# On local Galois deformation rings:
generalised reductive groups

Vytautas Paškūnas and Julian Quast

(Date: .)

###### Abstract.

We study deformation theory of mod $p$ Galois representations of $p$-adic
fields with values in generalised reductive group schemes, such as $L$-groups
and $C$-groups. We show that the corresponding deformation rings are complete
intersections of expected dimension. We determine their irreducible components
in many cases and show that they and their special fibres are normal and
complete intersection.

###### Contents

  1. 1 Introduction
    1. 1.1 The conjecture of Böckle–Juschka
    2. 1.2 Deformation problems with fixed partial determinant
    3. 1.3 Deformation space of Lafforgue’s pseudocharacters
    4. 1.4 Complete intersection
      1. 1.4.1 Definition
      2. 1.4.2 Bounding the dimension
    5. 1.5 Irreducible components
    6. 1.6 Overview by section
    7. 1.7 Notation
    8. 1.8 Acknowledgements
  2. 2 $\mathcal{O}$-group schemes
    1. 2.1 R-parabolics
    2. 2.2 $G$-semisimplification
    3. 2.3 Extensions
  3. 3 Deformation rings
    1. 3.1 Central extensions
  4. 4 Continuity
    1. 4.1 Continuity via algebra
    2. 4.2 Continuity via condensed sets
    3. 4.3 $R$-condensed representations
  5. 5 The space of generic matrices
    1. 5.1 Definition of $X^{\mathrm{gen}}_{\operatorname{GL}_{d}}$ and $X^{\mathrm{gen},\tau}_{G}$
    2. 5.2 Completion at maximal ideals and deformation problems
  6. 6 GIT quotients
    1. 6.1 Points in $X^{\mathrm{gen},\tau}_{G}$
    2. 6.2 Definition of $X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$
  7. 7 Lafforgue’s $G$-pseudocharacters
    1. 7.1 Definition of $G$-pseudocharacter
    2. 7.2 Deformations of $G$-pseudocharacters
    3. 7.3 Comparison of $X^{\mathrm{git},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ with $X^{\mathrm{ps}}_{G}$
  8. 8 Functoriality
    1. 8.1 Independence of $\tau$ and functoriality in $G$
    2. 8.2 Finite maps
    3. 8.3 Products
    4. 8.4 Functoriality in $\Gamma$
  9. 9 Absolutely irreducible locus
    1. 9.1 Trimming the component group
    2. 9.2 Absolutely irreducible locus
  10. 10 Bounds for the dimensions of reductive subgroups
  11. 11 R-Levi subgroups of codimension $2$
  12. 12 Dimension of fibres
  13. 13 Complete intersection
    1. 13.1 Induction hypothesis
    2. 13.2 $W$-special locus
    3. 13.3 Bounding the dimension of the space
    4. 13.4 Consequences
  14. 14 Non-special locus
  15. 15 Irreducible components and normality
    1. 15.1 Dimension
    2. 15.2 Generalised tori
    3. 15.3 Serre’s criterion for normality
    4. 15.4 Irreducible components
    5. 15.5 Conjecture of Böckle–Juschka
    6. 15.6 Labelling irreducible components
  16. 16 Deformations into $L$-groups and $C$-groups
  17. A Condensed sets
    1. A.1 Condensed structure on points

##  1\. Introduction

Let $p$ denote any prime number, let $F$ be a finite extension of
$\mathbb{Q}_{p}$, and let $\Gamma_{F}$ denote its absolute Galois group. Let
$L$ be another finite extension of $\mathbb{Q}_{p}$ with ring of integers
$\mathcal{O}$, uniformizer $\varpi$ and residue field $k=\mathcal{O}/\varpi$.

Let $G$ be a smooth affine group scheme over $\mathcal{O}$, such that its
neutral component $G^{0}$ is reductive and the component group $G/G^{0}$ is a
finite group scheme over $\mathcal{O}$. We call such group schemes
_generalised reductive_. We do not make any assumptions on the prime $p$
regarding $G$.

We fix a continuous representation $\overline{\rho}:\Gamma_{F}\rightarrow
G(k)$ and denote by
$D^{\square}_{\overline{\rho}}:\mathfrak{A}_{\mathcal{O}}\rightarrow\operatorname{Set}$
the functor from the category $\mathfrak{A}_{\mathcal{O}}$ of local artinian
$\mathcal{O}$-algebras with residue field $k$ to the category of sets, such
that for $(A,\mathfrak{m}_{A})\in\mathfrak{A}_{\mathcal{O}}$,
$D^{\square}_{\overline{\rho}}(A)$ is the set of continuous representations
$\rho_{A}:\Gamma_{F}\rightarrow G(A)$, such that
$\rho_{A}(\gamma)\equiv\overline{\rho}(\gamma)\pmod{\mathfrak{m}_{A}}$, for
all $\gamma\in\Gamma_{F}.$ The functor $D^{\square}_{\overline{\rho}}$ of
framed deformations of $\overline{\rho}$ is pro-represented by a complete
local noetherian $\mathcal{O}$-algebra $R^{\square}_{\overline{\rho}}$ with
residue field $k$.

Böckle, Iyengar and VP have studied in [8] ring theoretic properties of
$R^{\square}_{\overline{\rho}}$, when $G=\operatorname{GL}_{d}$. In this paper
we extend the results of [8] to an arbitrary generalised reductive group $G$.

######  Theorem 1.1 (Corollaries 13.27, 15.29).

The ring $R^{\square}_{\overline{\rho}}$ is a local complete intersection,
flat over $\mathcal{O}$ and of relative dimension $\dim
G_{k}([F:\mathbb{Q}_{p}]+1)$. In particular, every continuous representation
$\overline{\rho}:\Gamma_{F}\rightarrow G(k)$ has a lift to characteristic
zero. Moreover, $R^{\square}_{\overline{\rho}}$ is reduced and
$R^{\square}_{\overline{\rho}}[1/p]$ is normal.

Obstruction theory provides a presentation
$R^{\square}_{\overline{\rho}}=\mathcal{O}\llbracket
x_{1},\dots,x_{r}\rrbracket/(f_{1},\ldots,f_{s})$ with $r$ equal to the
dimension of the tangent space and $s$ equal to
$\dim_{k}H^{2}(\Gamma_{F},\operatorname{ad}\overline{\rho})$. The
Euler–Poincaré characteristic formula from local class field theory gives

(1) $r-s=\dim G_{k}([F:\mathbb{Q}_{p}]+1).$

Our theorem proves that $\dim R^{\square}_{\overline{\rho}}/\varpi$ is given
by this cohomological quantity, the _expected dimension_ in the spirit of the
_Dimension Conjecture_ of Gouvêa from [31, Lecture 4]. Having the expected
dimension implies that $\varpi,f_{1},\ldots,f_{s}$ is a regular sequence and
that $R^{\square}_{\overline{\rho}}$ is a local complete intersection. It also
implies (see [29, Lemma 7.5]) that the derived deformation ring of
$\overline{\rho}$ as introduced by Galatius and Venkatesh in [29] is homotopy
discrete, which means the derived deformation theory of $\overline{\rho}$ does
not contain more information than the usual deformation theory of
$\overline{\rho}$.

We expect that our results will play an important role in the categorical
$p$-adic local Langlands correspondence. Conjecturally [27, Conjecture 6.1.14]
on the Galois side one should consider a derived category of coherent sheaves
(or some $\infty$-category version of it) on the Emerton–Gee stack. The local
rings considered in 1.1 should arise as versal rings at finite type points of
the Emerton–Gee stack, when $G$ is an $L$-group or a $C$-group of a connected
reductive group defined over $F$ (see [26, Proposition 3.6.3], when
$G=\operatorname{GL}_{d}$). In other words, they are expected to describe the
local properties of the Emerton–Gee stack. An instance of this is a paper of
Yu Min [42], where he uses the analogue of 1.1 proved in [8] for
$\operatorname{GL}_{d}$ to show that if $G=\operatorname{GL}_{d}$ then the
derived version of the Emerton–Gee stack is equivalent to the classical
Emerton–Gee stack, obtaining an analogue of the result explained above, that
derived deformation rings are homotopy discrete, over the whole of the stack.
The Emerton–Gee stack has been recently defined and studied for
$\mathrm{GSp}_{4}$ in [38] and for tame groups in [39]. We infinitesimally
guarantee that the results of [42] will extend to this context.

In the paper we introduce and study a scheme
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$; its definition is
reviewed in Section 1.4.1 below. Its $\overline{k}$-points correspond to
continuous representations $\rho:\Gamma_{F}\rightarrow G(\overline{k})$ with
$G$-semisimplification equal to $\overline{\rho}^{\mathrm{ss}}$. It carries an
action of $G^{0}$ and the quotient stack
$[X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}/G^{0}]$ is an analogue of
the stacks studied by Wang-Erickson in [57] for $G=\operatorname{GL}_{d}$. We
expect that the disjoint union of the stacks
$[X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}/G^{0}]$ (or rather their
formal versions) taken over all possible $\overline{\rho}^{\mathrm{ss}}$ will
be the largest substack of the Emerton–Gee stack on which the universal
$(\varphi,\Gamma)$-module can be realised as a representation of $\Gamma_{F}$,
i.e. the situation described in [26, Remark 6.7.4] for $\operatorname{GL}_{d}$
will continue to hold in a more general setting.

###  1.1. The conjecture of Böckle–Juschka

Our second main result proves a generalisation of a conjecture of
Böckle–Juschka concerning the irreducible components of
$R^{\square}_{\overline{\rho}}$. We assume that $G^{0}$ is split over
$\mathcal{O}$ and $G/G^{0}$ is a constant group scheme. This can be achieved
after replacing $L$ by a finite unramified extension. Let $\Gamma_{E}$ be the
kernel of the homomorphism
$\Gamma_{F}\overset{\overline{\rho}}{\longrightarrow}G(k)\rightarrow(G/G^{0})(k)$.
We assume that $\mathcal{O}$ contains $\mu_{p^{\infty}}(E)$. One may further
assume that this map is surjective, as replacing $G$ with a subgroup does not
change the deformation problem, so that
$(G/G^{0})(k)=\operatorname{Gal}(E/F)$.

Let $G^{\prime}$ be the derived group scheme of $G^{0}$, let
$G^{\prime}_{\operatorname{sc}}\rightarrow G^{\prime}$ be the simply connected
central cover of $G^{\prime}$ and let $\pi_{1}(G^{\prime})$ be the the kernel
of this map. Then $\pi_{1}(G^{\prime})$ is a finite diagonalisable subgroup
scheme of the centre of $G^{\prime}_{\operatorname{sc}}$ and is isomorphic to
$\prod_{i=1}^{r}\mu_{n_{i}}$ for some integers $n_{i}$. The group scheme
$\pi_{1}(G^{\prime})$ is étale if and only if $p$ does not divide $n_{i}$ for
all $i$.

Our assumptions on $G$ imply that $G^{0}/G^{\prime}$ is a split torus. The
action of $G/G^{\prime}$ on $G^{0}/G^{\prime}$ by conjugation induces an
action of $\operatorname{Gal}(E/F)$ on its _character lattice_
$M:=X^{*}(G^{0}/G^{\prime})$.

Let $\overline{\psi}:\Gamma_{F}\rightarrow(G/G^{\prime})(k)$ be the
representation $\overline{\psi}=\varphi\circ\overline{\rho}$, where
$\varphi:G\rightarrow G/G^{\prime}$ is the quotient map. Let
$R^{\square}_{\overline{\psi}}$ be the universal deformation ring of
$\overline{\psi}$.

######  Theorem 1.2 (Corollaries 15.28, 15.30).

The natural map $R^{\square}_{\overline{\psi}}\to
R^{\square}_{\overline{\rho}}$, induced by composing a deformation of
$\overline{\rho}$ with $\varphi:G\rightarrow G/G^{\prime}$, is flat and if
$\pi_{1}(G^{\prime})$ is étale then it induces a bijection between the sets of
irreducible components.

In a companion paper [43] we deal with the case, when $G^{0}$ is a torus. In
particular, we show that the irreducible components of
$R^{\square}_{\overline{\psi}}$ can be labelled by characters
$\chi:(\mu_{p^{\infty}}(E)\otimes
M)^{\operatorname{Gal}(E/F)}\rightarrow\mathcal{O}^{\times}$.

We assume that $\pi_{1}(G^{\prime})$ is étale until the end of this
subsection. Let $R^{\square,\chi}_{\overline{\rho}}$ be an irreducible
component of $R^{\square}_{\overline{\rho}}$ corresponding to a character
$\chi$ under 1.2.

######  Theorem 1.3 (Corollary 15.20).

If $\pi_{1}(G^{\prime})$ is étale then the rings
$R^{\square,\chi}_{\overline{\rho}}$,
$R^{\square,\chi}_{\overline{\rho}}/\varpi$ are complete intersection, regular
in codimension $[F:\mathbb{Q}_{p}]$, normal domains.

Theorem 1.3 and a theorem of Grothendieck on factoriality of complete
intersections imply:

######  Corollary 1.4 (Corollary 15.22).

If $\pi_{1}(G^{\prime})$ is étale and $[F:\mathbb{Q}_{p}]\geq 3$ then
$R^{\square,\chi}_{\overline{\rho}}$ and
$R^{\square,\chi}_{\overline{\rho}}/\varpi$ are factorial.

If a further hypothesis on $\overline{\rho}$ and $G$ is satisfied (for
example, if $G/Z(G^{0})$ does not have $\operatorname{PGL}_{2}$ as a factor)
then we show in Corollary 15.20 that $R^{\square,\chi}_{\overline{\rho}}$ and
$R^{\square,\chi}_{\overline{\rho}}/\varpi$ are regular in codimension
$2[F:\mathbb{Q}_{p}]-1$ and hence 1.4 holds if $[F:\mathbb{Q}_{p}]=2$.

###  1.2. Deformation problems with fixed partial determinant

In fact, Theorems 1.1, 1.2, 1.3 and Corollary 1.4 hold in a more general
setting, which might be called deformation problems with _‘fixed partial
determinant’_. Assume that we have a $\operatorname{Gal}(E/F)$-invariant
decomposition $X^{*}(G^{0}/G^{\prime})=M_{1}\oplus M_{2}$ and let $H_{1}$ be
the quotient of $G/G^{\prime}$ such that their component groups coincide and
the character lattice of $H_{1}^{0}$ is equal to $M_{1}$. We fix a continuous
representation $\psi_{1}:\Gamma_{F}\rightarrow H_{1}(\mathcal{O})$, such that
$\psi_{1}\equiv\varphi_{1}\circ\overline{\rho}\pmod{\varpi}$, where
$\varphi_{1}:G\rightarrow H_{1}$ is the composition of $\varphi$ with the
quotient map $G/G^{\prime}\twoheadrightarrow H_{1}$. Let
$D^{\square,\psi_{1}}_{\overline{\rho}}$ be a subfunctor of
$D^{\square}_{\overline{\rho}}$, such that

$D^{\square,\psi_{1}}_{\overline{\rho}}(A)=\\{\rho_{A}\in
D^{\square}_{\overline{\rho}}(A):\varphi_{1}\circ\rho_{A}=\psi_{1}\otimes_{\mathcal{O}}A\\}.$

The functor $D^{\square,\psi_{1}}_{\overline{\rho}}$ is pro-represented by a
quotient $R^{\square,\psi_{1}}_{\overline{\rho}}$ of
$R^{\square}_{\overline{\rho}}$. Then the above results hold with
$R^{\square}_{\overline{\rho}}$ replaced with
$R^{\square,\psi_{1}}_{\overline{\rho}}$.

If $\pi_{1}(G^{\prime})$ is étale then the irreducible components of
$R^{\square,\psi_{1}}_{\overline{\rho}}$ can be indexed by characters
$\chi:(\mu_{p^{\infty}}(E)\otimes
M_{2})^{\operatorname{Gal}(E/F)}\rightarrow\mathcal{O}^{\times}$. There are
always two interesting cases: if $M_{1}=0$ then we impose an empty condition
and $R^{\square,\psi_{1}}_{\overline{\rho}}=R^{\square}_{\overline{\rho}}$; if
$M_{1}=X^{*}(G^{0}/G^{\prime})$ then $R^{\square,\psi_{1}}_{\overline{\rho}}$
parameterises deformations with ‘fixed determinant’ equal to $\psi_{1}$ and it
follows from above that $R^{\square,\psi_{1}}_{\overline{\rho}}$ and
$R^{\square,\psi_{1}}_{\overline{\rho}}/\varpi$ are integral domains. If
$G=\operatorname{GL}_{d}$ then $X^{*}(G^{0}/G^{\prime})=\mathbb{Z}$ and these
are the only cases that can occur. However, in general there are further
interesting cases naturally appearing in the Langlands program, as we discuss
in Section 16.

If $H$ is a connected reductive group over $F$ (or more generally over a
number field) which splits over a finite Galois extension $E$ of $F$ then it
is expected in [14] that the Galois representations attached to $C$-algebraic
automorphic forms on $H$ take values in the $C$-group ${}^{C}H$, which is a
generalised reductive group scheme over $\mathcal{O}$ with component group the
constant group scheme $\operatorname{Gal}(E/F)$. Moreover, they should satisfy
the condition $d\circ\rho=\chi_{\mathrm{cyc}}$, where $\chi_{\mathrm{cyc}}$ is
the $p$-adic cyclotomic character and $d$ is the canonical map
$d:{}^{C}H\rightarrow\mathbb{G}_{m}$. We explain in Theorem 16.5 that the
functor $D^{\square,\psi_{1}}_{\overline{\rho}}$ with $G={}^{C}H$,
$H_{1}=\operatorname{Gal}(E/F)\times\mathbb{G}_{m}$ and
$\psi_{1}:\Gamma_{F}\rightarrow H_{1}(\mathcal{O})$,
$\gamma\mapsto(\gamma|_{E},\chi_{\mathrm{cyc}}(\gamma))$ parameterises
deformations $\rho_{A}$ of $\overline{\rho}:\Gamma_{F}\rightarrow{}^{C}H(k)$
satisfying $d\circ\rho_{A}=\chi_{\mathrm{cyc}}\otimes_{\mathcal{O}}A$. We show
that in this case, under the assumption that $\pi_{1}(\widehat{H}^{\prime})$
is étale, the irreducible components are canonically in bijection with
characters of $p$-power torsion subgroup of $Z(H)^{0}(F)$, where $Z(H)^{0}$ is
the neutral component of the centre of $H$.

###  1.3. Deformation space of Lafforgue’s pseudocharacters

We refer the reader to Section 7 for the definition of V. Lafforgue’s
$G$-pseudocharacters. We single out some properties for the purpose of this
introduction. To every representation $\rho:\Gamma_{F}\rightarrow G(A)$ one
may attach an $A$-valued $G$-pseudocharacter $\Theta_{\rho}$. Moreover,
conjugation of $\rho$ with elements of $G^{0}(A)$ does not change the
associated $G$-pseudocharacter. If $\Theta$ is a $G$-pseudocharacter with
values in an algebraically closed field $\kappa$ then there exists a
representation $\rho:\Gamma_{F}\rightarrow G(\kappa)$, uniquely determined up
to $G^{0}(\kappa)$-conjugation, such that $\rho$ is $G$-semisimple (Definition
2.22) and $\Theta_{\rho}=\Theta$. If $A$ and $\kappa$ carry a topology then
there is a notion of continuous $G$-pseudocharacters, which interacts well
with continuous representations.

Let $\overline{\Theta}$ be the $G$-pseudocharacter attached to
$\overline{\rho}$. JQ has shown in [44] as part of his PhD thesis under the
direction of Gebhard Böckle that the functor
$D^{\mathrm{ps}}:\mathfrak{A}_{\mathcal{O}}\rightarrow\operatorname{Set}$,
such that $D^{\mathrm{ps}}(A)$ is the set of continuous $A$-valued
$G$-pseudocharacters deforming $\overline{\Theta}$ is pro-represented by a
complete local noetherian $\mathcal{O}$-algebra $R^{\mathrm{ps}}_{G}$ with
residue field $k$. The proof that $R^{\mathrm{ps}}_{G}$ is noetherian uses the
fact that $\Gamma_{F}$ is topologically finitely generated.

If $G=\operatorname{GL}_{d}$ then Emerson–Morel have shown in [25] that there
is a natural bijection between $G$-pseudocharacters and Chenevier’s
$d$-dimensional determinant laws defined in [16]. Thus in this case the ring
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$ coincides with the deformation ring
studied by Böckle–Juschka in [9]. The results of that paper are an important
input in [8]. In our paper we extend the main result [9, Theorem 5.5.1] to
$G$-pseudocharacters, also obtaining a new proof of it, when
$G=\operatorname{GL}_{d}$.

######  Theorem 1.5 (13.31).

The rings $R^{\mathrm{ps}}_{G}$ and $R^{\mathrm{ps}}_{G}/\varpi$ are
equidimensional of dimension $d+1$ and $d$, respectively, where $d=\dim
G_{k}[F:\mathbb{Q}_{p}]+\dim Z(G)_{k}$.

Let
$U^{\mathrm{ps}}:=(\operatorname{Spec}R^{\mathrm{ps}}_{G})\setminus\\{\text{the
closed point}\\}$. The non-special absolutely irreducible locus
$U^{\operatorname{n-spcl}}$ is an open subscheme of $U^{\mathrm{ps}}$ such
that a geometric point $x:\operatorname{Spec}\kappa\rightarrow
U^{\mathrm{ps}}$ lies in $U^{\operatorname{n-spcl}}$ if and only if the
representation $\rho_{x}:\Gamma_{F}\rightarrow G(\kappa)$ associated to the
specialisation of the universal $G$-pseudocharacter at $x$ is $G$-irreducible
(2.26) and
$H^{0}(\Gamma_{F},(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})_{x}^{*}(1))=0$,
where $\Gamma_{F}$ acts on
$(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})_{x}$ via $\rho_{x}$
composed with the adjoint action. If $\pi_{1}(G^{\prime})$ is étale or
$\mathrm{char}(\kappa)=0$ then
$(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})_{x}=(\operatorname{Lie}G^{\prime})_{x}$
and the last condition is equivalent to
$H^{0}(\Gamma_{F},(\operatorname{ad}^{0}\rho_{x})^{*}(1))=0$.

######  Theorem 1.6 (13.32).

$U^{\operatorname{n-spcl}}$ is Zariski dense in
$\operatorname{Spec}R^{\mathrm{ps}}_{G}$ and its special fibre is Zariski
dense in $\operatorname{Spec}R^{\mathrm{ps}}_{G}/\varpi$.

We also show in 15.25 that the reduced subscheme
$(\operatorname{Spec}R^{\mathrm{ps}}_{G}[1/p])^{\mathrm{red}}$ is normal. If
$\pi_{1}(G^{\prime})$ is étale then we compute the set of irreducible
components of $\operatorname{Spec}R^{\mathrm{ps}}_{G}$ in 15.27, which
together with 1.2 implies that
$\operatorname{Spec}R^{\square}_{\overline{\rho}}\rightarrow\operatorname{Spec}R^{\mathrm{ps}}_{G}$
induces a bijection between the sets of irreducible components.

###  1.4. Complete intersection

We will now sketch the proof of 1.1. Let
$\operatorname{Rep}^{\Gamma_{F}}_{G}:\mathcal{O}\text{-}\mathrm{alg}\rightarrow\operatorname{Set}$
be the functor, which maps an $\mathcal{O}$-algebra $A$ to the set of group
homomorphisms $\Gamma_{F}\rightarrow G(A)$. The scheme $G^{0}$ acts on
$\operatorname{Rep}^{\Gamma_{F}}_{G}$ by conjugation. We construct an affine
scheme
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}=\operatorname{Spec}A^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$
over $\mathcal{O}$, which is a $G^{0}$-invariant subfunctor of
$\operatorname{Rep}^{\Gamma_{F}}_{G}$ and the following desiderata hold:

  * (D1)

the specialisation of the universal representation
$\rho^{\mathrm{univ}}:\Gamma_{F}\rightarrow
G(A^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}})$ at $x\in
X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}(\overline{k})$ induces a
bijection $x\mapsto\rho_{x}$ between
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}(\overline{k})$ and the set
of continuous representations $\rho:\Gamma_{F}\rightarrow G(\overline{k})$
such that the $G$-semisimplification (2.21) of $\rho$ is equal to
$\overline{\rho}^{\mathrm{ss}}$;

  * (D2)

the completion of a local ring at $x\in
X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}(\overline{k})$ is
isomorphic to $R^{\square}_{\rho_{x}}$;

  * (D3)

the GIT-quotient $X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}\sslash
G^{0}$ is the spectrum of a complete local noetherian $\mathcal{O}$-algebra
$R^{\mathrm{git}}_{G,\overline{\rho}^{\mathrm{ss}}}$ with residue field $k$.

We then bound the dimension of the special fibre
$\overline{X}^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ from above

(2) $\dim\overline{X}^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}\leq\dim
G_{k}([F:\mathbb{Q}_{p}]+1).$

Once we can do this the first part of Theorem 1.1 follows immediately as (D2)
implies that $\dim R^{\square}_{\overline{\rho}}/\varpi\leq\dim
G_{k}([F:\mathbb{Q}_{p}]+1)$ and the result follows from (1) and a little
commutative algebra (3.9). So the overall strategy is similar to the strategy
in [8], however its execution requires substantially new ideas.

####  1.4.1. Definition

The difficulty in defining
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ stems from the fact that
we would like to work with continuous representations of $\Gamma_{F}$, but for
arbitrary $A$ there seems to be no sensible topology to put on $G(A)$ which
allows us to express the desired continuity condition by simply asking the map
$\Gamma_{F}\to G(A)$ to be continuous. The solution to this problem is
inspired by the analogous definition by Fargues–Scholze [28] in the $\ell\neq
p$ case, which uses condensed mathematics.

One may define $X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ as a
functor from
$R^{\mathrm{ps}}_{G}\text{-}\mathrm{alg}\rightarrow\operatorname{Set}$ such
that $X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}(A)$ is the set of
representations $\rho:\Gamma_{F}\rightarrow G(A)$, such that $\Theta_{\rho}$
is the specialisation of the universal $G$-pseudocharacter $\Theta^{u}$ along
$R^{\mathrm{ps}}_{G}\rightarrow A$ and $\rho$ extends to a homomorphism of
condensed groups

$\tilde{\rho}:\underline{\Gamma_{F}}\rightarrow
G(A_{\operatorname{disc}}\otimes_{(R^{\mathrm{ps}}_{G})_{\operatorname{disc}}}\underline{R^{\mathrm{ps}}_{G}}).$

We refer the reader to Section 4.2 for the notation. In Lemma 4.8 we show that
the condensed condition can be reformulated as follows: for some (equivalently
any) closed immersion $\tau:G\hookrightarrow\mathbb{A}^{n}$ the image
$\tau(\rho(\Gamma_{F}))$ is contained in a finitely generated
$R^{\mathrm{ps}}_{G}$-submodule $M\subseteq A^{n}=\mathbb{A}^{n}(A)$ such that
the map $\tau\circ\rho:\Gamma_{F}\rightarrow M$ is continuous for the
canonical topology on $M$ as an $R^{\mathrm{ps}}_{G}$-module. We call such
representations $R^{\mathrm{ps}}_{G}$-_condensed_. This definition of
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ is aesthetically pleasing
and helpful to prove that $X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$
is functorial in $G$. However, it is not immediately clear (at least to the
authors) from the definition that the functor is representable by a finite
type $R^{\mathrm{ps}}_{G}$-algebra.

If $G=\operatorname{GL}_{d}$ then one may linearise representations and work
instead with Cayley–Hamilton homomorphisms of
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebras $E^{u}\rightarrow
M_{d}(A)$, where $E^{u}$ is a universal Cayley–Hamilton quotient of the
completed group algebra
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}\llbracket\Gamma_{F}\rrbracket$. We
refer the reader to Section 5.1 for more details. Wang-Erickson has shown in
[57] that the algebra $E^{u}$ is a finitely generated
$R^{\mathrm{ps}}$-module. This result is the reason why
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-condensed representations do not
explicitly appear in [8]. In Lemma 5.2 we show that
$X^{\mathrm{gen}}_{\operatorname{GL}_{d},\overline{\rho}^{\mathrm{ss}}}$
coincides with the scheme $X^{\mathrm{gen}}$ defined in [8]. In particular, it
is represented by a finite type
$R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra
$A^{\mathrm{gen}}_{\operatorname{GL}_{d}}$.

In the general case, we fix a closed immersion
$\tau:G\hookrightarrow\operatorname{GL}_{d}$ of $\mathcal{O}$-group schemes
and define $X^{\mathrm{gen},\tau}_{G}$ as a closed subscheme of
$X^{\mathrm{gen}}_{\operatorname{GL}_{d}}:=X^{\mathrm{gen}}_{\operatorname{GL}_{d},(\tau\circ\overline{\rho})^{\mathrm{ss}}}$,
such that $\rho\in X^{\mathrm{gen}}_{\operatorname{GL}_{d}}(A)$ lies in
$X^{\mathrm{gen},\tau}_{G}(A)$ if and only if $\rho(\Gamma_{F})$ is contained
in $\tau(G(A))$. The functor $X^{\mathrm{gen},\tau}_{G}$ is representable by a
finite type $R^{\mathrm{ps}}_{\operatorname{GL}_{d}}$-algebra
$A^{\mathrm{gen},\tau}_{G}$. We show in Proposition 8.3 that the connected
component of $X^{\mathrm{gen},\tau}_{G}$ containing the point corresponding to
$\overline{\rho}:\Gamma_{F}\rightarrow G(k)$ is canonically isomorphic to the
functor $X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ defined via
condensed mathematics as above. In particular, the connected component is
independent of the chosen embedding $\tau$. Using the construction via
$X^{\mathrm{gen},\tau}_{G}$ we show that desiderata (D1), (D2), (D3) hold. The
condensed description allows to show in Proposition 8.5 that
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ is functorial in $G$.

A key input in the above argument is the following fundamental finiteness
theorem proved by Vinberg [56], when $S$ is a spectrum of a field of
characteristic zero, by Martin [40], when $S$ is a spectrum of a field of
characteristic $p$, and by Cotner [22] in general.

######  Theorem 1.7 (Cotner, [22]).

Let $S$ be a locally noetherian scheme and let $f:H\rightarrow G$ be a finite
morphism of generalised reductive smooth affine $S$-group schemes. Then the
induced map on the GIT-quotients

$H^{N}\sslash H\rightarrow G^{N}\sslash G$

is finite, where the action on both sides is given by conjugation.

The homomorphism $R^{\mathrm{ps}}_{G}\rightarrow
A^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ is $G^{0}$-equivariant for
the trivial action on the source. Hence, its image is contained in
$G^{0}$-invariants $R^{\mathrm{git}}_{G,\overline{\rho}^{\mathrm{ss}}}$.

######  Proposition 1.8 (7.4).

The morphism $R^{\mathrm{ps}}_{G}\rightarrow
R^{\mathrm{git}}_{G,\overline{\rho}^{\mathrm{ss}}}$ is a finite universal
homeomorphism.

We expect that the morphism is adequate in the sense of Alper [3], i.e. that
moreover $R^{\mathrm{ps}}_{G}[1/p]\cong
R^{\mathrm{git}}_{G,\overline{\rho}^{\mathrm{ss}}}[1/p]$, and hope to return
to this question in future work. If this were true then one could show that
$\operatorname{Spec}R^{\mathrm{ps}}_{G}[1/p]$ is reduced and hence normal, as
explained in Section 1.3.

We define $X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ for any
profinite group $\Gamma$ satisfying Mazur’s $p$-finiteness condition. However,
starting with Section 12 we use in an essential way that $\Gamma=\Gamma_{F}$:
besides Euler–Poincaré characteristic formula, it is crucial to our argument
that using local Tate duality we may transfer obstructions to lifting from
$H^{2}$ to $H^{0}$.

####  1.4.2. Bounding the dimension

We will now sketch how we obtain the bound (2). Let
$Y_{\overline{\rho}^{\mathrm{ss}}}$ be the preimage of the closed point of
$X^{\mathrm{ps}}_{G}:=\operatorname{Spec}R^{\mathrm{ps}}_{G}$ in
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$. We bound the dimension
of $Y_{\overline{\rho}^{\mathrm{ss}}}$ in 12.18. Its complement $V$ and the
special fibre $\overline{{V}}$ of $V$ are Jacobson schemes and residue fields
of closed points $x\in V$ are local fields. Moreover, in 5.17 we relate the
completions of local rings $\mathcal{O}_{V,x}$ to the deformation rings
$R^{\square}_{G,\rho_{x}}$, and completions of local rings
$\mathcal{O}_{\overline{{V}},x}$ to $R^{\square}_{G,\rho_{x}}/\varpi$. The
upshot is that if we control the dimension of
$R^{\square}_{G,\rho_{x}}/\varpi$ then we control the dimension of
$\mathcal{O}_{\overline{{V}},x}$ and if we can do this at every closed point
then we can bound the dimension of $\overline{{V}}$ and the dimension of its
closure in $\overline{X}^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$. In
Proposition 3.6 we establish a presentation

(3)
$R^{\square}_{G,\rho_{x}}\cong\frac{R^{\square}_{G/G^{\prime},\varphi\circ\rho_{x}}\llbracket
x_{1},\ldots,x_{r}\rrbracket}{(f_{1},\ldots,f_{s})},$

where $s=\dim_{\kappa(x)}H^{2}(\Gamma_{F},\operatorname{ad}^{0}\rho_{x})$ and
$r-s=\dim G^{\prime}_{k}([F:\mathbb{Q}_{p}]+1)$. We deduce from the results of
[43] that if $x\in\overline{{V}}$ is a closed point then
$R^{\square}_{G/G^{\prime},\varphi\circ\rho_{x}}/\varpi$ is formally smooth
over the group algebra $\kappa(x)[\mu]$ of dimension
$\dim(G/G^{\prime})_{k}([F:\mathbb{Q}_{p}]+1)$, where
$\mu=(\mu_{p^{\infty}}(E)\otimes M)^{\operatorname{Gal}(E/F)}$. This is easy
if $G=\operatorname{GL}_{d}$ or more generally if $G$ is connected, but
requires a non-trivial argument in the non-connected case, which we carry out
in [43]. This allows us to conclude that the closure in
$\overline{X}^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ of the locus
in $\overline{{V}}$, where $H^{2}(\Gamma_{F},\operatorname{ad}^{0}\rho_{x})$
vanishes, has dimension $\dim G_{k}([F:\mathbb{Q}_{p}]+1)$, if non-empty. This
locus will be referred to as the
$(\operatorname{Lie}G^{\prime}_{k})^{*}$_-non-special locus_ later on.

One is left to understand the locus, where
$H^{2}(\Gamma_{F},\operatorname{ad}^{0}\rho_{x})\neq 0$. By local Tate duality
this is equivalent to
$H^{0}(\Gamma_{F},(\operatorname{ad}^{0}\rho_{x})^{*}(1))\neq 0$. Let us
suppose that $\rho_{x}$ is absolutely $G$-irreducible. If
$G=\operatorname{GL}_{d}$ then Böckle–Juschka show in [9] that non-vanishing
of $H^{0}(\Gamma_{F},(\operatorname{ad}^{0}\rho_{x})^{*}(1))$ implies that
$\rho_{x}$ is induced from $\Gamma_{F^{\prime}}$, where $F^{\prime}$ is a
proper abelian extension of $F$. We know the dimension of
$\overline{X}^{\mathrm{ps},\Gamma_{F^{\prime}}}_{\operatorname{GL}_{d^{\prime}}}$,
where $d^{\prime}=\frac{d}{[F^{\prime}:F]}$, inductively and using this it is
shown in [8] that the closure of this locus has positive codimension. Such
arguments might work for classical groups (JQ has studied the case
$G=\operatorname{Sp}_{2n}$ under the assumption $p>2$ in [44, Section 7]), but
it seems impossible to get a characterisation like this for an arbitrary
generalised reductive group.

Our key observation is that instead of having such a precise description of
$\rho_{x}$, it is enough to observe that $\rho_{x}$ has _‘small’ image_.
Namely, let $H$ be the Zariski closure of $\rho_{x}(\Gamma_{F})$ in
$G(\kappa)$, where $\kappa$ is the algebraic closure of $\kappa(x)$. Since
$\rho_{x}$ is absolutely $G$-irreducible, its image is not contained in any
parabolic subgroup of $G_{\kappa}$. A theorem of Martin [40] asserts that $H$
is again generalised reductive. In Section 10 we prove two results:

######  Proposition 1.9 (10.2).

If a generalised reductive subgroup $H$ of $G_{\kappa}$ does not contain
$G^{\prime}_{\kappa}$ then $\dim G_{\kappa}-\dim H\geq 2$.

The main input in our proof of the above result is Matsushima’s theorem.

######  Lemma 1.10 (10.5).

The $G^{\prime}_{\kappa}$-invariants in
$(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})^{*}_{\kappa}$ are zero.

We note that the above Lemma would be false in small characteristics if we did
not take the dual. We first prove it for $\operatorname{SL}_{2}$ and then use
this together with arguments with roots to prove it in the general case.

Let us go back to the situation at hand, where $H$ is the Zariski closure of
$\rho_{x}(\Gamma_{F})$ in $G(\kappa)$. If $x$ lies in the special fibre, then
the twist by the cyclotomic character vanishes after restriction to
$\Gamma_{F(\zeta_{p})}$. Using this we show that non-vanishing of
$H^{0}(\Gamma_{F},(\operatorname{ad}\rho_{x})^{*}(1))$ implies that the
neutral component $H^{0}$ of $H$ has non-zero invariants in
$(\operatorname{Lie}G^{\prime})^{*}_{\kappa}$. If $\pi_{1}(G^{\prime})$ is
étale or $\mathrm{char}(\kappa)=0$ then
$(\operatorname{Lie}G^{\prime})^{*}_{\kappa}=(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})^{*}_{\kappa}$
and Proposition 1.9 and Lemma 1.10 imply that $\dim G_{\kappa}-\dim H^{0}\geq
2$ and hence $\dim G_{\kappa}-\dim H\geq 2$. The fact that the difference can
be bounded below by $2$ (and not by $1$) plays an important role in verifying
Serre’s criterion for normality in the proof of 1.3. As you can see there is
an induction argument shaping up. To explain it we need two further
ingredients.

Let $W$ be an algebraic representation of $G$ on a finite free
$\mathcal{O}$-module, and let $\rho:\Gamma_{F}\rightarrow G(A)$ be a
representation, where $A$ is a noetherian $\mathcal{O}$-algebra. We show in
Lemma 13.8 that there is a closed subscheme $X_{W,j}$ of
$X=\operatorname{Spec}A$ such that

$x\in
X_{W,j}\iff\dim_{\kappa(x)}H^{0}(\Gamma_{F},W\otimes_{\mathcal{O}}\kappa(x)(1))\geq
j+1.$

We refer to $X_{W,j}$ as _$W$ -special locus of level $j$_ and just _$W$
-special locus_ when $j=0$. The complement of the $W$-special locus is called
_$W$ -non-special_. The proof amounts to reformulating these conditions in
terms of some finitely generated $A$-module and a little of commutative
algebra. We apply this notion to
$W=(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})^{*}$ and if the image of
$\rho$ is contained in $P(A)$, where $P$ is a parabolic subgroup scheme of $G$
with unipotent radical $U$, then we apply it to $W=(\operatorname{Lie}U)^{*}$.
When working with the special fibre we consider representations $W$ over $k$
instead of $\mathcal{O}$.

The second ingredient, which uses 1.7 as an input, is the following finiteness
result proved in 8.6. If $H\rightarrow G_{k}$ is a finite morphism of
generalised reductive $k$-group schemes which maps an $H$-pseudocharacter
$\overline{\Psi}$ to $\overline{\Theta}$ then the natural map
$\overline{X}^{\mathrm{ps}}_{H,\overline{\Psi}}\rightarrow\overline{X}^{\mathrm{ps}}_{G}$
is finite. Moreover, there are only finitely many $H$-pseudocharacters
$\overline{\Psi}$ mapping to $\overline{\Theta}$. In particular, the dimension
of the union $\overline{X}^{\mathrm{ps}}_{HG}$ of the scheme theoretic images
can be bounded by the largest
$\dim\overline{X}^{\mathrm{ps}}_{H,\overline{\Psi}}$.

We are now in a position to sketch the proof of the bound in (2) when
$\overline{\rho}$ is absolutely $G$-irreducible. From (3) and the relation
between $R^{\square}_{G,\rho_{x}}/\varpi$ and the completions of
$\mathcal{O}_{\overline{{V}},x}$ we obtain a lower bound

(4) $\dim\overline{X}^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}\geq\dim
G_{k}([F:\mathbb{Q}_{p}]+1).$

We then construct a pair $(G_{1},\overline{\rho}_{1})$ such that $\dim
G_{1}\leq\dim G$, $\pi_{1}(G^{\prime}_{1})$ is étale and if (2) holds for
$(G_{1},\overline{\rho}_{1})$ then it also holds for $(G,\overline{\rho})$.
For example, if $G=\operatorname{PGL}_{d}$ then
$G_{1}=\operatorname{SL}_{d}/\mu_{m}$, where $d=p^{e}m$ and $p\nmid m$, and
$\overline{\rho}_{1}=\overline{\rho}$ as $G(k)=G_{1}(k)$. In general, the
presence of the component group of $G$ complicates the construction of
$G_{1}$, see Section 2.3. This reduction step allows us to assume that
$\pi_{1}(G^{\prime})$ is étale, which implies that
$\operatorname{Lie}G^{\prime}_{\operatorname{sc}}=\operatorname{Lie}G^{\prime}$.
We then show that there are finitely many generalised reductive subgroup
schemes $H_{i}$ of $G_{k}$, defined over some finite extension of $k$, such
that $\dim G_{k}-\dim H_{i}\geq 2$ and the
$(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})^{*}_{k}$-special locus in
$\overline{X}^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ is contained
in the preimage of the union of $\overline{X}^{\mathrm{ps}}_{H_{i}G}$. The
assumption that $\overline{\rho}$ is absolutely $G$-irreducible implies that
every fibre $X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}\rightarrow
X^{\mathrm{ps}}_{G}$ has dimension $\dim G_{k}-\dim Z(G)_{k}$. Inductively, we
may bound the dimension of $\overline{X}^{\mathrm{ps}}_{H_{i}G}$ by $\dim
H_{i}[F:\mathbb{Q}_{p}]+\dim Z(H_{i})$. Using this we show that the
codimension of the
$(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})^{*}_{k}$-special locus in
$\overline{X}^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ is at least
$2[F:\mathbb{Q}_{p}]$. It follows from (4) that the
$(\operatorname{Lie}G^{\prime})^{*}_{k}$-non-special locus in
$\overline{X}^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ is non-empty
and as explained above its closure has dimension $\dim
G_{k}([F:\mathbb{Q}_{p}]+1)$. This lets us conclude that (4) is an equality.

Let us briefly indicate how we find the subgroup schemes $H_{i}$ in the above
argument. The
$(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})^{*}_{k}$-special locus of
level $j$ is non-empty only for finitely many $j$. The group schemes $H_{i}$
correspond to the generic points of these loci for $j\geq 0$. Given such a
generic point $\eta$ we let $H$ be the Zariski closure of
$\rho_{\eta}(\Gamma_{F})$ in $G(\overline{\kappa(\eta)})$. Since
$\overline{\rho}^{\mathrm{ss}}$ is assumed to be absolutely $G$-irreducible
$H$ is not contained in any parabolic subgroup of
$G_{\overline{\kappa(\eta)}}$ and hence is generalised reductive. A conjugate
of $H$ by an element of $G(\overline{\kappa(\eta)})$ can be defined over
$\overline{k}$ and hence over a finite extension of $k$. Our arguments with
generalised reductive groups over algebraically closed fields use results of
Martin [40] and Bate–Martin–Röhrle [5] in an essential way.

The general case is an elaboration of this inductive argument. If
$\overline{\rho}^{\mathrm{ss}}$ is not absolutely $G$-irreducible then the
dimensions of the fibres
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}\rightarrow
X^{\mathrm{ps}}_{G}$ can vary. We bound their dimensions in Section 12. We
stratify $\overline{X}^{\mathrm{ps}}_{G}$ by
$\overline{X}^{\mathrm{ps}}_{LG}$, where $L$ runs over the Levi subgroup
schemes of $G$ containing a fixed maximal split torus. If
$y\in\overline{X}^{\mathrm{ps}}_{LG}$, but does not lie in any
$\overline{X}^{\mathrm{ps}}_{L^{\prime}G}$ for a proper Levi
$L^{\prime}\subsetneq L$ then for typical $y$ the fibre will have dimension
$\dim G_{k}-\dim Z(L)_{k}+\dim U_{k}[F:\mathbb{Q}_{p}]$, where $U$ is the
unipotent radical of any parabolic subgroup with Levi $L$. By induction we
know that $\dim\overline{X}^{\mathrm{ps}}_{LG}$ is at most $\dim
L_{k}[F:\mathbb{Q}_{p}]+\dim Z(L)_{k}$. The sum of these numbers does not
exceed $\dim G_{k}([F:\mathbb{Q}_{p}]+1)$. In fact, the difference is equal to
$\dim U_{k}[F:\mathbb{Q}_{p}]$. We deal with the fibres, which have dimension
exceeding $\dim G_{k}-\dim Z(L)_{k}+\dim U_{k}[F:\mathbb{Q}_{p}]$ by
considering the $(\operatorname{Lie}U_{k})^{\ast}$-special locus inside
$X^{\mathrm{ps}}_{L}$. The inductive argument is carried out in Section 13.

If $X^{\mathrm{ps}}_{LG}$ is non-empty for a Levi subgroup $L$ satisfying
$\dim G-\dim L=2$ then the bound $\dim U_{k}[F:\mathbb{Q}_{p}]$ is not good
enough to apply the Serre’s criterion for normality, when $F=\mathbb{Q}_{p}$,
as the assumption on $L$ forces $\dim U_{k}=1$. We analyse the
$(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})^{\ast}$-non-special locus
in the preimage of $X^{\mathrm{ps}}_{LG}$ in
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ in this case in detail in
Section 11. In Section 14 we show that the
$(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})^{\ast}$-special locus in
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ and also in
$\overline{X}^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ has
codimension of at least $1+[F:\mathbb{Q}_{p}]$ and of at least
$2[F:\mathbb{Q}_{p}]$ if $X^{\mathrm{ps}}_{LG}$ is empty for all Levi
subgroups $L$ of codimension $2$.

###  1.5. Irreducible components

We will sketch the proof of 1.3. Using the results of [43] we show that

(5)
$A^{\mathrm{gen}}_{G/G^{\prime},\overline{\psi}^{\mathrm{ss}}}\cong\mathcal{O}[\mu]\llbracket
x_{1},\ldots,x_{m}\rrbracket[t_{1}^{\pm 1},\ldots,t_{l}^{\pm 1}],$

where $l=\dim(G/G^{\prime})_{k}-\dim Z(G/G^{\prime})_{k}$,
$m+l=\dim(G/G^{\prime})_{k}([F:\mathbb{Q}_{p}]+1)$ and
$\mu=(\mu_{p^{\infty}}(E)\otimes M)^{\operatorname{Gal}(E/F)}$. Let
$\mathrm{X}(\mu)$ be the group of characters
$\chi:\mu\rightarrow\mathcal{O}^{\times}$. If $A$ is an
$A^{\mathrm{gen}}_{G/G^{\prime},\overline{\psi}^{\mathrm{ss}}}$-algebra then
we use (5) to define $A^{\chi}:=A\otimes_{\mathcal{O}[\mu],\chi}\mathcal{O}$
and if $X=\operatorname{Spec}A$ we write
$X^{\chi}:=\operatorname{Spec}A^{\chi}$. It follows from (5) that the
irreducible components of
$X^{\mathrm{gen}}_{G/G^{\prime},\overline{\psi}^{\mathrm{ss}}}$ are given by
$X^{\mathrm{gen},\chi}_{G/G^{\prime},\overline{\psi}^{\mathrm{ss}}}$ for
$\chi\in\mathrm{X}(\mu)$ and are regular. In particular, the completions of
local rings of
$X^{\mathrm{gen},\chi}_{G/G^{\prime},\overline{\psi}^{\mathrm{ss}}}$ are also
regular. Thus the presentation (3) gives us a presentation

(6)
$R^{\square,\chi}_{G,\rho_{x}}\cong\frac{R^{\square,\chi}_{G/G^{\prime},\varphi\circ\rho_{x}}\llbracket
x_{1},\ldots,x_{r}\rrbracket}{(f_{1},\ldots,f_{s})},$

where $R^{\square,\chi}_{G/G^{\prime},\varphi\circ\rho_{x}}$ is a regular
ring. Further, if $x$ is in the $(\operatorname{Lie}G^{\prime})^{\ast}$-non-
special locus then $s$ in the above presentation is $0$ and we can conclude
that $R^{\square,\chi}_{G,\rho_{x}}$ is a regular ring. We deduce that the
$(\operatorname{Lie}G^{\prime})^{\ast}$-non-special locus in
$X^{\mathrm{gen},\chi}_{G,\overline{\rho}^{\mathrm{ss}}}$ is contained in the
regular locus. In general, the $(\operatorname{Lie}G^{\prime})^{\ast}$-non-
special locus might be empty, for example this happens if
$G=\operatorname{PGL}_{p}$. On the other hand the complement of
$(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})^{\ast}$-non-special locus
has codimension of at least $1+[F:\mathbb{Q}_{p}]$ in
$X^{\mathrm{gen},\chi}_{G,\overline{\rho}^{\mathrm{ss}}}$. If
$\pi_{1}(G^{\prime})$ is étale then these loci coincide and this implies that
$X^{\mathrm{gen},\chi}_{G,\overline{\rho}^{\mathrm{ss}}}$ is regular in
codimension $[F:\mathbb{Q}_{p}]$. Since
$X^{\mathrm{gen},\chi}_{G,\overline{\rho}^{\mathrm{ss}}}$ is excellent the
same applies to the completions of its local rings, and we may deduce that the
deformation rings $R^{\square,\chi}_{G,\rho_{x}}$ are regular in codimension
$[F:\mathbb{Q}_{p}]$. To show that they are complete intersection it is enough
to bound their dimension from above and then use (6). Since $k[\mu]$ is a
local artinian algebra the underlying reduced subschemes of the special fibres
of $X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ and
$X^{\mathrm{gen},\chi}_{G,\overline{\rho}^{\mathrm{ss}}}$ coincide and hence
they have the same dimension. Serre’s criterion for normality implies that
$R^{\square,\chi}_{G,\rho_{x}}$ is normal. Since the ring is local, it is a
domain.

Since
$(\operatorname{Lie}G^{\prime})_{L}=(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})_{L}$
the $(\operatorname{Lie}G^{\prime})^{\ast}$-non-special and
$(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})^{\ast}$-non-special loci
in the generic fibre coincide. This allows us to prove that
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}[1/p]$ is normal without
any assumption on $\pi_{1}(G^{\prime})$.

A similar argument works in the ‘fixed partial determinant’ setting described
in Section 1.2. However, there is one new idea not present in [8]. To bound
the dimension of $X^{\mathrm{gen},\psi_{1}}_{G,\overline{\rho}^{\mathrm{ss}}}$
we show that there is a finite morphism
$X^{\mathrm{gen},\psi_{1}}_{G,\overline{\rho}^{\mathrm{ss}}}\rightarrow
X^{\mathrm{gen}}_{G/Z_{1},\overline{\rho}^{\mathrm{ss}}}$, where $Z_{1}$ is a
central subtorus of $G^{0}$ such that the composition $Z_{1}\rightarrow
G\rightarrow H_{1}$ is an isogeny. We can then bound the dimension of various
subschemes of $X^{\mathrm{gen},\psi_{1}}_{G,\overline{\rho}^{\mathrm{ss}}}$ by
dimensions of the corresponding subschemes in
$X^{\mathrm{gen}}_{G/Z_{1},\overline{\rho}^{\mathrm{ss}}}$. So for example, to
prove the results in the fixed determinant case, when
$G=\operatorname{GL}_{d}$, we use the results proved for
$G=\operatorname{PGL}_{d}$. In [8] one twists by characters to unfix the
determinant instead. This argument can be made to work, when $G$ is connected,
but will not work in general.

###  1.6. Overview by section

In Section 2 we review some facts about the generalised reductive
$\mathcal{O}$-group schemes and their parabolic subgroup schemes, which we
call R-parabolic. We also review $G$-semisimplification and
$G$-irreducibility. In Section 3 we recall deformation theory of
representations of a profinite group satisfying Mazur’s $p$-finiteness
condition. Proposition 3.6 is a key result of that section. In Section 4 we
discuss and compare notions of continuity via algebra and via condensed sets.
As explained in Section 1.4.1 this is used to define
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ later on. In Section 5 we
construct the scheme $X^{\mathrm{gen},\tau}_{G}$ and study the relation
between the completion of its local rings and deformation rings of Galois
representations. In Section 6 we study the GIT quotient
$X^{\mathrm{git},\tau}_{G}:=X^{\mathrm{gen},\tau}_{G}\sslash G^{0}$ and define
$X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ and its GIT quotient
$X^{\mathrm{git}}_{G,\overline{\rho}^{\mathrm{ss}}}$. In Section 7 we
introduce Lafforgue’s $G$-pseudocharacters and their deformation spaces
$X^{\mathrm{ps}}_{G}$ and show that the natural map
$X^{\mathrm{git}}_{G,\overline{\rho}^{\mathrm{ss}}}\rightarrow
X^{\mathrm{ps}}_{G}$ is a finite universal homeomorphism. In Section 8 we give
a condensed description of
$X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ as explained in
Section 1.4.1. This allows us to conclude that
$X^{\mathrm{gen},\tau}_{G,\overline{\rho}^{\mathrm{ss}}}$ is independent of
$\tau$, which we then omit from notation, and is functorial in $G$. In Section
9 we show that without loss of generality one may assume that the map
$\Gamma_{F}\overset{\overline{\rho}}{\longrightarrow}G(k)\rightarrow(G/G^{0})(\overline{k})$
is surjective. In this case we define the absolutely $G$-irreducible locus in
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}$ and show that if it is
non-empty then the difference between its dimension and the dimension of its
GIT quotient is $\dim G_{k}-\dim Z(G)_{k}$. In Section 10 we bound the
codimension of generalised reductive subgroups of generalised reductive groups
over algebraically closed fields. The results of this section are used to
bound the dimensions of
$(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})^{\ast}$-special and
$(\operatorname{Lie}U)^{\ast}$-special loci, as explained in Section 1.4.2. In
Section 11 we show that if $G$ is a generalised reductive group over an
algebraically closed field with an R-Levi $L$ of codimension $2$ such that
$L/L^{0}\rightarrow G/G^{0}$ is an isomorphism then $G/Z(G^{0})\cong
G_{1}\times\operatorname{PGL}_{2}$ and $L/Z(G)\cong
G_{1}\times\mathbb{G}_{m}$. Proposition 11.5 is used to bound the dimension of
the $(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})^{\ast}$-special locus
in Section 14. This section can be omitted if one is only interested in the
complete intersection property and does not care for normality and irreducible
components. In Section 12 we bound the dimensions of fibres of
$X^{\mathrm{gen}}_{G,\overline{\rho}^{\mathrm{ss}}}\rightarrow
X^{\mathrm{ps}}_{G}$. In Section 13, which is the core of the paper, we carry
out the induction argument described in Section 1.4.2. In Section 14 we bound
the dimension of the
$(\operatorname{Lie}G^{\prime}_{\operatorname{sc}})^{*}$-special locus. In
Section 15 we study the irreducible components of
$X^{\mathrm{gen},\psi_{1}}_{G,\overline{\rho}^{\mathrm{ss}}}$. In Section 16
we explain that our results apply, when $G$ is an $L$-group or a $C$-group,
for the convenience of the reader. The appendix recalls some background on
condensed mathematics, which is used in Section 4.

###  1.7. Notation

Let $F$ be a finite extension of $\mathbb{Q}_{p}$. We fix an algebraic closure
$\overline{F}$ over $F$. Let $\Gamma_{F}:=\operatorname{Gal}(\overline{F}/F)$
be the absolute Galois group of $F$. We will denote by $\zeta_{p}$ a primitive
$p$-th root of unity in $\overline{F}$.

Let $L$ be another finite extension of $\mathbb{Q}_{p}$, such that
$\operatorname{Hom}_{\mathbb{Q}_{p}\text{-}\mathrm{alg}}(F,L)$ has cardinality
$[F:\mathbb{Q}_{p}]$. Let $\mathcal{O}$ be the ring of integers in $L$,
$\varpi$ a uniformiser, and $k$ the residue field. All our schemes are defined
over $\mathcal{O}$ (unless stated otherwise). We omit
$\operatorname{Spec}\mathcal{O}$ from the notation, when we take fibre
products over it, so that $X\times
Y:=X\times_{\operatorname{Spec}\mathcal{O}}Y$.

If $A$ is a commutative ring and
$\rho:\Gamma\rightarrow\operatorname{GL}_{d}(A)$ is a representation, we let

$\rho^{\mathrm{lin}}:A[\Gamma]\rightarrow
M_{d}(A),~{}\sum_{\gamma\in\Gamma}a_{\gamma}\gamma\mapsto\sum_{\gamma\in\Gamma}a_{\gamma}\rho(\gamma)$

be its _linearisation_.

We denote by $A\text{-}\mathrm{alg}$ the category of commutative $A$-algebras.
For a commutative ring $R$ we let
$P_{1}R=\\{\mathfrak{p}\in\operatorname{Spec}R:\dim R/\mathfrak{p}=1\\}$.

###  1.8. Acknowledgements

The authors would like to thank Jarod Alper and Sean Cotner for the
correspondence regarding their work; Tasho Kaletha for a very useful
discussion regarding Section 11; Vincent Pilloni for sharing his notes of the
talk he gave on [8]; Sophie Morel for making a preliminary version of [25]
available to them; Toby Gee, James Newton and Sean Cotner for their comments.
The authors would like to especially thank Brian Conrad for pointing out an
error in an earlier version of the paper and for the subsequent
correspondence.

VP would like to thank his colleagues Ulrich Görtz, Daniel Greb and Jochen
Heinloth for several stimulating discussions regarding various aspects of the
paper.

This paper builds on the collaboration [8] of VP with Gebhard Böckle and
Ashwin Iyengar, where the case $G=\operatorname{GL}_{d}$ is treated. The paper
[7] by Gebhard Böckle, Michael Harris, Chandrashekhar Khare and Jack Thorne
also played an important role for us, especially at the beginning of the
project.

JQ would like to thank Gebhard Böckle for introducing him to Lafforgue’s
theory of $G$-pseudocharacters and their deformation theory during his PhD
studies at the University of Heidelberg.

Parts of the paper were written during the research stay of VP at the
Hausdorff Research Institute for Mathematics in Bonn for the Trimester Program
_The Arithmetic of the Langlands Program_. VP would like to thank the
organisers Frank Calegari, Ana Caraiani, Laurent Fargues and Peter Scholze for
the invitation and a stimulating research environment. The research stay of VP
was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813.

The research of JQ was funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) – project number 517234220.

##  2\. $\mathcal{O}$-group schemes

The category of $\mathcal{O}$-group schemes embeds fully faithfully in the
category of fppf group sheaves on
$(\mathcal{O}\text{-}\mathrm{alg})^{\mathrm{op}}$. Whenever an exact sequence
is considered it is understood as an exact sequence of fppf group sheaves. If
$G$ is a flat $\mathcal{O}$-group scheme of finite presentation and $N\leq G$
is a flat closed normal subgroup, then the quotient group sheaf $G/N$ is
representable by a scheme [4, Théorème 4.C]. If $G$ and $N$ are generalised
reductive then the quotient is affine by [3, Theorem 9.4.1].

######  Lemma 2.1.

Let $G$ be a flat affine group scheme of finite presentation over
$\mathcal{O}$. Then the functor
$\underline{Z}(G):\mathcal{O}\text{-}\mathrm{alg}\to\operatorname{Set}$ given
by

$\underline{Z}(G)(A):=\\{g\in G(A)\mid\forall A\to A^{\prime},\forall h\in
G(A^{\prime}),g=hgh^{-1}\\}$

is representable by a closed subgroup scheme of $G$, which we will denote by
$Z(G)$ and call the _centre_ of $G$.

###### Proof.

This is a special case of [1, Exposé VIII, Théorème 6.4] as explained in [1,
Exposé VIII, Exemples 6.5 (e)]. ∎

######  Lemma 2.2.

Let $G$ be a flat affine group scheme of finite presentation over
$\mathcal{O}$ and let $A\to B$ be a map of $\mathcal{O}$-algebras. Then
$\underline{Z}(G_{A})$ and $\underline{Z}(G_{B})$ as in 2.1 are representable
and we have $Z(G_{A})_{B}=Z(G_{B})$.

###### Proof.

By 2.1 $\underline{Z}(G)$ is representable by an $\mathcal{O}$-scheme $Z(G)$.
By definition $\underline{Z}(G_{A})$ is the restriction of $\underline{Z}(G)$
to the category of $A$-algebras and is thus represented by $Z(G)_{A}$. The
same argument shows, that $\underline{Z}(G_{B})$ is representable by
$Z(G)_{B}$ and the claim follows. ∎

######  Lemma 2.3.

Let $0\to G_{1}\to G_{2}\to G_{3}\to 0$ be a short exact sequence of affine
$\mathcal{O}$-group schemes of finite presentation. If $G_{1}$ and $G_{3}$ are
flat, then $G_{2}\to G_{3}$ is faithfully flat and in particular $G_{2}$ is
flat.

###### Proof.

Since $G_{1}$ is flat and of finite presentation, the map of
$\mathcal{O}$-schemes

$G_{1}\times G_{2}\to G_{2}\times G_{2},~{}(g,h)\mapsto(gh,h)$

is an fppf equivalence relation. The projection map onto the second factor is
clearly faithfully flat and of finite presentation. The first projection map
is faithfully flat and of finite presentation, since multiplication is
isomorphic to projection onto the second factor, as can be seen by considering
the automorphism $(g,h)\mapsto(g,g^{-1}h)$ of $G_{1}\times G_{2}$.

By definition of exactness $G_{3}$ is the quotient sheaf of $G_{2}$ by this
equivalence relation. Since $G_{3}$ is already representable by a scheme, we
know that the equivalence relation is effective by [30, Exposé IV, Corollaire
6.3.3]. By definition of effectivity [30, Exposé IV, Déf. 3.4.3] this means,
that the map $G_{2}\to G_{3}$ is faithfully flat and of finite presentation. ∎

######  Lemma 2.4.

If $G$ is a flat $\mathcal{O}$-group scheme of finite presentation then $\dim
G_{k}=\dim G_{L}$.

###### Proof.

See [30, Exposé VIB, Corollaire 4.3]. ∎

######  Definition 2.5.

A _generalised reductive group scheme_ over a scheme $S$ is a smooth affine
$S$-group scheme $G$, such that the geometric fibres of $G^{0}$ are reductive
and $G/G^{0}\rightarrow S$ is finite.

######  Remark 2.6.

It follows from [20, Proposition 3.1.3], that for a generalised reductive
$\mathcal{O}$-group scheme $G$ the quotient $G/G^{0}$ is étale. Sean Cotner
has shown in [22] using the results of Alper that a smooth affine $S$-group
scheme $G$ is generalised reductive if and only if it is geometrically
reductive in the sense of [3, Definition 9.1.1]. In particular, this holds
over $S=\operatorname{Spec}\mathcal{O}$.

######  Lemma 2.7.

Let $G$ be an affine group scheme of finite type over a perfect field
$\kappa$. If $G$ is reduced then $G$ is smooth over $\kappa$. Moreover, if we
additionally assume that the unipotent radical of $G$ is trivial then $G$ is
generalised reductive.

###### Proof.

The first assertion follows from [54, Tag 047N, Tag 047P]. Since $\kappa$ is
perfect, if the unipotent radical of $G$ is trivial then also the unipotent
radical of $G_{K}$ is trivial, for any field extension $K$ of $\kappa$ by [21,
Proposition 1.1.9]. Thus $G^{0}$ is reductive and $G/G^{0}$ is finite by [20,
Proposition 3.1.3]. ∎

######  Proposition 2.8.

Let $G$ be a generalised reductive $\mathcal{O}$-group scheme. Then there
exists a finite unramified extension $L^{\prime}/L$ with ring of integers
$\mathcal{O}^{\prime}$, such that the following hold:

  1. (1)

$(G/G^{0})_{\mathcal{O}^{\prime}}=G_{\mathcal{O}^{\prime}}/G_{\mathcal{O}^{\prime}}^{0}$
is a constant group scheme;

  2. (2)

$G(\mathcal{O}^{\prime})\rightarrow(G/G^{0})(\mathcal{O}^{\prime})$ is
surjective;

  3. (3)

$G^{0}_{\mathcal{O}^{\prime}}$ is split.

###### Proof.

By definition $G/G^{0}$ is finite étale. Using [54, Tag 0BND] and the fact
that the étale fundamental group of $\mathcal{O}$ is the Galois group of the
maximal unramified extension of $L$ we find a finite unramified extension
$L^{\prime}$, such that $(G/G^{0})_{\mathcal{O}^{\prime}}$ is constant. Since
the extension $\mathcal{O}\rightarrow\mathcal{O}^{\prime}$ is flat the
sequence of group schemes $0\rightarrow G^{0}\rightarrow G\rightarrow
G/G^{0}\rightarrow 0$ remains exact after base change to
$\mathcal{O}^{\prime}$.

Let $k^{\prime}$ be the residue field of $\mathcal{O}^{\prime}$. After further
enlarging $k^{\prime}$ we may ensure that
$G(k^{\prime})\rightarrow(G/G^{0})(k^{\prime})$ is surjective. Since $G$ is
smooth and $\mathcal{O}^{\prime}$ is complete the map
$G(\mathcal{O}^{\prime})\rightarrow G(k^{\prime})$ is surjective. Since
$G/G^{0}$ is a constant group scheme
$(G/G^{0})(k^{\prime})=(G/G^{0})(\mathcal{O}^{\prime})$ and hence we obtain
part (2).

Part (3) follows by the same argument as in part (1) from [20, Lemma 5.1.3]. ∎

######  Proposition 2.9.

Let $G$ be a generalised reductive $\mathcal{O}$-group scheme. Then $Z(G)\cap
G^{0}$ is flat over $\mathcal{O}$. In particular, $\dim Z(G)_{k}=\dim
Z(G)_{L}$.

###### Proof.

By 2.1 we see that $Z(G)$ exists as a closed subgroup scheme of $G$. For
flatness and the equality of dimensions, by 2.8 we may extend $\mathcal{O}$,
such that $G/G^{0}$ is constant and the map $G\to G/G^{0}$ has a section
$s:G/G^{0}\to G$ as $\mathcal{O}$-schemes. We write
$\underline{\Delta}=G/G^{0}$ for some finite group $\Delta$. The $s$ is
determined by a map $\Delta\to G(\mathcal{O})$ and for every
$\mathcal{O}$-algebra $A$ the map $G(A)\to(G/G^{0})(A)$ is surjective.

By [20, Theorem 3.3.4] $Z(G^{0})$ is diagonalizable, i.e.
$Z(G^{0})=\operatorname{Spec}(\mathcal{O}[M])$ for some finitely generated
abelian group $M$. We have an action of $\Delta$ by conjugation on $Z(G^{0})$.
Let $Z(G^{0})^{\Delta}$ be the maximal closed subgroup scheme of $Z(G^{0})$ on
which the action of $\Delta$ is trivial. The kernel of
$\mathcal{O}[Z(G^{0})]\to\mathcal{O}[Z(G^{0})^{\Delta}]$ is generated by
$x-\delta x$ for $x\in M$ and $\delta\in\Delta$. Hence,
$\mathcal{O}[Z(G^{0})^{\Delta}]=\mathcal{O}[M_{\Delta}]$.

The group $Z(G^{0})^{\Delta}$ is equal to $Z(G)\cap G^{0}$, since for every
$\mathcal{O}$-algebra $A$, the group $G(A)$ is generated by $G^{0}(A)$ and
$s((G/G^{0})(A))$. In particular, $Z(G)\cap G^{0}$ is flat over $\mathcal{O}$.
We have a short exact sequence

(7) $1\to Z(G)\cap G^{0}\to Z(G)\to H\to 1$

where $H$ is the image of $Z(G)$ in $G/G^{0}$, in particular $H$ is finite. It
follows from (7) that special (resp. generic) fibres of $Z(G)\cap G^{0}$ and
$Z(G)$ will have the same neutral component, and hence the same dimension.
Since $(Z(G)\cap G^{0})_{k}=\dim(Z(G)\cap G^{0})_{L}$ by 2.4, we deduce that
$\dim Z(G)_{k}=\dim Z(G)_{L}$. ∎

######  Lemma 2.10.

Let $G$ be a generalised reductive group scheme over $\mathcal{O}$. Then the
natural map $\pi:G\to G/G^{\prime}$ is smooth.

###### Proof.

By [20, Theorem 5.3.1] $G^{\prime}$ is a semisimple $\mathcal{O}$-group scheme
and in particular smooth. Since $G^{\prime}$ is the fppf sheafification of the
commutator subgroup functor $A\mapsto[G^{0}(A),G^{0}(A)]$, it is preserved
under any automorphism of $G^{0}$. So $G^{\prime}$ is a closed normal subgroup
scheme of $G$. By 2.3 the map $\pi$ is faithfully flat and of finite
presentation. So by [54, Tag 01V8] we need to see, that all fibres of $\pi$
are smooth. Let $\kappa$ be any field and let
$x:\operatorname{Spec}(\kappa)\to G/G^{\prime}$ be a $\kappa$-point; this also
defines a map $\mathcal{O}\to\kappa$. So we get a natural map
$x^{\prime}:\operatorname{Spec}(\kappa)\to(G/G^{\prime})_{\kappa}$, where
$(G/G^{\prime})_{\kappa}=G/G^{\prime}\times\operatorname{Spec}(\kappa)$. The
fibre of $\pi$ above $x$ coincides with the fibre of
$\pi_{\kappa}:G_{\kappa}\to(G/G^{\prime})_{\kappa}$ above $x^{\prime}$. By
translation in $(G/G^{\prime})_{k}$ this fibre is isomorphic to
$G_{\kappa}^{\prime}$, which is smooth. ∎

######  Lemma 2.11.

Let $G$ be a generalised reductive group scheme over $\mathcal{O}$. Let
$G^{\prime}_{\operatorname{sc}}\rightarrow G^{\prime}$ be the simply connected
central cover of $G^{\prime}$. Then there is a natural action of $G$ on
$\operatorname{Lie}(G^{\prime}_{\operatorname{sc}})$ such that the map
$\operatorname{Lie}(G^{\prime}_{\operatorname{sc}})\rightarrow\operatorname{Lie}G^{\prime}$
is $G$-equivariant.

###### Proof.

Existence and uniqueness of the simply connected central cover is ensured by
[20, Exercise 6.5.2 (i)]. For any $\mathcal{O}$-algebra $A$, the map
$(G^{\prime}_{\operatorname{sc}})_{A}\to(G^{\prime})_{A}$ is a simply
connected central cover of $(G^{\prime})_{A}$. By [20, Exercise 6.5.2 (iii)]
an automorphism of $(G^{\prime})_{A}$ lifts to a unique automorphism of
$(G^{\prime}_{\operatorname{sc}})_{A}$, so the conjugation action of $G(A)$ on
$(G^{\prime})_{A}$ by $A$-group scheme automorphisms extends to an action of
$G(A)$ on $(G^{\prime}_{\operatorname{sc}})_{A}$. This action is functorial in
$A$ and the map $(G^{\prime}_{\operatorname{sc}})_{A}\to(G^{\prime})_{A}$ is
$G(A)$-equivariant. So $G(A)$ acts functorially on
$\operatorname{Lie}(G^{\prime}_{\operatorname{sc}})\otimes A$, which gives
$\operatorname{Lie}(G^{\prime}_{\operatorname{sc}})$ the structure of a
rational $G$-module. This action is clearly compatible with the action of $G$
on $\operatorname{Lie}G^{\prime}$. ∎

###  2.1. R-parabolics

We recall some facts about parabolic subgroups of a generalised reductive
$\mathcal{O}$-group scheme $G$. The main issue is that we do not assume that
$G$ is connected. The letter R stands for Richardson. Let $A$ be an
$\mathcal{O}$-algebra and let $\lambda:\mathbb{G}_{m,A}\rightarrow G_{A}$ be a
cocharacter. We then define $P_{\lambda}$ (resp. $U_{\lambda}$) to be the
closed $A$-group scheme of $G_{A}$ representing the functor of points $g$ of
$G$ such that the limit $\lim_{t\rightarrow 0}\lambda(t)g\lambda(t)^{-1}$
exists (resp. exists and is equal to the identity), see [21, Lemmas 2.1.4,
2.1.5]. Let $L_{\lambda}=P_{\lambda}\cap P_{-\lambda}$, where $-\lambda$ is
the cocharacter defined by $-\lambda(t):=\lambda(1/t)=\lambda(t)^{-1}$. We
call subgroup schemes of $G$ the form $P_{\lambda}$ _R-parabolic_ subgroup
schemes and $L_{\lambda}$ _R-Levi_ subgroup schemes of $P_{\lambda}$. We will
only consider R-parabolic and R-Levi subgroup schemes over a connected base.

######  Lemma 2.12.

The following hold

  1. (1)

$L_{\lambda}$ is the scheme theoretic centraliser of $\lambda$;

  2. (2)

$U_{\lambda}$ is a closed normal subgroup scheme of $P_{\lambda}$;

  3. (3)

the multiplication map $L_{\lambda}\ltimes U_{\lambda}\rightarrow P_{\lambda}$
is an isomorphism of $R$-groups;

  4. (4)

$L_{\lambda}$, $U_{\lambda}$, $P_{\lambda}$ are smooth over $\mathcal{O}$.

###### Proof.

The assertions follow from Lemma 2.1.5 and Proposition 2.1.8(2) of [21]. The
last assertion is given by Theorem 4.1.7 (4) in [20]. ∎

######  Lemma 2.13.

Let $P$ be an R-parabolic subgroup of $G$ with R-Levi subgroup $L$, both
defined over $\mathcal{O}$. Then $P\cap G^{0}$ (resp. $L\cap G^{0})$ is the
neutral component of $P$ (resp. $L$) and is a parabolic (resp. Levi) subgroup
of $G^{0}$. In particular, $L$ is a generalised reductive $\mathcal{O}$-group
scheme. Moreover, there are only finitely many R-parabolic subgroups $Q$ of
$G$ such that $Q^{0}=P^{0}$.

###### Proof.

Let $\lambda:\mathbb{G}_{m}\rightarrow G$ be a cocharacter such that
$P=P_{\lambda,G}$ and $L=L_{\lambda,G}$. Since $\mathbb{G}_{m}$ is connected
$\lambda$ factors through $\lambda:\mathbb{G}_{m}\rightarrow G^{0}$. It
follows from [21, Lemma 2.1.5] that $P\cap G^{0}=P_{\lambda,G^{0}}$ and $L\cap
G^{0}=L_{\lambda,G^{0}}$. Let $\bar{x}$ be a geometric point of
$\operatorname{Spec}\mathcal{O}$. Then
$(P_{\lambda,G^{0}})_{\bar{x}}=P_{\lambda_{\bar{x}},G^{0}_{\bar{x}}}$ and
$(L_{\lambda,G^{0}})_{\bar{x}}=L_{\lambda_{\bar{x}},G^{0}_{\bar{x}}}$. It
follows from [21, Proposition 2.2.9] that
$P_{\lambda_{\bar{x}},G^{0}_{\bar{x}}}$ is a parabolic subgroup of
$G^{0}_{\bar{x}}$ with Levi $L_{\lambda_{\bar{x}},G^{0}_{\bar{x}}}$. Moreover,
both groups are connected and $L_{\lambda_{\bar{x}},G^{0}_{\bar{x}}}$ is
reductive by classical theory of reductive groups over algebraically closed
fields. Thus $P\cap G^{0}$ is the neutral component of $P$, $L\cap G^{0}$ is
the neutral component of $L$. Moreover, $P^{0}$ is a parabolic subgroup of
$G^{0}$ in the sense of [20, Definition 5.2.1] and it follows from Lemma 2.12
(3) that $L^{0}$ is a Levi of $P^{0}$ in the sense of [20, Definition 5.4.2].
Further, $L^{0}$ is reductive in the sense of [20, Definition 3.1.1]. The map
$L\rightarrow G/G^{0}$ identifies $L/L^{0}$ with a closed subgroup of
$G/G^{0}$. Hence $L/L^{0}$ is finite over $\mathcal{O}$. We deduce that $L$ is
generalised reductive and $L/L^{0}$ is finite étale over $\mathcal{O}$ by
Remark 2.6.

Since $U$ is contained in $P^{0}$ the map $L\rightarrow P/P^{0}$ induces an
isomorphism $L/L^{0}\cong P/P^{0}$. Thus $P/P^{0}$ is a closed subgroup of
$G/G^{0}$, which is finite étale over $\mathcal{O}$. Since $G/G^{0}$ is finite
étale over $\mathcal{O}$ by Remark 2.6, there are only finitely many closed
subgroup schemes of $G/G^{0}$, which are finite étale over $\mathcal{O}$. This
implies that there are only finitely many R-parabolic subgroups $Q$ such that
$Q^{0}=P^{0}$. ∎

######  Example 2.14.

Let $G=(\mathbb{G}_{m}\times\mathbb{G}_{m})\rtimes\mathbb{Z}/2\mathbb{Z}$ with
$\mathbb{Z}/2\mathbb{Z}$ acting on $\mathbb{G}_{m}\times\mathbb{G}_{m}$ by
permuting the coordinates. Let $\lambda,\mu:\mathbb{G}_{m}\rightarrow G$ be
the cocharacters $\lambda(t)=((t,1),0)$ and $\mu(t)=((1,1),0)$. Then
$P_{\lambda}=G^{0}$ and $P_{\mu}=G$, but their neutral components are the
same.

######  Remark 2.15.

If $G\rightarrow S$ is a connected reductive group scheme and $S$ is affine
then the notions of R-parabolic and parabolic subgroup schemes coincide; see
the discussion at the end of Section 5.4 in [20].

######  Proposition 2.16.

Let $G$ be a generalised reductive group over a field, let $L$ be an R-Levi
factor of an R-parabolic subgroup $P$, let $U$ be the unipotent radical of $P$
and assume that the natural map $L/L^{0}\to G/G^{0}$ is an isomorphism. Then

(8) $\displaystyle\dim L+\dim Z(L)\leq\dim G+\dim Z(G)-\dim U$

###### Proof.

We may assume throughout the proof that the base field is algebraically
closed. Let $T\subseteq L^{0}$ be a maximal torus of $G^{0}$. Since $G^{0}$ is
generated by $L^{0}$, the root subgroups of $U$ and their opposites, we have
$Z(G^{0})=Z(L^{0})\cap\bigcap_{a}\ker(a)$, where $a:T\to\mathbb{G}_{m}$ varies
over all roots, which generate root subgroups $U_{a}\subseteq U$. The groups
$U_{\pm a}$ commute with $\ker(a)$ and $\dim\ker(a)=\dim T-1$, so

(9) $\displaystyle\dim Z(L^{0})\leq\dim Z(G^{0})+\dim U.$

Since $\Delta:=G/G^{0}$ is constant, we have a scheme theoretic splitting
$s:G/G^{0}\to G$ and $\Delta$ acts on $Z(G^{0})$ by conjugation. Since we are
assuming, that $L/L^{0}=G/G^{0}$, we have a compatible action of $\Delta$ on
$Z(L^{0})$. We have a diagram

${1}$${Z(G^{0})}$${Z(L^{0})}$${Z(L^{0})/Z(G^{0})}$${1}$${1}$${Z(G^{0})^{\Delta}}$${Z(L^{0})^{\Delta}}$${Z(L^{0})^{\Delta}/Z(G^{0})^{\Delta}}$${1}$

and the right vertical map is injective, since
$Z(G^{0})^{\Delta}=Z(L^{0})^{\Delta}\cap Z(G^{0})$. As in the proof of 2.9, we
have $Z(G^{0})^{\Delta}=Z(G)\cap G^{0}$ and $Z(L^{0})^{\Delta}=Z(L)\cap
L^{0}$, so $\dim Z(G^{0})^{\Delta}=\dim Z(G)$ and $\dim Z(L^{0})^{\Delta}=\dim
Z(L)$. Injectivity of the right vertical map gives

(10) $\displaystyle\dim Z(L)-\dim Z(G)\leq\dim Z(L^{0})-\dim Z(G^{0})$

We conclude from (10) and (9) that

$\displaystyle\dim L+\dim Z(L)$ $\displaystyle\leq\dim L^{0}+\dim
Z(L^{0})-\dim Z(G^{0})+\dim Z(G)$ $\displaystyle\leq\dim L^{0}+\dim U+\dim
Z(G)=\dim G+\dim Z(G)-\dim U,$

where we use $\dim G^{0}=\dim L^{0}+2\dim U$ in the last step. ∎

For the rest of the subsection we assume that $G$ is split and $G/G^{0}$ is
constant. We fix a maximal split torus $T$ of $G$ defined over $\mathcal{O}$.
Let $s$ be the closed point of $\operatorname{Spec}\mathcal{O}$ and let
$\bar{s}$ be a geometric point above $s$.

######  Lemma 2.17.

Base change induces a bijection between the following sets:

  1. (1)

R-parabolics of $G$ containing $T$;

  2. (2)

R-parabolics of $G_{s}$ containing $T_{s}$;

  3. (3)

R-parabolics of $G_{\bar{s}}$ containing $T_{\bar{s}}$.

###### Proof.

We will first establish that $P\mapsto P_{\bar{s}}$ induces the bijection
between sets in parts (1) and (3). If $\lambda:\mathbb{G}_{m}\rightarrow G$ is
a cocharacter defined over $\mathcal{O}$ then
$(P_{\lambda})_{\bar{s}}=P_{\lambda_{\bar{s}}}$ by [21, Lemma 2.1.4].
Moreover, if $P$ contains $T$ then $P_{\bar{s}}$ will contain $T_{\bar{s}}$.
Hence, the map is well defined.

If $P$ and $Q$ are R-parabolics of $G$ such that $P_{\bar{s}}=Q_{\bar{s}}$
then $P^{0}_{\bar{s}}=Q^{0}_{\bar{s}}$. It follows from [20, Corollary 5.2.7
(2)] that there exists a Zariski open subset $U$ of
$\operatorname{Spec}\mathcal{O}$ containing $s$ such that
$P^{0}_{U}=Q^{0}_{U}$. Since $\mathcal{O}$ is a DVR we have
$U=\operatorname{Spec}\mathcal{O}$ and hence $P^{0}=Q^{0}$. It follows from
Lemma 2.13 that $P\cap G^{0}=P^{0}=Q^{0}=Q\cap G^{0}$. Thus it is enough to
show that the images of $P$ and $Q$ in $G/G^{0}$ coincide. Since $G/G^{0}$ is
a constant group scheme, we may check this after base change to $\bar{s}$,
which holds as $P_{\bar{s}}=Q_{\bar{s}}$.

Let $\bar{\lambda}:\mathbb{G}_{m}\rightarrow G_{\bar{s}}$ be a cocharacter.
Its image is contained in a maximal split torus $T^{\prime}$ of $G_{\bar{s}}$.
Moreover, $T^{\prime}$ is contained in $P_{\bar{\lambda}}$. If
$P_{\bar{\lambda}}$ contains $T_{\bar{s}}$ then there is $g\in
P_{\bar{\lambda}}(\kappa(\bar{s}))$ such that $T_{\bar{s}}=gT^{\prime}g^{-1}$
by [11, Proposition 11.19]. We thus may assume that the image of
$\bar{\lambda}$ is contained in $T_{\bar{s}}$. Since both $\mathbb{G}_{m}$ and
$T$ are split tori defined over $\mathcal{O}$ there is a cocharacter
$\lambda:\mathbb{G}_{m}\rightarrow T$, such that
$\lambda_{\bar{s}}=\bar{\lambda}$. Then $P_{\lambda}$ contains $T$ and
satisfies $(P_{\lambda})_{\bar{s}}=P_{\bar{\lambda}}$.

The proof that base change induces a bijection between the sets in parts (2)
and (3) is the same. Since $P_{\bar{s}}=(P_{s})_{\bar{s}}$ this also implies
that base change induces a bijection between the sets in parts (1) and (2). ∎

######  Lemma 2.18.

There are only finitely many R-parabolic subgroups of $G$ containing $T$. Each
of them has a unique R-Levi subgroup containing $T$.

###### Proof.

Lemma 2.17 implies that it is enough to show that there are only finitely many
R-parabolics of $G_{\bar{s}}$ containing $T_{\bar{s}}$. Lemma 2.13 implies
that we may assume that $G_{\bar{s}}$ is connected. Proposition 11.19 (b) in
[11] says that there are only finitely many Borel subgroups of
$G^{0}_{\bar{s}}$ containing $T_{\bar{s}}$. Proposition 14.18 in [11] implies
that there are only finitely many parabolic subgroups of $G^{0}_{\bar{s}}$
containing a given Borel subgroup. This proves the first assertion.

Let $M$ be the unique Levi of $P^{0}$ containing $T$, [20, Proposition 5.4.5].
Let $\lambda:\mathbb{G}_{m}\rightarrow G$ be a cocharacter such that
$P=P_{\lambda}$. Then $L^{0}_{\lambda}$ is also a Levi of $P^{0}$ by Lemma
2.13. The scheme of Levi subgroups of $P^{0}$ is a $U_{\lambda}$-torsor, [20,
Corollary 5.4.6]. Thus there is $g\in U_{\lambda}(\mathcal{O})$ such that
$M=gL^{0}_{\lambda}g^{-1}$. Then $gL_{\lambda}g^{-1}$ is an R-Levi of $P$
containing $T$.

If $L$ and $M$ are R-Levi subgroups of $P$ containing $T$ then
$M_{\bar{x}}=L_{\bar{x}}$ for every geometric point $\bar{x}$ of
$\operatorname{Spec}\mathcal{O}$ by [5, Corollary 6.5]. This implies that
$L=M$. ∎

######  Lemma 2.19.

Let $\bar{x}$ be a geometric point of $\operatorname{Spec}\mathcal{O}$ and let
$Q$ be an R-parabolic subgroup of $G_{\bar{x}}$ with R-Levi $M$. Then there is
an R-parabolic subgroup $P$ of $G$ defined over $\mathcal{O}$ with R-Levi $L$
containing $T$ and $g\in G^{0}(\kappa(\bar{x}))$, such that
$gQg^{-1}=P_{\bar{x}}$ and $gMg^{-1}=L_{\bar{x}}$.

###### Proof.

Let $\lambda:\mathbb{G}_{m,\bar{x}}\rightarrow G_{\bar{x}}$ be a cocharacter
such that $Q=P_{\lambda}$ and $M=L_{\lambda}$. There is a maximal torus
$T^{\prime}$ of $G_{\bar{x}}$, such that the image of $\lambda$ is contained
in $T^{\prime}$. Since tori are connected, both $T^{\prime}$ and $T_{\bar{x}}$
are contained in $G^{0}_{\bar{x}}$. Since $\kappa(\bar{x})$ is algebraically
closed any two maximal tori in $G^{0}_{\bar{x}}$ are conjugate, and hence
after conjugating $Q$ we may assume that
$\lambda:\mathbb{G}_{m,\bar{x}}\rightarrow T_{\bar{x}}$. Since $T$ is split
over $\mathcal{O}$, there is a cocharacter $\mu:\mathbb{G}_{m}\rightarrow T$
defined over $\mathcal{O}$ such that $\lambda=\mu_{\bar{x}}$. We have
$P_{\mu}\times_{\operatorname{Spec}\mathcal{O}}\bar{x}=P_{\mu_{\bar{x}}}$ by
[21, Lemma 2.1.4] and
$L_{\mu}\times_{\operatorname{Spec}\mathcal{O}}\bar{x}=L_{\mu_{\bar{x}}}$ by
[21, Lemma 2.1.5]. ∎

###  2.2. $G$-semisimplification

In this section let $G$ be a generalised reductive group over an algebraically
closed field $\kappa$. Let $\Gamma$ be a group and let $\rho:\Gamma\rightarrow
G(\kappa)$ be a representation. Let $P$ be an R-parabolic of $G$ minimal with
respect to the property that $P(\kappa)$ contains $\rho(\Gamma)$. Let $L$ be
an R-Levi of $P$ and let $U$ be the unipotent radical of $P$. The composition
$L\rightarrow P\rightarrow P/U$ is an isomorphism, and we define
$c_{P,L}:P(\kappa)\rightarrow G(\kappa)$ as the composition
$P(\kappa)\rightarrow(P/U)(\kappa)\overset{\cong}{\longrightarrow}L(\kappa)\hookrightarrow
G(\kappa)$.

######  Proposition 2.20.

The $G^{0}(\kappa)$-conjugacy class of $c_{P,L}\circ\rho$ is independent of
the choice of $P$ and $L$.

###### Proof.

If $G$ is connected then this is proved in [51, Proposition 3.3 (b)]. The same
argument is carried out in our more general setting in [44, Proposition 2.15].
∎

######  Definition 2.21.

The $G$-semisimplification of $\rho$ is the $G^{0}(\kappa)$-conjugacy class of
the representation $c_{P,L}\circ\rho$. It will be denoted by
$\rho^{\mathrm{ss}}$.

In [51, Section 3.2.1] Serre introduced the notion of $G$-complete
reducibility in the connected case. We extend this definition to generalised
reductive group schemes verbatim.

######  Definition 2.22.

A subgroup $H$ of $G(\kappa)$ is $G$-_completely reducible_ if for every
R-parabolic $P\subseteq G$ with $H\subseteq P(\kappa)$, there exists an R-Levi
subgroup $L\subseteq P$ with $H\subseteq L(\kappa)$. We say, that a
homomorphism $\rho:\Gamma\rightarrow G(\kappa)$ is $G$-_completely reducible_
, if its image is $G$-completely reducible. We will also use the term
$G$-_semisimple_ synonymously for $G$-completely reducible.

######  Proposition 2.23.

Let $\rho:\Gamma\to G(\kappa)$ be a homomorphism. Then $\rho$ is
$G$-completely reducible if and only if $\rho^{\mathrm{ss}}$ is the
$G^{0}(\kappa)$-conjugacy class of $\rho$.

###### Proof.

Let $P$ be a minimal R-parabolic, such that $\rho(\Gamma)\subseteq P(\kappa)$.
If $\rho$ is $G$-completely reducible, there exists some R-Levi $L$ of $P$,
such that $\rho(\Gamma)\subseteq L(\kappa)$. In particular
$\rho=c_{P,L}\circ\rho$. We can apply 2.20 to conclude, $\rho^{\mathrm{ss}}$
is the $G^{0}(\kappa)$-conjugacy class of $\rho$. Conversely, suppose that
$\rho\in\rho^{\mathrm{ss}}$, let $L$ be any R-Levi of $P$. The assumption
implies that $\rho$ is conjugate to $c_{P,L}\circ\rho$ by an element of $g\in
G^{0}(\kappa)$. Minimality of $P$ and [5, Corollary 6.4] imply that the image
of $c_{P,L}\circ\rho$ is $L$-irreducible, which means that it is not contained
in any proper R-parabolic of $L$. Hence, $\rho(\Gamma)$ is
$gLg^{-1}$-irreducible. It follows from the equivalence of parts (iv) and (v)
of [5, Corollary 3.5], which as explained in [5, Section 6.3] also holds for
non-connected groups, that $\rho(\Gamma)$ is $G$-completely reducible. ∎

######  Proposition 2.24.

Let $\rho:\Gamma\to G(\kappa)$ be a homomorphism. Then the determinant laws
attached to $(\tau\circ\rho)^{\mathrm{lin}}$ and
$(\tau\circ\rho^{\mathrm{ss}})^{\mathrm{lin}}$ agree.

###### Proof.

Let $P$ be an R-parabolic of $G$, minimal with respect to the property that
$P(\kappa)$ contains $\rho(\Gamma)$. Let $U$ be the unipotent radical of $P$
and let let $\lambda$ be a cocharacter, such that $P=P_{\lambda}$. Then
$\lim\nolimits_{t\to 0}\lambda(t)U\lambda(t)^{-1}=\\{1\\}$ and hence
$c_{P,L_{\lambda}}\circ\rho=\lim\nolimits_{t\to
0}\lambda(t)\rho\lambda(t)^{-1}$. Let
$D_{\tau\circ\rho}:\kappa[\Gamma]\to\kappa$ be the determinant law attached to
$(\tau\circ\rho)^{\mathrm{lin}}$ and let $D_{\tau\circ\rho^{\mathrm{ss}}}$ be
the determinant law attached to $(\tau\circ
c_{P,L_{\lambda}}\circ\rho)^{\mathrm{lin}}$. We note that replacing
$c_{P,L_{\lambda}}\circ\rho$ by a conjugate with an element of $G^{0}(\kappa)$
does not change the determinant law, as they are invariant under conjugation.

We also have a family of determinant laws
$D:\kappa[\Gamma]\to\kappa[t,t^{-1}]$ over $\mathbb{G}_{m}$ given by

$D_{A}:A[\Gamma]\to A[t,t^{-1}],\quad
r\mapsto\det(((\tau(\lambda(t))(\tau\circ\rho)\tau(\lambda(t))^{-1})\otimes\operatorname{id}_{A})(r)),$

which is actually constant in $t$ and equal to $D_{\tau\circ\rho}$. So this
family extends uniquely to a family over $\mathbb{A}^{1}$. Since the limit of
$\lambda(t)\rho\lambda(t)^{-1}$ as $t\to 0$ exists and formation of the
determinant is algebraic, we obtain $D^{t=0}=D_{\tau\circ\rho^{\mathrm{ss}}}$
and hence $D_{\tau\circ\rho}=D_{\tau\circ\rho^{\mathrm{ss}}}$. ∎

######  Remark 2.25.

In general $(\tau\circ\rho)^{\mathrm{ss}}$ is not isomorphic to
$\tau\circ\rho^{\mathrm{ss}}$. But it follows from 2.24, that
$D_{\tau\circ\rho^{\mathrm{ss}}}=D_{\tau\circ\rho}$ and in particular from
[16, Theorem 2.12] that $(\tau\circ\rho^{\mathrm{ss}})^{\mathrm{ss}}$ is
isomorphic to $(\tau\circ\rho)^{\mathrm{ss}}$.

######  Definition 2.26.

A subgroup $H$ of $G(\kappa)$ is $G$-_irreducible_ if $H$ is not contained in
any proper R-parabolic subgroup of $G$. We say, that a homomorphism
$\rho:\Gamma\rightarrow G(\kappa)$ is $G$-_irreducible_ , if its image is
$G$-irreducible.

######  Proposition 2.27.

Let $L$ and $M$ be R-Levi subgroups of $G$ and let $H$ be a closed subgroup of
$L\cap M$ such that $H$ is both $L$-irreducible and $M$-irreducible. Then
there is $g\in Z_{G}(H)^{0}(\kappa)$ such that $L=gMg^{-1}$.

###### Proof.

Let $\lambda,\mu:\mathbb{G}_{m}\rightarrow G$ be cocharacters such that
$L=L_{\lambda}$ and $M=L_{\mu}$. Since $H$ is contained in $L$ and in $M$,
Lemma 2.12 (1) implies that the both $\lambda$ and $\mu$ factor through the
inclusion $Z_{G}(H)\subseteq G$. Let $S$ (resp. $T$) be a maximal torus of
$Z_{G}(H)$ containing the image of $\lambda$ (resp. $\mu$). It follows from
the proof of [11, Proposition 8.18] that $Z_{G}(S)$ is an R-Levi of $G$
containing $H$ and contained in $L$. Since $H$ is $L$-irreducible by
assumption we get that $L=Z_{G}(S)$. Similarly, we obtain that $M=Z_{G}(T)$.
There is $g\in Z_{G}(H)^{0}(\kappa)$ such that $S=gTg^{-1}$ by [11,
Proposition 11.19]. Thus $L=Z_{G}(gTg^{-1})=gZ_{G}(T)g^{-1}=gMg^{-1}$. ∎

###  2.3. Extensions

Let $N$ and $\Delta$ be abstract groups. A _rigidified extension of $N$ by
$\Delta$_ is a group law $\ast$ on the set $N\times\Delta$, such that the maps
$\iota:N\rightarrow N\times\Delta$, $g\mapsto(g,1)$ and
$\pi:N\times\Delta\rightarrow\Delta$, $(g,\delta)\mapsto\delta$ are group
homomorphisms.

######  Definition 2.28.

A _generalised $2$-cocycle_ of $\Delta$ with coefficients in $N$ is a pair
$(\omega,c)$, where $\omega:\Delta\rightarrow\operatorname{Aut}(N)$ and
$c:\Delta\times\Delta\rightarrow N$ are maps (of sets) satisfying the
following:

  1. (1)

$\omega(\delta_{1})\circ\omega(\delta_{2})=\operatorname{Int}_{N}(c(\delta_{1},\delta_{2}))\circ\omega(\delta_{1}\delta_{2})$
in $\operatorname{Aut}(N)$;

  2. (2)

$\omega(1)=\operatorname{id}_{N}$;

  3. (3)

$c(\delta_{1},\delta_{2})c(\delta_{1}\delta_{2},\delta_{3})=\omega(\delta_{1})(c(\delta_{2},\delta_{3}))c(\delta_{1},\delta_{2}\delta_{3})$
in $N$;

  4. (4)

$c(\delta_{1},1)=c(1,\delta_{2})=1$;

where for $g\in N$, $\operatorname{Int}_{N}(g)$ is the inner automorphism
$x\mapsto gxg^{-1}$ of $N$.

We emphasise that, since $N$ is not assumed to be abelian, $\omega$ is not
going to be a group homomorphism in general. Let $(N\times\Delta,\ast)$ be a
rigidified extension of $N$ by $\Delta$. It is a pleasant exercise in Algebra
to verify that if we let

$\omega^{\ast}(\delta)(x):=s(\delta)xs(\delta)^{-1},\quad
c^{\ast}(\delta_{1},\delta_{2}):=s(\delta_{1})s(\delta_{2})s(\delta_{1}\delta_{2})^{-1},$

for all $x\in N$ and all $\delta,\delta_{1},\delta_{2}\in\Delta$, where
$s(\delta)=(1,\delta)$, then the group axioms for $\ast$ imply that
$(\omega^{\ast},c^{\ast})$ is a generalised $2$-cocycle.

######  Lemma 2.29.

Mapping $(N\times\Delta,\ast)$ to $(\omega^{\ast},c^{\ast})$ induces a
bijection from the set of rigidified extensions of $N$ by $\Delta$ and the set
of generalised $2$-cocycles of $\Delta$ with values in $N$.

###### Proof.

If $N$ is abelian then $\operatorname{Int}_{N}=\operatorname{id}_{N}$ and the
result is well known. In general, the assertion follows from [49, Satz I], a
nice exposition is given in [60, Section 2.2] and Lemma follows from [60,
Lemma 2.11]. One could also deduce the assertion from [24]: property (3)
implies that the element denoted by $f_{3}(x,y,z)$ in [24, Section 7] is
$1_{N}$. Theorem 8.1 in [24] then implies that every generalised $2$-cocycle
comes from a rigidified extension. ∎

######  Proposition 2.30.

Let $G$ be a generalised reductive group over a perfect field $\kappa$ of
characteristic $p$, such that $G^{0}$ is split semisimple,
$G/G^{0}=\underline{\Delta}$ is constant and the map
$G(\kappa)\rightarrow(G/G^{0})(\kappa)$ is surjective. Then there exists a
surjection of generalised reductive groups $H\to G$ over $\kappa$ with kernel
$\mu$, such that the following hold:

  1. (1)

$H\to G$ induces an isomorphism $H/H^{0}\cong G/G^{0}$;

  2. (2)

$0\to\mu\to H^{0}\to G^{0}\to 0$ is a central extension with
$\mu(\overline{\kappa})=1$;

  3. (3)

$\pi_{1}(H^{0})$ is étale.

Moreover, $H(\kappa)\cong G(\kappa)$.

###### Proof.

We may choose a set-theoretic section $s:\Delta\to G(\kappa)$ such that
$s(1)=1$. It induces an isomorphism of $\kappa$-schemes

(11) $\displaystyle
G^{0}\times\underline{\Delta}\xrightarrow{\sim}G,\quad(g,\delta)\mapsto
gs(\delta).$

We define a map $\omega:\Delta\to\operatorname{Aut}(G^{0})$ by
$\omega(\delta)(g):=s(\delta)gs(\delta)^{-1}$. We further define a map
$c:\Delta\times\Delta\to G^{0}(\kappa)$ by
$c(\delta_{1},\delta_{2}):=s(\delta_{1})s(\delta_{2})s(\delta_{1}\delta_{2})^{-1}$.

For a $\kappa$-algebra $A$, we get a map
$\omega_{A}:\Delta\to\operatorname{Aut}(G^{0}(A))$ by composing with the
homomorphism
$\operatorname{Aut}(G^{0})\to\operatorname{Aut}((G^{0})_{A})\to\operatorname{Aut}(G^{0}(A))$
with $\omega$ and a map $c_{A}:\Delta\times\Delta\rightarrow G^{0}(A)$ by
composing $c$ with the map $G^{0}(\kappa)\to G^{0}(A)$.

The natural map $\Delta\to\underline{\Delta}(A)$ is a group homomorphism. The
isomorphism (11) induces a bijection
$G^{0}(A)\times\underline{\Delta}(A)\xrightarrow{\sim}G(A)$. We use the
bijection to transport the group structure on
$G^{0}(A)\times\underline{\Delta}(A)$. When $A\neq 0$, 2.29 implies that
$(\omega_{A},c_{A})$ is a generalised $2$-cocycle of $\Delta$ with values in
$G^{0}(A)$ corresponding to the restriction of this group law to the subset
$G^{0}(A)\times\Delta$. Thus, for all
$\delta_{1},\delta_{2},\delta_{3}\in\Delta$ we obtain the following
identities:

  * (C1)

$\omega(\delta_{1})\circ\omega(\delta_{2})=\operatorname{Int}_{G^{0}}(c(\delta_{1},\delta_{2}))\circ\omega(\delta_{1}\delta_{2})$
in $\operatorname{Aut}(G^{0})$,

  * (C2)

$\omega(1)=\operatorname{id}_{G^{0}}$ in $\operatorname{Aut}(G^{0})$,

  * (C3)

$c(\delta_{1},\delta_{2})c(\delta_{1}\delta_{2},\delta_{3})=\omega_{\kappa}(\delta_{1})(c(\delta_{2},\delta_{3}))c(\delta_{1},\delta_{2}\delta_{3})$
in $G^{0}(\kappa)$, and

  * (C4)

$c(1,\delta_{1})=c(\delta_{1},1)=1$ in $G^{0}(\kappa)$,

where for $g\in G^{0}(\kappa)$, $\operatorname{Int}_{G^{0}}(g)$ is the inner
automorphism $x\mapsto gxg^{-1}$ of $G^{0}$. Here (C1) and (C2) follow by
considering all $A$. Identities (C3) and (C4) follow by considering the case
$A=\kappa$.

The simply connected central cover
$\pi_{\operatorname{sc}}:(G^{0})_{\operatorname{sc}}\to G^{0}$ has finite
multiplicative kernel $K$. Let $K_{0}$ be the prime to $p$ part of $K$ and
define $H_{0}:=(G^{0})_{\operatorname{sc}}/K_{0}$. The kernel $\mu$ of
$\pi:H_{0}\to G^{0}$ is isomorphic to $K/K_{0}$ and is thus a connected finite
multiplicative $\kappa$-group scheme. Since $G^{0}$ is split, $\mu$ is a
product of group schemes of the form $\mu_{p^{r}}$. We have an exact sequence
of non-abelian fppf cohomology groups

$0\to\mu(\kappa)\to H_{0}(\kappa)\to G^{0}(\kappa)\to
H^{1}_{\mathrm{fppf}}(\operatorname{Spec}(\kappa),\mu).$

Since
$H^{1}_{\mathrm{fppf}}(\operatorname{Spec}(\kappa),\mu_{p^{n}})=\kappa^{\times}/(\kappa^{\times})^{p^{n}}$
by [54, Tag 040Q] and $\kappa$ is a perfect field of characteristic $p$, we
have $\mu(\kappa)=1$ and
$H^{1}_{\mathrm{fppf}}(\operatorname{Spec}(\kappa),\mu)=0$. So we can see $c$
as a map to $H_{0}(\kappa)$, which we will denote by $\tilde{c}$. If $A$ is a
$\kappa$-algebra we let $\tilde{c}_{A}$ be the composition of $c$ with the map
$H_{0}(\kappa)\to H_{0}(A)$.

By [20, Exercise 6.5.2 (iii)] any automorphism $\varphi:G^{0}\to G^{0}$ lifts
uniquely to an automorphism
$\varphi_{\operatorname{sc}}:(G^{0})_{\operatorname{sc}}\to(G^{0})_{\operatorname{sc}}$
such that
$\pi_{\operatorname{sc}}\circ\varphi_{\operatorname{sc}}=\varphi\circ\pi_{\operatorname{sc}}$
and $\varphi_{\operatorname{sc}}$ preserves $K_{0}$, so $\varphi$ lifts
uniquely to an automorphism $L(\varphi):H_{0}\to H_{0}$ such that $\pi\circ
L(\varphi)=\varphi\circ\pi$ and we obtain an injective group homomorphism
$L:\operatorname{Aut}(G^{0})\hookrightarrow\operatorname{Aut}(H_{0})$.

Let $\tilde{\omega}:=L\circ\omega$, i.e. for all $\delta\in\Delta$, we have
$\omega(\delta)\circ\pi=\pi\circ\tilde{\omega}(\delta)$. We claim that the
pair $(\tilde{\omega},\tilde{c})$ satisfies the identities (C1), (C2), (C3)
and (C4) above. Clearly the map $\tilde{c}$ satisfies (C4). Via the
isomorphism $H^{0}(\kappa)\cong G^{0}(\kappa)$ the maps $\omega_{\kappa}$ and
$\tilde{\omega}_{\kappa}$ can be identified, so (C3) holds for
$(\tilde{\omega},\tilde{c})$. Property (C2) holds, as $L$ is a homomorphism.
By applying $L$ to the identity (C1) for $G^{0}$, we get

(12)
$\tilde{\omega}(\delta_{1})\circ\tilde{\omega}(\delta_{2})=L(\operatorname{Int}_{G^{0}}(c(\delta_{1},\delta_{2})))\circ\tilde{\omega}(\delta_{1}\delta_{2})\quad\text{
in }\operatorname{Aut}(H_{0}).$

Since under the isomorphism $H_{0}(\kappa)\xrightarrow{\sim}G^{0}(\kappa)$ the
element $\tilde{c}(\delta_{1},\delta_{2})$ is mapped to
$c(\delta_{1},\delta_{2})$, we have
$\pi\circ\operatorname{Int}_{H_{0}}(\tilde{c}(\delta_{1},\delta_{2}))=\operatorname{Int}_{G^{0}}(c(\delta_{1},\delta_{2}))\circ\pi$.
Thus
$L(\operatorname{Int}_{G^{0}}(c(\delta_{1},\delta_{2})))=\operatorname{Int}_{H_{0}}(\tilde{c}(\delta_{1},\delta_{2}))$
by definition of $L$ and we deduce from (12) that (C1) holds for $H_{0}$,
which finishes the proof of the claim. We deduce that for every
$\kappa$-algebra $A$ the pair $(\tilde{\omega}_{A},\tilde{c}_{A})$ is a
generalised $2$-cocycle of $\Delta$ with values in $H_{0}(A)$, which by Lemma
2.29 defines a group law $\ast$ on $H_{0}(A)\times\Delta$, natural in $A$.

The functor $A\mapsto(H_{0}(A)\times\Delta,\ast)$ defines a presheaf of
groups, which we denote by $H^{\mathrm{pre}}$. The map $H^{\mathrm{pre}}(A)\to
G(A)$, $(h,\delta)\mapsto\pi(h)s(\delta)$ is a group homomorphism natural in
$A$, as $\pi_{A}\circ\tilde{c}_{A}=c_{A}$ and for any $\delta\in\Delta$, we
have $\omega(\delta)\circ\pi=\pi\circ\tilde{\omega}(\delta)$. We let $H$ be
the Zariski sheafification of $H^{\mathrm{pre}}$. Since $\underline{\Delta}$
is the sheafification of the constant presheaf $\Delta$,
$H=H_{0}\times\underline{\Delta}$ and $H^{\mathrm{pre}}\rightarrow G$ induces
a map of $\kappa$-group schemes $H\rightarrow G$. Properties (1) and (2)
claimed in the Lemma hold by construction as $H^{0}=H_{0}$ and (3) holds as
$\pi_{1}(H^{0})$ is isomorphic to $K_{0}$. ∎

##  3\. Deformation rings

Let $\Gamma$ be a profinite group such that
$H^{1}(\Gamma^{\prime},\mathbb{F}_{p})$ is finite for all open subgroups
$\Gamma^{\prime}$ of $\Gamma$. This $p$-finiteness condition introduced by
Mazur holds, when $\Gamma=\Gamma_{F}$.

Let $\rho:\Gamma\rightarrow G(\kappa)$ be a continuous representation, where
$G$ is a smooth affine $\mathcal{O}$-group scheme and $\kappa$ is either a
finite extension of $k$, a finite extension of $L$ or a local field of
characteristic $p$ containing $k$ equipped with natural topology. We will
refer to such fields $\kappa$ as _a finite or a local $\mathcal{O}$-field_. In
this section we will study the deformation theory of $\rho$. We first define a
ring of coefficients $\Lambda$ over which the deformation problem is defined
following [8, Section 3.5].

  1. (1)

If $\kappa$ is a finite field then pick an unramified extension $L^{\prime}$
of $L$ with residue field $\kappa$ and let $\Lambda:=\mathcal{O}_{L^{\prime}}$
denote the ring of integers in $L^{\prime}$.

  2. (2)

If $\kappa$ is a finite extension of $L$ then let $\Lambda:=\kappa$, let
$\Lambda^{0}$ be the ring of integers in $\Lambda$ and let $t=\varpi$.

  3. (3)

If $\kappa$ is a local field of characteristic $p$ then let
$\mathcal{O}_{\kappa}$ be the ring of integers in $\kappa$ and let
$k^{\prime}$ be its residue field. Since $\mathrm{char}(\kappa)=p$ by choosing
a uniformizer we obtain an isomorphism $\mathcal{O}_{\kappa}\cong
k^{\prime}\llbracket t\rrbracket$. Let $L^{\prime}$ be an unramified extension
of $L$ with residue field $k^{\prime}$, let
$\Lambda^{0}:=\mathcal{O}_{L^{\prime}}\llbracket t\rrbracket$ and let
$\Lambda$ be the $p$-adic completion of $\Lambda^{0}[1/t]$. Then $\Lambda$ is
a complete DVR with uniformiser $\varpi$ and residue field $\kappa$. We equip
$\Lambda^{0}$ with its $(\varpi,t)$-adic topology, this induces a topology on
$\Lambda^{0}[1/t]$ and $\Lambda^{0}[1/t]/p^{n}\Lambda^{0}[1/t]$ for all $n\geq
1$. We equip $\Lambda=\varprojlim_{n}\Lambda^{0}[1/t]/p^{n}\Lambda^{0}[1/t]$
with the projective limit topology.

Let $\mathfrak{A}_{\Lambda}$ be the category of local artinian
$\Lambda$-algebras with residue field $\kappa$. If
$(A,\mathfrak{m}_{A})\in\mathfrak{A}_{\Lambda}$ then we define the topology on
$A$ in the cases (1), (2) and (3) above as follows:

  1. (1)

$A$ is a finite $\mathcal{O}/\varpi^{n}$-module for some $n\geq 1$ with the
discrete topology on $A$;

  2. (2)

$A$ is a finite dimensional $L$-vector space with the $p$-adic topology on
$A$;

  3. (3)

$A$ is a $\Lambda^{0}[1/t]/\varpi^{n}\Lambda^{0}[1/t]$-module of finite length
for some $n\geq 1$ and we put the induced topology on $A$.

Since $G$ is smooth the map $G(A)\rightarrow G(\kappa)$ is surjective for all
$A\in\mathfrak{A}_{\Lambda}$. Let
$D^{\square}_{\rho,G}:\mathfrak{A}_{\Lambda}\rightarrow\operatorname{Set}$ be
the functor such that $D^{\square}_{\rho,G}(A)$ is the set of continuous group
homomorphisms $\rho_{A}:\Gamma\rightarrow G(A)$ satisfying
$\rho_{A}\pmod{\mathfrak{m}_{A}}=\rho$.

######  Lemma 3.1.

Let $V$ be a continuous representation of $\Gamma$ on a finite dimensional
$\kappa$-vector space $V$. Then

(13)
$\dim_{\kappa}Z^{1}(\Gamma,V)=h^{1}(\Gamma,V)+\dim_{\kappa}V-h^{0}(\Gamma,V),$

where $h^{i}(\Gamma,V):=\dim_{\kappa}H^{i}(\Gamma,V)$. Moreover, if
$\Gamma=\Gamma_{F}$ then

(14)
$\begin{split}\dim_{\kappa}Z^{1}(\Gamma_{F},V)&=\dim_{\kappa}V([F:\mathbb{Q}_{p}]+1)+h^{0}(\Gamma_{F},V^{*}(1))\\\
&=\dim_{\kappa}V([F:\mathbb{Q}_{p}]+1)+h^{2}(\Gamma_{F},V).\end{split}$

###### Proof.

Mapping $v\in V$ to a coboundary $\gamma\mapsto(\gamma-1)v$ induces an exact
sequence:

$0\rightarrow V^{\Gamma}\rightarrow V\rightarrow Z^{1}(\Gamma,V)\rightarrow
H^{1}(\Gamma,V)\rightarrow 0,$

which implies (13). The second part follows from local Euler–Poincaré
characteristic formula and local Tate duality, see for example [9, Theorem
3.4.1]. ∎

Let $\kappa[\varepsilon]$ be the ring of dual numbers over $\kappa$. The Lie
algebra $\operatorname{Lie}G_{\kappa}$ of $G_{\kappa}$ is a finite dimensional
$\kappa$-vector space, which sits in the exact sequence of groups

$1\rightarrow 1+\varepsilon\operatorname{Lie}G_{\kappa}\rightarrow
G(\kappa[\varepsilon])\rightarrow G(\kappa)\rightarrow 1.$

The action of $G(\kappa)$ on $G(\kappa[\varepsilon])$ by conjugation induces a
$\kappa$-linear action of $G(\kappa)$ on $\operatorname{Lie}G_{\kappa}$. We
let $\operatorname{ad}\rho$ be the representation of $\Gamma$ on
$\operatorname{Lie}G_{\kappa}$ defined by

$\operatorname{ad}(\rho)(\gamma)(X):=\rho(\gamma)X\rho(\gamma)^{-1},\quad\forall\gamma\in\Gamma,\quad\forall
X\in\operatorname{Lie}G_{\kappa}.$

Then $\operatorname{ad}\rho$ is a continuous representation of $\Gamma$ for
the topology on $\operatorname{Lie}G_{\kappa}$ induced by the topology on
$\kappa$.

######  Lemma 3.2.

Let $\rho:\Gamma\to G(\kappa)$ be a continuous representation. Then
$D^{\square}_{\rho,G}$ is pro-representable by a complete local noetherian
$\Lambda$-algebra $R^{\square}_{\rho,G}$ with residue field $\kappa$.

###### Proof.

If $A,B,C\in\mathfrak{A}_{\Lambda}$, then a homomorphism $\Gamma\to
G(A\times_{C}B)$ is continuous if and only if its projections to $G(A)$ and
$G(B)$ are continuous. It follows that the functor
$\mathfrak{A}_{\Lambda}\to\operatorname{Set},~{}A\mapsto\operatorname{Hom}^{\mathrm{cont}}_{\mathrm{Group}}(\Gamma,G(A))$
preserves finite limits. The deformation functor $D^{\square}_{\rho,G}$ is
defined as a pullback of the latter, so $D^{\square}_{\rho,G}$ also preserves
finite limits. By Grothendieck’s criterion, $D^{\square}_{\rho,G}$ is pro-
representable by a projective limit of objects in $\mathfrak{A}_{\Lambda}$.

The map $Z^{1}(\Gamma,\operatorname{ad}\rho)\rightarrow
D^{\square}_{\rho,G}(\kappa[\varepsilon])$,
$\Phi\mapsto[\gamma\mapsto\rho(\gamma)(1+\varepsilon\Phi(\gamma))]$ is an
isomorphism of $\kappa$-vector spaces. Since $\Gamma$ satisfies Mazur’s
$p$-finiteness condition [9, Theorem 3.4.1] implies that
$h^{1}(\Gamma,\operatorname{ad}\rho)$ is finite. Lemma 3.1 implies that
$Z^{1}(\Gamma,\operatorname{ad}\rho)$ is finite dimensional, thus
$R^{\square}_{\rho,G}$ is noetherian. ∎

######  Lemma 3.3.

Let $f:G\to H$ be a continuous surjective open homomorphism between locally
profinite groups. Then there is a continuous map $s:H\to G$ with $f\circ
s=\operatorname{id}_{H}$.

###### Proof.

If $G$ and $H$ are profinite, this is [50, Section I.1, Proposition 1]. In
general, let $N$ be an open profinite subgroup of $G$. Then $N^{\prime}:=f(N)$
is an open profinite subgroup of $H$ and we find a continuous map
$s:N^{\prime}\to N$ with $f|_{N}\circ s=\operatorname{id}_{N^{\prime}}$. Let
$R\subseteq H$ be a set of $N^{\prime}$-coset representatives so that
$H=\bigcup_{r\in R}rN^{\prime}$ and choose $g_{r}\in G$ with $f(g_{r})=r$ for
each $r\in R$. We can extend $s$ to a continuous map $\tilde{s}:H\to G$ by
defining $\tilde{s}(rn^{\prime}):=g_{r}s(n^{\prime})$ for all $n^{\prime}\in
N^{\prime}$ and see, that $f\circ\tilde{s}=\operatorname{id}_{H}$, as desired.
∎

######  Lemma 3.4.

Let $\kappa$ be either a finite or a local $\mathcal{O}$-field, let $\Lambda$
be a coefficient ring for $\kappa$ and let
$(A,\mathfrak{m}_{A})\in\mathfrak{A}_{\Lambda}$. Then there is a module-finite
$\Lambda^{0}$-subalgebra $R$ of $A$ with $R\otimes_{\Lambda^{0}}\Lambda=A$. In
particular, $R$ is a profinite open subring of $A$.

###### Proof.

Let $\pi_{A}:A\to\kappa$ be the projection to the residue field. Pick
$\Lambda$-module generators $x_{1},\dots,x_{n}$ of $A$ and we may assume by
rescaling with $t$, that $\pi_{A}(x_{i})\in\mathcal{O}_{\kappa}$ for all $i$.
Since $\Lambda^{0}$ surjects onto $\mathcal{O}_{\kappa}$, we can pick
$y_{i}\in\Lambda^{0}$ with $\pi_{A}(y_{i})=\pi_{A}(x_{i})$ for all $i$. Since
$\mathfrak{m}_{A}$ is nilpotent, the finitely many nonzero products of the
elements $x_{i}-y_{i}\in\mathfrak{m}_{A}$ generate a multiplicatively closed
finitely generated $\Lambda^{0}$-submodule $I$ of $\mathfrak{m}_{A}$. In
particular $R:=\Lambda^{0}+I$ is a module-finite $\Lambda^{0}$-subalgebra of
$A$ with $R[1/t]=A$. Since $R$ is a finitely generated $\Lambda^{0}$-module,
$\Lambda^{0}$ is open in $\Lambda$ and $\Lambda$ is Hausdorff, $R$ is
profinite and open. ∎

######  Lemma 3.5.

Let $\kappa$ be either a finite or a local $\mathcal{O}$-field and let
$\Lambda$ be a coefficient ring for $\kappa$. Let $G$ be a smooth affine
$\mathcal{O}$-group.

  1. (1)

For every $A\in\mathfrak{A}_{\Lambda}$, the topological group $G(A)$ is
locally profinite.

  2. (2)

Let $f:A\to B$ be a surjective morphism in $\mathfrak{A}_{\Lambda}$. Then the
map $\varphi:G(A)\to G(B)$ induced by $f$ is surjective and there is a
continuous map $s:G(B)\to G(A)$ with $\varphi\circ
s=\operatorname{id}_{G(B)}$.

###### Proof.

When $\kappa$ is finite, both assertions are clear, so we assume that $\kappa$
is local. Let $R$ be a profinite open subalgebra of $A$ with
$R\otimes_{\Lambda^{0}}\Lambda=A$ as constructed in 3.4.

Part (1). We claim, that $G(R)\subseteq G(A)$ is a profinite open subgroup.
Choose a closed immersion $G\hookrightarrow\mathbb{A}^{n}$ over $\mathcal{O}$.
By definition $G(A)$ carries the subspace topology of $\mathbb{A}^{n}(A)$.
Since $\mathbb{A}^{n}(R)$ is open in $\mathbb{A}^{n}(A)$, $G(R)$ is open in
$G(A)$. Since $R$ is Hausdorff, $G(R)$ is closed in $\mathbb{A}^{n}(R)$ and
since $\mathbb{A}^{n}(R)$ is profinite, $G(R)$ is profinite.

Part (2). Surjectivity of $\varphi$ follows from smoothness of $G$. Then
$R^{\prime}:=f(R)$ is a compact subalgebra of $B$ with
$R^{\prime}\otimes_{\Lambda^{0}}\Lambda=B$. Since the topology of $B$ is by
definition induced by an arbitrary open $\Lambda^{0}$-lattice, we see that
$R^{\prime}$ is open in $B$. Since $G$ is smooth, we have a surjection
$G(R)\to G(R^{\prime})$ of open profinite subgroups of $G(A)$ and $G(B)$
respectively. Since any surjective continuous homomorphism between profinite
groups is open111Indeed, any continuous surjective homomorphism from a compact
group onto a Hausdorff group is open by Remark 1 after Proposition 24 in [12,
Ch. III, §2, no. 8]., we see that $\varphi$ is open. The assertion follows
from 3.3. ∎

######  Proposition 3.6.

Let $\varphi:G\to H$ be a morphism of smooth affine $\mathcal{O}$-group
schemes such that the induced morphism on neutral components $G^{0}\rightarrow
H^{0}$ is smooth and surjective. Let $\rho:\Gamma\to G(\kappa)$ be a
continuous representation and let
$\operatorname{ad}^{0,\varphi}=\ker(\operatorname{ad}\rho\rightarrow\operatorname{ad}\varphi\circ\rho)$.
Then there is a natural isomorphism of
$R^{\square}_{\varphi\circ\rho,H}$-algebras

(15) $R^{\square}_{\varphi\circ\rho,H}\llbracket
x_{1},\ldots,x_{r}\rrbracket/(f_{1},\ldots,f_{t})\cong R^{\square}_{\rho,G}$

where $r=\dim_{\kappa}Z^{1}(\Gamma,\operatorname{ad}^{0,\varphi}\rho)$,
$t=h^{2}(\Gamma,\operatorname{ad}^{0,\varphi}\rho)$. Moreover, if
$\Gamma=\Gamma_{F}$ then

(16) $r-t=(\dim G_{\kappa}-\dim H_{\kappa})\cdot([F:\mathbb{Q}_{p}]+1).$

###### Proof.

If $\kappa$ is a finite field then the argument of [8, Proposition 4.3], where
$G=\operatorname{GL}_{d}$, $H=\operatorname{GL}_{1}$, $\varphi=\det$, goes
through verbatim. If $\kappa$ is a local field then one needs to additionally
verify that obstruction to lifting can be realised by a continuous
$2$-cocycle, as noted in [8, Remark 4.4]. The solution to this problem is
explained in detail in [18, Lecture 6], when $\kappa$ is a local field of
characteristic zero and $H$ is trivial. We sketch how these arguments fit
together.

The exact sequence
$0\rightarrow\operatorname{ad}^{0,\varphi}\rho\rightarrow\operatorname{ad}\rho\rightarrow\operatorname{ad}\varphi\circ\rho\rightarrow
0$ of Galois representations induces an exact sequence of abelian groups:

$0\rightarrow Z^{1}(\Gamma,\operatorname{ad}^{0,\varphi}\rho)\rightarrow
Z^{1}(\Gamma,\operatorname{ad}\rho)\rightarrow
Z^{1}(\Gamma,\operatorname{ad}\varphi\circ\rho)$

and hence $r=\dim_{\kappa}\ker(Z^{1}(\operatorname{ad}\varphi))$. The map

$Z^{1}(\operatorname{ad}\varphi):Z^{1}(\Gamma,\operatorname{ad}\rho)\to
Z^{1}(\Gamma,\operatorname{ad}\varphi\circ\rho)$

is the induced map on Zariski tangent spaces of the map of deformation rings
$R_{\varphi\circ\rho}^{\square}\to R^{\square}_{\rho}$, and thus lifts to a
surjection

$\widetilde{\phi}:\tilde{R}:=R_{\varphi\circ\rho}^{\square}\llbracket
x_{1},\ldots,x_{r}\rrbracket\twoheadrightarrow R^{\square}_{\rho}.$

We set $J:=\ker\tilde{\phi}$. By Nakayama’s lemma, we need to show, that
$\dim_{\kappa}J/\tilde{\mathfrak{m}}J\leq t$. The module
$J/\tilde{\mathfrak{m}}J$ appears as the kernel of the sequence

(17) $\displaystyle 0\to
J/\tilde{\mathfrak{m}}J\to\tilde{R}/\tilde{\mathfrak{m}}J\to\tilde{R}/J\cong
R_{\rho,G}^{\square}\to 0$

In view of the above sequence, it is enough construct a homomorphism

(18)
$\alpha:\operatorname{Hom}_{\kappa}(J/\tilde{\mathfrak{m}}J,\kappa)\to\ker(H^{2}(\operatorname{ad}\varphi):H^{2}(\Gamma,\operatorname{ad}\rho)\to
H^{2}(\Gamma,\operatorname{ad}\varphi\circ\rho))$

and show, that $\ker(\alpha)$ injects into
$\operatorname{coker}(H^{1}(\operatorname{ad}\varphi))$. This will imply the
existence of the presentation in the statement of the Proposition, since then

(19)
$\displaystyle\dim_{\kappa}J/\tilde{\mathfrak{m}}J\leq\dim_{\kappa}\ker(H^{2}(\operatorname{ad}\varphi))+\dim_{\kappa}\operatorname{coker}(H^{1}(\operatorname{ad}\varphi))=h^{2}(\Gamma,\operatorname{ad}^{0,\varphi}\rho)$

where the last equality comes from the long exact cohomology sequence that
arises from
$0\to\operatorname{ad}^{0,\varphi}\rho\to\operatorname{ad}\rho\to\operatorname{ad}\varphi\circ\rho\to
0$.

For all $i>0$, we define $\tilde{R}_{i}:=\tilde{R}/\tilde{\mathfrak{m}}^{i}$,
$\tilde{\mathfrak{m}}_{i}:=\tilde{\mathfrak{m}}/\tilde{\mathfrak{m}}^{i}$,
$J_{i}:=(J+\tilde{\mathfrak{m}}^{i})/\tilde{\mathfrak{m}}^{i}$ and
$R_{\rho,G,i}^{\square}:=R_{\rho,G}^{\square}/\tilde{\mathfrak{m}}^{i}R_{\rho,G}^{\square}$.
The Artin–Rees lemma implies that
$J\cap(\tilde{\mathfrak{m}}J+\tilde{\mathfrak{m}}^{i})=\tilde{\mathfrak{m}}J$
for sufficiently large $i\gg 0$, thus the map $J/\tilde{\mathfrak{m}}J\to
J_{i}/\tilde{\mathfrak{m}}_{i}J_{i}$ is an isomorphism. We will fix such $i$
for the rest of the proof. It is therefore enough to work with objects of
$\mathfrak{A}_{\Lambda}$ whose maximal ideal has vanishing $i$-th power, we
will denote this category by $\mathfrak{A}_{\Lambda,i}$. The inclusion functor
$\mathfrak{A}_{\Lambda,i}\to\mathfrak{A}_{\Lambda}$ has a left adjoint given
by modding out the $i$-th power of the maximal ideal. The ring
$R_{\rho,G,i}^{\square}$ represents the restriction of $D^{\square}_{\rho,G}$
to $\mathfrak{A}_{\Lambda,i}$ and we will denote the universal lift by
$\rho^{\square}_{i}:\Gamma\to G(R_{\rho,G,i}^{\square})$. We obtain an exact
sequence

(20) $\displaystyle 0\to
J_{i}/\tilde{\mathfrak{m}}_{i}J_{i}\to\tilde{R}_{i}/\tilde{\mathfrak{m}}_{i}J_{i}\to\tilde{R}_{i}/J_{i}\cong
R_{\rho,G,i}^{\square}\to 0$

where $\tilde{R}_{i}/\tilde{\mathfrak{m}}_{i}J_{i}$ and
$R_{\rho,G,i}^{\square}$ are objects of $\mathfrak{A}_{\Lambda,i}$. If
$u\in\operatorname{Hom}_{\kappa}(J_{i}/\tilde{\mathfrak{m}}_{i}J_{i},\kappa)$
then by modding out $\ker(u)$ in (20) we obtain an exact sequence

(21) $\displaystyle 0\to I_{u}\to
R_{u}\xrightarrow{\phi_{u}}R^{\square}_{\rho,G,i}\to 0,$

The surjection of locally profinite groups $G(R_{u})\twoheadrightarrow
G(R^{\square}_{\rho,G,i})$ has a continuous section by 3.5. Once we have this
the proof is identical to the proof of [8, Proposition 4.3]. Moreover, if
$\Gamma=\Gamma_{F}$ then (16) follows from Lemma 3.1. ∎

######  Remark 3.7.

We will apply the above proposition in the setting of 2.10.

######  Corollary 3.8.

There is an isomorphism of local $\Lambda$-algebras

(22) $R^{\square}_{\rho,G}\cong\Lambda\llbracket
x_{1},\ldots,x_{r}\rrbracket/(f_{1},\ldots,f_{s})$

with $r=\dim_{\kappa}Z^{1}(\Gamma,\operatorname{ad}\rho)$ and
$s=\dim_{\kappa}H^{2}(\Gamma,\operatorname{ad}\rho)$. Further, if
$\Gamma=\Gamma_{F}$ then

$r-s=\dim G_{\kappa}([F:\mathbb{Q}_{p}]+1).$

###### Proof.

The assertion follows from Lemma 3.2 and Proposition 3.6 applied with $H$
equal to the trivial group. ∎

3.6 will be often used together with the following lemma.

######  Lemma 3.9.

Let $A$ be a complete local noetherian $\mathcal{O}$-algebra with residue
field $k$ and let $B=A\llbracket
x_{1},\ldots,x_{r}\rrbracket/(f_{1},\ldots,f_{t})$. If $A$ is complete
intersection and $\dim B\leq\dim A+r-t$ then $B$ is complete intersection and
$\dim B=\dim A+r-t$. Moreover, if $\dim k\otimes_{A}B\leq r-t$ then $B$ is a
flat $A$-algebra and $\dim k\otimes_{A}B=r-t$.

###### Proof.

Since $A$ is complete intersection we may write $A=S/(g_{1},\ldots,g_{s})$,
where $S$ is a complete local regular ring and $g_{1},\ldots,g_{s}$ is a
regular sequence in $S$. We choose $\tilde{f}_{1},\ldots,\tilde{f}_{t}\in
S\llbracket x_{1},\ldots,x_{r}\rrbracket$ that map to $f_{1},\ldots,f_{t}$.
Since when we quotient out by a relation the dimension either goes down by one
or stays then same, the assumption on the dimension of $B$ implies that $\dim
B=\dim A+r-t$ and $g_{1},\ldots,g_{s},\tilde{f}_{1},\ldots,\tilde{f}_{t}$ can
be extended to a system of parameters of $S\llbracket
x_{1},\ldots,x_{r}\rrbracket$, and hence is a regular sequence in the regular
ring $S\llbracket x_{1},\ldots,x_{r}\rrbracket$. Thus $B$ is complete
intersection and also $C:=S\llbracket
x_{1},\ldots,x_{r}\rrbracket/(\tilde{f}_{1},\ldots,\tilde{f}_{t})$ is complete
intersection. Since $k\otimes_{S}B=k\otimes_{A}B$, it follows from miracle
flatness [41, Theorem 23.1] that $C$ is flat over $S$, and hence
$B=C/(g_{1},\ldots,g_{s})$ is flat over $A=S/(g_{1},\ldots,g_{s})$. ∎

######  Lemma 3.10.

Let $\rho:\Gamma\rightarrow G(\kappa)$ be a continuous representation, where
$\kappa$ is either local or finite $\mathcal{O}$-field. Let $\kappa^{\prime}$
be a finite extension of $\kappa$ and let $\rho^{\prime}:\Gamma\rightarrow
G(\kappa^{\prime})$ be the representation obtained by composing $\rho$ with
the inclusion $G(\kappa)\subseteq G(\kappa^{\prime})$. Then there is a natural
isomorphism of local $\Lambda^{\prime}$-algebras:

$\Lambda^{\prime}\otimes_{\Lambda}R^{\square}_{G,\rho}\cong
R^{\square}_{G,\rho^{\prime}},$

where $\Lambda$ and $\Lambda^{\prime}$ are coefficient rings for $\kappa$ and
$\kappa^{\prime}$, respectively.

###### Proof.

The argument explained on page 457 of [59] also applies in our setting. ∎

######  Lemma 3.11.

Let $\kappa$ be a local $\mathcal{O}$-field with ring of integers
$\mathcal{O}_{\kappa}$ and residue field $k$. Let $\rho:\Gamma\rightarrow
G(\kappa)$ be a continuous representation, such that its image is contained in
$G(\mathcal{O}_{\kappa})$, and let $\overline{\rho}:\Gamma\rightarrow G(k)$ be
the composition of $\rho$ with the reduction map
$G(\mathcal{O}_{\kappa})\rightarrow G(k)$. Let
$\varphi:R^{\square}_{G,\overline{\rho}}\rightarrow\mathcal{O}_{\kappa}\hookrightarrow\kappa$
be the map induced by considering $\rho$ as a deformation of $\overline{\rho}$
to $\mathcal{O}_{\kappa}$. Let $\mathfrak{q}$ be the kernel of the natural map

$\Lambda\otimes_{\mathcal{O}}R^{\square}_{G,\overline{\rho}}\rightarrow\kappa,\quad\lambda\otimes
a\mapsto\bar{\lambda}\varphi(a),$

where $\Lambda$ is the coefficient ring for $\kappa$. Then there is a natural
isomorphism of local $\Lambda$-algebras between $R^{\square}_{G,\rho}$ and the
completion $\Lambda\otimes_{\mathcal{O}}R^{\square}_{G,\overline{\rho}}$ with
respect to $\mathfrak{q}$.

###### Proof.

The proof is the same as the proof of [9, Theorem 3.3.1] for
$\operatorname{GL}_{d}$, which in turn is based on [35, Proposition 9.5],
where the case $G=\operatorname{GL}_{d}$ and $\mathrm{char}(\kappa)=0$ is
treated. ∎

###  3.1. Central extensions

In this section, we assume that $G$ is generalised reductive and let $H=G/Z$,
where $Z$ is a flat closed subgroup scheme of $Z(G^{0})$, which is normal in
$G$. Part (i) of [20, Exercise 5.5.9] implies that $H$ is also generalised
reductive. We let $\varphi:G\rightarrow H$ be the quotient map. We fix a
continuous representation $\rho:\Gamma\rightarrow G(\kappa)$ and drop it from
the notation. We will write $R^{\square}_{G}$ for $R^{\square}_{\rho,G}$,
$R^{\square}_{H}$ for $R^{\square}_{\varphi\circ\rho,H}$, etc.

######  Corollary 3.12.

If $Z$ is finite étale then $R^{\square}_{H}=R^{\square}_{G}$.

###### Proof.

Since $Z$ is finite étale its Lie algebra is zero. We thus get
$\operatorname{ad}^{0,\varphi}\rho=0$ and the assertion follows from
Proposition 3.6. ∎

######  Proposition 3.13.

If $Z$ is a torus such that $Z\cap G^{\prime}$ is étale then there is a
natural isomorphism of local $\Lambda$-algebras
$R^{\square}_{G/G^{\prime}}\operatorname{\widehat{\otimes}}_{R^{\square}_{H/H^{\prime}}}R^{\square}_{H}\xrightarrow{\sim}R^{\square}_{G}$.

###### Proof.

Let $D:\mathfrak{A}_{\Lambda}\rightarrow\operatorname{Set}$ be the functor
such that $D(A)$ is the set of pairs $(\psi,\xi)\in
D^{\square}_{G/G^{\prime}}(A)\times D^{\square}_{H}(A)$ such that
$\pi\circ\psi=\theta\circ\xi$, where $\theta:H\rightarrow H/H^{\prime}$ and
$\pi:G/G^{\prime}\rightarrow H/H^{\prime}$ are the quotient maps. The
Proposition amounts to the claim that the map

(23) $D^{\square}_{G}(A)\rightarrow
D(A),\quad\rho_{A}\mapsto(\varphi\circ\rho_{A},q\circ\rho_{A})$

is bijective. Since $Z\cap G^{\prime}$ is étale by assumption, using 3.12 we
may assume that $Z\cap G^{\prime}$ is trivial. The map $G^{0}\to H^{0}$ is
surjective with kernel $Z$ and the map $G^{\prime}\to H^{\prime}$ is
surjective with kernel $Z\cap G^{\prime}$. The map of group schemes

(24) $G\rightarrow G/G^{\prime}\times_{H/H^{\prime}}H$

has kernel $Z\cap G^{\prime}$, so it is injective. If $([g],h)\in
G(A)/G^{\prime}(A)\times_{H(A)/H^{\prime}(A)}H(A)$ then, since $G\rightarrow
H$ is surjective, there exists a finite flat extension $A\hookrightarrow B$
and $\tilde{g}\in G(B)$, which maps to $h\in H(A)\subseteq H(B)$. Then
$\tilde{g}g^{-1}\in H^{\prime}(B)=G^{\prime}(B)$, so $[\tilde{g}]=[g]$ in
$G(B)/G^{\prime}(B)$. By sheafification this implies, that (24) is bijective.

This implies that (23) is bijective, since then given $(\psi,\xi)\in D(A)$ for
every $\gamma\in\Gamma$ there is a unique $\rho^{\prime}(\gamma)\in G(A)$ such
that $\varphi(\rho^{\prime}(\gamma))=\psi(\gamma)$ and
$\pi(\rho^{\prime}(\gamma))=\xi(\gamma)$. Uniqueness implies that
$\rho^{\prime}\in D^{\square}_{G}(A)$, which maps to $(\psi,\xi)$. ∎

######  Corollary 3.14.

If $\Gamma=\Gamma_{F}$, $\kappa$ is finite and we are in the setting of 3.13
then $\dim R^{\square}_{G}/\varpi=\dim
R^{\square}_{H}/\varpi+([F:\mathbb{Q}_{p}]+1)(\dim G_{k}-\dim H_{k})$.

###### Proof.

Let $A^{\prime}=(R^{\square}_{H/H^{\prime}}/\varpi)^{\mathrm{red}}$,
$B^{\prime}=(R^{\square}_{G/G^{\prime}}/\varpi)^{\mathrm{red}}$,
$A=(R^{\square}_{H}/\varpi)^{\mathrm{red}}$, where the superscript
$\mathrm{red}$ indicates reduced rings. We have shown in [43, Corollary 9.9]
that $A^{\prime}\rightarrow B^{\prime}$ is flat and the dimension $d$ of the
fibre $\kappa\otimes_{A^{\prime}}B^{\prime}$ is equal to

$(\dim(G/G^{\prime})_{k}-\dim(H/H^{\prime})_{k})([F:\mathbb{Q}_{p}]+1)=(\dim
G_{k}-\dim H_{k})([F:\mathbb{Q}_{p}]+1).$

Hence, $A\rightarrow B^{\prime}\operatorname{\widehat{\otimes}}_{A^{\prime}}A$
is flat and the fibre has dimension $d$. Thus $\dim A+d=\dim
B^{\prime}\operatorname{\widehat{\otimes}}_{A^{\prime}}A$ by [54, Tag 00ON].
3.13 implies that
$(B^{\prime}\operatorname{\widehat{\otimes}}_{A^{\prime}}A)^{\mathrm{red}}\cong(R^{\square}_{G}/\varpi)^{\mathrm{red}}$.
Hence, $\dim R^{\square}_{G}/\varpi=\dim A+d=\dim R^{\square}_{H}/\varpi+d$. ∎

For the rest of the section we let $Z$ be a finite connected diagonalisable
subgroup scheme of $Z(G^{0})$. If $\mathrm{char}(\kappa)=0$ then $Z$ is
trivial so we assume that $\mathrm{char}(\kappa)=p$. In this case there is a
finite abelian $p$-group $\mathrm{m}$ such that
$Z(A)=\operatorname{Hom}(\mathrm{m},A^{\times})$ for
$A\in\mathcal{O}\text{-}\mathrm{alg}$.

######  Lemma 3.15.

$R^{\square}_{H}\rightarrow R^{\square}_{G}$ is finite.

###### Proof.

Let $A:=\kappa\otimes_{R^{\square}_{H}}R^{\square}_{G}$ and let $\mathfrak{p}$
be a prime ideal of $A$. Since $A/\mathfrak{p}$ is a domain of characteristic
$p$, we have $Z(A/\mathfrak{p})=0$ and thus $G(A/\mathfrak{p})\subseteq
H(A/\mathfrak{p})$. Let $\rho_{\mathfrak{p}}$ be the specialisation of the
universal deformation $\Gamma\rightarrow G(R^{\square}_{G})$ at
$\mathfrak{p}$. Since $\varphi\circ\rho_{\mathfrak{p}}=\varphi\circ\rho$ by
definition of $A$, we get that $\rho_{\mathfrak{p}}(\Gamma)$ is contained in
$G(\kappa)$. Hence, $\mathfrak{p}$ is the maximal ideal of $A$. We conclude
that $A$ is zero dimensional and, since $R^{\square}_{G}$ is noetherian, $A$
is a finite dimensional $\kappa$-vector space. The claim follows from
Nakayama’s Lemma. ∎

######  Proposition 3.16.

$\dim R^{\square}_{H}/\varpi=\dim R^{\square}_{G}/\varpi$.

###### Proof.

For any local $\kappa$-algebra $A$ we have an exact sequence of non-abelian
fppf cohomology groups

(25) $0\rightarrow Z(A)\rightarrow G(A)\rightarrow H(A)\rightarrow
H^{1}_{\mathrm{fppf}}(\operatorname{Spec}(A),Z)$

Let $e\geq 0$ be an integer such that $p^{e}\cdot\mathrm{m}=0$. Let
$A=R^{\square}_{H}/\varpi$ and let $\mathfrak{m}$ be the maximal ideal of $A$.
We choose elements $a_{1},\ldots,a_{r}\in\mathfrak{m}$ such that their images
build a basis of $\mathfrak{m}/\mathfrak{m}^{2}$ as a $\kappa$-vector space.
Let $B=A[x_{1},\ldots,x_{n}]/I$, where $I$ is the ideal generated by
$x_{i}^{p^{e}}-a_{i}$ for $1\leq i\leq r$. Then $B$ is is finite flat over $A$
and every $x\in 1+\mathfrak{m}$ is a $p^{e}$-th power in $B^{\times}$. Let
$A^{\mathrm{red}}$ and $B^{\mathrm{red}}$ be the maximal reduced quotients of
$A$ and $B$, respectively. Then $A^{\mathrm{red}}$ is a subring of
$B^{\mathrm{red}}$ and every $x\in 1+\mathfrak{m}_{A^{\mathrm{red}}}$ is a
$p^{e}$-th power in $B^{\mathrm{red}}$. Let $\rho^{\prime}:\Gamma\rightarrow
H(A^{\mathrm{red}})$ be the specialisation of the universal deformation
$\Gamma\rightarrow H(R^{\square}_{H})$ along $R^{\square}_{H}\rightarrow
A^{\mathrm{red}}$.

Since $Z$ is a finite product of $\mu_{p^{n}}$ with $n\leq e$ and
$B^{\mathrm{red}}$ is reduced and of characteristic $p$, we have
$Z(B^{\mathrm{red}})=1$. Since
$H^{1}_{\mathrm{fppf}}(\operatorname{Spec}(A),\mu_{p^{n}})=A^{\times}/(A^{\times})^{p^{n}}$
by [54, Tag 040Q], it follows from (25) that there exists a continuous
representation $\tilde{\rho}:\Gamma\rightarrow G(B^{\mathrm{red}})$ deforming
$\rho$ such that
$\varphi\circ\tilde{\rho}=\rho^{\prime}\otimes_{A^{\mathrm{red}}}B^{\mathrm{red}}$.
This induces a map of $A$-algebras $R^{\square}_{G}/\varpi\rightarrow
B^{\mathrm{red}}$. This map is finite as $B$ is a finite $A$-algebra. Hence
$\dim(R^{\square}_{G}/\varpi)\geq\dim B^{\mathrm{red}}=\dim B=\dim A$, where
the last equality follows as $B$ is finite and flat over $A$. Lemma 3.15
implies that $\dim(R^{\square}_{G}/\varpi)\leq\dim A$. ∎

##  4\. Continuity

In this section let $\mathcal{O}$ be any ring, let $R\rightarrow A$ be a map
of $\mathcal{O}$-algebras, with $R$ noetherian and profinite. Let
$\rho:\Gamma\rightarrow G(A)$ be a group homomorphism, where $\Gamma$ is a
profinite group and $G$ is an affine $\mathcal{O}$-group scheme of finite
type.

In Section 4.1 we define an algebraic notion of continuity for such
representations, such that whenever $A$ is a topological $R$-algebra the
algebraic notion of continuity implies that the representation
$\rho:\Gamma\rightarrow G(A)$ is continuous for the topology on the target
induced by the topology of $A$. Our construction is based on the fact that if
$M$ is a finitely generated $R$-module then since $R$ is noetherian and
profinite there is a unique topology on $M$ making $M$ into a Hausdorff
topological $R$-module; this topology is profinite. For example, if
$G=\mathbb{G}_{a}$ then our algebraic notion of continuity is the requirement
that $\rho(\Gamma)$ is contained in a finitely generated $R$-submodule
$M\subseteq A=\mathbb{G}_{a}(A)$ and the map $\rho:\Gamma\rightarrow M$ is
continuous. We extend this example following Conrad’s exposition in [19] on
how to topologise $X(A)$, when $A$ is a topological ring.

In Section 4.2 we reformulate this notion in terms of condensed sets. In fact,
one could reprove the results of Section 4.1 entirely within the framework of
condensed sets. However, we have chosen not to do so as the arguments in
Section 4.1 are elementary and suffice for the paper.

###  4.1. Continuity via algebra

The following is the key lemma.

######  Lemma 4.1.

Let $f\in A[y_{1},\ldots,y_{n}]$ be a polynomial, which we view as a function
on $A^{n}$ and let $V(f):=\\{\underline{a}\in A^{n}:f(\underline{a})=0\\}$.
Let $M$ be a finitely generated $R$-submodule of $A^{n}$. Then $f|_{M}:M\to A$
is a continuous map with values in a finitely generated $R$-submodule of $A$.
In particular, $V(f)\cap M$ is a closed subset of $M$ for the natural topology
on $M$ as an $R$-module.

###### Proof.

We first focus on the case, when $f$ is a monomial of degree $d\geq 1$. There
exists an $A$-multilinear form $\beta:A^{\otimes_{A}d}\to A$, such that
$f(a)=\beta(a\otimes\dots\otimes a)$ for all $a\in A$. Write $\iota:M\to A$
for the inclusion map. We have a natural $R$-linear map
$\iota^{\otimes}:M^{\otimes_{R}d}\to A^{\otimes_{A}d}$ and we denote by $N$
the image of $M^{\otimes_{R}d}$ under
$\tilde{\beta}:=\beta\circ\iota^{\otimes}$ in $A$. In particular $f|_{M}$
takes values in $N$ and arises as the composition

$M\xrightarrow{\Delta}M^{d}\xrightarrow{\mathrm{can}}M^{\otimes_{R}d}\xrightarrow{\tilde{\beta}}N.$

The diagonal $\Delta$ and $\tilde{\beta}$ are $R$-linear, hence continuous. We
are left to show, that the universal $d$-multilinear form $M^{d}\to
M^{\otimes_{R}d}$ is continuous. Choose an $R$-linear surjection
$R^{s}\twoheadrightarrow M$. We have a commutative square

(26) ${(R^{s})^{d}}$${(R^{s})^{\otimes_{R}d}}$${M^{d}}$${M^{\otimes_{R}d}}$

where the top and the bottom maps are the universal $d$-multilinear forms. The
vertical maps are quotient maps. The top map is a polynomial map over $R$,
hence continuous and thus the bottom map is continuous, which finishes the
proof in this case.

Now assume, that $f$ is a general polynomial and let us write
$f=\sum_{i}a_{i}f_{i}$ for monomials $f_{i}$ and $a_{i}\in A$. As in the
previous step, we obtain multilinear maps $\beta_{i}:A^{\otimes_{A}d}\to A$
and we define $\tilde{\beta}_{i}:=\beta_{i}\circ\iota^{\otimes}$,
$N_{i}:=\operatorname{Im}(\tilde{\beta}_{i})$ and $N:=\sum_{i}a_{i}N_{i}$. The
multiplication maps $N_{i}\to N,~{}x\mapsto a_{i}x$ are $R$-linear, thus $N$
is a finitely generated $R$-module. So $f|_{M}=\sum_{i}a_{i}f_{i}|_{M}$ is a
sum of continuous maps and hence continuous, which finishes the proof.

Since $R$ is Hausdorff, so is $N$ and we get that $V(f)\cap M$ is closed. ∎

Let $X$ be an affine scheme of finite type over $\mathcal{O}$. A point $x\in
X(A)$ corresponds to an $\mathcal{O}$-algebra homomorphism $x:B\rightarrow A$.
We thus have a canonical map

$X(A)\times\Gamma(X,\mathcal{O}_{X})\rightarrow
A,\quad(x,b)\mapsto\mathrm{ev}_{b}(x):=x(b).$

This induces a canonical injection

$\iota_{X}:=\prod_{b\in B}\mathrm{ev}_{b}:X(A)\hookrightarrow
A^{B}:=\prod_{b\in B}A.$

We will denote the projection map onto the $b$-th component by
$p_{b}:A^{B}\rightarrow A$. As explained above any finitely generated
$R$-module $M$ has a unique topology, which makes it into a Hausdorff
topological $R$-module. Moreover, $M$ is profinite with respect to this
topology.

######  Lemma 4.2.

Let $S$ be a profinite set and let $f:S\rightarrow X(A)$ be a map of sets.
Then the following are equivalent:

  1. (1)

for each $b\in B$ there exists a finitely generated $R$-submodule $M_{b}$ of
$A$, such that $\iota_{X}(f(S))$ is contained in $\prod_{b\in B}M_{b}$ and the
map $\iota_{X}\circ f:S\rightarrow\prod_{b\in B}M_{b}$ is continuous for the
product topology on the target.

  2. (2)

for all $n\geq 1$ and all closed immersions
$\tau:X\hookrightarrow\mathbb{A}^{n}$ of $\mathcal{O}$-schemes there exists a
finitely generated $R$-submodule $M\subseteq\mathbb{A}^{n}(A)=A^{n}$ such that
$\tau(f(S))\subseteq M$ and $\tau\circ f:S\rightarrow M$ is continuous for a
profinite topology on the target.

  3. (3)

there exists a closed immersion $\tau:X\hookrightarrow\mathbb{A}^{n}$ of
$\mathcal{O}$-schemes and a finitely generated $R$-submodule
$M\subseteq\mathbb{A}^{n}(A)=A^{n}$ such that $\tau(f(S))\subseteq M$ and
$\tau\circ f:S\rightarrow M$ is continuous for a profinite topology on the
target.

###### Proof.

A closed immersion $\tau:X\hookrightarrow\mathbb{A}^{n}$ corresponds to a
surjective homomorphism of $\mathcal{O}$-algebras
$\mathcal{O}[x_{1},\ldots,x_{n}]\twoheadrightarrow B$. Let $b_{1},\ldots
b_{n}$ be the images of $x_{1},\ldots x_{n}$ in $B$. Then the map
$X(A)\overset{\tau}{\longrightarrow}\mathbb{A}^{n}(A)=A^{n}$ factors as
$X(A)\rightarrow
A^{B}\overset{\prod_{i=1}^{n}p_{b_{i}}}{\longrightarrow}A^{n}$. The image of
$\prod_{b}M_{b}$ under this map is $\prod_{i=1}^{n}M_{b_{i}}$, which is a
finitely generated $R$-module. Moreover, this projection is continuous. Hence,
part (1) implies part (2).

Part (2) trivially implies part (3).

Let us assume that (3) holds. Let $M_{i}\subseteq A$ be the image of
$M\subseteq\prod_{i=1}^{n}A$ under the projection onto the $i$-th component.
Then $M\subseteq\prod_{i=1}^{n}M_{i}$ and $\prod_{i=1}^{n}M_{i}$ is a finitely
generated $R$-module. Hence, we may replace $M$ with $\prod_{i=1}^{n}M_{i}$.
Let $b_{i}\in B$ be the images of $x_{i}$ as before. If $b\in B$ then there is
$g\in\mathcal{O}[x_{1},\ldots,x_{n}]$ such that $b=g(b_{1},\ldots,b_{n})$. The
image of $\prod_{i=1}^{n}M_{i}$ under the map $\varphi_{g}:A^{n}\rightarrow
A$, $(a_{1},\ldots,a_{n})\mapsto g(a_{1},\ldots,a_{n})$ is contained in a
finitely generated $R$-module $M_{b}$ by the arguments in Lemma 4.1. Since
$\tau(f(S))\subseteq\prod_{i=1}^{n}M_{i}$ we have
$\varphi_{g}(\tau(f(S)))\subseteq M_{b}$. Moreover, the induced map
$S\rightarrow M_{b}$ is continuous since polynomial maps are continuous. The
composition $\varphi_{g}\circ\tau:X(A)\rightarrow A$ is the map given by
evaluation at $b$. Hence, (3) implies (1). ∎

######  Lemma 4.3.

Let $g:X\rightarrow Y$ be a morphism of affine schemes of finite type over
$\mathcal{O}$. Let $S$ be a profinite set and let $f:S\rightarrow X(A)$ be a
map of sets. If the equivalent conditions of Lemma 4.2 hold for $f$ then they
also hold for $g\circ f$.

###### Proof.

Let us assume that $\iota_{X}(f(S))$ is contained in
$\prod_{b\in\Gamma(X,\mathcal{O}_{X})}M_{b}$. We claim that
$\iota_{Y}(g(f(S)))$ is contained in
$\prod_{c\in\Gamma(Y,\mathcal{O}_{Y})}M_{g^{\sharp}(c)}$. Indeed, if
$c\in\Gamma(Y,\mathcal{O}_{Y})$ and $x\in X(A)$ then
$\mathrm{ev}_{c}(g(x))=\mathrm{ev}_{g^{\sharp}(c)}(x)$. ∎

######  Lemma 4.4.

Let $S$ be a profinite set and let $f_{1}:S\rightarrow X_{1}(A)$ and
$f_{2}:S\rightarrow X_{2}(A)$ be maps of sets, where $X_{1}$ and $X_{2}$ are
affine schemes of finite type over $\mathcal{O}$. If the equivalent conditions
of Lemma 4.2 hold for $f_{1}$ and $f_{2}$ then they also hold for the product
$f_{1}\times f_{2}:S\rightarrow X_{1}(A)\times X_{2}(A)=(X_{1}\times
X_{2})(A)$.

###### Proof.

We pick closed embeddings $X_{1}\hookrightarrow\mathbb{A}^{n_{1}}$,
$X_{2}\hookrightarrow\mathbb{A}^{n_{2}}$. This induces a closed embedding
$X_{1}\times
X_{2}\hookrightarrow\mathbb{A}^{n_{1}}\times\mathbb{A}^{n_{2}}\cong\mathbb{A}^{n_{1}+n_{2}}$.
If $f_{1}(S)\subseteq M_{1}\subseteq A^{n_{1}}$ and $f_{2}(S)\subseteq
M_{2}\subseteq A^{n_{2}}$ then $(f_{1}\times f_{2})(S)\subseteq M_{1}\times
M_{2}\subseteq A^{n_{1}}\times A^{n_{2}}$. Moreover, if $f_{1}:S\rightarrow
M_{1}$ and $f_{2}:S\rightarrow M_{2}$ are continuous then $f_{1}\times
f_{2}:S\rightarrow M_{1}\times M_{2}$ is continuous. ∎

######  Lemma 4.5.

Let $R^{\prime}$ be a noetherian profinite $R$-algebra together with a map of
$R$-algebras $R^{\prime}\rightarrow A$. Let $S$ be a profinite set and let
$f:S\rightarrow X(A)$ be a map of sets. If the equivalent conditions of Lemma
4.2 hold for $f$ with respect to $R$ then they also hold with respect to
$R^{\prime}$.

###### Proof.

It is immediate that part (1) of Lemma 4.2 holds with $M^{\prime}_{b}$ equal
to the $R^{\prime}$-submodule of $A$ generated by $M_{b}$. ∎

###  4.2. Continuity via condensed sets

We will now express the equivalent conditions in Lemma 4.2 in terms of
condensed sets. Recall that these are certain functors from the category of
profinite sets to the category of sets, see [48] and Appendix A. We may view
profinite sets as condensed sets via the Yoneda embedding
$S\mapsto\underline{S}$, where $\underline{S}(T):=\mathcal{C}(T,S)$. The
underlying set of a condensed set is just that functor evaluated at a point.

Recall further, that every profinite group (ring) is naturally a condensed
group (ring), as the functor from topological spaces to condensed sets
commutes with finite limits [48, Proposition 1.7]. We can also equip a
profinite ring $R$ with the discrete topology and write
$R_{\operatorname{disc}}$ to denote the associated discrete condensed ring. We
have a natural map of condensed rings
$R_{\operatorname{disc}}\to\underline{R}$. An abstract $R$-module $M$ can be
seen as a discrete condensed $R_{\operatorname{disc}}$-module
$M_{\operatorname{disc}}$, so that
$M_{\operatorname{disc}}(T)=\mathcal{C}(T,M)$, where the target $M$ is given
the discrete topology. The tensor product of condensed
$R_{\operatorname{disc}}$-modules