}_{M}$. We moreover set
$N\vcentcolon=(2d+1)M+d$.

######  Proposition 2.4.5.

For every ${\underline{\mu}}\in X({\mathrm{\bf T}})$, there exist
${\underline{\nu}}\in{\Delta}_{N}$ and $C\in\varepsilon^{{\mathbb{N}}_{0}}$
such that, for all non-empty $I\subset\\{0,\ldots,d\\}$, there is an
isomorphism

$\varphi^{I}_{{\underline{\mu}},{\underline{\nu}}}\colon{\mathcal{E}}(U_{I})_{\underline{\mu}}\overset{\cong}{\longrightarrow}{\mathcal{E}}(U_{I})_{\underline{\nu}}$

of $K$-vector spaces satisfying
${\lvert{\varphi^{I}_{{\underline{\mu}},{\underline{\nu}}}(v)}\rvert}=C\,{\lvert{v}\rvert}$,
for all $v\in{\mathcal{E}}(U_{I,\varepsilon})_{\underline{\mu}}$, and such
that these isomorphisms are compatible with the restriction homomorphisms,
i.e. for all $I\subset J\subset\\{0,\ldots,d\\}$, the following diagram
commutes:

${{\mathcal{E}}(U_{I})_{\underline{\mu}}}$${{\mathcal{E}}(U_{J})_{\underline{\mu}}}$${{\mathcal{E}}(U_{I})_{\underline{\nu}}}$${{\mathcal{E}}(U_{J})_{\underline{\nu}}.}$$\scriptstyle{\varphi^{I}_{{\underline{\mu}},{\underline{\nu}}}}$$\scriptstyle{\varphi^{J}_{{\underline{\mu}},{\underline{\nu}}}}$

###### Proof.

We proceed by induction on
$\lVert{\underline{\mu}}\rVert\vcentcolon=\sum_{j=0}^{d}{\lvert{\mu_{j}}\rvert}$.
Note that for ${\underline{\mu}}\in{\Delta}_{N}$, we may take
${\underline{\nu}}\vcentcolon={\underline{\mu}}$, $C\vcentcolon=1$, and the
identity homomorphisms to obtain the assertion of the proposition. This also
deals with the base case $\lVert{\underline{\mu}}\rVert=0$.

Hence we now suppose that ${\underline{\mu}}\notin{\Delta}_{N}$. We let
$u,v\in\\{0,\ldots,d\\}$ such that $\mu_{u}$ is maximal among the entries
$\mu_{0},\ldots,\mu_{d}$ and the entry $\mu_{v}$ is minimal. We set
${\underline{\mu}}-\alpha_{u,v}\vcentcolon={\underline{\mu}}-\epsilon_{u}+\epsilon_{v}$.

For fixed $I\subset\\{0,\ldots,d\\}$ with $i\in I$, let
$v_{1},\ldots,v_{n}\in{\mathcal{E}}(x_{i})$ be a $K$-basis of weight vectors
of weights ${\underline{\lambda}}_{1},\ldots,{\underline{\lambda}}_{n}$. We
then have the following $K$-basis for the weight space

${\mathcal{E}}(U_{I})_{\underline{\mu}}=\bigoplus_{l\in
L_{{\underline{\mu}},I}}K\cdot
X^{{\underline{\mu}}-{\underline{\lambda}}_{l}}\,v_{l}$

where
$L_{{\underline{\mu}},I}\vcentcolon=\big{\\{}l\in\\{1,\ldots,n\\}\,\big{|}\,X^{{\underline{\mu}}-{\underline{\lambda}}_{l}}\in{\mathcal{O}}(U_{I})\big{\\}}$.

######  Lemma 2.4.6.

For $l\in\\{1,\ldots,n\\}$, we have

$X^{{\underline{\mu}}-{\underline{\lambda}}_{l}}\in{\mathcal{O}}(U_{I})\quad\text{
if and only if }\quad
X^{({\underline{\mu}}-\alpha_{u,v})-{\underline{\lambda}}_{l}}\in{\mathcal{O}}(U_{I}).$

If this is the case, for some $l\in\\{1,\ldots,n\\}$, we moreover have
$\mu_{u}>M+1$ and $\mu_{v}<-(M+1)$ so that in particular $u\neq v$.

###### Proof.

We first note that
$X^{{\underline{\mu}}-{\underline{\lambda}}_{l}}\in{\mathcal{O}}(U_{I})$ if
and only if $\sum_{j=0}^{d}\mu_{j}-\lambda_{l,j}=0$ and, for all
$j\in\\{0,\ldots,d\\}\setminus I$, we have $\mu_{j}-\lambda_{l,j}\geq 0$. For
$X^{({\underline{\mu}}-\alpha_{u,v})-{\underline{\lambda}}_{l}}$ the analogous
statement holds. To show the claimed equivalence it thus suffices to focus on
the exponents of $X_{u}$ and $X_{v}$.

Because ${\underline{\mu}}\notin{\Delta}_{N}$, there exists
$j\in\\{0,\ldots,d\\}$ such that ${\lvert{\mu_{j}}\rvert}>N$. We distinguish
the two cases that $\mu_{j}>0$ and that $\mu_{j}<0$. If $\mu_{j}>0$, then
$\mu_{u}>N\geq M+1$ by the maximality of $\mu_{u}$. Using
${\lvert{\lambda_{l,u}}\rvert}\leq M$ we find that

$\mu_{u}-1-\lambda_{l,u}>N-1-M\geq 0$

and also $\mu_{u}-\lambda_{l,u}\geq 0$, so that the exponent of $X_{u}$ is not
an obstacle in this case. Concerning the exponent of $X_{v}$, since
$({\underline{\mu}}-\alpha_{u,v})_{v}-\lambda_{l,v}\geq\mu_{v}-\lambda_{l,v}$,
we find that if
$X^{{\underline{\mu}}-{\underline{\lambda}}_{l}}\in{\mathcal{O}}(U_{I})$, then
$X^{({\underline{\mu}}-\alpha_{u,v})-{\underline{\lambda}}_{l}}\in{\mathcal{O}}(U_{I})$.
Conversely suppose that
$X^{({\underline{\mu}}-\alpha_{u,v})-{\underline{\lambda}}_{l}}\in{\mathcal{O}}(U_{I})$
so that in particular
$\sum_{j=0}^{d}\big{(}({\underline{\mu}}-\alpha_{u,v})_{j}-\lambda_{l,j}\big{)}=0$
and $({\underline{\mu}}-\alpha_{u,v})_{v}-\lambda_{l,v}\geq 0$. We compute

$\displaystyle\bigg{\lvert}\sum_{j=0}^{d}\mu_{j}\bigg{\rvert}$
$\displaystyle=\bigg{\lvert}\sum_{j=0}^{d}(\mu_{j}-\lambda_{l,j})+\sum_{j=0}^{d}\lambda_{l,j}\bigg{\rvert}=\bigg{\lvert}\underset{=0}{\underbrace{\sum_{j=0}^{d}\big{(}({\underline{\mu}}-\alpha_{u,v})_{j}-\lambda_{l,j}\big{)}}}+\sum_{j=0}^{d}\lambda_{l,j}\bigg{\rvert}$
$\displaystyle=\bigg{\lvert}\sum_{j=0}^{d}\lambda_{l,j}\bigg{\rvert}\leq\sum_{j=0}^{d}{\lvert{\lambda_{l,j}}\rvert}\leq(d+1)M.$

Then

$(d+1)M\geq\bigg{\lvert}\sum_{j=0}^{d}\mu_{j}\bigg{\rvert}=\bigg{\lvert}\mu_{u}+\sum_{\begin{subarray}{c}j=0\\\
j\neq u\end{subarray}}^{d}\mu_{j}\bigg{\rvert}$

together with $\mu_{u}>N=(2d+1)M+d$ implies that
$\sum_{\begin{subarray}{c}j=0\\\ j\neq u\end{subarray}}^{d}\mu_{j}<-d(M+1)$.
Hence there exists $j^{\prime}\in\\{0,\ldots,d\\}$, $j^{\prime}\neq u$, such
that $\mu_{j^{\prime}}<-(M+1)$. By the minimality of $\mu_{v}$, it follows
that $\mu_{v}<-(M+1)$. But using ${\lvert{\lambda_{l,v}}\rvert}\leq M$ we find
that

$\displaystyle({\underline{\mu}}-\alpha_{u,v})_{v}-\lambda_{l,v}$
$\displaystyle=\mu_{v}+1-\lambda_{l,v}<-(M+1)+1+M=0.$

Therefore
$X^{({\underline{\mu}}-\alpha_{u,v})-{\underline{\lambda}}_{l}}\in{\mathcal{O}}(U_{I})$
can only occur if $v\in I$. We conclude that
$X^{{\underline{\mu}}-{\underline{\lambda}}_{l}}\in{\mathcal{O}}(U_{I})$ as
well which finishes the proof in this case.

Now we consider the case that $\mu_{j}<0$. Here we find that
$\mu_{v}<-N\leq-(M+1)$ by the minimality of $\mu_{v}$. Like before, we have

$(d+1)M\geq\bigg{\lvert}\sum_{j=0}^{d}\mu_{j}\bigg{\rvert}=\bigg{\lvert}\mu_{v}+\sum_{\begin{subarray}{c}j=0\\\
j\neq v\end{subarray}}^{d}\mu_{j}\bigg{\rvert}.$

From this and $\mu_{v}<-N$ we conclude that there exists
$j^{\prime}\in\\{0,\ldots,d\\}$, $j^{\prime}\neq v$, such that
$\mu_{j^{\prime}}>M+1$. As $\mu_{u}$ is maximal, it follows that $\mu_{u}>M+1$
as well. Therefore

$\displaystyle({\underline{\mu}}-\alpha_{u,v})_{u}-\lambda_{l,u}$
$\displaystyle=\mu_{u}-1-\lambda_{l,u}>(M+1)-1-M=0,$

and moreover $\mu_{u}-\lambda_{l,u}\geq 0$. This shows that the exponent of
$X_{u}$ is not an obstacle in this case. For the exponent of $X_{v}$, we
compute

$\displaystyle({\underline{\mu}}-\alpha_{u,v})_{v}-\lambda_{l,v}$
$\displaystyle=\mu_{v}+1-\lambda_{l,v}<-N+1+M<0,$

and also $\mu_{v}-\lambda_{l,v}<0$. Therefore either of
$X^{{\underline{\mu}}-{\underline{\lambda}}_{l}}\in{\mathcal{O}}(U_{I})$ or
$X^{({\underline{\mu}}-\alpha_{u,v})-{\underline{\lambda}}_{l}}\in{\mathcal{O}}(U_{I})$
implies that $v\in I$. This shows the assertion in the case $\mu_{j}<0$. ∎

Using the above lemma, we see that

${\mathcal{E}}(U_{I})_{{\underline{\mu}}-\alpha_{u,v}}=\bigoplus_{l\in
L_{{\underline{\mu}},I}}K\cdot
X^{({\underline{\mu}}-\alpha_{u,v})-{\underline{\lambda}}_{l}}\,v_{l},$

and we define the following isomorphism of $K$-vector spaces

$\varphi^{I}_{{\underline{\mu}},{\underline{\mu}}-\alpha_{u,v}}\colon{\mathcal{E}}(U_{I})_{{\underline{\mu}}}\longrightarrow{\mathcal{E}}(U_{I})_{{\underline{\mu}}-\alpha_{u,v}}\,,\quad
X^{{\underline{\mu}}-{\underline{\lambda}}_{l}}\,v_{l}\longmapsto
X^{({\underline{\mu}}-\alpha_{u,v})-{\underline{\lambda}}_{l}}\,v_{l}.$

For $l\in L_{{\underline{\mu}},I}$, it follows from $\mu_{u}>M+1$,
$\mu_{v}<-(M+1)$, and ${\underline{\lambda}}_{l}\in{\Delta}_{M}$ that

$\displaystyle({\underline{\mu}}-\alpha_{u,v})_{u}-\lambda_{l,u}$
$\displaystyle\geq 0,$ $\displaystyle\mu_{u}-\lambda_{l,u}$ $\displaystyle\geq
0,$ $\displaystyle({\underline{\mu}}-\alpha_{u,v})_{v}-\lambda_{l,v}$
$\displaystyle\leq 0,$ $\displaystyle\mu_{v}-\lambda_{l,v}$ $\displaystyle\leq
0.$

Therefore

$\displaystyle\max\big{(}0,({\underline{\mu}}-\alpha_{u,v})_{u}-\lambda_{l,u}\big{)}$
$\displaystyle=\mu_{u}-1-\lambda_{l,u}$
$\displaystyle=\max\big{(}0,\mu_{u}-\lambda_{l,u}\big{)}-1,$
$\displaystyle\max\big{(}0,({\underline{\mu}}-\alpha_{u,v})_{v}-\lambda_{l,v}\big{)}$
$\displaystyle=0$ $\displaystyle=\max\big{(}0,\mu_{v}-\lambda_{l,v}\big{)},$

and we conclude that

$\displaystyle\big{\lvert}X^{({\underline{\mu}}-\alpha_{u,v})-{\underline{\lambda}}_{l}}\,v_{l}\big{\rvert}$
$\displaystyle=\bigg{(}\frac{1}{\varepsilon}\bigg{)}^{\sum_{j=0}^{d}\max\big{(}0,({\underline{\mu}}-\alpha_{u,v})_{j}-\lambda_{l,j}\big{)}}$
$\displaystyle=\bigg{(}\frac{1}{\varepsilon}\bigg{)}^{\big{(}\sum_{j=0}^{d}\max(0,\mu_{j}-\lambda_{l,j})\big{)}-1}=\varepsilon\,\big{\lvert}X^{{\underline{\mu}}-{\underline{\lambda}}_{l}}\,v_{l}\big{\rvert}.$

Hence we have
${\lvert{\varphi^{I}_{{\underline{\mu}},{\underline{\mu}}-\alpha_{u,v}}(v)}\rvert}=\varepsilon\,{\lvert{v}\rvert}$,
for all $v\in{\mathcal{E}}(U_{I})_{\underline{\mu}}$.

Next we verify that the above family of isomorphisms
$\varphi^{I}\vcentcolon=\varphi^{I}_{{\underline{\mu}},{\underline{\mu}}-\alpha_{u,v}}$
is compatible with the restriction maps. For this consider $I\subset
J\subset\\{0,\ldots,d\\}$ with $i\in I$ and $j\in J$. Let

$\displaystyle{\mathcal{E}}(U_{I})=\bigoplus_{l=1}^{n}{\mathcal{O}}(U_{I})\,v_{l}\quad\text{and}\quad{\mathcal{E}}(U_{J})=\bigoplus_{k=1}^{n}{\mathcal{O}}(U_{J})\,w_{k},$

for a $K$-basis consisting of weight vectors $v_{1},\ldots,v_{n}$ of
${\mathcal{E}}(x_{i})$ of weights
${\underline{\lambda}}_{1},\ldots,{\underline{\lambda}}_{n}$, and a $K$-basis
consisting of weight vectors $w_{1},\ldots,w_{n}$ of ${\mathcal{E}}(x_{j})$ of
weights ${\underline{\kappa}}_{1},\ldots,{\underline{\kappa}}_{n}$ which yield
the isomorphisms $\varphi^{I}$ and $\varphi^{J}$ respectively. Let ${\rm
res}\colon{\mathcal{E}}(U_{I})\hookrightarrow{\mathcal{E}}(U_{J})$ denote the
restriction map which is ${\mathcal{O}}(U_{I})$-linear and injective.
Furthermore, there are $a_{l,k}\in K$ such that

${\rm
res}(v_{l})=\sum_{k=1}^{n}a_{l,k}\,X^{{\underline{\lambda}}_{l}-{\underline{\kappa}}_{k}}\,w_{k}\quad\text{
, for all $l=1,\ldots,n$,}$

because ${\rm res}(v_{l})\in{\mathcal{E}}(U_{J})$ is of weight
${\underline{\lambda}}_{l}$. For $l\in\\{1,\ldots,n\\}$ such that
$X^{{\underline{\mu}}-{\underline{\lambda}}_{l}}\in{\mathcal{O}}(U_{I})$ we
now compute

$\displaystyle{\rm
res}\Big{(}\varphi^{I}\big{(}X^{{\underline{\mu}}-{\underline{\lambda}}_{l}}\,v_{l}\big{)}\Big{)}$
$\displaystyle={\rm
res}\Big{(}X^{({\underline{\mu}}-\alpha_{u,v})-{\underline{\lambda}}_{l}}\,v_{l}\Big{)}=X^{({\underline{\mu}}-\alpha_{u,v})-{\underline{\lambda}}_{l}}\,{\rm
res}(v_{l})$
$\displaystyle=X^{({\underline{\mu}}-\alpha_{u,v})-{\underline{\lambda}}_{l}}\sum_{k=1}^{n}a_{l,k}\,X^{{\underline{\lambda}}_{l}-{\underline{\kappa}}_{k}}\,w_{k}=\sum_{k=1}^{n}a_{l,k}\,X^{({\underline{\mu}}-\alpha_{u,v})-{\underline{\kappa}}_{k}}\,w_{k}$
$\displaystyle=\varphi^{J}\bigg{(}\sum_{k=1}^{n}a_{l,k}\,X^{{\underline{\mu}}-{\underline{\kappa}}_{k}}\,w_{k}\bigg{)}=\varphi^{J}\bigg{(}X^{{\underline{\mu}}-{\underline{\lambda}}_{l}}\sum_{k=1}^{n}a_{l,k}\,X^{{\underline{\lambda}}_{l}-{\underline{\kappa}}_{k}}\,w_{k}\bigg{)}$
$\displaystyle=\varphi^{J}\big{(}X^{{\underline{\mu}}-{\underline{\lambda}}_{l}}\,{\rm
res}(v_{l})\big{)}=\varphi^{J}\Big{(}{\rm
res}\big{(}X^{{\underline{\mu}}-{\underline{\lambda}}_{l}}\,v_{l}\big{)}\Big{)}$

which shows the compatibility for $I\subset J$.

Finally, we want to apply the induction hypothesis. In the case that, for all
$I\subset\\{0,\ldots,d\\}$ with $I\neq\emptyset$, the set
$L_{{\underline{\mu}},I}$ is empty, we find that
${\mathcal{E}}(U_{I})_{\underline{\mu}}=\\{0\\}$, for all such $I$. We may for
example take ${\underline{\nu}}\vcentcolon=(M+1,\ldots,M+1)\in{\Delta}_{N}$,
and see that, for all such $I$ and all $l=1,\ldots,n$, we have
${\underline{\nu}}-{\underline{\mu}}_{l}\notin\Lambda_{I}$. It follows that
${\mathcal{E}}(U_{I})_{\underline{\nu}}=\\{0\\}$, and we obtain the sought
isomorphisms trivially.

On the other hand, if there is some $I\subset\\{0,\ldots,d\\}$ with
$I\neq\emptyset$ for which $L_{{\underline{\mu}},I}$ is non-empty, the second
assertion of Lemma 2.4.6 implies that

$\displaystyle\lVert{\underline{\mu}}-\alpha_{u,v}\rVert$
$\displaystyle={\lvert{\mu_{u}-1}\rvert}+{\lvert{\mu_{v}+1}\rvert}+\sum_{\begin{subarray}{c}j=0\\\
j\neq
u,v\end{subarray}}^{d}{\lvert{\mu_{j}}\rvert}=(\mu_{u}-1)-(\mu_{v}+1)+\sum_{\begin{subarray}{c}j=0\\\
j\neq
u,v\end{subarray}}^{d}{\lvert{\mu_{j}}\rvert}=\lVert{\underline{\mu}}\rVert-2.$

Applying the induction hypothesis to ${\underline{\mu}}-\alpha_{u,v}$, we
obtain ${\underline{\nu}}\in{\Delta}_{N}$ and the family of isomorphisms
$\varphi^{I}_{{\underline{\mu}}-\alpha_{u,v},{\underline{\nu}}}$, for
$I\subset\\{0,\ldots,d\\}$, as specified. Then the statement follows for
${\underline{\mu}}$ by taking the compositions

$\varphi^{I}_{{\underline{\mu}},{\underline{\nu}}}\vcentcolon=\varphi^{I}_{{\underline{\mu}}-\alpha_{u,v},{\underline{\nu}}}\circ\varphi^{I}_{{\underline{\mu}},{\underline{\mu}}-\alpha_{u,v}}.$

∎

###### Proof of Theorem 2.4.3.

Recall that in (2.19), for every non-empty $I\subset\\{0,\ldots,d\\}$, we
fixed a norm on ${\mathcal{E}}(U_{I})$. This endows the space of $q$-th Čech
cochains, for $q\geq 0$,

$C^{q}({\mathcal{U}},{\mathcal{E}})=\bigoplus_{(i_{0},\ldots,i_{q})\in\\{{r}+1,\ldots,d\\}^{q+1}}{\mathcal{E}}\big{(}U_{\\{i_{0},\ldots,i_{q}\\}}\big{)}$

with a norm as well. We also have a decomposition of
$C^{q}({\mathcal{U}},{\mathcal{E}})$ into weight spaces

$C^{q}({\mathcal{U}},{\mathcal{E}})_{\underline{\lambda}}=\bigoplus_{(i_{0},\ldots,i_{q})\in\\{{r}+1,\ldots,d\\}^{q+1}}{\mathcal{E}}\big{(}U_{\\{i_{0},\ldots,i_{q}\\}}\big{)}_{\underline{\lambda}}\quad\text{,
for ${\underline{\lambda}}\in X({\mathrm{\bf T}})$,}$

under the induced ${\mathrm{\bf T}}$-action. Note that if we have a weight
decomposition of $f\in C^{q}({\mathcal{U}},{\mathcal{E}})$, it follows from
(2.19) that

${\lvert{f}\rvert}=\sup_{{\underline{\lambda}}\in X({\mathrm{\bf
T}})}{\lvert{f_{\lambda}}\rvert}\quad\text{, for
$f=\sum_{{\underline{\lambda}}\in X({\mathrm{\bf
T}})}f_{\underline{\lambda}}$, with $f_{\underline{\lambda}}\in
C^{q}({\mathcal{U}},{\mathcal{E}})_{\underline{\lambda}}$.}$

Since the restriction maps ${\rm
res}_{I,J}\colon{\mathcal{E}}(U_{I})\hookrightarrow{\mathcal{E}}(U_{J})$, for
$I\subset J\subset\\{{r}+1,\ldots,d\\}$, are ${\mathrm{\bf T}}$-equivariant,
so is the differential

$\displaystyle d^{q}\colon C^{q}({\mathcal{U}},{\mathcal{E}})$
$\displaystyle\longrightarrow C^{q+1}({\mathcal{U}},{\mathcal{E}}),$
$\displaystyle\big{(}f_{(i_{0},\ldots,i_{q})}\big{)}_{(i_{0},\ldots,i_{q})}$
$\displaystyle\longmapsto\bigg{(}\sum_{k=0}^{q+1}(-1)^{k}{\rm
res}_{I,J}\big{(}f_{(j_{0},\ldots,\widehat{j_{k}},\ldots,j_{q+1})}\big{)}\bigg{)}_{(j_{0},\ldots,j_{q+1})},$

of the Čech complex. Therefore we may restrict $d^{q}$ to the individual
weight spaces

$d^{q}_{{\underline{\lambda}}}\colon
C^{q}({\mathcal{U}},{\mathcal{E}})_{\underline{\lambda}}\longrightarrow
C^{q+1}({\mathcal{U}},{\mathcal{E}})_{\underline{\lambda}}\quad\text{, for
${\underline{\lambda}}\in X({\mathrm{\bf T}})$.}$

Now let $N\in{\mathbb{N}}$ be defined as before Proposition 2.4.5, and
consider

$d^{q}_{N}\colon\bigoplus_{{\underline{\lambda}}\in{\Delta}_{N}}C^{q}({\mathcal{U}},{\mathcal{E}})_{\underline{\lambda}}\longrightarrow\bigoplus_{{\underline{\lambda}}\in{\Delta}_{N}}C^{q+1}({\mathcal{U}},{\mathcal{E}})_{\underline{\lambda}}\,,\quad\big{(}f_{\underline{\lambda}}\big{)}_{{\underline{\lambda}}\in{\Delta}_{N}}\longmapsto\big{(}d^{q}_{\underline{\lambda}}(f_{\underline{\lambda}})\big{)}_{{\underline{\lambda}}\in{\Delta}_{N}}.$

Since this homomorphism takes values in a finite-dimensional $K$-vector space,
$d^{q}_{N}$ is strict by [11, I.2.3. Cor.]. Hence there exists $R>0$ such that

$B_{R}(0)\cap{\mathrm{Im}}(d^{q}_{N})\subset d_{N}^{q}\big{(}B_{1}(0)\big{)}$
(2.20)

where $B_{R}(0)$ and $B_{1}(0)$ denote the “closed” balls of radius $R$ and
$1$ respectively.

To see that $d^{q}$ itself is strict, it suffices to show that

$B_{R}(0)\cap{\mathrm{Im}}(d^{q})\subset d^{q}\big{(}B_{1}(0)\big{)}$ (2.21)

by scaling. For this, let $g\in{\mathrm{Im}}(d^{q})\cap B_{R}(0)$, and let
$g=\sum_{{\underline{\lambda}}\in X({\mathrm{\bf T}})}g_{\underline{\lambda}}$
be a weight decomposition with $g_{\underline{\lambda}}\in
C^{q+1}({\mathcal{U}},{\mathcal{E}})_{\underline{\lambda}}$. As
${\lvert{g}\rvert}=\sup_{{\underline{\lambda}}\in X({\mathrm{\bf
T}})}{\lvert{g_{\underline{\lambda}}}\rvert}$, we have
${\lvert{g_{\underline{\lambda}}}\rvert}\leq R$, for all
${\underline{\lambda}}\in X({\mathrm{\bf T}})$. We moreover have
$g_{\underline{\lambda}}\in{\mathrm{Im}}(d^{q}_{\underline{\lambda}})$ since
$d^{q}$ is ${\mathrm{\bf T}}$-equivariant. For
${\underline{\lambda}}\in{\Delta}_{N}$, we can therefore conclude by (2.20)
that there exists $f_{\underline{\lambda}}\in
C^{q}({\mathcal{U}},{\mathcal{E}})_{\underline{\lambda}}$ such that
${\lvert{f_{\underline{\lambda}}}\rvert}\leq 1$ and
$d^{q}_{\underline{\lambda}}(f_{\underline{\lambda}})=g_{\underline{\lambda}}$.

For ${\underline{\lambda}}\notin{\Delta}_{N}$ in turn, we apply Proposition
2.4.5 to find ${\underline{\nu}}\in{\Delta}_{N}$,
$C\in\varepsilon^{{\mathbb{N}}_{0}}$, and a compatible family of isomorphisms

$\varphi^{I}_{{\underline{\lambda}},{\underline{\nu}}}\colon{\mathcal{E}}(U_{I})_{\underline{\lambda}}\overset{\cong}{\longrightarrow}{\mathcal{E}}(U_{I})_{\underline{\nu}}\quad\text{,
for all non-empty $I\subset\\{0,\ldots,d\\}$,}$

such that
$\lvert\varphi^{I}_{{\underline{\lambda}},{\underline{\nu}}}(v)\rvert=C\,{\lvert{v}\rvert}$,
for all $v\in{\mathcal{E}}(U_{I})_{\underline{\lambda}}$. We take the direct
sum of these isomorphisms to obtain isomorphisms

$\varphi^{q}_{{\underline{\lambda}},{\underline{\nu}}}\colon
C^{q}({\mathcal{U}},{\mathcal{E}})_{\underline{\lambda}}\overset{\cong}{\longrightarrow}C^{q}({\mathcal{U}},{\mathcal{E}})_{\underline{\nu}}\quad\text{,
for all $q\geq 0$,}$

such that
$\lvert\varphi^{q}_{{\underline{\lambda}},{\underline{\nu}}}(v)\rvert=C\,{\lvert{v}\rvert}$,
for all $v\in C^{q}({\mathcal{U}},{\mathcal{E}})_{\underline{\lambda}}$.
Moreover, the compatibility of the
$\varphi^{I}_{{\underline{\lambda}},{\underline{\nu}}}$ with the canonical
restriction maps implies that the diagram

${C^{q}({\mathcal{U}},{\mathcal{E}})_{\underline{\lambda}}}$${C^{q+1}({\mathcal{U}},{\mathcal{E}})_{\underline{\lambda}}}$${C^{q}({\mathcal{U}},{\mathcal{E}})_{\underline{\nu}}}$${C^{q+1}({\mathcal{U}},{\mathcal{E}})_{\underline{\nu}}}$$\scriptstyle{d^{q}_{\underline{\lambda}}}$$\scriptstyle{\varphi^{q}_{{\underline{\lambda}},{\underline{\nu}}}}$$\scriptstyle{\varphi^{q+1}_{{\underline{\lambda}},{\underline{\nu}}}}$$\scriptstyle{d^{q}_{{\underline{\nu}}}}$

commutes. Since $\varepsilon\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$, we may assume that $\varepsilon={\lvert{a}\rvert}$, for some
$a\in K$, after an extension of scalars by a finite extension of $K$. Then by
scaling (2.20) by an appropriate power of $a$ and restricting to the vectors
which are rational with respect to the original $K$, we see that

$B_{CR}(0)\cap{\mathrm{Im}}(d^{q}_{N})\subset
d^{q}_{N}\big{(}B_{C}(0)\big{)}.$

Because we have
$\lvert\varphi^{q+1}_{{\underline{\lambda}},{\underline{\nu}}}(g_{\underline{\lambda}})\rvert=C\,{\lvert{g_{\underline{\lambda}}}\rvert}\leq
CR$, we find $h\in C^{q}({\mathcal{U}},{\mathcal{E}})_{\underline{\nu}}$ such
that ${\lvert{h}\rvert}\leq C$ and
$d^{q}_{\underline{\nu}}(h)=\varphi^{q+1}_{{\underline{\lambda}},{\underline{\nu}}}(g_{\underline{\lambda}})$.
Then
$f_{\underline{\lambda}}\vcentcolon=\big{(}\varphi^{q}_{{\underline{\lambda}},{\underline{\nu}}}\big{)}^{-1}(h)$
satisfies
$d^{q}_{\underline{\lambda}}(f_{\underline{\lambda}})=g_{\underline{\lambda}}$
and ${\lvert{f_{\underline{\lambda}}}\rvert}=C^{-1}{\lvert{h}\rvert}\leq 1$.

In total we obtain, for every ${\underline{\lambda}}\in X({\mathrm{\bf T}})$,
an element $f_{\underline{\lambda}}\in
C^{q}({\mathcal{U}},{\mathcal{E}})_{\underline{\lambda}}$ such that
$d^{q}_{\underline{\lambda}}(f_{\underline{\lambda}})=g_{\underline{\lambda}}$
and ${\lvert{f_{\underline{\lambda}}}\rvert}\leq 1$. Taking the sum

$f\vcentcolon=\sum_{{\underline{\lambda}}\in X({\mathrm{\bf
T}})}f_{\underline{\lambda}}$

we find that $d^{q}(f)=g$ and that
${\lvert{f}\rvert}=\sup_{{\underline{\lambda}}\in X({\mathrm{\bf
T}})}{\lvert{f_{\underline{\lambda}}}\rvert}\leq 1$. This shows (2.21) and
therefore the strictness of $d^{q}$. ∎

###  2.5. Local Cohomology with respect to Tubes around Schubert Varieties

We begin by fixing some notation. For ${r}\in\\{0,\ldots,d-1\\}$ and
$0<\varepsilon<1$ with $\varepsilon\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$, we set

$\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\vcentcolon={\mathrm{Ker}}\Big{(}H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow
H^{i}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\Big{)}$

and endow it with the subspace topology. We define
$\widetilde{H}^{i}_{({\mathbb{P}}_{K}^{r})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$,
$\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$,
and
$\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
analogously.

######  Lemma 2.5.1.

Let $0<\varepsilon<\varepsilon^{\prime}<1$ with
$\varepsilon,\varepsilon^{\prime}\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$, and $I\subset\\{0,\ldots,d\\}$. Then the transition map

${\mathcal{E}}(U_{I,\varepsilon})\longrightarrow{\mathcal{E}}(U_{I,\varepsilon^{\prime}})$

is a compact homomorphism between $K$-Banach spaces with dense image.

###### Proof.

If $I=\emptyset$, we have
$U_{I,\varepsilon}=U_{I,\varepsilon^{\prime}}={\mathbb{P}}_{K}^{d}$ and
${\mathcal{E}}({\mathbb{P}}_{K}^{d})$ is a finite dimensional $K$-vector
space. Therefore, the homomorphism in question is compact by [67, Lemma 16.4].

Now suppose that $I\neq\emptyset$. Our method here is similar to the proof of
Lemma 2.3.3. For $I=\\{i_{0},\ldots,i_{m}\\}$, we consider the affinoid
$K$-algebra

$\displaystyle A_{\varepsilon}$
$\displaystyle\vcentcolon=K\big{\langle}\varepsilon
T_{(j,i_{k})}\,\big{|}\,j=0,\ldots,d,k=0,\ldots,m\text{, with }j\neq
i_{k}\big{\rangle}$
$\displaystyle=\Bigg{\\{}\sum_{{\underline{\mu}}\in{\mathbb{N}}^{d(m+1)}_{0}}a_{{\underline{\mu}}}\,{\underline{T}}^{{\underline{\mu}}}\in
K\mkern-3.0mu\left[\mkern-2.5mu\left[{\underline{T}}\right]\mkern-2.5mu\right]\,\Bigg{|}\,{\lvert{a_{{\underline{\mu}}}}\rvert}\left(\frac{1}{\varepsilon}\right)^{{\lvert{{\underline{\mu}}}\rvert}}\rightarrow
0\text{ as ${\lvert{{\underline{\mu}}}\rvert}\rightarrow\infty$}\Bigg{\\}},$

cf. [7, 6.1.5 Thm. 4]. Like in the proof of Lemma 2.4.1, we have
$U_{I,\varepsilon}={\mathrm{Sp}}(A_{\varepsilon}/{\mathfrak{a}}A_{\varepsilon})$,
for the ideal

${\mathfrak{a}}=\big{(}T_{(i_{j},i_{k})}T_{(i_{k},i_{j})}-1\,\big{|}\,j,k=0,\ldots,m\text{,
with }j\neq k\big{)}.$

With similar definitions for $U_{I,\varepsilon^{\prime}}$, we obtain a
commutative diagram of affinoid $K$-algebras

${A_{\varepsilon}}$${A_{\varepsilon^{\prime}}}$${A_{\varepsilon}/{\mathfrak{a}}A_{\varepsilon}}$${A_{\varepsilon^{\prime}}/{\mathfrak{a}}A_{\varepsilon^{\prime}}}$$\scriptstyle{\psi}$$\scriptstyle{\tau}$$\scriptstyle{\tau^{\prime}}$$\scriptstyle{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}}$

where the vertical maps are strict epimorphisms, and $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}$
is the transition homomorphism
${\mathcal{O}}(U_{I,\varepsilon})\rightarrow{\mathcal{O}}(U_{I,\varepsilon^{\prime}})$.
We claim that $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}$
is an inner homomorphism of affinoid $K$-algebras in the sense of [5, Def.
2.5.1]. Indeed we have, for $T=T_{(j,i_{k})}\in A_{\varepsilon}$,

$\displaystyle\sup_{x\in U_{I,\varepsilon^{\prime}}}{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}(\tau(T))(x)}\rvert}=\sup_{x\in
U_{I,\varepsilon^{\prime}}}{\lvert{\tau^{\prime}(\psi(T))(x)}\rvert}\leq{\lvert{\psi(T)}\rvert}_{\sup}={\lvert{\psi(T)}\rvert}_{A_{\varepsilon^{\prime}}}=\frac{1}{\varepsilon^{\prime}}<\frac{1}{\varepsilon}.$

It follows from [70, §1 Lemma 5] that $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{}$
is a compact homomorphism of locally convex $K$-vector spaces.

For a general ${\mathrm{\bf G}}$-equivariant vector bundle ${\mathcal{E}}$, we
again may apply Kiehl’s theorem [7, 9.4.3 Thm. 3] to find strict epimorphisms
${\mathcal{O}}(U_{I,\varepsilon})^{\oplus
k}\twoheadrightarrow{\mathcal{E}}(U_{I,\varepsilon})$ and
${{\mathcal{O}}(U_{I,\varepsilon^{\prime}})^{\oplus
k}\twoheadrightarrow{\mathcal{E}}(U_{I,\varepsilon^{\prime}})}$ which are
compatible, for some $k\in{\mathbb{N}}_{0}$. Then Lemma A.3 (ii) and (iii)
applied to

${{\mathcal{O}}(U_{I,\varepsilon})^{\oplus
k}}$${{\mathcal{O}}(U_{I,\varepsilon^{\prime}})^{\oplus
k}}$${{\mathcal{E}}(U_{I,\varepsilon})}$${{\mathcal{E}}(U_{I,\varepsilon^{\prime}})}$$\scriptstyle{{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{}}^{\oplus k}}$

show that
${\mathcal{E}}(U_{I,\varepsilon})\rightarrow{\mathcal{E}}(U_{I,\varepsilon^{\prime}})$
is compact.

Finally it follows from Lemma 2.4.1 that the image of the transition map is
dense. ∎

######  Proposition 2.5.2.

Let $0<\varepsilon<1$ with $\varepsilon\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$. Let $(\varepsilon_{m})_{m\in{\mathbb{N}}}$ be some strictly
decreasing sequence with $\varepsilon_{m}\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$, $\varepsilon<\varepsilon_{m}<1$, and
$\varepsilon_{m}\rightarrow\varepsilon$. Then the $K$-Fréchet space
$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
is the projective limit of the $K$-Banach spaces

$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\cong\varprojlim_{\varepsilon_{m}\searrow\varepsilon}H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}).$
(2.22)

The transition homomorphisms

$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow
H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\quad\text{,
for $\varepsilon_{m}<\varepsilon_{n}$,}$

which are induced from the inclusion
${\mathbb{P}}^{r}_{K}(\varepsilon_{m})^{-}\subset{\mathbb{P}}^{r}_{K}(\varepsilon_{n})^{-}$
are compact and have dense image so that
$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
is nuclear. Moreover, the local cohomology group
$H^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$ of the
algebraic variety ${\mathbb{P}}_{K}^{d}$ constitutes a dense subspace of
$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$.

The analogous statements for
$\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
hold as well.

###### Proof.

We first focus on the $K$-Banach spaces associated with the “closed”
neighbourhoods of ${\mathbb{P}}_{K}^{r}$. For now we fix $0<\varepsilon<1$
with $\varepsilon\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$ and consider the Čech complex
$C^{\bullet}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})$ associated with the
covering (2.12) which computes the cohomology groups
$H^{i}({\mathbb{P}}^{d}_{K}\setminus{\mathbb{P}}_{K}^{r}(\varepsilon)^{-},{\mathcal{E}})$.
For this Čech complex, we let

$\displaystyle Z^{i}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})$
$\displaystyle\vcentcolon={\mathrm{Ker}}\big{(}C^{i}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})\overset{d^{i}_{\varepsilon}}{\longrightarrow}C^{i+1}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})\big{)},$
$\displaystyle B^{i}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})$
$\displaystyle\vcentcolon={\mathrm{Im}}\big{(}C^{i-1}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})\overset{d^{i-1}_{\varepsilon}}{\longrightarrow}C^{i}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})\big{)}$

denote the space of $i$-th Čech cocycles respectively the space of $i$-th Čech
coboundaries. We use the analogous notation for further Čech complexes that
will occur.

Recall from Lemma 2.4.1 that the cochains
$C^{i}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})$ have the structure of a
$K$-Banach space, and the cochains
$C^{i}({\mathcal{U}},{\mathcal{E}}^{\mathrm{alg}})$ of algebraic sections
constitute a dense subspace therein. As a first step we want to see that the
induced locally convex topology on
$H^{i}({\mathbb{P}}^{d}_{K}\setminus{\mathbb{P}}_{K}^{r}(\varepsilon)^{-},{\mathcal{E}})$,
$\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})$,
and
$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})$
gives those spaces the structure of $K$-Banach spaces as well, and that their
“algebraic” counterparts
$H^{i}({\mathbb{P}}^{d}_{K}\setminus{\mathbb{P}}_{K}^{r},{\mathcal{E}}^{\mathrm{alg}})$,
$\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}^{d}_{K},{\mathcal{E}}^{\mathrm{alg}})$,
and
$H^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}^{d}_{K},{\mathcal{E}}^{\mathrm{alg}})$
embed as dense subspaces respectively.

To this end we consider $C^{i}({\mathcal{U}},{\mathcal{E}}^{\mathrm{alg}})$
with the subspace topology coming from
$C^{i}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})$ so that
$C^{i}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})$ is the completion of
$C^{i}({\mathcal{U}},{\mathcal{E}}^{\mathrm{alg}})$. We will endow all
“algebraic” terms with the topologies induced from
$C^{i}({\mathcal{U}},{\mathcal{E}}^{\mathrm{alg}})$, and all “analytic” terms
with the ones induced from $C^{i}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})$.

From Theorem 2.4.3 we know that the differential $d^{i}\colon
C^{i}({\mathcal{U}},{\mathcal{E}}^{\mathrm{alg}})\rightarrow
C^{i+1}({\mathcal{U}},{\mathcal{E}}^{\mathrm{alg}})$ is strict, and the
differential $d^{i}_{\varepsilon}\colon
C^{i}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})\rightarrow
C^{i+1}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})$ is the completion of
$d^{i}$. Therefore [7, 1.1.9 Prop. 5] implies that
$Z^{i}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})$ is the completion of
$Z^{i}({\mathcal{U}},{\mathcal{E}}^{\mathrm{alg}})$, and
$B^{i+1}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})$ is the completion of
$B^{i+1}({\mathcal{U}},{\mathcal{E}}^{\mathrm{alg}})$. Moreover note that
$d^{i}_{\varepsilon}$ is strict by [7, 1.1.9 Prop. 4], i.e.
$B^{i+1}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})$ is a closed subspace of
$C^{i+1}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})$ so that
$H^{i}({\mathbb{P}}^{d}_{K}\setminus{\mathbb{P}}_{K}^{r}(\varepsilon)^{-},{\mathcal{E}})$
is a $K$-Banach space.

We also have the short strictly exact sequence

$0\longrightarrow
B^{i}({\mathcal{U}},{\mathcal{E}}^{\mathrm{alg}})\longrightarrow
Z^{i}({\mathcal{U}},{\mathcal{E}}^{\mathrm{alg}})\longrightarrow
H^{i}({\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{r},{\mathcal{E}}^{\mathrm{alg}})\longrightarrow
0.$

The completion of this sequence yields the short strictly exact sequence [7,
1.1.9 Cor. 6]

$0\longrightarrow
B^{i}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})\longrightarrow
Z^{i}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})\longrightarrow
H^{i}({\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{r},{\mathcal{E}}^{\mathrm{alg}})^{{\,\widehat{}}}\longrightarrow
0.$

Therefore the map
$H^{i}({\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{r},{\mathcal{E}}^{\mathrm{alg}})^{\,\widehat{}}\rightarrow
H^{i}({\mathbb{P}}^{d}_{K}\setminus{\mathbb{P}}_{K}^{r}(\varepsilon)^{-},{\mathcal{E}})$
obtained by the universal property of the completion is a topological
isomorphism, and
$H^{i}({\mathbb{P}}^{d}_{K}\setminus{\mathbb{P}}_{K}^{r}(\varepsilon)^{-},{\mathcal{E}})$
is the completion of
$H^{i}({\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{r},{\mathcal{E}}^{\mathrm{alg}})$.

Now recall the long exact sequence of local cohomology

$\displaystyle\ldots\longrightarrow
H^{i-1}({\mathbb{P}}^{d}_{K},{\mathcal{E}}^{\mathrm{alg}})$
$\displaystyle\xrightarrow{\alpha^{i-1}}H^{i-1}({\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}^{r}_{K},{\mathcal{E}}^{\mathrm{alg}})$
(2.23)
$\displaystyle\xrightarrow{\partial^{i-1}}H^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}^{d}_{K},{\mathcal{E}}^{\mathrm{alg}})\xrightarrow{\beta^{i}}H^{i}({\mathbb{P}}^{d}_{K},{\mathcal{E}}^{\mathrm{alg}})\longrightarrow\ldots.$

We endow the local cohomology group
$H^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}^{d}_{K},{\mathcal{E}}^{\mathrm{alg}})$
with the locally convex final topology with respect to $\partial^{i-1}$. As
seen in Remark 2.1.1 this makes $\partial^{i-1}$ strict and $\beta^{i}$
continuous. Furthermore, the homomorphisms $\alpha^{i}$ and $\beta^{i}$ are
strict as well since
$H^{i}({\mathbb{P}}_{K}^{d},{\mathcal{E}}^{\mathrm{alg}})$ is a finite-
dimensional locally convex $K$-vector space [59, Cor. 3.4.25]. The analogous
situation arises for
${\mathbb{P}}^{r}_{K}(\varepsilon)^{-}\subset{\mathbb{P}}_{K}^{d}$.

From this long strictly exact sequence, we obtain the following commutative
diagram with strictly exact rows

${0}$${{\mathrm{Ker}}(\alpha^{i-1})}$${H^{i-1}({\mathbb{P}}_{K}^{d},{\mathcal{E}}^{\mathrm{alg}})}$${{\mathrm{Im}}(\alpha^{i-1})}$${0}$${0}$${{\mathrm{Ker}}(\alpha^{i-1}_{\varepsilon})}$${H^{i-1}({\mathbb{P}}_{K}^{d},{\mathcal{E}})}$${{\mathrm{Im}}(\alpha^{i-1}_{\varepsilon})}$${0.}$$\scriptstyle{\alpha^{i-1}}$$\scriptstyle{\alpha^{i-1}_{\varepsilon}}$
(2.24)

Because $\alpha^{i-1}_{\varepsilon}$ is a strict epimorphism, it follows that
${\mathrm{Im}}(\alpha^{i-1})\rightarrow{\mathrm{Im}}(\alpha^{i-1}_{\varepsilon})$
is a strict epimorphism [76, Prop. 1.1.8].

From (2.23) we moreover obtain the commutative diagram

${0}$${{\mathrm{Im}}(\alpha^{i-1})}$${H^{i-1}({\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{r},{\mathcal{E}}^{\mathrm{alg}})}$${\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}^{\mathrm{alg}})}$${0}$${0}$${{\mathrm{Im}}(\alpha^{i-1}_{\varepsilon})}$${H^{i-1}({\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{r}(\varepsilon)^{-},{\mathcal{E}})}$${\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})}$${0}$$\scriptstyle{\partial^{i-1}}$$\scriptstyle{\partial^{i-1}_{\varepsilon}}$
(2.25)

with strictly exact rows. Here we have used the exactness of (2.23) for the
identification ${\mathrm{Ker}}(\partial^{i-1})={\mathrm{Im}}(\alpha^{i-1})$
and
${\mathrm{Im}}(\partial^{i-1})={\mathrm{Ker}}(\beta^{i})=\vcentcolon\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}^{\mathrm{alg}})$,
and likewise for the “analytic” terms. The composition
${\mathrm{Im}}(\alpha^{i-1})\rightarrow
H^{i-1}({\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{r},{\mathcal{E}}^{\mathrm{alg}})\rightarrow
H^{i-1}({\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{r}(\varepsilon)^{-},{\mathcal{E}})$
is a strict monomorphism [76, Prop. 1.1.7]. Therefore
${\mathrm{Im}}(\alpha^{i-1})\rightarrow{\mathrm{Im}}(\alpha^{i-1}_{\varepsilon})$
is a strict monomorphism as well [76, Prop. 1.1.8], and we conclude that
${\mathrm{Im}}(\alpha^{i-1})\cong{\mathrm{Im}}(\alpha_{\varepsilon}^{i-1})$.
By taking the completion of the first row of (2.25) and reasoning similarly to
before, we find that
$\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
is the completion of
$\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}^{\mathrm{alg}})$.

From (2.24) we also can conclude that
${\mathrm{Ker}}(\alpha^{i-1})={\mathrm{Ker}}(\alpha^{i-1}_{\varepsilon})$ by
applying the appropriate version of the snake lemma A.13. Since
${\mathrm{Im}}(\beta^{i})={\mathrm{Ker}}(\alpha^{i})$ and
${\mathrm{Im}}(\beta^{i}_{\varepsilon})={\mathrm{Ker}}(\alpha^{i}_{\varepsilon})$
by the exactness of (2.23), we arrive at the short strictly exact sequence

$0\longrightarrow\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}^{\mathrm{alg}})\longrightarrow
H^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}^{\mathrm{alg}})\longrightarrow{\mathrm{Ker}}(\alpha^{i})\longrightarrow
0.$

The completion of this sequence yields the following commutative diagram with
strictly exact rows [7, 1.1.9 Prop. 4, Cor. 6]

${0}$${\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}^{\mathrm{alg}})^{\,\widehat{}}}$${H^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}^{\mathrm{alg}})^{\,\widehat{}}}$${{\mathrm{Ker}}(\alpha^{i})}$${0}$${0}$${\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})}$${H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})}$${{\mathrm{Ker}}(\alpha^{i}_{\varepsilon})}$${0.}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$
(2.26)

Via the snake lemma A.13 it follows that the vertical homomorphism in the
middle is a topological isomorphism, too. Therefore
$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
is the completion of
$H^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}^{\mathrm{alg}})$.

Fixing $0<\varepsilon<1$ with $\varepsilon\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$ and a strictly decreasing sequence
$(\varepsilon_{m})_{m\in{\mathbb{N}}}$ as specified, we now consider
$\varepsilon<\varepsilon_{m}<\varepsilon_{n}<1$ with
$\varepsilon_{m},\varepsilon_{n}\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$. Using the commutativity of

${\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}^{\mathrm{alg}})}$${\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})}$${\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})}$

it follows from the statements about the density of the “algebraic” subspaces
which we have just shown that the transition homomorphism
$\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\rightarrow\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
has dense image. Analogously one argues for
$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\rightarrow
H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$.

To show that the transition homomorphisms are compact, we start at the
commutative diagram

${0}$${Z^{i}({\mathcal{U}}_{\varepsilon_{m}},{\mathcal{E}})}$${C^{i}({\mathcal{U}}_{\varepsilon_{m}},{\mathcal{E}})}$${B^{i+1}({\mathcal{U}}_{\varepsilon_{m}},{\mathcal{E}})}$${0}$${0}$${Z^{i}({\mathcal{U}}_{\varepsilon_{n}},{\mathcal{E}})}$${C^{i}({\mathcal{U}}_{\varepsilon_{n}},{\mathcal{E}})}$${B^{i+1}({\mathcal{U}}_{\varepsilon_{n}},{\mathcal{E}})}$${0}$$\scriptstyle{d^{i}_{\varepsilon_{m}}}$$\scriptstyle{d^{i}_{\varepsilon_{n}}}$

with strictly exact rows. We have seen in Lemma 2.5.1 that the vertical
homomorphism in the middle is compact. Hence Lemma A.3 (i) implies that
$Z^{i}({\mathcal{U}}_{\varepsilon_{m}},{\mathcal{E}})\rightarrow
Z^{i}({\mathcal{U}}_{\varepsilon_{n}},{\mathcal{E}})$ is compact. Using Lemma
A.3 (ii) one argues in the analogous way to conclude that
$H^{i-1}({\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-},{\mathcal{E}})\rightarrow
H^{i-1}({\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-},{\mathcal{E}})$
and
$\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\rightarrow\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
are compact. For

${0}$${\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})}$${H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})}$${{\mathrm{Ker}}(\alpha^{i}_{\varepsilon_{m}})}$${0}$${0}$${\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})}$${H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})}$${{\mathrm{Ker}}(\alpha^{i}_{\varepsilon_{n}})}$${0}$

the short strictly exact sequences of locally convex $K$-vector spaces in both
rows split compatibly as
${\mathrm{Ker}}(\alpha_{\varepsilon_{m}}^{i})\cong{\mathrm{Ker}}(\alpha^{i}_{\varepsilon_{n}})$
is finite-dimensional [59, Cor. 3.4.27]. Therefore the transition homomorphism
$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\rightarrow
H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
is compact by Lemma A.3 (iii).

######  Remark 2.5.3.

Having seen that the transition maps
$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\rightarrow
H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
have dense image, we can apply [56, Prop. 1.3.3] to obtain the topological
isomorphism (2.22) at this point. However, we will additionally present the
alternative way mentioned in [56, Rmk. 1.3.6] to do so. To this end, one needs
the following lemma.

######  Lemma 2.5.4 (cf. [56, Lemma 1.3.7]).

Consider a projective system of strictly exact sequences

$0\longrightarrow V^{\prime}_{n}\longrightarrow V_{n}\longrightarrow
V^{\prime\prime}_{n}\longrightarrow 0\quad\text{, for $n\in{\mathbb{N}}$,}$

of $K$-Fréchet spaces. If the transition maps $V^{\prime}_{n+1}\rightarrow
V^{\prime}_{n}$, for all $n\in{\mathbb{N}}$, have dense image, then the
sequence

$0\longrightarrow\varprojlim_{n\in{\mathbb{N}}}V^{\prime}_{n}\longrightarrow\varprojlim_{n\in{\mathbb{N}}}V_{n}\longrightarrow\varprojlim_{n\in{\mathbb{N}}}V^{\prime\prime}_{n}\longrightarrow
0$ (2.27)

is strictly exact, too.

###### Proof.

We may view $V^{\prime}_{n}$ as the kernel of $V_{n}\rightarrow
V^{\prime\prime}_{n}$. Since taking the (projective) limit commutes with
kernels, we see that $\varprojlim_{n\in{\mathbb{N}}}V^{\prime}_{n}$ is the
kernel of
$\varprojlim_{n\in{\mathbb{N}}}V_{n}\rightarrow\varprojlim_{n\in{\mathbb{N}}}V^{\prime\prime}_{n}$.

Moreover, as the transition homomorphisms $V^{\prime}_{n+1}\rightarrow
V^{\prime}_{n}$ have dense image, the topological Mittag-Leffler condition is
fulfilled for this inverse system. It follows that (2.27) is a short exact
sequence of vector spaces [34, 13.2.4 (i)]. Finally, the open mapping theorem
[67, Prop. 8.6] implies that
$\varprojlim_{n\in{\mathbb{N}}}V_{n}\rightarrow\varprojlim_{n\in{\mathbb{N}}}V^{\prime\prime}_{n}$
is strict, too. ∎

The differential $d^{i}_{\varepsilon}\colon
C^{i}({\mathcal{U}}_{\varepsilon}^{-},{\mathcal{E}})\rightarrow
C^{i+1}({\mathcal{U}}_{\varepsilon}^{-},{\mathcal{E}})$ is the projective
limit of the differentials $d^{i}_{\varepsilon_{m}}\colon
C^{i}({\mathcal{U}}_{\varepsilon_{m}},{\mathcal{E}})\rightarrow
C^{i+1}({\mathcal{U}}_{\varepsilon_{m}},{\mathcal{E}})$. Therefore we have
$C^{i}({\mathcal{U}}_{\varepsilon}^{-},{\mathcal{E}})\cong\varprojlim_{\varepsilon_{m}\searrow\varepsilon}C^{i}({\mathcal{U}}_{\varepsilon_{m}},{\mathcal{E}})$
and
$Z^{i}({\mathcal{U}}_{\varepsilon}^{-},{\mathcal{E}})\cong\varprojlim_{\varepsilon_{m}\searrow\varepsilon}Z^{i}({\mathcal{U}}_{\varepsilon_{m}},{\mathcal{E}})$
because the (projective) limit commutes with taking kernels. But by Lemma
2.5.4, there is the short strictly exact sequence

$0\longrightarrow\varprojlim_{\varepsilon_{m}\searrow\varepsilon}Z^{i}({\mathcal{U}}_{\varepsilon_{m}},{\mathcal{E}})\longrightarrow\varprojlim_{\varepsilon_{m}\searrow\varepsilon}C^{i}({\mathcal{U}}_{\varepsilon_{m}},{\mathcal{E}})\longrightarrow\varprojlim_{\varepsilon_{m}\searrow\varepsilon}B^{i+1}({\mathcal{U}}_{\varepsilon_{m}},{\mathcal{E}})\longrightarrow
0$

so that we can conclude
$B^{i+1}({\mathcal{U}}_{\varepsilon}^{-},{\mathcal{E}})\cong\varprojlim_{\varepsilon_{m}\searrow\varepsilon}B^{i+1}({\mathcal{U}}_{\varepsilon_{m}},{\mathcal{E}})$.
In a similar way we obtain
$H^{i}({\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{r}(\varepsilon),{\mathcal{E}})\cong\varprojlim_{\varepsilon_{m}\searrow\varepsilon}H^{i}({\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-},{\mathcal{E}})$.

Likewise arguing for the homomorphisms
$\beta^{i}_{\varepsilon}=\varprojlim_{\varepsilon_{m}\searrow\varepsilon}\beta^{i}_{\varepsilon_{m}}$
and
$\alpha^{i}_{\varepsilon}=\varprojlim_{\varepsilon_{m}\searrow\varepsilon}\alpha^{i}_{\varepsilon_{m}}$,
we find that
$\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\cong\varprojlim_{\varepsilon_{m}\searrow\varepsilon}\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
and
${\mathrm{Ker}}(\alpha_{\varepsilon}^{i})\cong\varprojlim_{\varepsilon_{m}\searrow\varepsilon}{\mathrm{Ker}}(\alpha_{\varepsilon_{m}}^{i})$.
We now take the projective limit over the projective system

$0\longrightarrow\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow
H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow{\mathrm{Ker}}(\alpha^{i}_{\varepsilon_{m}})\longrightarrow
0$

of short strictly exact sequences to arrive at the following commutative
diagram with strictly exact rows

${0}$${\varprojlim\limits_{\varepsilon_{m}\searrow\varepsilon}\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})}$${\varprojlim\limits_{\varepsilon_{m}\searrow\varepsilon}H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})}$${\varprojlim\limits_{\varepsilon_{m}\searrow\varepsilon}{\mathrm{Ker}}(\alpha^{i}_{\varepsilon_{m}})}$${0}$${0}$${\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)}({\mathbb{P}}_{K}^{d},{\mathcal{E}})}$${H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)}({\mathbb{P}}_{K}^{d},{\mathcal{E}})}$${{\mathrm{Ker}}(\alpha^{i}_{\varepsilon})}$${0.}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

Since the outer vertical maps are topological isomorphisms, it follows from
the snake lemma A.13 that
$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\cong\varprojlim_{\varepsilon_{m}\searrow\varepsilon}H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$.
∎

The same reasoning shows the following statement in the extreme case
$\varepsilon=0$.

######  Corollary 2.5.5.

For any strictly decreasing sequence
$(\varepsilon_{m})_{m\in{\mathbb{N}}}\subset{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$ with $0<\varepsilon_{m}<1$ and $\varepsilon_{m}\rightarrow 0$,
the $K$-Fréchet space
$H^{i}_{({\mathbb{P}}_{K}^{r})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
is the projective limit of the $K$-Banach spaces

$H^{i}_{({\mathbb{P}}_{K}^{r})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\cong\varprojlim_{\varepsilon_{m}\searrow
0}H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}).$

with compact transition maps which have dense image. Moreover, the local
cohomology group
$H^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$ of the
algebraic variety ${\mathbb{P}}_{K}^{d}$ constitutes a dense subspace of
$H^{i}_{({\mathbb{P}}_{K}^{r})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$.

For
$\widetilde{H}^{i}_{({\mathbb{P}}_{K}^{r})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
the analogous assertions hold.

There also is a result for the projective limit of the local cohomology groups
$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
with respect to the “open” $\varepsilon$-neighbourhoods.

######  Corollary 2.5.6.

For any strictly decreasing sequence
$(\varepsilon_{m})_{m\in{\mathbb{N}}}\subset{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$ with $0<\varepsilon_{m}<1$ and $\varepsilon_{m}\rightarrow 0$,
there is a topological isomorphism

$H^{i}_{({\mathbb{P}}_{K}^{r})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\cong\varprojlim_{\varepsilon_{m}\searrow
0}H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
(2.28)

of $K$-Fréchet spaces. Moreover, the transition homomorphisms

$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow
H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\quad\text{,
for $\varepsilon_{m}<\varepsilon_{n}$,}$

are compact and have dense image.

Again, the analogous statements are true for
$\widetilde{H}^{i}_{({\mathbb{P}}_{K}^{r})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$.

###### Proof.

For the topological isomorphism (2.28) one argues analogously to the last part
of the proof of the preceeding proposition. To show that the topological
Mittag-Leffler condition for Lemma 2.5.4 is fulfilled one uses the statement
of Proposition 2.5.2 about the density of
$H^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\subset
H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$.

To see that the transition homomorphisms are compact, note that, for
$\varepsilon_{m}<\varepsilon_{n}$, the transition map factors as

$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{m})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow
H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow
H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}}).$
(2.29)

The first homomorphism is a continuous linear map from a nuclear locally
convex $K$-vector space to a $K$-Banach space and therefore is compact by [67,
Prop. 19.5]. It follows from [67, Rmk. 16.7 (i)] that the composition (2.29)
is compact as well.

For
$\widetilde{H}^{i}_{({\mathbb{P}}_{K}^{r})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
one argues analogously. ∎

######  Proposition 2.5.7.

We have the following description of the local cohomology groups

$H^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})=\begin{cases}0&\text{
, for $i<d-{r}$ or $i>d$,}\\\
H^{d-{r}}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})&\text{ ,
for $i=d-{r}$,}\\\ H^{i}({\mathbb{P}}_{K}^{d},{\mathcal{E}})&\text{ , for
$i>d-{r}$.}\end{cases}$

For $0<\varepsilon<1$ with $\varepsilon\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$, we have

$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)}({\mathbb{P}}_{K}^{d},{\mathcal{E}})=\begin{cases}0&\text{
, for $i<d-{r}$ or $i>d$,}\\\
H^{d-{r}}_{{\mathbb{P}}_{K}^{r}(\varepsilon)}({\mathbb{P}}_{K}^{d},{\mathcal{E}})&\text{
, for $i=d-{r}$,}\\\ H^{i}({\mathbb{P}}_{K}^{d},{\mathcal{E}})&\text{ , for
$i>d-{r}$,}\end{cases}$

and similarly for
$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
and
$H^{i}_{({\mathbb{P}}_{K}^{r})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$.

###### Proof.

The statement for
$H^{i}_{{\mathbb{P}}_{K}^{r}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$ is shown in
[56, pp. 595–597] (and the reasoning there is independent of the field $K$).
For
$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$,
$H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$,
and
$H^{i}_{({\mathbb{P}}_{K}^{r})^{\mathrm{rig}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
the assertion then follows from the density of the “algebraic” local
cohomology groups. ∎

###  2.6. Local Cohomology Groups with respect to Schubert Varieties as
Locally Analytic Representations

We let ${\mathrm{\bf B}}\subset{\mathrm{\bf G}}={\mathrm{GL}}_{d+1,K}$ denote
the Borel subgroup of lower triangular matrices, and ${\mathrm{\bf
T}}\subset{\mathrm{\bf G}}$ the maximal torus of diagonal matrices. For
$i=0,\ldots,d$, let $\epsilon_{i}\colon{\mathrm{\bf
T}}\rightarrow{\mathbb{G}}_{m}$ be the character defined via
$\epsilon_{i}({\rm diag}(t_{0},\ldots,t_{d}))=t_{i}$, and set
$\alpha_{i,j}\vcentcolon=\epsilon_{i}-\epsilon_{j}$, for $i\neq j$, and
$\alpha_{i}\vcentcolon=\alpha_{i+1,i}$, for $i=0,\ldots,d-1$. Then the roots
of ${\mathrm{\bf G}}$ with respect to ${\mathrm{\bf T}}$ are

$\Phi=\\{\alpha_{i,j}\mid 0\leq i\neq j\leq d\\}$

and its simple roots with respect to ${\mathrm{\bf T}}\subset{\mathrm{\bf B}}$
are

$\Delta=\\{\alpha_{0},\ldots,\alpha_{d-1}\\}.$

Moreover, for $I\subset\Delta$, we let ${\mathrm{\bf
P}}_{I}\subset{\mathrm{\bf G}}$ denote the (lower) standard parabolic subgroup
associated with $I$, i.e. the subgroup generated by ${\mathrm{\bf B}}$ and the
root subgroups ${\mathrm{\bf U}}_{-\alpha}$, for $\alpha\in I$. For example,
we have ${\mathrm{\bf P}}_{\emptyset}={\mathrm{\bf B}}$ and ${\mathrm{\bf
P}}_{\Delta}={\mathrm{\bf G}}$. We write $P_{I}\vcentcolon={\mathrm{\bf
P}}_{I}(K)$ which is a locally $K$-analytic subgroup of $G$. We set
$P_{I,0}\vcentcolon={\mathrm{\bf P}}_{I,{\mathbb{Z}}}({\mathcal{O}}_{K})$
which is a compact open subgroup of $P_{I}$. Here ${\mathrm{\bf
P}}_{I,{\mathbb{Z}}}$ denotes the respective standard parabolic subgroup of
${\mathrm{GL}}_{d+1,{\mathbb{Z}}}$. Additionally, consider the canonical
reduction homomorphism

$p_{n}\colon
G_{0}\longrightarrow{\mathrm{GL}}_{d+1}({\mathcal{O}}_{K}/{\pi}^{n}),$

for $n\in{\mathbb{N}}$. We define

$P^{n}_{I}\vcentcolon=p_{n}^{-1}\big{(}{\mathrm{\bf
P}}_{I,{\mathbb{Z}}}({\mathcal{O}}_{K}/{\pi}^{n})\big{)}$

which is an open compact subgroup of $G_{0}$ containing $P_{I,0}$.

We now fix ${r}\in\\{0,\ldots,d-1\\}$. Then the maximal parabolic subgroup
${\mathrm{\bf P}}_{\Delta\setminus\\{\alpha_{r}\\}}$ stabilizes the subvariety
${\mathbb{P}}_{K}^{r}$ of ${\mathbb{P}}_{K}^{d}$. Consequently
$P_{\Delta\setminus\\{\alpha_{r}\\}}$ stabilizes
$({\mathbb{P}}_{K}^{r})^{\mathrm{rig}}$ under the group action of $G$ on
$({\mathbb{P}}_{K}^{d})^{\mathrm{rig}}$.

We claim that, for any $n\in{\mathbb{N}}$ and
$\varepsilon\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$ with ${\lvert{{\pi}}\rvert}^{n}\leq\varepsilon<1$, the
subgroup $P^{n}_{\Delta\setminus\\{\alpha_{{r}}\\}}\subset G_{0}$ stabilizes
${\mathbb{P}}^{r}_{K}(\varepsilon)$. Indeed, for
$[z_{0}{\,:\,}\ldots{\,:\,}z_{d}]\in{\mathbb{P}}^{r}_{K}(\varepsilon)$ and
$g=(g_{ij})\in P^{n}_{\Delta\setminus\\{\alpha_{{r}}\\}}$, let us write
$g^{-1}z=\vcentcolon[w_{0}{\,:\,}\ldots{\,:\,}w_{d}]$. We thus have, for
$j={r}+1,\ldots,d$,

$\displaystyle{\lvert{w_{j}}\rvert}=\bigg{\lvert}\sum_{i=0}^{d}z_{i}g_{ij}\bigg{\rvert}\leq\max\Big{(}\max_{i=0}^{r}{\lvert{z_{i}g_{ij}}\rvert},\max_{i={r}+1}^{d}{\lvert{z_{i}g_{ij}}\rvert}\Big{)}\leq\max\Big{(}\max_{i=0}^{r}1\cdot{\lvert{{\pi}^{n}}\rvert},\max_{i={r}+1}^{d}\varepsilon\cdot
1\Big{)}\leq\varepsilon.$

Analogously one computes that ${\mathbb{P}}^{r}_{K}(\varepsilon)^{-}$ is
stabilized by $P^{n}_{\Delta\setminus\\{\alpha_{{r}}\\}}$, for any
$\varepsilon\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$ and $n\in{\mathbb{N}}$ with
${\lvert{{\pi}}\rvert}^{n}<\varepsilon<1$.

######  Proposition 2.6.1 (cf. [56, Cor. 1.3.9]).

Let $\varepsilon\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$ and $n\in{\mathbb{N}}$ with
${\lvert{{\pi}}\rvert}^{n}<\varepsilon<1$. Then the representation

$P^{n}_{\Delta\setminus\\{\alpha_{{r}}\\}}\times\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon)}({\mathbb{P}}^{d}_{K},{\mathcal{E}})^{\prime}_{b}\longrightarrow\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon)}({\mathbb{P}}^{d}_{K},{\mathcal{E}})^{\prime}_{b}\,,\quad(g,\ell)\longmapsto\ell(g^{-1}{\,\\_\,}),$

is locally analytic. Moreover, for any strictly decreasing sequence
$(\varepsilon_{m})_{m\in{\mathbb{N}}}\subset{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$ with $\varepsilon_{m}\rightarrow\varepsilon$ and
$\varepsilon<\varepsilon_{m}<1$, the canonical map

$\varinjlim_{\varepsilon_{m}\searrow\varepsilon}\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{m})^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})^{\prime}\longrightarrow\Big{(}\varprojlim_{\varepsilon_{m}\searrow\varepsilon}\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{m})^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})\Big{)}^{\prime}_{b}=\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon)}({\mathbb{P}}^{d}_{K},{\mathcal{E}})^{\prime}_{b}$

is a topological isomorphism, and
$\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon)}({\mathbb{P}}^{d}_{K},{\mathcal{E}})^{\prime}_{b}$
is of compact type this way, i.e. it is the inductive limit of the $K$-Banach
spaces

${\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{1})^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})^{\prime}}$${\ldots}$${\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{m})^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})^{\prime}}$${\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{m+1})^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})^{\prime}}$${\ldots}$

with compact, injective transition homomorphisms.

For the extreme case $\varepsilon=0$, the representation

$P_{\Delta\setminus\\{\alpha_{{r}}\\}}\times\widetilde{H}^{i}_{({\mathbb{P}}^{r}_{K})^{\mathrm{rig}}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})^{\prime}_{b}\longrightarrow\widetilde{H}^{i}_{({\mathbb{P}}^{r}_{K})^{\mathrm{rig}}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})^{\prime}_{b}\,,\quad(g,\ell)\longmapsto\ell(g^{-1}{\,\\_\,}),$

is locally analytic, and the underlying locally convex $K$-vector space
$\widetilde{H}^{i}_{({\mathbb{P}}^{r}_{K})^{\mathrm{rig}}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})^{\prime}_{b}$
is of compact type analogously to (i).

Like in Section 2.3, we proceed step by step.

######  Lemma 2.6.2.

Let $0<\varepsilon<1$ with $\varepsilon\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$. For $n\in{\mathbb{N}}$ with
${\lvert{{\pi}}\rvert}^{n}\leq\varepsilon$ (respectively,
${\lvert{{\pi}}\rvert}^{n}<\varepsilon$), the group
$P^{n}_{\Delta\setminus\\{\alpha_{{r}}\\}}$ acts on
$H^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon)}({\mathbb{P}}^{d}_{K},{\mathcal{E}})$
(respectively, on
$H^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon)^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})$)
by continuous endomorphisms, and the topological isomorphism (2.22) is
$P^{n}_{\Delta\setminus\\{\alpha_{{r}}\\}}$-equivariant. Moreover, the
analogous assertions are true for
$\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon)}({\mathbb{P}}^{d}_{K},{\mathcal{E}})$
and
$\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon)^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})$.

In the extreme case $\varepsilon=0$, $P_{\Delta\setminus\\{\alpha_{{r}}\\}}$
acts on
$H^{i}_{({\mathbb{P}}^{r}_{K})^{\mathrm{rig}}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})$
and
$\widetilde{H}^{i}_{({\mathbb{P}}^{r}_{K})^{\mathrm{rig}}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})$
by continuous endomorphisms, and the topological isomorphism (2.22) is
$P_{\Delta\setminus\\{\alpha_{{r}}\\},0}$-equivariant.

###### Proof.

For the local cohomology groups this follows from the discussion in Section
2.2. The fact that the long exact sequence of local cohomology is equivariant
for the respective group actions, implies the assertion for the kernels
$\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon)}({\mathbb{P}}^{d}_{K},{\mathcal{E}})$,
$\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon)^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})$,
and
$\widetilde{H}^{i}_{({\mathbb{P}}^{r}_{K})^{\mathrm{rig}}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})$.
∎

######  Lemma 2.6.3.

For $n\in{\mathbb{N}}$ and $\varepsilon\in{\lvert{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{K}}\rvert}$ with ${\lvert{{\pi}}\rvert}^{n}<\varepsilon<1$, the
representation

$P^{n}_{\Delta\setminus\\{\alpha_{{r}}\\}}\times\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon)^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})\longrightarrow\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon)^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})\,,\quad(g,v)\longmapsto
g.v,$

on the $K$-Banach space
$\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon)^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})$
is locally analytic.

###### Proof.

Like in Lemma 2.3.4 the only assertion left to show is that the orbit maps

$P^{n}_{\Delta\setminus\\{\alpha_{{r}}\\}}\longrightarrow\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon)^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})\,,\quad
g\longmapsto g.v,$

are locally analytic, for every
$v\in\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon)^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})$.
To do so, we first show that the orbit maps for the
$P^{n}_{\Delta\setminus\\{\alpha_{{r}}\\}}$-action on
$H^{i}({\mathbb{P}}^{d}_{K}\setminus{\mathbb{P}}^{r}_{K}(\varepsilon)^{-},{\mathcal{E}})$
are locally analytic.

Fix $g\in P^{n}_{\Delta\setminus\\{\alpha_{{r}}\\}}$ and consider the affinoid
subdomain $gD_{n}\subset{\mathrm{GL}}_{d+1}(C)$, where we set
$D_{n}\vcentcolon=1+{\pi}^{n}M_{d+1}({\mathcal{O}}_{C})$, with the rigid
analytic chart

$\iota_{g}\colon D_{n}\longrightarrow gD_{n}\,,\quad h\longmapsto gh.$

Moreover, we claim that, for every fixed
$v\in\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon)^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})$,
the orbit map

$P^{n}_{\Delta\setminus\\{\alpha_{{r}}\\}}\longrightarrow
H^{i}({\mathbb{P}}^{d}_{K}\setminus{\mathbb{P}}^{r}_{K}(\varepsilon)^{-},{\mathcal{E}})\,,\quad
h\longmapsto h.v,$

restricted to $gD_{n}(K)$ is given by a convergent power series.

To this end, we consider the admissible covering ${\mathcal{U}}_{\varepsilon}$
of ${\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}^{r}_{K}(\varepsilon)^{-}$ whose
Čech complex $C^{i}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})$ computes
$H^{i}({\mathbb{P}}^{d}_{K}\setminus{\mathbb{P}}^{r}_{K}(\varepsilon)^{-},{\mathcal{E}})$.
We fix a non-empty subset $I\subset\\{{r}+1,\ldots,d\\}$ and write
$U\vcentcolon=U_{I,\varepsilon}$ with the notation from (2.15). We then have
$h(U)=g(U)$, for all $h\in gD_{n}$. Indeed, let
$z=[z_{0}{\,:\,}\ldots{\,:\,}z_{d}]\in U$ and $(1+h^{\prime})\in D_{n}$ so
that

$[z_{0}{\,:\,}\ldots{\,:\,}z_{d}]\cdot(1+h^{\prime})=\vcentcolon[w_{0}{\,:\,}\ldots{\,:\,}w_{d}]\quad\text{,
with $w_{i}=z_{i}+\sum_{j=0}^{d}z_{j}h^{\prime}_{ji}$.}$

For $i\in I$, we have

$\displaystyle\Big{\lvert}\sum_{j=0}^{d}z_{j}h^{\prime}_{ji}\Big{\rvert}\leq\max_{j=0}^{d}{\lvert{z_{j}h^{\prime}_{ji}}\rvert}\leq\max_{j=0}^{d}{\lvert{z_{j}}\rvert}\,{\lvert{{\pi}}\rvert}^{n}<\max_{j=0}^{d}\,\varepsilon\,{\lvert{z_{j}}\rvert}\leq{\lvert{z_{i}}\rvert}$

where the last inequality holds because $z\in U\subset U_{i,\varepsilon}$.
This implies ${\lvert{w_{i}}\rvert}={\lvert{z_{i}}\rvert}$. Now we compute,
for $i\in I$, $j\in\\{0,\ldots,d\\}$:

$\displaystyle\varepsilon\,{\lvert{w_{j}}\rvert}$
$\displaystyle\leq\max\Big{(}\varepsilon\,{\lvert{z_{j}}\rvert},\max_{k=0}^{d}\,\varepsilon\,{\lvert{z_{k}h^{\prime}_{kj}}\rvert}\Big{)}\leq\max\Big{(}{\lvert{z_{i}}\rvert},\max_{k=0}^{d}\,\varepsilon\,{\lvert{{\pi}}\rvert}^{n}\,{\lvert{z_{k}}\rvert}\Big{)}\leq{\lvert{z_{i}}\rvert}={\lvert{w_{i}}\rvert}$

which implies that $(1+h^{\prime})^{-1}.z\in U$.

Since the above non-empty subset $I\subset\\{{r}+1,\ldots,d\\}$ was arbitrary,
we obtain a map

$gD_{n}(K)\times
C^{i}(g({\mathcal{U}}_{\varepsilon}),{\mathcal{E}})\longrightarrow
C^{i}({\mathcal{U}}_{\varepsilon},{\mathcal{E}})$

that affords the $P^{n}_{\Delta\setminus\\{\alpha_{{r}}\\}}$-action on
$H^{i}({\mathbb{P}}^{d}_{K}\setminus{\mathbb{P}}^{r}_{K}(\varepsilon)^{-},{\mathcal{E}})$
restricted to $gD_{n}(K)$, cf. (2.5). Here $g({\mathcal{U}}_{\varepsilon})$
denotes the translated covering
${\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}^{r}_{K}(\varepsilon)^{-}=\bigcup_{i={r}+1}^{d}g(U_{i,\varepsilon})$.
Consequently it suffices to show that, for every $v\in{\mathcal{E}}(g(U))$
with $U\vcentcolon=U_{I,\varepsilon}$, for non-empty
$I\subset\\{{r}+1,\ldots,d\\}$, the map
$gD_{n}(K)\rightarrow{\mathcal{E}}(U)$, $h\mapsto h.v$, is given by a
convergent power series.

For this we proceed similarly to the proof or Lemma 2.3.4. Using that
$h(U)=g(U)$, for all $h\in gD_{n}$, the group action
$\sigma\colon{\mathrm{GL}}_{d+1,K}\times_{K}{\mathbb{P}}_{K}^{d}\rightarrow{\mathbb{P}}_{K}^{d}$
induces the following commutative diagram:

${D_{n}\times_{K}U}$${gD_{n}\times_{K}U}$${g(U)}$${U}$.$\scriptstyle{\iota_{g}\times\mathrm{id}}$$\scriptstyle{{\mathrm{pr}}_{2}}$$\scriptstyle{\sigma}$$\scriptstyle{{\mathrm{pr}}_{2}}$

Let $F_{v}\in{\mathcal{E}}(U)\langle T_{1},\ldots,T_{(d+1)^{2}}\rangle$ be the
power series to which $v\in{\mathcal{E}}(g(U))$ is mapped under

${{\mathcal{E}}(g(U))}$${(\iota_{g}\times{\mathrm{id}})^{\ast}\sigma^{\ast}{\mathcal{E}}(D_{n}\times_{K}U)}$${(\iota_{g}\times{\mathrm{id}})^{\ast}{\mathrm{pr}}_{2}^{\ast}{\mathcal{E}}(D_{n}\times_{K}U)}$${\cong{\mathrm{pr}}_{2}^{\ast}{\mathcal{E}}(D_{n}\times_{K}U)}$${\cong\big{(}{\mathcal{O}}(D_{n}){\,\widehat{\otimes}_{K}\,}{\mathcal{O}}(U)\big{)}\otimes_{{\mathcal{O}}(U)}{\mathcal{E}}(U)}$${\cong{\mathcal{E}}(U)\langle
T_{1},\ldots,T_{(d+1)^{2}}\rangle.}$$\scriptstyle{(\iota_{g}\times\mathrm{id})^{\ast}\Phi(D_{n}\times_{K}U)}$

Now consider, for $h\in D_{n}(K)$,

${D_{n}\times_{K}U}$${gD_{n}\times_{K}U}$${g(U)}$${U}$.$\scriptstyle{\iota_{g}\times{\mathrm{id}}}$$\scriptstyle{\sigma}$$\scriptstyle{h\times{\mathrm{id}}}$$\scriptstyle{gh}$

In terms of $K$-affinoid algebras the morphism $h\times{\mathrm{id}}\colon
U\rightarrow D_{n}\times_{K}U$ is given by the evaluation homomorphism of
power series

$\displaystyle\mathrm{ev}_{h}\colon{\mathcal{O}}(D_{n}){\,\widehat{\otimes}_{K}\,}{\mathcal{O}}(U)\cong{\mathcal{O}}(U)\langle
T_{1},\ldots,T_{(d+1)^{2}}\rangle$
$\displaystyle\longrightarrow{\mathcal{O}}(U),$ $\displaystyle F$
$\displaystyle\longmapsto F(h).$

Hence we arrive at the commutative diagram

${{\mathcal{E}}(g(U))}$${(\iota_{g}\times{\mathrm{id}})^{\ast}\sigma^{\ast}{\mathcal{E}}(D_{n}\times_{K}U)}$${{\mathcal{E}}(U)\langle
T_{1},\ldots,T_{(d+1)^{2}}\rangle}$${(gh)^{\ast}{\mathcal{E}}(U)}$${{\mathcal{E}}(U)}$$\scriptstyle{(h\times{\mathrm{id}})^{\ast}}$$\scriptstyle{\mathrm{ev}_{h}}$$\scriptstyle{\Phi_{gh}(U)}$

which shows that $gh.v=F_{v}(h)$.

Having seen that the $P^{n}_{\Delta\setminus\\{\alpha_{r}\\}}$-representation
$H^{i}({\mathbb{P}}^{d}_{K}\setminus{\mathbb{P}}^{r}_{K}(\varepsilon)^{-},{\mathcal{E}})$
is locally analytic, we now consider the long exact sequence of local
cohomology

$\ldots\longrightarrow
H^{i-1}({\mathbb{P}}^{d}_{K}\setminus{\mathbb{P}}^{r}_{K}(\varepsilon)^{-},{\mathcal{E}})\xrightarrow{\partial^{i-1}_{\varepsilon}}H^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow
H^{i}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow\ldots.$

Since this sequence is $P^{n}_{\Delta\setminus\\{\alpha_{r}\\}}$-equivariant,
Proposition 1.3.8 (ii) implies that the kernel
$\widetilde{H}^{i}_{{\mathbb{P}}_{K}^{r}(\varepsilon)^{-}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
is a locally analytic representation, too. ∎

###### Proof of Proposition 2.6.1.

We argue similarly to the proof of Proposition 2.3.1. We have seen in
Proposition 2.5.2 that the transition maps
$\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{m+1})^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})\rightarrow\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{m})^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})$
are compact and have dense image.

Like in the proof of Corollary 2.3.5 one deduces from Lemma 2.6.3 that the
contragredient representation on
$\widetilde{H}^{i}_{{\mathbb{P}}^{r}_{K}(\varepsilon_{m})^{-}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})^{\prime}$
is locally analytic, too. The proposition then follows analogously. ∎

######  Remark 2.6.4.

The cohomology groups $H^{i}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$ are finite-
dimensional algebraic ${\mathrm{\bf G}}$-representations. It follows from
Corollary 1.7.7 that induced homomorphism
$G\rightarrow{\mathrm{GL}}(H^{i}({\mathbb{P}}_{K}^{d},{\mathcal{E}}))$ on
$K$-valued points is a homomorphism of locally $K$-analytic Lie groups.
Therefore the $H^{i}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$ are locally analytic
$G$-representations by Proposition 1.3.10.

##  3\. The ${\mathrm{GL}}_{d+1}(K)$-Representation
$H^{0}({\mathcal{X}},{\mathcal{E}})$

Let $K$ be a non-archimedean local field, and ${\mathcal{E}}$ a ${\mathrm{\bf
G}}$-equivariant vector bundle on ${\mathbb{P}}_{K}^{d}$. Here we write
${\mathrm{\bf G}}={\mathrm{GL}}_{d+1,K}$, and $G={\mathrm{GL}}_{d+1}(K)$ for
its associated locally $K$-analytic Lie group.

###  3.1. Orlik’s Fundamental Complex and the Associated Spectral Sequence

In this section we want to recapitulate Orlik’s method [56] of using the
geometric structure of the divisor at infinity
${\mathcal{Y}}\vcentcolon={\mathbb{P}}^{d}_{K}\setminus{\mathcal{X}}$ to
obtain a filtration by locally analytic $G$-subrepresentations of
$H^{0}({\mathcal{X}},{\mathcal{E}})^{\prime}_{b}$, and to express the
respective subquotients as extensions of certain locally analytic
$G$-representations. Since the reasoning introduced there for a $p$-adic field
$K$ carries over to the case of a general non-archimedean local field
verbatim, we only present an overview. At times we give some additional
details but at others we refer to [56] for the full proofs.

The space of global sections $H^{0}({\mathcal{X}},{\mathcal{E}})$ that we are
interested in relates to the complement ${\mathcal{Y}}$ via the long exact
sequence of local cohomology

$0\longrightarrow H^{0}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow
H^{0}({\mathcal{X}},{\mathcal{E}})\longrightarrow
H^{1}_{\mathcal{Y}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})\longrightarrow
H^{1}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow 0.$ (3.1)

Here the higher cohomology groups $H^{i}({\mathcal{X}},{\mathcal{E}})$, for
$i>0$, vanish as ${\mathcal{X}}$ is quasi-Stein. Because the
$H^{i}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$, for $i=0,1$, are finite-
dimensional algebraic $G$-representations, the main difficulty lies in
understanding $H^{1}_{\mathcal{Y}}({\mathbb{P}}^{d}_{K},{\mathcal{E}})$.

In this regard the strategy of [56] unfolds as follows. Let
$({\mathbb{P}}_{K}^{d})^{\mathrm{ad}}$ and ${\mathcal{X}}^{\mathrm{ad}}$
denote the adic spaces attached to ${\mathbb{P}}_{K}^{d}$ and ${\mathcal{X}}$
respectively. Then one considers the complement

${\mathcal{Y}}^{\mathrm{ad}}\vcentcolon=({\mathbb{P}}_{K}^{d})^{\mathrm{ad}}\setminus{\mathcal{X}}^{\mathrm{ad}}$

which is a closed pseudo-adic subspace of
$({\mathbb{P}}_{K}^{d})^{\mathrm{ad}}$ by [57, Lemma 3.2], cf. [39, Ch. 1.10].
Since the Zariski topoi of ${\mathcal{X}}$ and ${\mathcal{X}}^{\mathrm{ad}}$,
and the ones of ${\mathbb{P}}_{K}^{d}=({\mathbb{P}}_{K}^{d})^{\mathrm{rig}}$
and $({\mathbb{P}}_{K}^{d})^{\mathrm{ad}}$ are equivalent (see [38, Prop. 4.5
(i)]), it follows that
$H^{i}_{{\mathcal{Y}}^{\mathrm{ad}}}\big{(}({\mathbb{P}}_{K}^{d})^{\mathrm{ad}},{\mathcal{E}}\big{)}=H^{i}_{{\mathcal{Y}}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$,
for all $i\geq 0$.

Recall that, for a subset $I$ of the set of simple roots
$\Delta=\\{\alpha_{0},\ldots,\alpha_{d-1}\\}$ of ${\mathrm{\bf
G}}\vcentcolon={\mathrm{GL}}_{d+1,K}$, we denote the associated (lower)
standard parabolic subgroup by ${\mathrm{\bf P}}_{I}$. We write
$P_{I}\vcentcolon={\mathrm{\bf P}}_{I}(K)$, $P_{I,0}\vcentcolon={\mathrm{\bf
P}}_{I,{\mathbb{Z}}}({\mathcal{O}}_{K})$, and
$P^{n}_{I}\vcentcolon=p^{-1}_{n}({\mathrm{\bf
P}}_{I,{\mathbb{Z}}}({\mathcal{O}}_{K}/{\pi}^{n}))$ where

$p_{n}\colon
G_{0}\vcentcolon={\mathrm{GL}}_{d+1}({\mathcal{O}}_{K})\longrightarrow{\mathrm{GL}}_{d+1}({\mathcal{O}}_{K}/{\pi}^{n})$

is the canonical reduction homomorphism. Here ${\mathrm{\bf
P}}_{I,{\mathbb{Z}}}$ denotes the respective standard parabolic subgroup of
${\mathrm{GL}}_{d+1,{\mathbb{Z}}}$.

For a subset $I\subsetneq\Delta$ with $\Delta\setminus
I=\\{\alpha_{i_{1}},\ldots,\alpha_{i_{s}}\\}$,
$i_{1}{\,<\,}\ldots{\,<\,}i_{s}$, we define the closed subvariety

$Y_{I}\vcentcolon={\mathbb{P}}_{K}\bigg{(}\bigoplus_{j=0}^{i_{1}}K\cdot
e_{j}\bigg{)}={\mathbb{P}}_{K}^{i_{1}}=V_{+}(X_{i_{1}+1},\ldots,X_{d})\subset{\mathbb{P}}_{K}^{d}$

so that

${\mathcal{Y}}^{\mathrm{ad}}=\bigcup_{I\subsetneq\Delta}\bigcup_{g\in
G/P_{I}}gY_{I}^{\mathrm{ad}}.$

Moreover, for a compact open subset $W\subset G/P_{I}$, we consider

$Z_{I}^{W}\vcentcolon=\bigcup_{g\in W}gY_{I}^{\mathrm{ad}}$

which is a closed pseudo-adic subspace of
$({\mathbb{P}}_{K}^{d})^{\mathrm{ad}}$, see [57, Lemma 3.2]. In particular, we
have
${\mathcal{Y}}^{\mathrm{ad}}=Z^{G/P_{\Delta\setminus\\{\alpha_{d-1}\\}}}_{\Delta\setminus\\{\alpha_{d-1}\\}}$.

Next one defines on ${\mathcal{Y}}^{\mathrm{ad}}$ certain étale sheaves of
locally constant sections supported on $Z_{I}^{G/P_{I}}$. To this end, let

$\displaystyle\Phi_{I,g}$ $\displaystyle\colon gY_{I}^{\mathrm{ad}}$
$\displaystyle\longrightarrow{\mathcal{Y}}^{\mathrm{ad}}$
$\displaystyle\Psi_{I,W}$ $\displaystyle\colon Z_{I}^{W}$
$\displaystyle\longrightarrow{\mathcal{Y}}^{\mathrm{ad}}$

be the embeddings of closed pseudo-adic spaces and consider the étale sheaves

$\displaystyle{\mathbb{Z}}_{g,I}$
$\displaystyle\vcentcolon=(\Phi_{I,g})_{\ast}(\Phi_{I,g})^{\ast}{\mathbb{Z}}_{{\mathcal{Y}}^{\mathrm{ad}}}$
$\displaystyle{\mathbb{Z}}_{Z^{W}_{I}}$
$\displaystyle\vcentcolon=(\Psi_{I,W})_{\ast}(\Psi_{I,W})^{\ast}{\mathbb{Z}}_{{\mathcal{Y}}^{\mathrm{ad}}}$

where ${\mathbb{Z}}_{{\mathcal{Y}}^{\mathrm{ad}}}$ denotes the constant étale
sheaf on ${\mathcal{Y}}^{\mathrm{ad}}$ with stalks equal to ${\mathbb{Z}}$.

Furthermore, we let ${\mathcal{C}}_{G/P_{I}}$ denote the category of disjoint
coverings of $G/P_{I}$ by compact open subsets with morphisms given by
refinement. For a covering $c\in{\mathcal{C}}_{G/P_{I}}$ of the form
$G/P_{I}=\bigcup_{j\in A}W_{j}$, let ${\mathbb{Z}}_{c}$ denote the image of
the sheaf homomorphism

$\bigoplus\limits_{j\in
A}{\mathbb{Z}}_{Z^{W_{j}}_{I}}\lhook\joinrel\longrightarrow\prod\limits_{g\in
G/P_{I}}{\mathbb{Z}}_{g,I}$

which is induced by the homomorphisms
${\mathbb{Z}}_{Z_{I}^{W_{j}}}\rightarrow{\mathbb{Z}}_{g,I}$, for $g\in W_{j}$,
cf. [56, p. 621]. Taking the inductive limit over all coverings of
${\mathcal{C}}_{G/P_{I}}$ one arrives at the aforementioned sheaf

$\varinjlim_{c\in{\mathcal{C}}_{G/P_{I}}}{\mathbb{Z}}_{c}$

of locally constant sections supported on $Z_{I}^{G/P_{I}}$ with values in
${\mathbb{Z}}$.

With the appropriate sheaf homomorphisms given by restriction, these sheaves
fit together to yield a complex of sheaves on ${\mathcal{Y}}^{\mathrm{ad}}$

$\displaystyle
0\longrightarrow{\mathbb{Z}}_{{\mathcal{Y}}^{\mathrm{ad}}}\longrightarrow\bigoplus_{\begin{subarray}{c}I\subset\Delta\\\
{\lvert{\Delta\setminus
I}\rvert}=1\end{subarray}}\varinjlim_{c\in{\mathcal{C}}_{G/P_{I}}}{\mathbb{Z}}_{c}$
$\displaystyle\longrightarrow\ldots\longrightarrow\bigoplus_{\begin{subarray}{c}I\subset\Delta\\\
{\lvert{\Delta\setminus
I}\rvert}=i\end{subarray}}\varinjlim_{c\in{\mathcal{C}}_{G/P_{I}}}{\mathbb{Z}}_{c}\longrightarrow\ldots$
(3.2) $\displaystyle\ldots$
$\displaystyle\longrightarrow\bigoplus_{\begin{subarray}{c}I\subset\Delta\\\
{\lvert{\Delta\setminus
I}\rvert}=d-1\end{subarray}}\varinjlim_{c\in{\mathcal{C}}_{G/P_{I}}}{\mathbb{Z}}_{c}\longrightarrow\varinjlim_{c\in{\mathcal{C}}_{G/P_{\emptyset}}}{\mathbb{Z}}_{c}\longrightarrow
0.$

######  Theorem 3.1.1 ([56, Thm. 2.1.1]).

The complex (3.2) is acyclic.

Now, let
$0\rightarrow{\mathcal{E}}\rightarrow{\mathcal{I}}^{0}\rightarrow{\mathcal{I}}^{1}\rightarrow\ldots$
be an injective resolution of the
${\mathcal{O}}_{{\mathbb{P}}_{K}^{d}}$-module ${\mathcal{E}}$. Let
$\iota\colon{\mathcal{Y}}^{\mathrm{ad}}\hookrightarrow({\mathbb{P}}_{K}^{d})^{\mathrm{ad}}$
denote the closed embedding. We want to consider the double complex obtained
by applying ${\mathrm{Hom}}(\iota_{\ast}({\,\\_\,}),{\mathcal{I}}^{q})$ to the
acyclic resolution (3.2) of ${\mathbb{Z}}_{{\mathcal{Y}}^{\mathrm{ad}}}$, i.e.

$E_{0}^{p,q}\vcentcolon=\begin{cases}\displaystyle{\mathrm{Hom}}\Bigg{(}\iota_{\ast}\bigg{(}\bigoplus\limits_{\begin{subarray}{c}I\subset\Delta\\\
{\lvert{\Delta\setminus
I}\rvert}=-p+1\end{subarray}}\varinjlim\limits_{c\in{\mathcal{C}}_{G/P_{I}}}{\mathbb{Z}}_{c}\bigg{)},{\mathcal{I}}^{q}\Bigg{)}&\text{,
if $-(d-1)\leq p\leq 0$, $q\geq 0$,}\\\ 0&\text{, else.}\end{cases}$ (3.3)

This double complex is concentrated in the upper left quadrant.

There is a natural action of $G$ on $E_{0}^{\bullet,\bullet}$ as follows: For
fixed $g\in G$ and $I\subsetneq\Delta$, we have a homomorphism by
functoriality of taking the inverse image under the automorphism $g$

${\mathrm{Hom}}\bigg{(}\iota_{\ast}\Big{(}\varinjlim\limits_{c\in{\mathcal{C}}_{G/P_{I}}}{\mathbb{Z}}_{c}\Big{)},{\mathcal{I}}^{q}\bigg{)}\longrightarrow{\mathrm{Hom}}\bigg{(}g^{-1}\iota_{\ast}\Big{(}\varinjlim\limits_{c\in{\mathcal{C}}_{G/P_{I}}}{\mathbb{Z}}_{c}\Big{)},g^{-1}{\mathcal{I}}^{q}\bigg{)}.$
(3.4)

Moreover, for the restriction $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{g}\colon{\mathcal{Y}}^{\mathrm{ad}}\rightarrow{\mathcal{Y}}^{\mathrm{ad}}$
of $g$ we have ${\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{g}}^{-1}{\mathbb{Z}}_{c}={\mathbb{Z}}_{g^{-1}c}$ where $g^{-1}c$ denotes
the translated covering $G/P_{I}=\bigcup_{j\in A}g^{-1}W_{j}$, for
$c=\\{W_{j}\mid j\in A\\}$. This yields

$g^{-1}\iota_{\ast}\Big{(}\varinjlim\limits_{c\in{\mathcal{C}}_{G/P_{I}}}{\mathbb{Z}}_{c}\Big{)}\cong\iota_{\ast}{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{g}}^{-1}\Big{(}\varinjlim\limits_{c\in{\mathcal{C}}_{G/P_{I}}}{\mathbb{Z}}_{c}\Big{)}\cong\iota_{\ast}\Big{(}\varinjlim\limits_{c\in{\mathcal{C}}_{G/P_{I}}}{\mathbb{Z}}_{g^{-1}c}\Big{)}.$

Together with the homomorphism $g^{-1}{\mathcal{I}}^{q}\rightarrow
g^{\ast}{\mathcal{I}}^{q}\rightarrow{\mathcal{I}}^{q}$ induced from
$\Phi_{g}\colon g^{\ast}{\mathcal{E}}\rightarrow{\mathcal{E}}$ in the second
component we obtain a homomorphism

${\mathrm{Hom}}\bigg{(}g^{-1}\iota_{\ast}\Big{(}\varinjlim\limits_{c\in{\mathcal{C}}_{G/P_{I}}}{\mathbb{Z}}_{c}\Big{)},g^{-1}{\mathcal{I}}^{q}\bigg{)}\longrightarrow{\mathrm{Hom}}\bigg{(}\iota_{\ast}\Big{(}\varinjlim\limits_{c\in{\mathcal{C}}_{G/P_{I}}}{\mathbb{Z}}_{g^{-1}c}\Big{)},{\mathcal{I}}^{q}\bigg{)}.$
(3.5)

Then $g$ acts via the endomorphism that arises as the composition of (3.4) and
(3.5).

Associated with the double complex (3.3) we have two spectral sequences
${}^{\rm h}\\!E^{p,q}_{r}$ and ${}^{\rm v}\\!E^{p,q}_{r}$. Since, for every
$n\in{\mathbb{Z}}$, there are only finitely many pairs
$(p,q)\in{\mathbb{Z}}^{2}$ with $p+q=n$ and $E^{p,q}_{0}\neq 0$, both spectral
sequences converge to the total cohomology of the double complex [77, Tag
0132]. As all rows of ${}^{\rm h}\\!E^{p,q}_{0}$ are exact apart from the
entry at $p=0$, we compute

${}^{\rm
h}\\!E_{1}^{p,q}=\begin{cases}{\mathrm{Hom}}(\iota_{\ast}{\mathbb{Z}}_{{\mathcal{Y}}^{\mathrm{ad}}},{\mathcal{I}}^{q})&\text{,
if $p=0$, $q\geq 0$,}\\\ 0&\text{, else.}\end{cases}$

Therefore ${}^{\rm h}\\!E_{2}^{p,q}$ reads as follows

${}^{\rm
h}\\!E_{2}^{p,q}=\begin{cases}{\mathrm{Ext}}^{q}(\iota_{\ast}{\mathbb{Z}}_{{\mathcal{Y}}^{\mathrm{ad}}},{\mathcal{E}})&\text{,
if $p=0$, $q\geq 0$,}\\\ 0&\text{, else,}\end{cases}$

and the spectral sequence ${}^{\rm h}\\!E_{r}^{p,q}$ collapses at the second
page. Moreover, we have the $G$-equivariant isomorphism

${}^{\rm
h}\\!E_{2}^{0,q}={\mathrm{Ext}}^{q}(\iota_{\ast}{\mathbb{Z}}_{{\mathcal{Y}}^{\mathrm{ad}}},{\mathcal{E}})=H^{q}_{{\mathcal{Y}}^{\mathrm{ad}}}\big{(}({\mathbb{P}}^{d}_{K})^{\mathrm{ad}},{\mathcal{E}}\big{)}=H^{q}_{\mathcal{Y}}({\mathbb{P}}^{d}_{K},{\mathcal{E}}),$

for all $q\geq 0$, by [33, Prop. 2.3 bis.].

We now turn to the spectral sequence ${}^{\rm v}\\!E_{r}^{p,q}$ and compute
that

${}^{\rm
v}\\!E^{p,q}_{1}=\begin{cases}\displaystyle{\mathrm{Ext}}^{q}\Bigg{(}\iota_{\ast}\bigg{(}\bigoplus\limits_{\begin{subarray}{c}I\subset\Delta\\\
{\lvert{\Delta\setminus
I}\rvert}=-p+1\end{subarray}}\varinjlim\limits_{c\in{\mathcal{C}}_{G/P_{I}}}{\mathbb{Z}}_{c}\bigg{)},{\mathcal{E}}\Bigg{)}&\text{,
if $-(d-1)\leq p\leq 0$, $q\geq 0$,}\\\ 0&\text{, else.}\end{cases}$

Furthermore, for all $I\subsetneq\Delta$, it is shown in [56, Prop. 2.2.1]
that there is an isomorphism

${\mathrm{Ext}}^{q}\bigg{(}\iota_{\ast}\Big{(}\varinjlim_{c\in{\mathcal{C}}_{G/P_{I}}}{\mathbb{Z}}_{c}\Big{)},{\mathcal{E}}\bigg{)}=\varprojlim_{n\in{\mathbb{N}}}\bigoplus_{g\in
G_{0}/P_{I}^{n}}H^{q}_{gY_{I}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}}),$
(3.6)

for all $q\geq 0$, with the definition of the “open”
$\varepsilon_{n}$-neighbourhood of $Y_{I}$ from (2.11). Here and in the
following, we abbreviate
$\varepsilon_{n}\vcentcolon={\lvert{{\pi}}\rvert}^{n}$, for
$n\in{\mathbb{N}}$.

######  Remark 3.1.2.

We want to explain how the $G$-action on the ${\mathrm{Ext}}$-groups on the
left hand side of (3.6) transfers to the right hand side. While doing so, we
will introduce some useful notation.

First note that $G/P_{I}\cong G_{0}/P_{I,0}$ by the Iwasawa decomposition (see
[16, §3.5]). The proof of the isomorphism (3.6) uses that the family of
coverings

$G/P_{I}=\bigcup_{g\in G_{0}/P^{n}_{I}}gP_{I}^{n}/P_{I}\quad\text{, with
$gP_{I}^{n}/P_{I}\vcentcolon=\big{\\{}gpP_{I}\in G/P_{I}\mid p\in
P^{n}_{I}\big{\\}}$, }$

for $n\in{\mathbb{N}}$, is cofinal in ${\mathcal{C}}_{G/P_{I}}$. This shows
that

$\varinjlim_{c\in{\mathcal{C}}_{G/P_{I}}}{\mathbb{Z}}_{c}=\varinjlim_{n\in{\mathbb{N}}}\bigoplus_{g\in
G_{0}/P_{I}^{n}}{\mathbb{Z}}_{Z_{I}^{gP^{n}_{I}}}.$

One proceeds by applying [33, Prop. 2.3 bis.] and arguing that certain higher
derived inverse limits vanish.

Then the isomorphism (3.6) is $G$-equivariant when the right hand side is
equipped with the following $G$-action by continuous endomorphisms: For fixed
$g\in G$, to give an endomorphism by which $g$ acts it suffices to define
compatible homomorphisms

$\varprojlim_{n\in{\mathbb{N}}}\bigoplus_{g\in
G_{0}/P_{I}^{n}}H^{q}_{gY_{I}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow\bigoplus_{i=1}^{s_{m}}H^{q}_{g_{i}Y_{I}(\varepsilon_{m})}({\mathbb{P}}_{K}^{d},{\mathcal{E}}),$
(3.7)

for all $m\in{\mathbb{N}}$. Here the $g_{1},\ldots,g_{s_{m}}$ are some coset
representatives of $G_{0}/P_{I}^{m}$ so that we have
$G/P_{I}=\bigcup_{i=1}^{s_{m}}g_{i}P_{I}^{m}/P_{I}$.

We choose $n=n(g,m)\geq m$ to be large enough such that the covering
$G/P_{I}=\bigcup_{j=1}^{s_{n}}h_{j}P^{n}_{I}/P_{I}$, for coset representatives
$h_{1},\ldots,h_{s_{n}}$ of $G_{0}/P^{n}_{I}$, is a refinement of the
translated covering $G/P_{I}=\bigcup_{i=1}^{s_{m}}gg_{i}P^{m}_{I}/P_{I}$. In
this way we obtain a surjection

$\sigma_{g}\colon\\{1,\ldots,s_{n}\\}\longrightarrow\\{1,\ldots,s_{m}\\}$

where $\sigma_{g}(j)$ is defined via $h_{j}P^{n}_{I}/P_{I}\subset
gg_{\sigma_{g}(j)}P^{m}_{I}/P_{I}$.

One computes that

$h_{j}Y_{I}(\varepsilon_{n})=h_{j}P^{n}_{I}.Y_{I}\subset
gg_{\sigma_{g}(j)}P^{m}_{I}.(P_{I}.Y_{I})=gg_{\sigma_{g}(j)}Y_{I}(\varepsilon_{m})$

Then the homomorphisms

$\varphi_{g}\colon
H^{q}_{h_{j}Y_{I}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow
H^{q}_{g_{\sigma_{g}(j)}Y_{I}(\varepsilon_{m})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$

from (2.6) give

$\displaystyle\bigoplus_{j=1}^{s_{n}}H^{q}_{h_{j}Y_{I}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
$\displaystyle\longrightarrow\bigoplus_{i=1}^{s_{m}}H^{q}_{g_{i}Y_{I}(\varepsilon_{m})}({\mathbb{P}}_{K}^{d},{\mathcal{E}}),$
$\displaystyle(v_{1},\ldots,v_{s_{n}})$
$\displaystyle\longmapsto\Big{(}\sum_{j\in\sigma_{g}^{-1}(\\{1\\})}\varphi_{g}(v_{j}),\ldots,\sum_{j\in\sigma_{g}^{-1}(\\{s_{m}\\})}\varphi_{g}(v_{j})\Big{)}.$

Combining this with the projection

$\varprojlim_{n\in{\mathbb{N}}}\bigoplus_{g\in
G_{0}/P_{I}^{n}}H^{q}_{gY_{I}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow\bigoplus_{j=1}^{s_{n}}H^{q}_{h_{j}Y_{I}(\varepsilon_{n(g,m)})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$

yields the sought homomorphism (3.7). One checks that these homomorphisms are
compatible and do not depend on the choice of $n$. ∎

Note that when $g\in G_{0}$, the $n$ for the action of $g$ can always be
chosen to be $n=m$. In this case, $g$ even acts on each constituent of
projective limit in (3.6) individually.

Using the isomorphisms

$\varphi_{g_{i}}\colon
H^{q}_{g_{i}Y_{I}(\varepsilon_{m})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow
H^{q}_{Y_{I}(\varepsilon_{m})}({\mathbb{P}}_{K}^{d},{\mathcal{E}}),$

for coset representatives $g_{1},\ldots,g_{s_{m}}$ of $G_{0}/P_{I}^{m}$, we
obtain a $G_{0}$-equivariant isomorphism

$\bigoplus_{i=1}^{s_{m}}H^{q}_{g_{i}Y_{I}(\varepsilon_{m})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\overset{\cong}{\longrightarrow}{\mathrm{Ind}}_{P_{I}^{m}}^{G_{0}}\Big{(}H^{q}_{Y_{I}(\varepsilon_{m})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\Big{)}\,,\quad(v_{1},\ldots,v_{s_{m}})\longmapsto\sum_{i=1}^{s_{m}}g_{i}\bullet\varphi_{g_{i}}(v_{i}).$

Consequently the spectral sequence reads as follows

${}^{\rm
v}\\!E_{1}^{p,q}=\bigoplus\limits_{\begin{subarray}{c}I\subset\Delta\\\
{\lvert{\Delta\setminus
I}\rvert}=-p+1\end{subarray}}\varprojlim_{n\in{\mathbb{N}}}\,{\mathrm{Ind}}^{G_{0}}_{P_{I}^{n}}\Big{(}H^{q}_{Y_{I}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\Big{)}\Rightarrow{}^{\rm
v}\\!E_{\infty}^{p+q}={}^{\rm
h}\\!E_{\infty}^{p+q}=H^{p+q}_{\mathcal{Y}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}).$

One continues analysing this spectral sequence by considering complexes
$K^{\bullet}_{q,n}$ defined as

$K^{p}_{q,n}\vcentcolon=\bigoplus\limits_{\begin{subarray}{c}I\subset\Delta\\\
{\lvert{\Delta\setminus
I}\rvert}=-p+1\end{subarray}}{\mathrm{Ind}}^{G_{0}}_{P_{I}^{n}}\Big{(}H^{q}_{Y_{I}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\Big{)},$

for $-(d-1)\leq p\leq 0$, $q\geq 0$, so that ${}^{\rm
v}\\!E_{1}^{\bullet,q}=\varprojlim_{n\in{\mathbb{N}}}K^{\bullet}_{q,n}$.
Applying Proposition 2.5.7 to the local cohomology groups with respect to
$Y_{I}(\varepsilon_{n})={\mathbb{P}}_{K}^{i_{1}}$, for $I\subsetneq\Delta$
with $\Delta\setminus I=\\{\alpha_{i_{1}},\ldots,\alpha_{i_{s}}\\}$,
$i_{1}{\,<\,}\ldots{\,<\,}i_{s}$, and $q\geq 0$, we find that

$H^{q}_{Y_{I}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})=\begin{cases}0&\text{,
if $\\{\alpha_{0},\ldots,\alpha_{d-q-1}\\}{\not\subset}I$,}\\\
H^{q}_{{\mathbb{P}}_{K}^{d-q}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})&\text{,
if $\\{\alpha_{0},\ldots,\alpha_{d-q-1}\\}\subset I$ and $\alpha_{d-q}\notin
I$,}\\\ H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}})&\text{, if
$\\{\alpha_{0},\ldots,\alpha_{d-q}\\}\subset I$.}\end{cases}$

In particular the complex $K^{\bullet}_{q,n}$ is concentrated in the degrees
$p=-q+1,\ldots,0$.

The next step is to write $K^{\bullet}_{q,n}$ as an extension of complexes
$K^{\prime\bullet}_{q,n}$ and $K^{\prime\prime\bullet}_{q,n}$ via the
$G$-equivariant, short strictly exact sequence

${0}$${0}$${K^{\prime
p}_{q,n}}$${\vcentcolon=\joinrel=\joinrel=}$${\displaystyle\bigoplus\limits_{\begin{subarray}{c}I\subset\Delta\\\
{\lvert{\Delta\setminus I}\rvert}=-p+1\\\ \alpha_{0},\ldots,\alpha_{d-q-1}\in
I\\\ \alpha_{d-q}\notin
I\end{subarray}}{\mathrm{Ind}}^{G_{0}}_{P_{I}^{n}}\Big{(}\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\Big{)}}$${K_{q,n}^{p}}$${=\joinrel=\joinrel=}$${\displaystyle\bigoplus\limits_{\begin{subarray}{c}I\subset\Delta\\\
{\lvert{\Delta\setminus I}\rvert}=-p+1\\\ \alpha_{0},\ldots,\alpha_{d-q-1}\in
I\end{subarray}}{\mathrm{Ind}}^{G_{0}}_{P_{I}^{n}}\Big{(}H^{q}_{Y_{I}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\Big{)}}$${K^{\prime\prime
p}_{q,n}}$${\vcentcolon=\joinrel=\joinrel=}$${\displaystyle\bigoplus\limits_{\begin{subarray}{c}I\subset\Delta\\\
{\lvert{\Delta\setminus I}\rvert}=-p+1\\\ \alpha_{0},\ldots,\alpha_{d-q-1}\in
I\end{subarray}}{\mathrm{Ind}}^{G_{0}}_{P_{I}^{n}}\big{(}H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\big{)}}$${0}$${0}$

of $K$-Fréchet spaces. This sequence is induced by the short strictly exact
sequences

$0\longrightarrow 0\longrightarrow
H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow
H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow 0,$

for $I\subsetneq\Delta$ with $\alpha_{0},\ldots,\alpha_{d-q}\in I$, and by

$0\longrightarrow\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow
H^{q}_{{\mathbb{P}}_{K}^{d-q}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow
H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}})\longrightarrow 0,$

for $I\subsetneq\Delta$ with $\alpha_{0},\ldots,\alpha_{d-q-1}\in I$,
$\alpha_{d-q}\notin I$. Here the second homomorphism in the latter sequence is
surjective as
$H^{q}({\mathbb{P}}_{K}^{d}\setminus{\mathbb{P}}_{K}^{d-q}(\varepsilon_{n}),{\mathcal{E}})=0$.

By Proposition 2.5.2 the projective systems $(K^{\prime
p}_{q,n})_{n\in{\mathbb{N}}}$ satisfy the topological Mittag-Leffler condition
of Lemma 2.5.4. Therefore we obtain the $G_{0}$-equivariant, short strictly
exact sequence

$0\longrightarrow\varprojlim_{n\in{\mathbb{N}}}K^{\prime\bullet}_{q,n}\longrightarrow{}^{\rm
v}\\!E_{1}^{\bullet,q}\longrightarrow\varprojlim_{n\in{\mathbb{N}}}K^{\prime\prime\bullet}_{q,n}\longrightarrow
0$ (3.8)

of complexes of $K$-Fréchet spaces.

One finds that the complexes $K^{\prime\bullet}_{q,n}$ and
$K^{\prime\prime\bullet}_{q,n}$, for all $q\geq 0$, $n\in{\mathbb{N}}$, are
acyclic aside from the very left and right position [56, Lemma 2.2.5]. After
checking the (topological) Mittag-Leffler conditions one then concludes that
the complexes $\varprojlim_{n\in{\mathbb{N}}}K^{\prime\bullet}_{q,n}$ and
$\varprojlim_{n\in{\mathbb{N}}}K^{\prime\prime\bullet}_{q,n}$ are acyclic
apart from the very left and right position as well [56, Rmk. 2.2.6]. To
compute ${}^{\rm v}\\!E_{2}^{p,q}$ we can consider the long exact sequence of
the cohomology of the complexes (3.8). It follows that the only non-vanishing
terms of ${}^{\rm
v}\\!E_{2}^{p,q}=H^{p}\big{(}\varprojlim_{n\in{\mathbb{N}}}K^{\bullet}_{p,n}\big{)}$
are the ones for $p=-q+1$, $q=1,\ldots,d$, and the ones for $p=0$, $q\geq 2$.
For these terms we obtain a short strictly exact sequence

$\displaystyle
0\longrightarrow{\mathrm{Ker}}\Big{(}\varprojlim_{n\in{\mathbb{N}}}$
$\displaystyle{\hskip
1.5pt}K^{\prime-q+1}_{q,n}\rightarrow\varprojlim_{n\in{\mathbb{N}}}K^{\prime-q+2}_{q,n}\Big{)}\longrightarrow{}^{\rm
v}\\!E_{2}^{-q+1,q}$ (3.9)
$\displaystyle\longrightarrow{\mathrm{Ker}}\Big{(}\varprojlim_{n\in{\mathbb{N}}}K^{\prime\prime-q+1}_{q,n}\rightarrow\varprojlim_{n\in{\mathbb{N}}}K^{\prime\prime-q+2}_{q,n}\Big{)}\longrightarrow
0,$

for $q=1,\ldots,d$, and one shows that

${}^{\rm v}\\!E_{2}^{0,q}=H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}}),$

for $q\geq 2$. Moreover, from this one concludes that the spectral sequence
${}^{\rm v}\\!E_{r}^{p,q}$ degenerates at the $E_{2}$-page [56, p. 633].

We want to investigate the strong dual of (3.9). We first look at
$\varprojlim_{n\in{\mathbb{N}}}K^{\prime\prime p}_{q,n}$. Since each
$K^{\prime\prime p}_{q,n}$ is finite-dimensional, the transition homomorphisms
of $\big{(}K^{\prime\prime p}_{q,n}\big{)}_{n\in{\mathbb{N}}}$ are compact
[67, Lemma 16.4] so that $\varprojlim_{n\in{\mathbb{N}}}K^{\prime\prime
p}_{q,n}$ is a nuclear $K$-Fréchet space [67, Prop. 19.9]. Then
${\mathrm{Ker}}\big{(}\varprojlim_{n\in{\mathbb{N}}}K^{\prime\prime-q+1}_{q,n}\rightarrow\varprojlim_{n\in{\mathbb{N}}}K^{\prime\prime-q+2}_{q,n}\big{)}$
is a nuclear $K$-Fréchet space, too [67, Prop. 19.4 (i)]. It follows that

$\displaystyle{\mathrm{Ker}}\Big{(}\varprojlim_{n\in{\mathbb{N}}}K^{\prime\prime-q+1}_{q,n}\rightarrow\varprojlim_{n\in{\mathbb{N}}}K^{\prime\prime-q+2}_{q,n}\Big{)}^{\prime}_{b}$
$\displaystyle\cong\Big{(}\varprojlim_{n\in{\mathbb{N}}}{\mathrm{Ker}}\big{(}K^{\prime\prime-q+1}_{q,n}\rightarrow\varprojlim_{n\in{\mathbb{N}}}K^{\prime\prime-q+2}_{q,n}\big{)}\Big{)}^{\prime}_{b}$
$\displaystyle\cong\varinjlim_{n\in{\mathbb{N}}}{\mathrm{Ker}}\big{(}K^{\prime\prime-q+1}_{q,n}\rightarrow
K^{\prime\prime-q+2}_{q,n}\big{)}^{\prime}_{b}$

by [67, Prop. 16.5]. As $K^{\prime\prime\bullet}_{q,n}$ is exact at
$K^{\prime\prime-q+2}_{q,n}$, the image of
$K^{\prime\prime-q+1}_{q,n}\rightarrow K^{\prime\prime-q+2}_{q,n}$ is closed
and this homomorphism is strict [11, IV. §4.2 Thm. 1]. Hence [11, IV. §4.1
Prop. 2] implies that there is topological isomorphism

${\mathrm{Ker}}\big{(}K^{\prime\prime-q+1}_{q,n}\rightarrow
K^{\prime\prime-q+2}_{q,n}\big{)}^{\prime}_{b}\cong{\mathrm{Coker}}\Big{(}\big{(}K^{\prime\prime-q+2}_{q,n}\big{)}^{\prime}_{b}\rightarrow\big{(}K^{\prime\prime-q+1}_{q,n}\big{)}^{\prime}_{b}\Big{)}.$

To simplify the notation we write ${\mathrm{\bf
Q}}_{d-q}\vcentcolon={\mathrm{\bf P}}_{I_{d-q}}$ for the standard parabolic
subgroup corresponding to the subset
$I_{d-q}=\\{\alpha_{0},\ldots,\alpha_{d-q-1}\\}\subset\Delta$. In Remark 1.5.5
(i) we have seen that taking the strong dual and finite induction commute with
each other. Therefore, we obtain a $G$-equivariant, topological isomorphism

$\displaystyle{\mathrm{Coker}}$
$\displaystyle\Big{(}\big{(}K^{\prime\prime-q+2}_{q,n}\big{)}^{\prime}_{b}\rightarrow\big{(}K^{\prime\prime-q+1}_{q,n}\big{)}^{\prime}_{b}\Big{)}$
$\displaystyle\cong{\mathrm{Ind}}^{G_{0}}_{Q^{n}_{d-q}}\big{(}H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}\big{)}\bigg{/}\sum_{i=d-q}^{d-1}{\mathrm{Ind}}^{G_{0}}_{P^{n}_{I_{d-q}\cup\\{\alpha_{i}\\}}}\big{(}H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}\big{)}$
$\displaystyle\cong\Big{(}H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}\otimes_{K}{\mathrm{Ind}}^{G_{0}}_{Q^{n}_{d-q}}(K)\Big{)}\bigg{/}\sum_{i=d-q}^{d-1}H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}\otimes_{K}{\mathrm{Ind}}^{G_{0}}_{P^{n}_{I_{d-q}\cup\\{\alpha_{i}\\}}}(K)$
$\displaystyle\cong
H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}\otimes_{K}v^{G_{0}}_{Q^{n}_{d-q}}$

where

$v^{G_{0}}_{Q^{n}_{d-q}}\vcentcolon={\mathrm{Ind}}^{G_{0}}_{Q^{n}_{d-q}}(K)\bigg{/}\sum_{i=d-q}^{d-1}{\mathrm{Ind}}^{G_{0}}_{P^{n}_{I_{d-q}\cup\\{\alpha_{i}\\}}}(K).$
(3.10)

We have used the push-pull formula from Remark 1.5.5 (ii), since
$H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}$ already is a
$G$-representation, and that taking the projective tensor product with
$H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}$ is exact [13, Lemma 2.1
(ii)].

Furthermore, [27, Prop. 1.1.32 (i)] yields

$\displaystyle\varinjlim_{n\in{\mathbb{N}}}\Big{(}H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}\otimes_{K}v^{G_{0}}_{Q^{n}_{d-q}}\Big{)}\cong
H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}{\,\widehat{\otimes}_{K}\,}\varinjlim_{n\in{\mathbb{N}}}\Big{(}v^{G_{0}}_{Q^{n}_{d-q}}\Big{)}.$

The second factor of this tensor product can be expressed as a smooth
generalized Steinberg representation (cf. [17])

$v^{G}_{P_{I}}\vcentcolon={\mathrm{Ind}}^{{\mathrm{sm}},G}_{P_{I}}(K)\bigg{/}\sum_{I\subsetneq
J\subset\Delta}{\mathrm{Ind}}^{{\mathrm{sm}},G}_{P_{J}}(K),$

for $I\subset\Delta$. Here ${\mathrm{Ind}}^{{\mathrm{sm}},G}_{P_{I}}(K)$
denotes the smooth induction of the trivial $P_{I}$-representation, i.e. the
space locally constant functions invariant under $P_{I}$ endowed with left
regular $G$-action:

${\mathrm{Ind}}^{{\mathrm{sm}},G}_{P_{I}}(K)\vcentcolon=\big{\\{}f\in
C^{\mathrm{sm}}(G,K)\,\big{|}\,\forall g\in G,p\in P_{I}:f(gp)=f(g)\big{\\}}.$

######  Lemma 3.1.3.

For $I\subset\Delta$, there is a $G$-equivariant, topological isomorphism

$\varinjlim_{n\in{\mathbb{N}}}\bigg{(}{\mathrm{Ind}}^{G_{0}}_{P^{n}_{I}}(K)\bigg{/}\sum_{I\subsetneq
J\subset\Delta}{\mathrm{Ind}}^{G_{0}}_{P_{J}^{n}}(K)\bigg{)}\cong
v^{G}_{P_{I}}$

where $v^{G}_{P_{I}}$ denotes the generalized smooth Steinberg representation
endowed with the finest locally convex topology.

###### Proof.

For all $J\subset\Delta$ and $n\in{\mathbb{N}}$, we have a well-defined,
$G_{0}$-equivariant inclusion

${\mathrm{Ind}}^{G_{0}}_{P_{J}^{n}}(K)\longrightarrow{\mathrm{Ind}}_{P_{J}}^{{\mathrm{sm}},G}(K)\,,\quad\sum_{i=1}^{s_{n}}g_{i}\bullet\lambda_{i}\longmapsto\big{[}g\mapsto\lambda_{i}\text{
, if $gP_{J}\in g_{i}P_{J}^{n}/P_{J}$}\big{]}.$ (3.11)

Here $g_{1},\ldots,g_{s_{n}}$ are coset representatives of $G_{0}/P^{n}_{J}$.
Since ${\mathrm{Ind}}^{G_{0}}_{P_{J}^{n}}(K)$ is finite dimensional, this is a
continuous homomorphism when the right hand side carries the finest locally
convex topology.

Moreover, given a locally constant function $f\colon G\rightarrow K$ in
${\mathrm{Ind}}^{{\mathrm{sm}},G}_{P_{J}}(K)$, there exists $n\in{\mathbb{N}}$
such that for the covering $G/P_{J}=\bigcup_{i=1}^{s_{n}}g_{i}P^{n}_{J}/P_{J}$
the function $f$ is constant on each open subset $g_{i}P^{n}_{J}\cdot P_{J}$.
Thus $f$ is contained in the image of (3.11). With the choice of the finest
locally convex topology on ${\mathrm{Ind}}^{{\mathrm{sm}},G}_{P_{J}}(K)$ it
follows that
${\mathrm{Ind}}^{{\mathrm{sm}},G}_{P_{J}}(K)\cong\varinjlim_{n\in{\mathbb{N}}}{\mathrm{Ind}}^{G_{0}}_{P^{n}_{J}}(K)$
is a topological isomorphism.

With $v^{G_{0}}_{P^{n}_{I}}$ defined in the obvious way, we take the inductive
limit over the diagrams

${\displaystyle\bigoplus_{I\subsetneq
J\subset\Delta}{\mathrm{Ind}}^{G_{0}}_{P_{J}^{n}}(K)}$${{\mathrm{Ind}}^{G_{0}}_{P^{n}_{I}}(K)}$${v^{G_{0}}_{P^{n}_{I}}}$${0}$${\displaystyle\bigoplus_{I\subsetneq
J\subset\Delta}{\mathrm{Ind}}^{{\mathrm{sm}},G}_{P_{J}}(K)}$${{\mathrm{Ind}}^{{\mathrm{sm}},G}_{P_{I}}(K)}$${v^{G}_{P_{I}}}$${0}$

which have strictly exact rows. The snake lemma A.13 then shows that the
induced $G_{0}$-equivariant homomorphism
$\varinjlim_{n\in{\mathbb{N}}}v^{G_{0}}_{P^{n}_{I}}\rightarrow v^{G}_{P_{I}}$
is a topological isomorphism. Moreover, this isomorphism is $G$-equivariant
when $\varinjlim_{n\in{\mathbb{N}}}v^{G_{0}}_{P^{n}_{I}}$ carries the
$G$-action induced from Remark 3.1.2. ∎

We now turn towards $\varprojlim_{n\in{\mathbb{N}}}K^{\prime p}_{q,n}$. Here
the transition homomorphisms of $\big{(}K^{\prime
p}_{q,n}\big{)}_{n\in{\mathbb{N}}}$ are compact by Corollary 2.5.6 so that
$\varprojlim_{n\in{\mathbb{N}}}K^{\prime p}_{q,n}$ is a nuclear $K$-Fréchet
space, too. Similarly to before one finds that

${\mathrm{Ker}}\Big{(}\varprojlim_{n\in{\mathbb{N}}}K^{\prime-q+1}_{q,n}\rightarrow\varprojlim_{n\in{\mathbb{N}}}K^{\prime-q+2}_{q,n}\Big{)}^{\prime}_{b}\cong\varinjlim_{n\in{\mathbb{N}}}{\mathrm{Coker}}\Big{(}\big{(}K^{\prime-q+1}_{q,n}\big{)}^{\prime}_{b}\rightarrow\big{(}K^{\prime-q+2}_{q,n}\big{)}^{\prime}_{b}\Big{)}$

with injective, compact transition homomorphisms in the inductive limit of the
right hand side.

We write ${\mathrm{\bf P}}_{d-q}\vcentcolon={\mathrm{\bf P}}_{J_{d-q}}$ for
the standard parabolic subgroup corresponding to the subset
$J_{d-q}\vcentcolon=\Delta\setminus\\{\alpha_{d-q}\\}\subset\Delta$. We recall
from Proposition 2.6.1 (ii) that
$W_{n}\vcentcolon=\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}$
is a locally analytic $P^{n}_{d-q}$-representation. Using the exactness of the
“finite” induction ${\mathrm{Ind}}^{G_{0}}_{P^{n}_{d-q}}({\,\\_\,})$, the
push-pull formula from Remark 1.5.5 (ii), and exactness of tensoring with
$W_{n}$ [13, Lemma 2.1 (ii)], we compute that

$\displaystyle{\mathrm{Coker}}$
$\displaystyle\Big{(}\big{(}K^{\prime-q+2}_{q,n}\big{)}^{\prime}_{b}\rightarrow\big{(}K^{\prime-q+1}_{q,n}\big{)}^{\prime}_{b}\Big{)}$
$\displaystyle\cong{\mathrm{Ind}}^{G_{0}}_{Q^{n}_{d-q}}(W_{n})\Bigg{/}\sum_{i=d-q+1}^{d-1}{\mathrm{Ind}}^{G_{0}}_{P^{n}_{I_{d-q}\cup\\{\alpha_{i}\\}}}(W_{n})$
$\displaystyle\cong{\mathrm{Ind}}^{G_{0}}_{P^{n}_{d-q}}\Bigg{(}{\mathrm{Ind}}^{P^{n}_{d-q}}_{Q^{n}_{d-q}}(W_{n})\Bigg{/}\sum_{i=d-q+1}^{d-1}{\mathrm{Ind}}^{P^{n}_{d-q}}_{P^{n}_{I_{d-q}\cup\\{\alpha_{i}\\}}}(W_{n})\Bigg{)}$
$\displaystyle\cong{\mathrm{Ind}}^{G_{0}}_{P^{n}_{d-q}}\Bigg{(}\Big{(}W_{n}\otimes_{K}{\mathrm{Ind}}^{P^{n}_{d-q}}_{Q^{n}_{d-q}}(K)\Big{)}\Bigg{/}\sum_{i=d-q+1}^{d-1}W_{n}\otimes_{K}{\mathrm{Ind}}^{P^{n}_{d-q}}_{P^{n}_{I_{d-q}\cup\\{\alpha_{i}\\}}}(K)\Bigg{)}$
$\displaystyle\cong{\mathrm{Ind}}^{G_{0}}_{P^{n}_{d-q}}\Big{(}W_{n}\otimes_{K}v^{P^{n}_{d-q}}_{Q^{n}_{d-q}}\Big{)}$

with the finite-dimensional representations

$v^{P^{n}_{d-q}}_{Q^{n}_{d-q}}\vcentcolon={\mathrm{Ind}}^{P^{n}_{d-q}}_{Q^{n}_{d-q}}(K)\Bigg{/}\sum_{i=d-q+1}^{d-1}{\mathrm{Ind}}^{P^{n}_{d-q}}_{P^{n}_{I_{d-q}\cup\\{\alpha_{i}\\}}}(K).$

######  Remark 3.1.4.

We want to describe the induced $G$-action on

${\mathrm{Ker}}\Big{(}\varprojlim_{n\in{\mathbb{N}}}K^{\prime-q+1}_{q,n}\rightarrow\varprojlim_{n\in{\mathbb{N}}}K^{\prime-q+2}_{q,n}\Big{)}^{\prime}_{b}\cong\varinjlim_{n\in{\mathbb{N}}}\,{\mathrm{Ind}}^{G_{0}}_{P^{n}_{d-q}}\Big{(}\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}\otimes_{K}v^{P^{n}_{d-q}}_{Q^{n}_{d-q}}\Big{)}.$

To ease the notation, we write ${\mathrm{\bf P}}={\mathrm{\bf P}}_{d-q}$, and
${\mathrm{\bf Q}}={\mathrm{\bf Q}}_{d-q}$ here. Fix $g\in G$ and consider an
element $v$ of the right hand side term. Let $m\in{\mathbb{N}}$ such that

$v=\sum_{i=1}^{s_{m}}g_{i}\bullet
v_{i}\in{\mathrm{Ind}}^{G_{0}}_{P^{m}}\Big{(}\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}(\varepsilon_{m})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}\otimes_{K}v^{P^{m}}_{Q^{m}}\Big{)},$

where $g_{1},\ldots,g_{s_{m}}$ are coset representatives of $G_{0}/P^{m}$.

Similarly to Remark 3.1.2, let $n\geq m$ such that
$G/Q=\bigcup_{j=1}^{s_{n}}h_{j}Q^{n}/Q$ is a refinement of the translated
covering $G/Q=\bigcup_{i=1}^{s_{m}}gg_{i}Q^{m}/Q$, for coset representatives
$h_{1},\ldots,h_{s_{n}}$ of $G_{0}/Q^{n}$. We can consider the induced
coverings of $G/P$ under the surjection $G/Q\rightarrow G/P$. After enlarging
$n$ we may assume that the induced covering
$G/P=\bigcup_{i=1}^{s_{n}}h_{j}P^{n}/P$ is a refinement of the induced
covering $G/P=\bigcup_{i=1}^{s_{m}}gg_{i}P^{m}/P$.

For $j=1,\ldots,s_{n}$, we have $h_{j}Q^{n}/Q\subset
gg_{\sigma_{g}(j)}Q^{m}/Q$ with the notation of Remark 3.1.2. We claim that
this implies $h_{j}P^{n}/Q\subset gg_{\sigma_{g}(j)}P^{m}/Q$. Indeed, using

$\displaystyle
h_{j}P^{n}/Q=\bigcup_{h_{j^{\prime}}P^{n}=h_{j}P^{n}}h_{j^{\prime}}Q^{n}/Q\subset\bigcup_{h_{j^{\prime}}P^{n}=h_{j}P^{n}}gg_{\sigma_{g}(j^{\prime})}Q^{m}/Q\subset\bigcup_{h_{j^{\prime}}P^{n}=h_{j}P^{n}}gg_{\sigma_{g}(j^{\prime})}P^{m}/Q$

it suffices to show that
$g_{\sigma_{g}(j)}P^{m}=g_{\sigma_{g}(j^{\prime})}P^{m}$ if
$h_{j}P^{n}=h_{j^{\prime}}P^{n}$. For this, we compute that
$h_{j}Q^{n}/P\subset h_{j}P^{n}/P\cap gg_{\sigma_{g}(j)}P^{m}/P$. By the
assumption on $n$ with respect to the covering of $G/P$ this shows that
$h_{j}P^{n}/P\subset gg_{\sigma_{g}(j)}P^{m}/P$ 999Recall that the function
$\sigma_{g}$ is defined with regard to the coverings of $G/Q$.. The, if
$h_{j}P^{n}=h_{j^{\prime}}P^{n}$, the sets $gg_{\sigma_{g}(j)}P^{m}/P$ and
$gg_{\sigma_{g}(j^{\prime})}P^{m}/P$ have non-empty intersection which implies
$g_{\sigma_{g}(j)}P^{m}=g_{\sigma_{g}(j^{\prime})}P^{m}$.

We now set $p_{g,j}\vcentcolon=h^{-1}_{j}gg_{\sigma_{g}(j)}$ so that we have
$P^{n}/Q\subset p_{g,j}P^{m}/Q$. Then $g$ induces continuous homomorphisms

$\varphi_{p_{g,j}}^{t}\colon\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}(\varepsilon_{m})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}\longrightarrow\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}.$

Furthermore, we have $p_{g,j}^{-1}P^{n}\subset P^{m}\cdot Q$. Written in terms
of locally constant functions $p_{g,j}$ gives a continuous homomorphism

$\displaystyle{\mathrm{Ind}}^{P^{m}}_{Q^{m}}(K)$
$\displaystyle\longrightarrow{\mathrm{Ind}}^{P^{n}}_{Q^{n}}(K),$
$\displaystyle f$ $\displaystyle\longmapsto
f(p_{g,j}^{-1}{\,\\_\,})=\big{[}p\mapsto f(p^{\prime})\text{ , if
$p_{g,j}^{-1}p=p^{\prime}q$, for $p^{\prime}\in P^{m}$, $q\in Q$}\big{]}.$

This in turn yields a continuous homomorphism $\psi_{p_{g,j}}\colon
v^{P^{m}}_{Q^{m}}\rightarrow v^{P^{n}}_{Q^{n}}$. We let $\tau_{p_{g,j}}$
denote the tensor product

$\tau_{p_{g,j}}\vcentcolon=\varphi^{t}_{p_{g,j}}\otimes\psi_{p_{g,j}}\colon\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}(\varepsilon_{m})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}{\,\otimes_{K}\,}v^{P^{m}}_{Q^{m}}\longrightarrow\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}{\,\otimes_{K}\,}v^{P^{n}}_{Q^{n}}.$

In total the homomorphism by which $g$ acts is given by

$g.\bigg{(}\sum_{i=1}^{s_{m}}g_{i}\bullet
v_{i}\bigg{)}=\sum_{j=1}^{s_{n}}h_{j}\bullet\tau_{p_{g,j}}(v_{\sigma_{g}(j)})\in{\mathrm{Ind}}^{G_{0}}_{P^{n}}\Big{(}\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}\otimes_{K}v^{P^{n}}_{Q^{n}}\Big{)}.$

∎

Since the spectral sequence ${}^{\rm v}\\!E_{r}^{p,q}$ degenerates at the
$E_{2}$-page, we obtain a filtration of
$H^{1}_{\mathcal{Y}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$ by $G$-invariant
subspaces whose successive subquotients are isomorphic to ${}^{\rm
v}\\!E_{2}^{-q+1,q}$. More precisely and taking into account the long exact
sequence (3.1) of local cohomology with respect to
${\mathcal{Y}}\subset{\mathbb{P}}_{K}^{d}$, we arrive at the following
theorem.

######  Theorem 3.1.5 (cf. [56, Thm. 2.2.8]).

Let ${\mathcal{E}}$ be a ${\mathrm{\bf G}}$-equivariant vector bundle on
${\mathbb{P}}_{K}^{d}$. Then there exist a filtration by closed
$D(G)$-submodules

$H^{0}({\mathcal{X}},{\mathcal{E}})=V^{d}\supset V^{d-1}\supset\ldots\supset
V^{1}\supset V^{0}=H^{0}({\mathbb{P}}_{K}^{d},{\mathcal{E}}),$

and, for $q=1,\ldots,d$, short strictly exact sequences of locally analytic
$G$-representations

$\displaystyle 0\longrightarrow$ $\displaystyle{\hskip
2.0pt}H^{q}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}{\,\otimes_{K}\,}v^{G}_{Q_{d-q}}\longrightarrow\big{(}V^{q}/V^{q-1}\big{)}^{\prime}_{b}$
(3.12)
$\displaystyle\longrightarrow\varinjlim_{n\in{\mathbb{N}}}{\mathrm{Ind}}^{G_{0}}_{P^{n}_{d-q}}\Big{(}\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}{\,\otimes_{K}\,}v^{P^{n}_{d-q}}_{Q^{n}_{d-q}}\Big{)}\longrightarrow
0.$

When $K$ is a $p$-adic field, Orlik analyses the right hand side term of
(3.12) further. In [56, p. 634] he shows that there is an isomorphism of
locally analytic $G$-representations

$\varinjlim_{n\in{\mathbb{N}}}{\mathrm{Ind}}^{G_{0}}_{P^{n}_{d-q}}\Big{(}\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}(\varepsilon_{n})}({\mathbb{P}}_{K}^{d},{\mathcal{E}})^{\prime}_{b}{\,\otimes_{K}\,}v^{P^{n}_{d-q}}_{Q^{n}_{d-q}}\Big{)}\cong{\mathrm{Ind}}^{{\mathrm{la}},G}_{P_{d-q}}\Big{(}N^{\prime}_{d-q}\otimes
v^{P_{d-q}}_{Q_{d-q}}\Big{)}^{{\mathfrak{d}}_{d-q}}.$ (3.13)

We explain the notation used here. Let $U({\mathfrak{g}})$ and
$U({\mathfrak{p}}_{d-q})$ denote the universal enveloping algebras of the Lie
algebras of ${\mathrm{\bf G}}$ and ${\mathrm{\bf P}}_{d-q}$ respectively. One
shows that
$\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
is a quotient of a generalized Verma module for $U({\mathfrak{g}})$. More
precisely, there exists a finite-dimensional $P_{d-q}$-subrepresentation
$N_{d-q}\subset\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
which generates it as a $U({\mathfrak{g}})$-module [56, Lemma 1.2.1], i.e.
there exists an epimorphism of $U({\mathfrak{g}})$-modules

$U({\mathfrak{g}})\otimes_{U({\mathfrak{p}}_{d-q})}N_{d-q}\relbar\joinrel\twoheadrightarrow\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}}({\mathbb{P}}_{K}^{d},{\mathcal{E}}).$

Let ${\mathfrak{d}}_{d-q}$ denote the kernel of this epimorphism. Then
${\mathrm{Ind}}^{{\mathrm{la}},G}_{P_{d-q}}\big{(}N^{\prime}_{d-q}\otimes
v^{P_{d-q}}_{Q_{d-q}}\big{)}^{{\mathfrak{d}}_{d-q}}$ indicates the subspace of
those functions in the locally analytic induction that are annihilated by
${\mathfrak{d}}_{d-q}$, cf. [56, p. 607]. In particular, the right hand side
of (3.13) does not depend on the choice of $N_{d-q}$.

However, when $K$ is of positive characteristic
$\widetilde{H}^{q}_{{\mathbb{P}}_{K}^{d-q}}({\mathbb{P}}_{K}^{d},{\mathcal{E}})$
in general is no longer finitely generated, even if we replace
$U({\mathfrak{g}})$ by the algebraic distribution algebra ${\rm
Dist}({\mathrm{\bf G}})\cong{\mathrm{hy}}(G)$ of ${\mathrm{\bf G}}$, see [50,
Ch. 2.3]. We tackle this problem by adapting a different description of
${\mathrm{Ind}}^{{\mathrm{la}},G}_{P_{d-q}}\big{(}N^{\prime}_{d-q}\otimes
v^{P_{d-q}}_{Q_{d-q}}\big{)}^{{\mathfrak{d}}_{d-q}}$ via the functors
${\mathcal{F}}_{P}^{G}$ defined by Orlik and Strauch in [58]. In Section 3.4
we then compare Orlik’s and our description for the case of a $p$-adic field.

###  3.2. The Subquotients of
$H^{0}({\mathcal{X}},{\mathcal{E}})^{\prime}_{b}$ as Locally Analytic
${\mathrm{GL}}_{d+1}({\mathcal{O}}_{K})$-Representations

We now want to analyse the $G$-representations that occur as a quotient of
$\big{(}{}^{\rm v}\\!E_{2}^{-q+1,q}\big{)}^{\prime}_{b}$ in (3.12). We change
the notation in as much as we fix ${r}\vcentcolon=d-q\in\\{0,\ldots,d-1\\}$,
and write ${\mathrm{\bf P}}\vcentcolon={\mathrm{\bf P}}_{d-q}$ as well as
${\mathrm{\bf Q}}\vcentcolon={\mathrm{\bf Q}}_{d-q}$. For $n\in{\mathbb{N}}$,
we have seen in the proof of Proposition 2.6.1 that