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
$\mathcal O$
, uniformiser
$\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 a torus and the component group
$G/G^0$
is a finite étale group scheme over
$\mathcal O$
. We call such group schemes generalised tori. We do not make any assumptions regarding p and the component group of G. After replacing L by a finite unramified extension, one may assume that
$G^0$
is a split torus and
$G/G^0$
is a constant group scheme. We will assume this for the rest of the introduction.
An important example of a generalised torus that we have in mind is the L-group of a torus defined over F: let H be a torus over F, which splits over a finite Galois extension E over F. Then the L-group of H is
${}^L H= \hat {H}\,{\rtimes}\, \underline {\operatorname {\mathrm {Gal}}(E/F)}$
, where
$\hat {H}$
is the split
$\mathcal O$
-torus, such that the character lattice of
$\hat {H}$
is equal to the cocharacter lattice of
$H_E$
and
$\underline {\operatorname {\mathrm {Gal}}(E/F)}$
is the constant group scheme associated to
$\operatorname {\mathrm {Gal}}(E/F)$
. In this example, the surjection
$G\twoheadrightarrow G/G^0$
has a section of group schemes. We do not assume this in general.
We fix a continuous representation
$\overline {\rho }:\Gamma _F\rightarrow G(k)$
and denote by
$D^{\square }_{\overline {\rho }}: \mathfrak A_{\mathcal O}\rightarrow \text { Set}$
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
$$ \begin{align*}\rho_A(\gamma) \equiv \overline{\rho}(\gamma) \quad\pmod{\mathfrak m_A}, \quad \forall\gamma\in \Gamma_F.\end{align*} $$
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.
Theorem 1.1. There is a finite extension
$L'$
of L with ring of integers
$\mathcal O'$
and a continuous representation
$\rho :\Gamma _F \rightarrow G(\mathcal O')$
lifting
$\overline {\rho }$
.
Remark 1.2. If
$G\cong G^0\,{\rtimes}\, (G/G^0)$
, then a lift as in Theorem 1.1 can be constructed over
$\mathcal O$
using the Teichmüller lift to define a section of
$G^0(\mathcal O)\twoheadrightarrow G^0(k)$
, which then induces a section
$\sigma : G(k)\rightarrow G(\mathcal O)$
. Then
$\sigma \circ \overline {\rho }$
is the required lift. In particular, in this case, Theorem 1.1 holds if we replace
$\Gamma _F$
with any profinite group. However, it seems nontrivial to prove Theorem 1.1 in general, and our argument uses that the Euler–Poincaré characteristic formula holds for
$\Gamma _F$
.
Let
$\Gamma _E$
be the kernel of the composition
$\Gamma _F \overset {\overline {\rho }}{\longrightarrow } G(k)\rightarrow (G/G^0)(k)$
and let
$\Delta $
be the image of this map. We identify
$\Delta =\operatorname {\mathrm {Gal}}(E/F)$
. Let M be the character lattice of
$G^0$
. The action of G on
$G^0$
by conjugation induces an action of
$\Delta $
on M. Let
$\Gamma _E^{\mathrm {ab}, p}$
be the maximal abelian pro-p quotient of
$\Gamma _E$
. Below, we will consider the diagonal action of
$\Delta $
on
$\Gamma _E^{\mathrm {ab},p}\otimes M$
and
$\mu _{p^{\infty }}(E)\otimes M$
, where
$\mu _{p^{\infty }}(E)$
is the subgroup of p-power roots of unity in E.
Theorem 1.3. Assume that Theorem 1.1 holds with
$\mathcal O=\mathcal O'$
. Then
where
$m= \mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M \cdot ([F:\mathbb {Q}_p]+1)$
.
Let
$\overline {\Theta }$
be the G-pseudocharacter associated to
$\overline {\rho }$
and let
$D^{\mathrm {ps}}_{\overline {\Theta }}: \mathfrak A_{\mathcal O} \rightarrow \text { Set}$
be the deformation functor such that
$D^{\mathrm {ps}}_{\overline {\Theta }}(A)$
is the set of continuous A-valued G-pseudocharacters deforming
$\overline {\Theta }$
. These notions are reviewed in Section 6. The functor
$D^{\mathrm {ps}}_{\overline {\Theta }}$
is pro-represented by a complete local noetherian
$\mathcal O$
-algebra
$R^{\mathrm {ps}}_{\overline {\Theta }}$
with residue field k, and we denote the universal deformation by
$\Theta ^u$
. Sending a deformation of
$\overline {\rho }$
to its G-pseudocharacter induces a natural transformation
$D^{\square }_{\overline {\rho }} \rightarrow D^{\mathrm {ps}}_{\overline {\Theta }}$
, and hence a map of local
$\mathcal O$
-algebras
$R^{\mathrm {ps}}_{\overline {\Theta }}\rightarrow R^{\square }_{\overline {\rho }}$
. If G is a torus or, more generally, when G is commutative, then representations and G-pseudocharacters coincide, and this map is an isomorphism.
Theorem 1.4. The map
$R^{\mathrm {ps}}_{\overline {\Theta }} \rightarrow R^{\square }_{\overline {\rho }}$
is formally smooth. Moreover, if Theorem 1.1 holds with
$\mathcal O'=\mathcal O$
, then
where
$r=\mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M \cdot [F:\mathbb {Q}_p]+ \mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M_{\Delta }$
.
We let
$X^{\mathrm {gen}}_{G, \overline {\Theta }}: R^{\mathrm {ps}}_{\overline {\Theta }}\text {-}\mathrm {alg}\rightarrow \text { Set}$
be the functor such that
$X^{\mathrm {gen}}_{G, \overline {\Theta }}(A)$
is the set of representations
$\rho : \Gamma _F\rightarrow G(A)$
such that its G-pseudocharacter
$\Theta _{\rho }$
satisfies
$\Theta _{\rho } =\Theta ^u \otimes _{R^{\mathrm {ps}}_{\overline {\Theta }}} A$
. This functor is representable by a finite type
$R^{\mathrm {ps}}_{\overline {\Theta }}$
-algebra
$A^{\mathrm {gen}}_{G, \overline {\Theta }}$
.
Theorem 1.5. The map
$R^{\mathrm {ps}}_{\overline {\Theta }}\rightarrow A^{\mathrm {gen}}_{G, \overline {\Theta }}$
is smooth. Moreover, if Theorem 1.1 holds with
$\mathcal O'=\mathcal O$
, then
$$ \begin{align*}A^{\mathrm{gen}}_{G, \overline{\Theta}}\cong R^{\mathrm{ps}}_{\overline{\Theta}}[t_1^{\pm 1}, \ldots, t_s^{\pm 1}],\end{align*} $$
where
$s= \mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M -\mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M_{\Delta }$
.
We note that
$\mu _{p^{\infty }}(E)$
is a finite p-group. This implies that 
is also a finite p-group, and we denote its order by
$p^m$
. We assume further that
$\mathcal O$
contains all
$p^m$
-th roots of
$1$
and let
$\mathrm X(\mu )$
be the group of characters
$\chi : \mu \rightarrow \mathcal O^{\times }$
. It then follows from Theorems 1.3, 1.4, 1.5 that the sets of irreducible components of
$R^{\square }_{\overline {\rho }}$
,
$R^{\mathrm {ps}}_{\overline {\Theta }}$
and
$A^{\mathrm {gen}}_{G, \overline {\Theta }}$
, respectively, are in
$\mathrm X(\mu )$
-equivariant bijection with
$\mathrm X(\mu )$
. Moreover, the irreducible components and their special fibres are regular. The identification of the set of components with
$\mathrm X(\mu )$
is non-canonical in general, as one has to distinguish one component, which corresponds to the trivial character. However, we explain in Section 7.4 that there is a canonical action of
$\mathrm X(\mu )$
on the set of irreducible components, which is faithful and transitive.
Remark 1.6. Let us point out that if
$G\cong G^0\,{\rtimes}\, G/G^0$
, then the lift constructed in Remark 1.2 is canonical, as it is minimally ramified. This distinguishes the irreducible component that it lies on. Thus, in this case, there is a canonical
$\mathrm X(\mu )$
-equivariant bijection between
$\mathrm X(\mu )$
and the set of irreducible components.
The above theorems are used in an essential way in a companion paper [Reference Paškūnas and Quast12], where we study deformations of Galois representations with values in generalised reductive
$\mathcal O$
-group schemes G, which means that the neutral component
$G^0$
is reductive and the component group
$G/G^0$
is finite étale. If we let
$G'$
be the derived subgroup scheme of
$G^0$
, then
$G^0/G'$
is a torus and
$G/G'$
is a generalised torus as considered in this paper. If
$\overline {\rho }: \Gamma _F \rightarrow G(k)$
is a continuous representation, then we show in [Reference Paškūnas and Quast12, Proposition 4.20] that its deformation ring
$R^{\square }_{\overline {\rho }}$
can be presented over the deformation ring of
$\varphi \circ \overline {\rho }: \Gamma _F \rightarrow (G/G')(k)$
, where
$\varphi : G\rightarrow G/G'$
is the quotient map. This allows us to split up the arguments into ‘torus part’, carried out in this paper, and ‘semisimple part’, carried out in [Reference Paškūnas and Quast12].
If
$\mu $
is any finite abelian p-group, then
$\mu \cong \prod _{i=1}^l \mathbb Z/p^{e_i}$
, and hence,
$$ \begin{align*}\mathcal O[\mu]\cong \bigotimes_{i=1}^l \frac{\mathcal O[z_i]}{( (1+z_i)^{p^{e_i}}-1)}\end{align*} $$
is complete intersection. It then follows from Theorems 1.3, 1.4, 1.5 that the rings
$R^{\square }_{\overline {\rho }}$
,
$R^{\mathrm {ps}}_{\overline {\Theta }}$
and
$A^{\mathrm {gen}}_{G, \overline {\Theta }}$
are locally complete intersections. To relate their dimensions to the dimensions of the deformation rings appearing in [Reference Paškūnas and Quast12], we note that
$\mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M= \dim G_k$
, and if
$\Delta =(G/G^0)(\bar {k})$
(which we may assume without changing the functors
$D^{\square }_{\overline {\rho }}$
,
$D^{\mathrm {ps}}_{\overline {\Theta }}$
and
$X^{\mathrm {gen}}_{G,\overline {\Theta }}$
), then
$\mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M_{\Delta } = \dim Z(G)_k$
. The scheme
$X^{\mathrm {gen}}_{G,\overline {\Theta }}$
coincides with the scheme denoted by
$X^{\mathrm {gen}}_{G, \bar {\rho }^{\mathrm {ss}}}$
in [Reference Paškūnas and Quast12], when G is a generalised torus. In [Reference Paškūnas and Quast12], in the definition of
$X^{\mathrm {gen}}_{G, \bar {\rho }^{\mathrm {ss}}}$
, an additional condition is imposed on representations, called
$R^{\mathrm {ps}}_{\overline {\Theta }}$
-condensed, and we show in Lemma 9.6 that when G is a generalised torus, this condition holds automatically.
1.1 Arithmetic setting
Let us first recall how one proves the result when
$G=\mathbb G_m$
. The first step is to observe that Theorem 1.1 holds with
$\mathcal O'=\mathcal O$
– for example, using the Teichmüller lift
$\sigma : k^{\times }\rightarrow \mathcal O^{\times }$
and letting
$\rho _0=\sigma \circ \overline {\rho }$
. The second step is to observe that the map
$\rho \mapsto \rho \rho ^{-1}_0$
induces a bijection between
$D^{\square }_{\overline {\rho }} \rightarrow D^{\square }_{\mathbf 1}$
, where
$\mathbf 1$
is the trivial representation. One may identify
$D^{\square }_{\mathbf 1}(A)$
with the set of continuous group homomorphisms
$\Gamma _F \rightarrow 1+\mathfrak m_A$
, and since the target is an abelian p-group, the functor is representable by 
. An interesting feature of this argument is that
$D^{\square }_{\mathbf 1}$
takes values in
$\operatorname {\mathrm {Ab}}$
as opposed to
$\text {Set}$
, and the map
$$ \begin{align*}D_{\mathbf 1}^{\square}(A) \times D_{\overline{\rho}}^{\square}(A)\rightarrow D^{\square}_{\overline{\rho}}(A), \quad (\Phi, \rho)\mapsto \Phi \rho\end{align*} $$
defines an action of an abelian group
$D_{\mathbf 1}^{\square }(A)$
on the set
$D^{\square }_{\overline {\rho }}(A)$
. Moreover, if the set is nonempty, then this action is faithful and transitive.
If G is connected, then
$G=\mathfrak D(M)$
, where M is the character lattice, and so
$G(A)=\operatorname {\mathrm {Hom}}(M, A^{\times })$
. The same argument goes through, and we only need to observe that
$$ \begin{align*}\operatorname{\mathrm{Hom}}^{\mathrm{cont}}_{\mathrm{Group}}(\Gamma_F, G(A))\cong \operatorname{\mathrm{Hom}}^{\mathrm{cont}}_{\mathrm{Group}}(\Gamma_F^{\mathrm{ab}}\otimes M, A^{\times}),\end{align*} $$
which implies that
$D^{\square }_{\overline {\rho }}$
is representable by 
.
Let us consider the general case now. The first issue, alluded to in Remark 1.2, is that it is not clear anymore that such a lift exists. However, let us postpone the explanation how we deal with this problem until Section 1.3 and let us give ourselves a lift
$\rho _0: \Gamma _F \rightarrow G(\mathcal O)$
of
$\overline {\rho }$
. Then the map
$\rho \mapsto [\gamma \mapsto \rho (\gamma ) \rho _0(\gamma )^{-1}]$
considered above induces a bijection between
$D^{\square }_{\overline {\rho }}(A)$
and the set
$\widehat {Z}^1(A)$
of continuous
$1$
-cocycles
$\Phi : \Gamma _F \rightarrow \operatorname {\mathrm {Hom}}(M, 1+\mathfrak m_A)$
. Further, one may interpret
$\widehat {Z}^1$
as
$D^{\square }_{\overline {\rho }}$
for the group
$G=\mathfrak D(M)\,{\rtimes}\, \underline {\Delta }$
and
$\overline {\rho }: \Gamma _F \rightarrow G(k)$
given by
$\overline {\rho }(\gamma )= (1, \pi (\gamma ))$
, where
$\pi : \Gamma _F\rightarrow \Delta $
is the quotient map. The functor
$\widehat {Z}^1$
is the analogue of
$D^{\square }_{\mathbf 1}$
, and we have to find the ring pro-representing the functor
$\widehat {Z}^1$
.
Before we describe our solution, let us point out one dead end. If
$\Phi \in \widehat {Z}^1(A)$
, then its restriction to
$\Gamma _E$
is just a continuous homomorphism
$$ \begin{align*}\Phi|_{\Gamma_E}: \Gamma_E \rightarrow \operatorname{\mathrm{Hom}}(M, 1+\mathfrak m_A),\end{align*} $$
as
$\Gamma _E$
acts trivially on M. This induces a continuous homomorphism
$$ \begin{align*}\varphi: \Gamma_E^{\mathrm{ab},p}\otimes M \rightarrow 1+\mathfrak m_A,\end{align*} $$
and the fact that the homomorphism is obtained by a restriction of a cocycle to
$\Gamma _E$
implies that
$\varphi $
factors through
$(\Gamma _E^{\mathrm {ab},p}\otimes M)_{\Delta }\rightarrow 1+\mathfrak m_A$
. However, in Theorem 1.3 we have
$\Delta $
-invariants and not
$\Delta $
-coinvariants appearing. Tate group cohomology gives us an exact sequence
$$ \begin{align*}0\rightarrow \widehat{H}^{-1}(\Delta, \Gamma_E^{\mathrm{ab},p}\otimes M)\rightarrow (\Gamma_E^{\mathrm{ab},p}\otimes M)_{\Delta} \rightarrow (\Gamma_E^{\mathrm{ab},p}\otimes M)^{\Delta}\rightarrow \widehat{H}^0(\Delta, \Gamma_E^{\mathrm{ab},p}\otimes M)\rightarrow 0.\end{align*} $$
One may show that the Tate group cohomology groups
$\widehat {H}^i(\Delta , \Gamma _E^{\mathrm {ab},p}\otimes M)$
are finite for all
$i\in \mathbb Z$
, but there is no reason why they should vanish, unless p does not divide
$|\Delta |$
. We conclude that the restriction to
$\Gamma _E$
will lose information in general.
Our solution is motivated by Langlands’ paper [Reference Langlands11] on his correspondence for tori, where he shows that the universal coefficient theorem induces an isomorphism
$$ \begin{align} H^1(W_{E/F}, \operatorname{\mathrm{Hom}}(M, \mathbb C^{\times}))\cong \operatorname{\mathrm{Hom}} (H_1(W_{E/F}, M), \mathbb C^{\times})), \end{align} $$
and there are natural isomorphisms
$$ \begin{align} H_1(W_{E/F}, M) \cong H_1(E^{\times}, M)^{\Delta} \cong (E^{\times}\otimes M)^{\Delta}, \end{align} $$
where the Weil group
$W_{E/F}$
fits into a short exact sequence
$$ \begin{align} 0\rightarrow E^{\times} \rightarrow W_{E/F}\rightarrow \Delta\rightarrow 0, \end{align} $$
corresponding to the fundamental class
$[u_{E/F}]\in H^2(\Delta , E^{\times })$
. Since every continuous representation
$\Gamma _F \rightarrow G(A)$
with image in
$(G/G^0)(A)$
equal to
$\Delta $
factors through the profinite completion of
$W_{E/F}$
, this motivates us to study the functor
$$ \begin{align*}\operatorname{\mathrm{CRing}} \rightarrow \operatorname{\mathrm{Ab}},\quad A\mapsto Z^1(W_{E/F}, G(A))\end{align*} $$
without imposing any continuity conditions on the cocycles and allowing any commutative ring A. We determine the ring representing this functor and show that the ring representing
$\widehat {Z}^1$
arises as a completion of it with respect to the maximal ideal corresponding to the trivial cocycle.
1.2 Abstract setting
We carry out this study in Section 4 in the following abstract setting: let
$\Gamma _1$
be an abstract group,
$\Gamma _2$
a normal subgroup of
$\Gamma _1$
of finite index and let 
. The exact sequence
$$ \begin{align} 0\rightarrow \Gamma_2^{\mathrm{ab}}\rightarrow \Gamma_1/[\Gamma_2, \Gamma_2]\rightarrow \Delta \rightarrow 0 \end{align} $$
defines a class in
$H^2(\Delta , \Gamma _2^{\mathrm {ab}})$
. Its image in
$\operatorname {\mathrm {Ext}}^1_{\mathbb Z[\Delta ]}(I_{\Delta }, \Gamma _2^{\mathrm {ab}})$
, where
$I_{\Delta }$
is the augmentation ideal in the group ring
$\mathbb Z[\Delta ]$
, defines an extension of
$\mathbb Z[\Delta ]$
-modules
$$ \begin{align} 0\rightarrow \Gamma_2^{\mathrm{ab}}\rightarrow \mathcal E\rightarrow I_{\Delta}\rightarrow 0. \end{align} $$
Theorem 1.7. The functor
$$ \begin{align*}\operatorname{\mathrm{CRing}} \rightarrow \operatorname{\mathrm{Ab}}, \quad A\mapsto Z^1(\Gamma_1, \operatorname{\mathrm{Hom}}(M, A^{\times}))\end{align*} $$
is represented by the group algebra
$\mathbb Z[(\mathcal E\otimes M)_{\Delta }]$
. The fpqc sheafification of the functor
$$ \begin{align*}\operatorname{\mathrm{CRing}} \rightarrow \operatorname{\mathrm{Ab}}, \quad A\mapsto H^1(\Gamma_1, \operatorname{\mathrm{Hom}}(M, A^{\times}))\end{align*} $$
is represented by the group algebra
$\mathbb Z[H_1(\Gamma _1, M)]$
.
We prove Theorem 1.7 by first replacing M with
$\mathbb Z[\Delta ]\otimes M$
. In this case,
$$ \begin{align*}\operatorname{\mathrm{Hom}}(\mathbb Z[\Delta]\otimes M, A^{\times})\cong \operatorname{\mathrm{Ind}}^{\Gamma_1}_{\Gamma_2} \operatorname{\mathrm{Hom}}(M, A^{\times})\end{align*} $$
and we can use Shapiro’s lemma to identify
$H^1$
with
$$ \begin{align*}\operatorname{\mathrm{Hom}}(\Gamma_2, \operatorname{\mathrm{Hom}}(M, A^{\times}))\cong \operatorname{\mathrm{Hom}}(\Gamma_2^{\mathrm{ab}}\otimes M, A^{\times}).\end{align*} $$
Further, after choosing a representative
$\overline {c}$
for each coset
$c\in \Gamma _1/\Gamma _2$
, we construct in Section 3 a homomorphism of abelian groups,
$$ \begin{align*}H^1(\Gamma_1,\operatorname{\mathrm{Ind}}_{\Gamma_2}^{\Gamma_1} V) \rightarrow Z^1(\Gamma_1, \operatorname{\mathrm{Ind}}_{\Gamma_2}^{\Gamma_1} V),\end{align*} $$
for any
$\Gamma _2$
-module V, which is functorial in V and becomes the identity, when composed with the natural quotient map. These ingredients allow us to show that the functor
$$ \begin{align} \operatorname{\mathrm{Ab}}\rightarrow \operatorname{\mathrm{Ab}}, \quad V\mapsto Z^1(\Gamma_1, \operatorname{\mathrm{Hom}}(\mathbb Z[\Delta]\otimes M, V)) \end{align} $$
is representable by
$(\Gamma _2^{\mathrm {ab}}\oplus I_{\Delta })\otimes M$
. We then define an action of
$\Delta $
on
$\mathbb Z[\Delta ]\otimes M$
, which commutes with the action of
$\Gamma _1$
, such that
$$ \begin{align*}Z^1(\Gamma_1, \operatorname{\mathrm{Hom}}(M, V)) \cong Z^1(\Gamma_1, \operatorname{\mathrm{Hom}}(\mathbb Z[\Delta]\otimes M, V))^{\Delta}.\end{align*} $$
To prove Theorem 1.7, we verify that the action of
$\Delta $
on
$(\Gamma _2^{\mathrm {ab}}\oplus I_{\Delta })\otimes M$
induced by its action on the functor in (1.6) is isomorphic to
$\mathcal E\otimes M$
. This calculation is carried out in Section 4. The last part of Theorem 1.7 follows by observing that (1.1) remains an isomorphism if
$\mathbb C^{\times }$
is replaced by any divisible group, and for any A, there is a faithfully flat map
$A\rightarrow B$
such that
$B^{\times }$
is divisible.
Let G be a generalised torus over some base ring
$\mathcal O$
. Let
$\mathrm {PC}^{\Gamma _1}_{G}: \mathcal O\text {-}\mathrm {alg}\rightarrow \text { Set}$
be the functor such that
$\mathrm {PC}^{\Gamma _1}_{G}(A)$
is the set of A-valued G-pseudocharacters of
$\Gamma _1$
. Let
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_G: \mathcal O\text {-}\mathrm {alg} \rightarrow \text { Set}$
be the functor such that
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_G(A)$
is the set of all representations
$\rho :\Gamma _1 \rightarrow G(A)$
.
Theorem 1.8. Mapping a representation to its G-pseudocharacter induces an isomorphism of schemes
The proof of Theorem 1.8 follows the arguments of Emerson–Morel [Reference Emerson and Morel8], where they prove an analogous result, when G is a connected reductive group over a field of characteristic
$0$
. Since in our case
$G^0$
is a torus, it is linearly reductive over
$\mathcal O$
, and their arguments carry over integrally.
Let us assume that
$G^0$
is split over
$\mathcal O$
and
$G/G^0$
is a constant group scheme. Let
$\rho _0: \Gamma _1\rightarrow G(\mathcal O)$
be a representation and let
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi }$
be the subfunctor of
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G}$
consisting of representations
$\rho $
such that
$\Pi \circ \rho = \Pi \circ \rho _0$
, where
$\Pi : G \rightarrow G/G^0$
is the quotient map. One may similarly define
$\mathrm {PC}^{\Gamma _1}_{G, \pi }$
. We show that the map
$$ \begin{align*}Z^1(\Gamma_1, G^0(A))\rightarrow \operatorname{\mathrm{Rep}}^{\Gamma_1}_{G,\pi}(A), \quad \Phi\mapsto \Phi \rho_0\end{align*} $$
is bijective and use Theorem 1.7 to prove:
Theorem 1.9. The functors
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi }$
and
$\mathrm {PC}^{\Gamma _1}_{G, \pi }$
are represented by the group algebras
$\mathcal O[(\mathcal E \otimes M)_{\Delta }]$
and
$\mathcal O[H_1(\Gamma _1, M)]$
, respectively.
If
$\Gamma _1=W_{E/F}$
, then
$H_1(\Gamma _1, M)\cong (E^{\times }\otimes M)^{\Delta }$
by (1.2). To prove Theorem 1.4, we relate the completions of the local rings of
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi }$
and
$\mathrm {PC}^{\Gamma _1}_{G, \pi }$
to deformation rings parameterising continuous deformations and G-pseudocharacters of the profinite completion
$\widehat {\Gamma }_1$
of
$\Gamma _1$
. These arguments are carried out in Section 7. We expect that the abstract setting will also be useful studying deformations of global Galois groups.
1.3 Producing a lift
Let us go back to the problem of exhibiting a lift of
$\overline {\rho }: \Gamma _F\rightarrow G(k)$
to characteristic zero as claimed in Theorem 1.1. Using
$\overline {\rho }$
instead of
$\rho _0$
, we may identify the restriction of
$D^{\square }_{\overline {\rho }}$
to
$\mathfrak A_k$
with the restriction of
$\widehat {Z}^1$
to
$\mathfrak A_k$
. This allows us to conclude that
where
$\mathcal E$
is defined by (1.5), where (1.4) is equal to (1.3), so that
$\Gamma _1=W_{E/F}$
and
$\Gamma _2=E^{\times }$
and
$(-)^{\wedge , p}$
denotes the pro-p completion.
In Section 8, we compute
$$ \begin{align*}\mathop{\mathrm{ rank}}\nolimits_{\mathbb Z_p} ((\mathcal E\otimes M)_{\Delta})^{\wedge, p}= \mathop{\mathrm{ rank}}\nolimits_{\mathbb Z} M \cdot ([F:\mathbb {Q}_p]+1),\end{align*} $$
which is then also equal to
$\dim R^{\square }_{\overline {\rho }}/\varpi $
. Moreover, using Mazur’s obstruction theory and the Euler–Poincaré characteristic formula, we show that
where
$r-s= \dim _k \operatorname {\mathrm {Lie}} G_k\cdot ([F:\mathbb {Q}_p]+1)$
. Since
$\mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M = \dim _k \operatorname {\mathrm {Lie}} G_k$
, a little commutative algebra implies that
$\varpi , f_1, \ldots , f_s$
is a regular sequence in 
, and hence,
$R^{\square }_{\overline {\rho }}$
is
$\mathcal O$
-torsion free. A closed point of the generic fibre gives us the required lift.
1.4 Overview by section
In Section 2, we define what a generalised torus is. In Section 3, we establish an explicit version of Shapiro’s lemma. In Section 4, we compute the ring representing the functor
$A\mapsto Z^1(\Gamma _1, \operatorname {\mathrm {Hom}}(M, A^{\times }))$
. This section is the technical heart of the paper. In Section 5, we relate the functor
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi }$
to the functor
$Z^1(\Gamma _1, G^0(-))$
. In Section 6, we introduce Lafforgue’s G-pseudocharacters and study the functor
$\mathrm {PC}^{\Gamma _1}_{G, \pi }$
. In Section 7, we transfer the results about abstract representations and G-pseudocharacters of
$\Gamma _1$
to continuous representations and continuous G-pseudocharacters of its profinite completion
$\widehat {\Gamma }_1$
and prove versions of Theorems 1.3, 1.4, 1.5 with
$\Gamma _F$
replaced by
$\widehat {\Gamma }_1$
. In Section 8, we compute
$\mathbb Z_p$
-ranks of certain pro-p completions appearing naturally in the arithmetic setting. We show in Lemma 8.7 that the pro-p completion of
$(E^{\times }\otimes M)^{\Delta }$
is isomorphic to
$(\Gamma _E^{\mathrm {ab},p}\otimes M)^{\Delta }$
. In Section 9, we use the results proved in the previous sections to prove Theorems 1.1, 1.3, 1.4, 1.5.
1.5 Notation
Let p denote any prime number. Let F be a finite extension of
$\mathbb {Q}_p$
. We fix an algebraic closure
$\overline {F}$
of F. Let 
be the absolute Galois group of F. Let L be another finite extension of
$\mathbb {Q}_p$
with ring of integers
$\mathcal O$
, uniformiser
$\varpi $
and residue field k. However,
$\mathcal O$
is allowed to be an arbitrary commutative ring such that
$\operatorname {\mathrm {Spec}} \mathcal O$
is connected in Sections 5 and 6.
Let
$\mathfrak A_{\mathcal O}$
be the category of local artinian
$\mathcal O$
-algebras with residue field k. We let
$\widehat {\mathfrak A}_{\mathcal O}$
be the category of pro-objects of
$\mathfrak A_{\mathcal O}$
. Concretely, one may identify
$\widehat {\mathfrak A}_{\mathcal O}$
with the category of pseudo-compact local
$\mathcal O$
-algebras with residue field k. Let
$\mathfrak A_k$
be the full subcategory of
$\mathfrak A_{\mathcal O}$
consisting of objects killed by
$\varpi $
. Then
$\widehat {\mathfrak A}_k$
is the full subcategory of
$\widehat {\mathfrak A}_{\mathcal O}$
consisting of objects killed by
$\varpi $
, which can be identified with the category of local pseudo-compact k-algebras with residue field k.
The groups denoted by
$\Gamma _1$
and
$\Gamma _2$
are abstract groups with no topology. We will denote by
$\widehat {\Gamma }_1$
the profinite completion of
$\Gamma _1$
. If
$\mathcal A$
is an abelian group, then we will denote its pro-p completion by
$\mathcal A^{\wedge , p}$
. If X is a scheme, then we denote the ring of its global sections by
$\mathcal O(X)$
. If
$\Gamma $
is a group, then we denote by
$\mathcal O[\Gamma ]$
its group algebra over a ring
$\mathcal O$
.
2 Generalised tori
Definition 2.1. Let S be a scheme. A generalised torus is a smooth affine S-group scheme G, such that the geometric fibres of
$G^0$
are tori and
$G/G^0 \to S$
is finite.
Remark 2.2. It follows from [Reference Conrad6, Proposition 3.1.3] that for a generalised torus G, the quotient
$G/G^0$
is étale. Sean Cotner has shown in [Reference Cotner7] that if S is locally noetherian, then for a smooth affine S-group scheme G, the identity component
$G^0$
has reductive geometric fibers and
$G/G^0$
is finite étale if and only if G is geometrically reductive in the sense of [Reference Alper1, Definition 9.1.1]. In particular, this holds over
$S=\operatorname {\mathrm {Spec}} \mathcal O$
.
Remark 2.3. When G is a generalised torus over
$\mathcal O$
, then
$G^0$
is linearly reductive (i.e., taking
$G^0$
-invariants is an exact functor on the category of
$\mathcal O(G^0)$
-comodules). It follows that if A is a commutative
$\mathcal O$
-algebra with trivial
$G^0$
-action, M is an A-module with
$G^0$
-action and N is an A-module with trivial
$G^0$
-action, then
$$ \begin{align*}{(M \otimes_A N)^{G^0} = M^{G^0} \otimes_A N}.\end{align*} $$
If M is an abelian group, then we denote by 
the diagonalisable group scheme. For any
$\mathcal O$
-algebra A, we have
$$ \begin{align*}\mathfrak D(M)(A) = \operatorname{\mathrm{Hom}}(M, A^{\times}).\end{align*} $$
If M is a finitely generated free abelian group, then
$\mathfrak D(M)$
is a split torus.
3 Shapiro’s lemma
We recall an explicit version of Shapiro’s lemma. Let
$\Gamma _1$
be a group and let
$\Gamma _2$
be a subgroup of
$\Gamma _1$
of finite index. Let V be an abelian group with a
$\Gamma _2$
-action. We let
$\operatorname {\mathrm {Ind}}_{\Gamma _2}^{\Gamma _1}{V}$
be the set of functions
$f: \Gamma _1 \rightarrow V$
, such that
$f(kg)= k f(g)$
for all
$k\in \Gamma _2$
and
$g\in \Gamma _1$
. Then
$\operatorname {\mathrm {Ind}}_{\Gamma _2}^{\Gamma _1}{V}$
is naturally an abelian group, isomorphic to a finite direct sum of copies of V, on which
$\Gamma _1$
acts by right translations; that is, 
for all
$g, h\in \Gamma _1$
. In this section, topology does not play a role, all cohomology is just abstract group cohomology, and there is no continuity condition on the cocycles.
Proposition 3.1. Let
$\Phi : \Gamma _1 \rightarrow \operatorname {\mathrm {Ind}}_{\Gamma _2}^{\Gamma _1}{V}$
be a
$1$
-cocycle. Then the function
$\varphi : \Gamma _2\rightarrow V$
, given by
$$ \begin{align*}\varphi(k)=[\Phi(k)](1),\end{align*} $$
is a
$1$
-cocycle. Moreover, the above map induces an isomorphism in cohomology
$$ \begin{align*}H^1(\Gamma_1, \operatorname{\mathrm{Ind}}_{\Gamma_2}^{\Gamma_1}{V})\cong H^1(\Gamma_2, V).\end{align*} $$
Proof. [Reference Lang10, Theorem 3.7].
We will construct an explicit inverse of the map above following [Reference Lang10]. For each right coset c of
$\Gamma _2$
in
$\Gamma _1$
, we fix a coset representative
$\overline {c}$
, so that the representative of the trivial coset is
$1$
. In particular,
$$ \begin{align*}\Gamma_1= \bigcup_c \Gamma_2 \overline{c}= \bigcup_c \overline{c}^{-1}\Gamma_2.\end{align*} $$
Since
$c g= \Gamma _2 \overline {cg}$
for every
$g\in \Gamma _1$
, we have
$$ \begin{align*}\overline{c} g \overline{cg}^{-1}\in \Gamma_2, \quad \forall g\in \Gamma_1.\end{align*} $$
Let
$\mathcal F_V$
be the subgroup of
$\operatorname {\mathrm {Ind}}_{\Gamma _2}^{\Gamma _1}{V}$
of functions with support in
$\Gamma _2$
. By evaluating at
$1$
, we obtain a canonical isomorphism of
$\Gamma _2$
-representations between
$\mathcal F_V$
and V; the inverse homomorphism is obtained by mapping v to the function
$g \mapsto g v$
. If
$\varphi \in Z^1(\Gamma _2, V)$
, then the isomorphism gives us a cocycle
$f_{\varphi }\in Z^1(\Gamma _2, \mathcal F_V)$
, where if
$h\in \Gamma _2$
, then
$f_{\varphi }(h)\in \mathcal F_V$
is the function given by
$$ \begin{align} [f_{\varphi}(h)](k)= k \varphi(h), \quad \forall k \in \Gamma_2. \end{align} $$
Lemma 3.2. Let
$\varphi :\Gamma _2\rightarrow V$
be a
$1$
-cocycle, and let
$f_{\varphi }: \Gamma _2\rightarrow \mathcal F_V$
be the
$1$
-cocycle corresponding to
$\varphi $
, via the canonical isomorphism
$\mathcal F_V\cong V$
and let
${\Phi _{\varphi }:\Gamma _1\rightarrow \operatorname {\mathrm {Ind}}_{\Gamma _2}^{\Gamma _1}{V}}$
be the function given by
$$ \begin{align*}\Phi_{\varphi}(g)=\sum_c \overline{c}^{-1} f_{\varphi}(\overline{c} g \overline{cg}^{-1}).\end{align*} $$
Then
$\Phi _{\varphi }$
is a
$1$
-cocycle, such that
$$ \begin{align*}[\Phi_{\varphi}(k)](1)=\varphi(k), \quad \forall k\in \Gamma_2.\end{align*} $$
Proof. We have to show that
$$ \begin{align*}\Phi_{\varphi}(gh)=\Phi_{\varphi}(g)+ g\Phi_{\varphi}(h),\quad \forall g, h\in \Gamma_1.\end{align*} $$
It is enough to show that the equality holds once we evaluate both sides at
$k \overline {c}$
, where
$\overline {c}$
is a representative of the coset c and
$k\in \Gamma _2$
. If
$f\in \mathcal F_V$
, then
$\operatorname {\mathrm {Supp}} \overline {c}^{-1} f\subseteq c$
; hence,
$$ \begin{align*}[\Phi_{\varphi}(gh)](k\overline{c})= [f_{\varphi}(\overline{c} gh \overline{cgh}^{-1})](k), \quad [\Phi_{\varphi}(g)](k\overline{c})= [f_{\varphi}(\overline{c} g \overline{cg}^{-1})](k),\end{align*} $$
$$ \begin{align*} \notag [g \Phi_{\varphi}(h)](k\overline{c})=[\Phi_{\varphi}(h)](k\overline{c}g)=& [f_{\varphi}(\overline{cg} h \overline{cgh}^{-1})](k \overline{c} g \overline{cg}^{-1})\\ =&[\overline{c}g \overline{cg}^{-1} f_{\varphi}(\overline{cg} h \overline{cgh}^{-1})](k). \end{align*} $$
Since
$f_{\varphi }$
is a
$1$
-cocycle, we have
$$ \begin{align*}\overline{c}g \overline{cg}^{-1} f_{\varphi}(\overline{cg} h \overline{cgh}^{-1})+f_{\varphi}(\overline{c} g \overline{cg}^{-1})=f_{\varphi}(\overline{c}g \overline{cg}^{-1}\overline{cg} h \overline{cgh}^{-1})=f_{\varphi}(\overline{c}g h \overline{cgh}^{-1}),\end{align*} $$
and hence,
$\Phi _{\varphi }$
is a
$1$
-cocycle. Since the representative of the trivial coset was chosen to be
$1$
, we have
$$ \begin{align*}[\Phi_{\varphi}(k)](1)= [f_{\varphi}(k)](1)=\varphi(k), \quad \forall k\in \Gamma_2.\\[-35pt] \end{align*} $$
Remark 3.3. The map
$\varphi \mapsto \Phi _{\varphi }$
depends on the choice of the coset representatives
$\bar {c}$
, and so it is not canonical. However, once these representatives have been fixed, it is immediate from the formulas that the map is functorial in V. If
$\alpha :V\rightarrow W$
is a
$\Gamma _2$
-equivariant homomorphism of abelian groups equipped with
$\Gamma _2$
-action and
$\psi \in Z^1(\Gamma _2, W)$
is the image of
$\varphi $
under the map
$Z^1(\Gamma _2, V)\rightarrow Z^1(\Gamma _2, W)$
induced by
$\alpha $
, then
$\Phi _{\varphi }$
maps to
$\Phi _{\psi }\in Z^1(\Gamma _1, \operatorname {\mathrm {Ind}}_{\Gamma _2}^{\Gamma _1}{W})$
.
4 The space of
$1$
-cocycles
We shall later identify our space of deformations with a space of
$1$
-cocycles. In this section, we study the space of
$1$
-cocycles in a more abstract situation. We fix the following notation:
-
•
$\Gamma _1$
is an abstract group. -
• M is an abelian group equipped with a linear left
$\Gamma _1$
-action with kernel
$\Gamma _2$
of finite index in
$\Gamma _1$
. -
•
. We denote the projection
$\pi : \Gamma _1 \to \Delta $
.
. We denote the projection 
We start by constructing an abelian group
$\mathcal E$
, which represents the functor
$$ \begin{align*}\operatorname{\mathrm{Ab}} \to \operatorname{\mathrm{Ab}}, ~V \mapsto Z^1(\Gamma_1, \operatorname{\mathrm{Ind}}^{\Gamma_1}_{\Gamma_2} V).\end{align*} $$
It turns out that
$\mathcal E$
carries an action of
$\Delta $
, such that
$(\mathcal E \otimes M)_{\Delta }$
represents the functor
$$ \begin{align*}\operatorname{\mathrm{Ab}} \rightarrow \operatorname{\mathrm{Ab}}, ~V \mapsto Z^1(\Gamma_1, \operatorname{\mathrm{Hom}}(M, V)).\end{align*} $$
Our strategy is to realize
$\operatorname {\mathrm {Hom}}(M, V)$
as
$\Delta $
-invariants in
$\operatorname {\mathrm {Ind}}^{\Gamma _1}_{\Gamma _2}\operatorname {\mathrm {Hom}}(M, V)$
, and then use the explicit version of Shapiro’s lemma of Section 3 to compute
$$ \begin{align*}Z^1(\Gamma_1, \operatorname{\mathrm{Ind}}^{\Gamma_1}_{\Gamma_2} \operatorname{\mathrm{Hom}}(M, V)).\end{align*} $$
At the end of this section, we study the functors
$\operatorname {\mathrm {CRing}}\rightarrow \operatorname {\mathrm {Ab}}$
defined by
$$ \begin{align*} A\mapsto Z^1(\Gamma_1, \operatorname{\mathrm{Hom}}(M, A^{\times})), \quad A \mapsto H^1(\Gamma_1, \operatorname{\mathrm{Hom}}(M, A^{\times})). \end{align*} $$
Let V be an abelian group equipped with a trivial
$\Gamma _2$
-action. We may identify
$\operatorname {\mathrm {Ind}}^{\Gamma _1}_{\Gamma _2} V$
with the space of functions
$f: \Delta \rightarrow V$
where the action is given by
$[g\cdot f](c)= f(cg)$
for all
$g\in \Gamma _1$
and
$c\in \Delta $
. We also have an action of
$\Delta $
on
$\operatorname {\mathrm {Ind}}_{\Gamma _2}^{\Gamma _1} V$
, which commutes with the action of
$\Gamma _1$
and is given by
$[d\cdot f](c)= f(d^{-1} c)$
for all
$d\in \Delta $
. This induces an action of
$\Delta $
on
$Z^1(\Gamma _1, \operatorname {\mathrm {Ind}}^{\Gamma _1}_{\Gamma _2} V)$
, which is given explicitly by
$$ \begin{align} [[d\ast \Phi](g)](c)= [\Phi(g)]( d^{-1} c). \end{align} $$
Since the action of
$\Gamma _2$
on V is trivial, we have
$$ \begin{align} Z^1(\Gamma_2, V)= H^1(\Gamma_2, V)= \operatorname{\mathrm{Hom}}(\Gamma_2, V), \end{align} $$
where
$\operatorname {\mathrm {Hom}}$
stands for group homomorphisms. For each
$c\in \Delta $
, we fix a coset representative
$\overline {c}\in \Gamma _1$
as in the previous section.
Corollary 4.1. Let
$\varphi \in \operatorname {\mathrm {Hom}}(\Gamma _2, V)$
. Then the
$1$
-cocycle
$\Phi _{\varphi }$
constructed in Lemma 3.2 is given by
$$ \begin{align*}\Phi_{\varphi}(g)=\sum_{c\in \Delta} \varphi(\overline{c} g \overline{cg}^{-1})\mathbf 1_c,\end{align*} $$
where
$\mathbf 1_c$
is the indicator function for the coset c.
Proof. Since the action of
$\Gamma _2$
on V is trivial, by assumption, it follows from (3.1) that
$[f_{\varphi }(h)](k)= \varphi (h)$
for all
$k, h\in \Gamma _2$
. The assertion then follows from Lemma 3.2.
Proposition 3.1 and (4.2) give us an exact sequence of abelian groups:
$$ \begin{align*}0\rightarrow B^1(\Gamma_1, \operatorname{\mathrm{Ind}}^{\Gamma_1}_{\Gamma_2} V) \rightarrow Z^1(\Gamma_1, \operatorname{\mathrm{Ind}}^{\Gamma_1}_{\Gamma_2} V)\rightarrow \operatorname{\mathrm{Hom}}(\Gamma_2, V)\rightarrow 0,\end{align*} $$
and Lemma 3.2 gives us a section so that
$$ \begin{align} Z^1(\Gamma_1, \operatorname{\mathrm{Ind}}^{\Gamma_1}_{\Gamma_2} V)\cong B^1(\Gamma_1, \operatorname{\mathrm{Ind}}^{\Gamma_1}_{\Gamma_2} V)\oplus \operatorname{\mathrm{Hom}}(\Gamma_2, V). \end{align} $$
We want to understand the action of
$\Delta $
on
$B^1(\Gamma _1, \operatorname {\mathrm {Ind}}^{\Gamma _1}_{\Gamma _2} V)\oplus \operatorname {\mathrm {Hom}}(\Gamma _2, V)$
via this isomorphism. It follows from (4.1) that
$[[d\ast \Phi ](g)](dc) = [\Phi (g)](c)$
. Hence,
$$ \begin{align*}[d\ast \Phi_{\varphi}](g)=\sum_{c\in \Delta} \varphi(\overline{c} g \overline{cg}^{-1})\mathbf 1_{dc}\end{align*} $$
for
$\varphi \in \operatorname {\mathrm {Hom}}(\Gamma _2, V)$
. Since
$\Gamma _2$
is normal in
$\Gamma _1$
,
$c k = c$
for all
$k\in \Gamma _2$
and all
$c\in \Delta $
. Hence,
$\overline {ck}= \overline {c}$
, and we conclude that
We note that
$\varphi ((\overline {d} h)^{-1} k (\overline {d} h))= \varphi (\overline {d}^{-1} k \overline {d})$
for all
$h,k\in \Gamma _2$
, which justifies the third equality, and we take the last equality as the definition
$d\cdot \varphi $
. Since
$[[\Phi _{d\cdot \varphi }](k)](1)=[d\cdot \varphi ](k)$
by Lemma 3.2, Proposition 3.1 implies that the cocycle
$d\ast \Phi _{\varphi } - \Phi _{d\cdot \varphi }$
is a
$1$
-coboundary, and thus, there is a function
$f:\Delta \rightarrow V$
depending on d such that for all
$g\in \Gamma _1$
, we have
$$ \begin{align} [d\ast \Phi_{\varphi}](g) - \Phi_{d\cdot \varphi}(g)= (g-1) f. \end{align} $$
After subtracting a constant function, we may assume that
$f(1)=0$
. By evaluating both sides of (4.4) at
$1$
, we obtain
$$ \begin{align} f(g) &= f(g)-f(1)= [[d\ast \Phi_{\varphi}](g)](1) - [\Phi_{d\cdot \varphi}(g)](1) \nonumber\\ &= \varphi( \overline{d^{-1}} g \overline{d^{-1}g}^{-1}) - [d\cdot\varphi](g \overline{g}^{-1}) \nonumber\\ &=\varphi( \overline{d^{-1}} g \overline{d^{-1}g}^{-1}) - \varphi( \overline{d^{-1}} g \overline{g}^{-1} \overline{d^{-1}}^{-1})\\ &=\varphi( (\overline{d^{-1}} g \overline{g}^{-1} \overline{d^{-1}}^{-1})^{-1} \overline{d^{-1}} g \overline{d^{-1}g}^{-1})\nonumber\\ &=\varphi( \overline{d^{-1}} \overline{g} \overline{d^{-1} g}^{-1}).\nonumber \end{align} $$
We note that in the above calculation,
$\overline {g}$
is the fixed coset representative of the coset
$\Gamma _2 g$
. In particular,
$\overline {g}\in \Gamma _1$
and not in
$\Delta $
. The function f satisfies
$f(kg)=f(g)$
for all
$k\in \Gamma _2$
, so we may consider as a function on
$\Delta $
. In this case, the formula (4.5) applied with
$g=\overline {c}$
gives us
$$ \begin{align} f(c)= \varphi( \overline{d^{-1}} \overline{c} \overline{d^{-1} c}^{-1}), \quad \forall c\in \Delta. \end{align} $$
The function f corresponds to a group homomorphism
$\alpha _f: \mathbb Z[\Delta ]\rightarrow V$
, given by
$\alpha _f(c)= f(c)$
. Since
$f(1)=0$
, the restriction of
$\alpha _f$
to the augmentation ideal
$I_{\Delta }$
is given by
$$ \begin{align} \alpha_f( c-1) = \varphi( \overline{d^{-1}} \overline{c} \overline{d^{-1} c}^{-1}), \quad \forall c\in \Delta. \end{align} $$
Since V is abelian, we have
$\operatorname {\mathrm {Hom}}(\Gamma _2, V)= \operatorname {\mathrm {Hom}}(\Gamma _2^{\mathrm {ab}}, V)$
, where
$\Gamma _2^{\mathrm {ab}}$
is the maximal abelian quotient of
$\Gamma _2$
. For
$d, c\in \Delta $
, we define
$\kappa (d, c)$
to be the image of
$\overline {d} \overline {c} \overline {dc}^{-1}$
in
$\Gamma _2^{\mathrm {ab}}$
. The map
$\kappa : \Delta \times \Delta \rightarrow \Gamma _2^{\mathrm {ab}}$
is the
$2$
-cocycle corresponding to the extension
$$ \begin{align*} 0\rightarrow \Gamma_2^{\mathrm{ab}}\rightarrow \Gamma_1/[\Gamma_2, \Gamma_2] \rightarrow \Delta\rightarrow 0. \end{align*} $$
We define an action of
$\Delta $
on
$\Gamma _2^{\mathrm {ab}}\oplus I_{\Delta }$
by letting
$$ \begin{align} d\ast ( g, c-1): = ( d g d^{-1} \cdot \kappa(d, c), d c - d). \end{align} $$
This action defines an action of
$\Delta $
on
$\operatorname {\mathrm {Hom}}(\Gamma _2^{\mathrm {ab}}\oplus I_{\Delta }, V)$
by
$$ \begin{align} [ d\ast (\varphi, \alpha)]( g, c-1)&= (\varphi, \alpha) ( d^{-1}\ast (g, c-1)) \nonumber\\ &= \varphi(d^{-1} g d) + \varphi(\kappa(d^{-1}, c))+ \alpha( d^{-1}c -d^{-1}). \end{align} $$
We obtain an exact sequence of
$\mathbb Z[\Delta ]$
-modules
$$ \begin{align} 0\rightarrow \Gamma_2^{\mathrm{ab}} \rightarrow \mathcal E \rightarrow I_{\Delta}\rightarrow 0 \end{align} $$
by letting
$\mathcal E= \Gamma _2^{\mathrm {ab}}\oplus I_{\Delta }$
with the
$\Delta $
-action defined as above.
Remark 4.2. Using the exact sequence
$0\rightarrow I_{\Delta }\rightarrow \mathbb Z[\Delta ]\rightarrow \mathbb Z\rightarrow 0$
of
$\mathbb Z[\Delta ]$
-modules, one obtains a canonical identification
$$ \begin{align*}\operatorname{\mathrm{Ext}}^1_{\mathbb Z[\Delta]}(I_{\Delta}, \Gamma_2^{\mathrm{ab}}) \cong \operatorname{\mathrm{Ext}}^2_{\mathbb Z[\Delta]}(\mathbb Z, \Gamma_2^{\mathrm{ab}})\cong H^2(\Delta, \Gamma_2^{\mathrm{ab}});\end{align*} $$
see [Reference Cartan and Eilenberg4, Chapter XIV, §4, Remark]. The extension class of (4.10) gives an element in
$\operatorname {\mathrm {Ext}}^1_{\mathbb Z[\Delta ]}( I_{\Delta },\Gamma _2^{\mathrm {ab}})$
. One can show that the image of this class in
$H^2(\Delta , \Gamma _2^{\mathrm {ab}})$
is equal to the class of
$\kappa $
.
To
$\alpha \in \operatorname {\mathrm {Hom}}(I_{\Delta }, V)$
, we may associate a function
$f_{\alpha }: \Delta \rightarrow V$
by letting 
. We note that
$f_{\alpha }(1)=0$
. We then define a
$1$
-coboundary
$b_{\alpha }\in B^1(\Gamma _1, \operatorname {\mathrm {Ind}}^{\Gamma _1}_{\Gamma _2} V)$
by 
. We may recover
$f_{\alpha }$
from
$b_{\alpha }$
as the unique function
$f:\Delta \rightarrow V$
satisfying
$f(1)=0$
and
$b_{\alpha }(g)= (g-1)f$
for all
$g\in \Gamma _1$
. Thus, the map
$$ \begin{align} \operatorname{\mathrm{Hom}}(I_{\Delta}, V) \rightarrow B^1(\Gamma_1, \operatorname{\mathrm{Ind}}^{\Gamma_1}_{\Gamma_2} V), \quad \alpha \mapsto b_{\alpha} \end{align} $$
is an isomorphism. We now record the consequence of our calculations:
Proposition 4.3. If V is an abelian group with the trivial
$\Gamma _2$
-action, then sending
$(\varphi , \alpha )$
to
$\Phi _{\varphi }+b_{\alpha }$
induces an isomorphism
$$ \begin{align*}\operatorname{\mathrm{Hom}} ( \Gamma_2^{\mathrm{ab}}\oplus I_{\Delta}, V) \overset{\cong}{\longrightarrow} Z^1(\Gamma_1, \operatorname{\mathrm{Ind}}^{\Gamma_1}_{\Gamma_2} V),\end{align*} $$
which is
$\Delta $
-equivariant for the actions defined in (4.9) and (4.1).
Proof. It follows from (4.11) and (4.3) that the map is an isomorphism of abelian groups. We have to check that it is
$\Delta $
-equivariant. We have
$$ \begin{align*} [[d\ast b_{\alpha}](g)](c)&= [b_{\alpha}(g)](d^{-1} c)= f_{\alpha}(d^{-1}c g)- f_{\alpha}(d^{-1}c)\\ &= \alpha( d^{-1}c g -1) - \alpha( d^{-1}c -1)=\alpha( d^{-1}c g - d^{-1} c), \end{align*} $$
$$ \begin{align*} [b_{d\ast \alpha}(g)](c)&=f_{d\ast \alpha}(cg) - f_{d\ast\alpha}(c)= [d\ast \alpha](cg-1) - [d\ast \alpha]( c-1)\\ &= \alpha(d^{-1} cg -d^{-1}) - \alpha(d^{-1}c -d^{-1})=\alpha(d^{-1} cg -d^{-1}c). \end{align*} $$
Hence,
$d\ast b_{\alpha } = b_{d\ast \alpha }$
. It follows from (4.4), (4.6) and (4.7) that
$d\ast \Phi _{\varphi } = \Phi _{d\cdot \varphi } + b_{\beta }$
, where
$\beta ( c-1)= \varphi (\kappa (d^{-1}, c))$
for all
$c\in \Delta $
. Moreover, it follows from (4.9) that
$d\ast (\varphi , 0) = (d\cdot \varphi , \beta )$
. Hence,
$d\ast (\varphi , 0)$
is mapped to
$\Phi _{d\cdot \varphi } + b_{\beta } = d\ast \Phi _{\varphi }$
.
Recall that M is an abelian group with an action of
$\Delta $
, so
$\Gamma _2$
acts trivially on
$\operatorname {\mathrm {Hom}}(M, V)$
. We have natural isomorphisms of
$\Gamma _1$
-representations
$$ \begin{align} \operatorname{\mathrm{Ind}}^{\Gamma_1}_{\Gamma_2} \operatorname{\mathrm{Hom}}( M, V) \cong \operatorname{\mathrm{Hom}} ( \mathbb Z[\Delta], \operatorname{\mathrm{Hom}}(M, V)) \cong \operatorname{\mathrm{Hom}}(\mathbb Z[\Delta]\otimes M, V), \end{align} $$
where the action of
$\Gamma _1$
on the last term is induced by its action on
$\mathbb Z[\Delta ]\otimes M$
, which is given by
$g \cdot (c \otimes m) = cg^{-1} \otimes m$
, so that
$[g \cdot f](x)=f(g^{-1}x)$
for all
$x\in \mathbb Z[\Delta ]\otimes M$
. The diagonal action of
$\Delta $
on
$\mathbb Z[\Delta ]\otimes M$
is given by
$d\cdot (c\otimes m) = dc \otimes dm$
, commutes with the action of
$\Gamma _1$
and induces a (left) action of
$\Delta $
on
$\operatorname {\mathrm {Ind}}^{\Gamma _1}_{\Gamma _2} \operatorname {\mathrm {Hom}}( M, V)$
, which commutes with the action of
$\Gamma _1$
. If
$M=\mathbb Z$
with the trivial
$\Delta $
-action, then the construction recovers the action of
$\Delta $
on
$\operatorname {\mathrm {Ind}}^{\Gamma _1}_{\Gamma _2} V$
considered earlier.
Lemma 4.4. As
$\Gamma _1$
-representations,
$(\operatorname {\mathrm {Ind}}^{\Gamma _1}_{\Gamma _2} \operatorname {\mathrm {Hom}}( M, V))^{\Delta }$
is naturally isomorphic to
$\operatorname {\mathrm {Hom}}(M, V)$
with
$\Gamma _1$
acting on it via
$\Delta $
.
Proof. The map
$\mathbb Z[\Delta ]\otimes M \rightarrow \mathbb Z[\Delta ]\otimes M$
,
$c\otimes m \mapsto c\otimes c^{-1}m$
is an isomorphism of
$\Delta \times \Gamma _1$
-representations, where on the source, the action is the one considered above, and on the target,
$\Delta $
acts by
$d \ast ( c\otimes m) = dc \otimes m$
and
$\Gamma _1$
acts by
$g \ast (c\otimes m) = c g^{-1} \otimes g m$
. It is immediate by considering the
$\ast $
-action that
$(\mathbb Z[\Delta ]\otimes M)_{\Delta } \cong M$
and
$\Gamma _1$
acts on M via
$\Delta $
. Using (4.12), we can then translate this statement to a statement about invariants.
Proposition 4.5. Let
$\mathcal E$
be the
$\mathbb Z[\Delta ]$
-module constructed above. Then for all abelian groups V, we have an isomorphism of abelian groups functorial in V:
$$ \begin{align} \operatorname{\mathrm{Hom}}( (\mathcal E\otimes M)_{\Delta}, V) \cong Z^1(\Gamma_1, \operatorname{\mathrm{Hom}}(M, V)), \end{align} $$
where the action of
$\Delta $
on
$\mathcal E\otimes M$
is diagonal and
$\Gamma _1$
acts on
$\operatorname {\mathrm {Hom}}(M, V)$
via
$\Delta $
.
Proof. Proposition 4.3 gives us
$\Delta $
-equivariant isomorphisms:
$$ \begin{align*} \operatorname{\mathrm{Hom}}( \mathcal E\otimes M, V) \cong \operatorname{\mathrm{Hom}} ( \mathcal E, \operatorname{\mathrm{Hom}}(M, V)) \cong Z^1(\Gamma_1, \operatorname{\mathrm{Ind}}^{\Gamma_1}_{\Gamma_2} \operatorname{\mathrm{Hom}}(M, V)). \end{align*} $$
The assertion follows after taking
$\Delta $
-invariants and applying Lemma 4.4.
The exact sequence
$0\rightarrow I_{\Delta }\rightarrow \mathbb Z[\Delta ]\rightarrow \mathbb Z\rightarrow 0$
remains exact after tensoring with M. Taking
$\Delta $
-coinvariants yields an exact sequence
$$ \begin{align*} 0\rightarrow H_1(\Delta, M)\rightarrow (I_{\Delta}\otimes M)_{\Delta} \rightarrow M \rightarrow M_{\Delta}\rightarrow 0. \end{align*} $$
This gives us a surjection
$(I_{\Delta }\otimes M)_{\Delta } \twoheadrightarrow I_{\Delta } M$
. By composing this map with the surjection
$(\mathcal E \otimes M)_{\Delta } \twoheadrightarrow (I_{\Delta }\otimes M)_{\Delta }$
induced by (4.10), we obtain a surjection
$(\mathcal E\otimes M)_{\Delta }\twoheadrightarrow I_{\Delta } M$
. This in turn yields an injection
$$ \begin{align*}\operatorname{\mathrm{Hom}}(I_{\Delta} M, V)\hookrightarrow \operatorname{\mathrm{Hom}}((\mathcal E\otimes M)_{\Delta}, V).\end{align*} $$
Lemma 4.6. The isomorphism (4.13) in Proposition 4.5 identifies the space of
$1$
-coboundaries
$B^1(\Gamma _1, \operatorname {\mathrm {Hom}}(M, V))$
with a subgroup of
$\operatorname {\mathrm {Hom}}(I_{\Delta } M, V)$
. Moreover, it induces an isomorphism between the two groups if V is divisible.
Proof. The isomorphism
$(\mathbb Z[\Delta ]\otimes M)_{\Delta }\cong M$
for the diagonal action of
$\Delta $
on
$\mathbb Z[\Delta ]\otimes M$
is realised by the map
$c\otimes m \mapsto c^{-1} m$
. The image of
$I_{\Delta }\otimes M$
under this map is equal to
$I_{\Delta }M$
. Thus, the isomorphism
$$ \begin{align*}\vartheta: \operatorname{\mathrm{Hom}}(M, V) \overset{\cong}{\longrightarrow} (\operatorname{\mathrm{Ind}}^{\Gamma_1}_{\Gamma_2} \operatorname{\mathrm{Hom}}(M,V))^{\Delta}\end{align*} $$
in Lemma 4.4 is given explicitly by
$$ \begin{align*}[\vartheta(\alpha)(c)](m)=\alpha(c^{-1}m), \quad \forall c\in \Delta, \quad \forall m\in M .\end{align*} $$
Let
$b\in B^1(\Gamma _1, \operatorname {\mathrm {Hom}}(M,V))$
be a boundary with
$b(g)=(g-1)\alpha $
for all
$g\in \Gamma _1$
. Its image in
$B^1(\Gamma _1, \operatorname {\mathrm {Ind}}^{\Gamma _1}_{\Gamma _2}\operatorname {\mathrm {Hom}}(M,V))$
is the boundary
$b'$
given by
$b'(g)= (g-1) \vartheta (\alpha )$
. The constant function
$f'(c)=\alpha $
for all
$c\in \Delta $
is
$\Gamma _1$
-invariant, and thus,
$$ \begin{align*}b'(g)= (g-1)( \vartheta(\alpha) - f'), \quad \forall g\in \Gamma_1.\end{align*} $$
Since
$(\vartheta (\alpha )-f')(1)= \alpha -\alpha =0$
, by Proposition 4.3,
$b'=b_{\beta }$
, where
$\beta : I_{\Delta }\rightarrow \operatorname {\mathrm {Hom}}(M, V)$
is given by
$$ \begin{align*}[\beta(c-1)](m)= [(\vartheta(\alpha)- f')(c)](m)= \alpha(c^{-1}m)-\alpha(m)= \alpha(c^{-1}m -m).\end{align*} $$
We conclude that the image of
$B^1(\Gamma _1, \operatorname {\mathrm {Hom}}(M,V))$
under the isomorphism in Proposition 4.5 is contained in
$\operatorname {\mathrm {Hom}}(I_{\Delta }M, V)$
.
Conversely, if we start with a homomorphism
$\beta ': I_{\Delta }M \rightarrow V$
, then by pulling it back under the surjection
$I_{\Delta }\otimes M \twoheadrightarrow I_{\Delta }M$
, we obtain a homomorphism
$\beta : I_{\Delta }\otimes M \rightarrow V$
given by
$\beta ( (c-1)\otimes m)= \beta '(c^{-1}m -m)$
. The corresponding coboundary
$b_{\beta } \in B^1(\Gamma _1, \operatorname {\mathrm {Ind}}^{\Gamma _1}_{\Gamma _2}\operatorname {\mathrm {Hom}}(M,V))$
is given by
$b_{\beta }(g)= (g-1)f_{\beta }$
, where
$$ \begin{align*}[f_{\beta}(c)](m)=\beta((c-1)\otimes m)= \beta'(c^{-1} m -m).\end{align*} $$
If V is divisible, then V is injective in
$\operatorname {\mathrm {Ab}}$
and the exact sequence
$$ \begin{align*}0\rightarrow I_{\Delta}M \rightarrow M \rightarrow M_{\Delta}\rightarrow 0\end{align*} $$
gives rise to an exact sequence
$$ \begin{align*}0\rightarrow \operatorname{\mathrm{Hom}}(M_{\Delta}, V)\rightarrow \operatorname{\mathrm{Hom}}(M, V)\rightarrow \operatorname{\mathrm{Hom}} (I_{\Delta}M, V)\rightarrow 0,\end{align*} $$
and we conclude that there is
$\alpha \in \operatorname {\mathrm {Hom}}(M, V)$
mapping to
$\beta '$
, so that
$$ \begin{align*}\beta'(c^{-1} m -m)=\alpha(c^{-1}m) - \alpha(m).\end{align*} $$
Let
$f'$
be the constant function
$f'(c)=\alpha $
for all
$c\in \Delta $
. Then the boundary
$g\mapsto (g-1)(f_{\beta }+f')$
is equal to
$b_{\beta }$
and
$f_{\beta }+f'= \vartheta (\alpha )$
.
By taking
$V= (\mathcal E\otimes M)_{\Delta }$
in Proposition 4.5, we obtain a natural cocycle
$$ \begin{align*}\Phi_{\mathrm{nat}}\in Z^1(\Gamma_1, \operatorname{\mathrm{Hom}}(M, (\mathcal E\otimes M)_{\Delta})),\end{align*} $$
which corresponds to the identity in
$\operatorname {\mathrm {Hom}}( (\mathcal E\otimes M)_{\Delta }, (\mathcal E\otimes M)_{\Delta })$
.
Lemma 4.7. If V is any abelian group, then for all
$\Gamma _1$
-modules N and all
$i\ge 0$
, we have a canonical map
$$ \begin{align} H^i(\Gamma_1, \operatorname{\mathrm{Hom}}(N, V)) \rightarrow \operatorname{\mathrm{Hom}}( H_i(\Gamma_1, N), V), \end{align} $$
which is an isomorphism if V is divisible.
Proof. If
$(C_{\bullet }, d_{\bullet })$
is a complex of abelian groups, then we let
$Z_n = \ker (d_n:C_n\rightarrow C_{n-1})$
and
$Z^n=\ker (d_{n+1}^{\ast }: \operatorname {\mathrm {Hom}}(C_n, V)\rightarrow \operatorname {\mathrm {Hom}}(C_{n+1}, V))$
. The evaluation pairing
$Z_n \times Z^n\rightarrow V$
induces a bilinear map
$H_n(C_{\bullet })\times H^n(\operatorname {\mathrm {Hom}}(C_{\bullet }, V))\rightarrow V$
, which induces a canonical map
$H^n(\operatorname {\mathrm {Hom}}(C_{\bullet }, V))\rightarrow \operatorname {\mathrm {Hom}} (H_n(C_{\bullet }), V)$
. If V is divisible, then V is injective in
$\operatorname {\mathrm {Ab}}$
and the map is an isomorphism.
Let
$P_{\bullet }\twoheadrightarrow \mathbb Z$
be a resolution of
$\mathbb Z$
by projective
$\mathbb Z[\Gamma _1]$
-modules. The complex
$\operatorname {\mathrm {Hom}}_{\Gamma _1}(P_{\bullet }, \operatorname {\mathrm {Hom}}(N, V))\cong \operatorname {\mathrm {Hom}} ( (P_{\bullet }\otimes N)_{\Gamma _1}, V)$
computes the cohomology groups
$H^i(\Gamma _1, \operatorname {\mathrm {Hom}}(N, V))$
. The complex
$(P_{\bullet }\otimes N)_{\Gamma _1}$
computes the homology groups
$H_i(\Gamma _1, N)$
. We apply the previous discussion to 
to obtain the required homomorphism.
Applying Lemma 4.7 with
$N=M$
and
$V=(\mathcal E\otimes M)_{\Delta }$
, we obtain a homomorphism
$\varphi _{\mathrm {nat}}: H_1(\Gamma _1, M)\rightarrow (\mathcal E \otimes M)_{\Delta }$
corresponding to the cohomology class
$[\Phi _{\mathrm {nat}}]$
.
Lemma 4.8. For all abelian groups V, the composition
$$ \begin{align} \operatorname{\mathrm{Hom}}((\mathcal E \otimes M)_{\Delta}, V) & \overset{({4.13})}{\longrightarrow} Z^1(\Gamma_1, \operatorname{\mathrm{Hom}}(M, V))\twoheadrightarrow \nonumber\\ &H^1(\Gamma_1, \operatorname{\mathrm{Hom}}(M, V)) \xrightarrow{({4.14})} \operatorname{\mathrm{Hom}}(H_1(\Gamma_1, M), V) \end{align} $$
is given by
$\psi \mapsto \psi \circ \varphi _{\mathrm {nat}}$
.
Proof. This follows from the fact that all our constructions are functorial in V. If we denote the four functors appearing in (4.15) with A, B, C, D, then for a homomorphism of abelian groups
$\psi :W\rightarrow V$
, we obtain a diagram:
with commutative squares. If we take
$W=(\mathcal E \otimes M)_{\Delta }$
, then the identity map in
$A((\mathcal E \otimes M)_{\Delta })$
maps to
$\varphi _{\mathrm {nat}}$
in
$D((\mathcal E \otimes M)_{\Delta })$
by construction and hence to
$\psi \circ \varphi _{\mathrm {nat}}$
in
$D(V)$
. Since the identity maps to
$\psi $
in
$A(V)$
, we obtain the assertion.
Proposition 4.9. There is an exact sequence of abelian groups
$$ \begin{align} 0\rightarrow H_1(\Gamma_1, M) \xrightarrow{\varphi_{\mathrm{nat}}} (\mathcal E \otimes M)_{\Delta} \xrightarrow{q} I_{\Delta} M\rightarrow 0 \end{align} $$
which is functorial in M, and for all abelian groups V, the diagram
commutes.
Proof. Lemmas 4.6 and 4.8 give us the commutative diagram above. Moreover, if V is divisible, then by Lemma 4.7, the top row is exact, the last arrow in the top row is surjective and the first vertical arrow is an isomorphism. We deduce that for all divisible V, the maps
$q:(\mathcal E\otimes M)_{\Delta }\twoheadrightarrow I_{\Delta }M$
and
$\varphi _{\mathrm {nat}}: H_1(\Gamma , M)\rightarrow (\mathcal E\otimes M)_{\Delta }$
induce an exact sequence
$$ \begin{align} 0\rightarrow \operatorname{\mathrm{Hom}}(I_{\Delta}M, V)\rightarrow \operatorname{\mathrm{Hom}}((\mathcal E\otimes M)_{\Delta}, V)\rightarrow \operatorname{\mathrm{Hom}}(H_1(\Gamma, M), V)\rightarrow 0. \end{align} $$
Since divisible groups are precisely injective objects in
$\operatorname {\mathrm {Ab}}$
, which has enough injectives, an abelian group A is zero if and only if
$\operatorname {\mathrm {Hom}}(A, V)=0$
for all divisible groups V. Using this and the exactness of (4.17), we obtain that
$\mathrm {Im}(q\circ \varphi _{\mathrm {nat}})=0$
; thus, (4.16) is a complex and a further application of the same argument shows that (4.16) is exact.
Corollary 4.10. The functor
$\operatorname {\mathrm {CRing}} \rightarrow \operatorname {\mathrm {Ab}}$
given by
$$ \begin{align*}A\mapsto Z^1(\Gamma_1, \operatorname{\mathrm{Hom}}(M, A^{\times}))\end{align*} $$
is represented by the group algebra
$\mathbb Z[(\mathcal E\otimes M)_{\Delta }]$
.
Proof. If W is an abelian group, then we may identify W with a subgroup of units of the group ring
$\mathbb Z[W]$
. The map
$$ \begin{align*} \operatorname{\mathrm{Hom}}_{\operatorname{\mathrm{CRing}}}(\mathbb Z[W], A) \rightarrow \operatorname{\mathrm{Hom}}( W, A^{\times}), \quad \psi \mapsto \psi|_W \end{align*} $$
is an isomorphism. Applying this observation to
$W=(\mathcal E \otimes M)_{\Delta }$
and using Proposition 4.5 yields the assertion.
Corollary 4.11. If M is a free
$\mathbb Z$
-module of finite rank, then
$$ \begin{align} \mathbb Z[(\mathcal E\otimes M)_{\Delta}]\cong \mathbb Z[H_1(\Gamma_1, M)] [t_1^{\pm 1}, \ldots, t_s^{\pm 1}], \end{align} $$
where
$s =\mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M -\mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M_{\Delta }$
.
Proof. Since
$I_{\Delta } M$
is a submodule of M, which is free of finite rank over
$\mathbb Z$
, we have an isomorphism
$I_{\Delta } M \cong \mathbb Z^{s}$
, where s is as above. By choosing a section to the surjection
$(\mathcal E\otimes M)_{\Delta }\twoheadrightarrow I_{\Delta }M$
in (4.16), we obtain an isomorphism
$(\mathcal E\otimes M)_{\Delta } \cong H_1(\Gamma _1, M) \times \mathbb Z^s$
, and this implies the assertion.
Proposition 4.12. The map (4.14) induces a map of presheaves
$\operatorname {\mathrm {CRing}} \to \mathrm { Set}$
$$ \begin{align} H^1(\Gamma_1, \operatorname{\mathrm{Hom}}(M, (-)^{\times})) \to \operatorname{\mathrm{Spec}}(\mathbb Z[H_1(\Gamma_1,M)]) \end{align} $$
which exhibits the right-hand side as the fpqc sheafification of the left-hand side.
Proof. The right-hand side is an fpqc sheaf since it is representable [Reference Project15, Tag 023Q]. So it suffices to find for every ring A and every homomorphism
${f : \mathbb Z[H_1(\Gamma _1,M)] \to A}$
an fpqc cover of A and a descent datum in the left-hand side, which maps to a descent datum in the right-hand side, which descends to f. By [Reference Zou17, Lemma 4.1.1], for every ring A, there is a faithfully flat map
$A \to B$
, such that
$B^{\times }$
is divisible. By Lemma 4.7, the map
$$ \begin{align*}H^1(\Gamma_1, \operatorname{\mathrm{Hom}}(M, B^{\times})) \rightarrow \operatorname{\mathrm{Hom}}_{\operatorname{\mathrm{CRing}}}(\mathbb Z[H_1(\Gamma_1, M)], B)\end{align*} $$
is bijective. So the canonical descent datum associated with f and the map
$A \to B$
come from a descent datum in the right-hand side.
5 Admissible representations
In this section, let
$\mathcal O$
be an arbitrary commutative ring, such that
$\operatorname {\mathrm {Spec}} \mathcal O$
is connected. Let G be a generalised torus over
$\mathcal O$
, such that
$G^0$
is split and such that
$G/G^0$
is constant. Let
$\Pi : G \to G/G^0$
be the projection map. We define the character lattice of
$G^0$
by
We may identify
$G^0$
with the split torus
$\mathfrak D(M)$
introduced in Section 2. We can write
$G/G^0 = \underline \Delta $
for a finite group
$\Delta $
, where 
. For any
$\mathcal O$
-algebra A, we have a natural map
$\Delta \to \underline \Delta (A) = \operatorname {\mathrm {Hom}}_{\mathcal O\text {-}\mathrm {alg}}(\mathrm {Map}(\Delta ,\mathcal O), A)$
defined by evaluation. The
$\mathcal O$
-group scheme G acts on
$G^0$
by conjugation, and this action factors over
$G/G^0 = \underline \Delta $
. So we have a well-defined action map
$G/G^0 \times G^0 \to G^0, ~(g,h) \mapsto ghg^{-1}$
with the property that for every
$\mathcal O$
-algebra A and every
$g \in (G/G^0)(A)$
, the map
$G^0(A) \to G^0(A), ~h \mapsto g h g^{-1}$
is a group automorphism of
$G^0(A)$
. This induces a (left) action of
$\Delta $
on M via (5.1).
Let
$\Gamma _1$
be an abstract group and let
$\pi : \Gamma _1 \twoheadrightarrow \Delta $
be a surjective homomorphism with kernel
$\Gamma _2$
. So we are in the situation of Section 4.
Definition 5.1. We say that a representation
$\rho : \Gamma _1\rightarrow G(A)$
is admissible if
$\Pi \circ \rho : \Gamma _1 \to \underline {\Delta }(A)$
sends
$\gamma \in \Gamma _1$
to the image of
$\pi (\gamma ) \in \Delta $
in
$\underline \Delta (A)$
.
This terminology is motivated by admissible Galois representations into L-groups appearing in the Langlands correspondence; see [Reference Borel3, §9].
Let
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(A)$
be the set of admissible representations
$\rho : \Gamma _1\rightarrow G(A)$
. The group
$\mathfrak D(M)(A)$
acts on
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(A)$
by conjugation. This defines a scheme-theoretic action
$$ \begin{align} \mathfrak D(M) \times \operatorname{\mathrm{Rep}}^{\Gamma_1}_{G,\pi}\rightarrow \operatorname{\mathrm{Rep}}^{\Gamma_1}_{G,\pi}. \end{align} $$
Proposition 5.2. Assume there is a representation
$\rho _0 \in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(\mathcal O)$
. Then
$$ \begin{align} Z^1(\Gamma_1, \mathfrak D(M)(A)) \to \operatorname{\mathrm{Rep}}^{\Gamma_1}_{G,\pi}(A), \quad \Phi \mapsto [\gamma \mapsto \Phi(\gamma) \rho_0(\gamma)] \end{align} $$
is a bijection, which is natural in
$A \in \mathcal O\text {-}\mathrm {alg}$
. In particular,
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }$
is representable by an
$\mathcal O$
-algebra, and (5.3) induces a natural isomorphism
$$ \begin{align} \mathcal O(\operatorname{\mathrm{Rep}}^{\Gamma_1}_{G,\pi}) \xrightarrow{\cong} \mathcal O[(\mathcal E\otimes M)_{\Delta}]. \end{align} $$
Moreover, (5.3) is
$\mathfrak D(M)(A)$
-equivariant and induces a natural bijection between the set of
$\mathfrak D(M)(A)$
-orbits in
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(A)$
and
$H^1(\Gamma _1, \mathfrak D(M)(A))$
. In particular, (5.4) is
$\mathfrak D(M)$
-equivariant.
Proof. For
$\Phi \in Z^1(\Gamma _1, \mathfrak D(M)(A))$
, we have
$$ \begin{align} \Phi(\gamma_1\gamma_2) \rho_0(\gamma_1\gamma_2) &= \Phi(\gamma_1){}^{\gamma_1}\Phi(\gamma_2)\rho_0(\gamma_1)\rho_0(\gamma_2) \nonumber\\ &= \Phi(\gamma_1) \rho_0(\gamma_1) \Phi(\gamma_2)\rho_0(\gamma_2), \end{align} $$
where in the last equality we use that
$\rho _0$
is admissible. Thus,
$\Phi \rho _0$
is a homomorphism. By applying the projection
$\Pi : G(A) \to \underline \Delta (A)$
, we verify that
$\Phi \rho _0 \in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(A)$
. It is clear that (5.3) is a bijection
$\mathrm {Map}(\Gamma _1, G(A)) \to \mathrm {Map}(\Gamma _1, G(A))$
with inverse
$\rho \mapsto \rho \rho _0^{-1}$
. If
$\rho \in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(A)$
, we see that 
is a
$1$
-cocycle by reverting the computation in (5.5). The claim about representability follows from Corollary 4.10.
The
$\mathfrak D(M)(A)$
-conjugacy classes are the
$B^1(\Gamma _1, \mathfrak D(M)(A))$
-orbits: if we write an admissible homomorphism
$\rho : \Gamma _1 \to G(A)$
as above as
$\rho (\gamma ) = \Phi (\gamma )\rho _0(\gamma )$
and
$g \in \mathfrak D(M)(A)$
, then
$$ \begin{align*} g\rho(\gamma) g^{-1} &= g\Phi(\gamma) ({}^{\gamma}g^{-1}) \rho_0(\gamma) \\ &= \beta_g(\gamma) \Phi(\gamma) \rho_0(\gamma), \end{align*} $$
where
$\beta _g \in B^1(\Gamma _1, \mathfrak D(M)(A))$
is the coboundary
$\beta _g(\gamma )= g ({}^{\gamma }g^{-1})$
. It follows that the
$\mathfrak D(M)(A)$
-orbits are the classes in
$H^1(\Gamma _1, \mathfrak D(M)(A))$
.
Remark 5.3. If
$G = G^0 \,{\rtimes}\, \underline \Delta $
, then the composition of
$\pi $
with the natural map
$\Delta \to \underline \Delta (\mathcal O) \to G^0(\mathcal O) \,{\rtimes}\, \underline \Delta (\mathcal O) = G(\mathcal O)$
is a canonical choice for a representation
$\rho _0 \in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(\mathcal O)$
. In general, the isomorphisms (5.3) and (5.4) depend on
$\rho _0$
.
Proposition 5.4. Assume there is a representation
$\rho _0 \in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(\mathcal O)$
. Under the isomorphism (5.3), the action (5.2) corresponds to the ring homomorphism
$$ \begin{align*}\mathcal O[(\mathcal E\otimes M)_{\Delta}]\rightarrow \mathcal O[(\mathcal E\otimes M)_{\Delta}]\otimes \mathcal O[M],\end{align*} $$
which sends
$x\in (\mathcal E\otimes M)_{\Delta }$
to
$x\otimes q(x)$
, where
$q: (\mathcal E\otimes M)_{\Delta }\twoheadrightarrow I_{\Delta }M$
is the map of (4.16).
Proof. We have seen in the proof of Proposition 5.2 that the action of
$\mathfrak D(M)$
on
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }$
corresponds under the isomorphism (5.3) to the action via the boundary map
$$ \begin{align*}\mathfrak D(M)(A) \to Z^1(\Gamma_1, \mathfrak D(M)(A))\end{align*} $$
and group multiplication. By commutativity of the left square in Proposition 4.9, this map is induced by the composition
$(\mathcal E \otimes M)_{\Delta } \xrightarrow {q} I_{\Delta }M \to M$
. As the comultiplication on
$\mathcal O[(\mathcal E \otimes M)_{\Delta }]$
is given by
$x \mapsto x \otimes x$
for all
$x \in (\mathcal E \otimes M)_{\Delta }$
, the claim follows by composing with
$\mathcal O[(\mathcal E\otimes M)_{\Delta }] \to \mathcal O[M]$
in the second factor.
Corollary 5.5. Assume that there is a representation
$\rho _0 \in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(\mathcal O)$
. The action of
$\mathfrak D(M)$
on
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }$
factors through the action of
$\mathfrak D(I_{\Delta }M)$
, which acts freely.
Proof. As in the proof of Proposition 5.4, we use the composition
$(\mathcal E \otimes M)_{\Delta } \xrightarrow {q} I_{\Delta }M \to M$
to see that the boundary map
$\mathfrak D(M)(A) \to Z^1(\Gamma _1, \mathfrak D(M)(A))$
factors through
$\mathfrak D(I_{\Delta }M)$
.
Let A be any
$\mathcal O$
-algebra and let
$H(A)$
be the group
$\operatorname {\mathrm {Hom}}( (\mathcal E \otimes M)_{\Delta }, A^{\times })$
. Since
$q: (\mathcal E \otimes M)_{\Delta } \rightarrow I_{\Delta }M$
is surjective, we may identify
$\mathfrak D(I_{\Delta }M)(A)$
with a subgroup of
$H(A)$
. The isomorphism (5.4) identifies
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(A)$
with
$H(A)$
, and the action of
$\mathfrak D(I_{\Delta }M)(A)$
on
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(A)$
is identified with the action of
$\mathfrak D(I_{\Delta }M)(A)$
on
$H(A)$
by multiplication, which is a free action. Hence, the action of
$\mathfrak D(I_{\Delta }M)$
on
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }$
is free.
Corollary 5.6. Assume that there is a representation
$\rho _0 \in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(\mathcal O)$
. The isomorphisms (4.19) and (5.3) induce an isomorphism
$$ \begin{align} \operatorname{\mathrm{Rep}}^{\Gamma_1}_{G,\pi} /\mathfrak D(M) \xrightarrow{\cong} \operatorname{\mathrm{Spec}}(\mathcal O[H_1(\Gamma_1, M)]), \end{align} $$
where
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi } /\mathfrak D(M)$
denotes the fpqc sheafification of the presheaf quotient. Moreover this quotient is a GIT quotient. In particular, (5.6) induces isomorphisms
$$ \begin{align} \mathcal O[H_1(\Gamma_1, M)] \xrightarrow{\cong} \mathcal O(\operatorname{\mathrm{Rep}}^{\Gamma_1}_{G,\pi})^{G^0} \xrightarrow{\cong} \mathcal O[(\mathcal E\otimes M)_{\Delta}]^{G^0} , \end{align} $$
and the composition of the maps in (5.7) is induced by
$\varphi _{\mathrm {nat}}$
.
Proof. The isomorphism (5.6) follows directly from Proposition 4.12 and Proposition 5.2. To see that
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi } /\mathfrak D(M)$
is also a GIT quotient, we observe that every
$\mathfrak D(M)$
-equivariant map from
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }$
to an affine scheme equipped with the trivial
$\mathfrak D(M)$
-action factors through
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi } /\mathfrak D(M)$
by the universal property of the fpqc sheaf quotient. Since
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi } /\mathfrak D(M)$
is affine, it is indeed the GIT quotient, and this gives the first isomorphism in (5.7). The second isomorphism in (5.7) comes from (5.4).
Let
$B \in \mathcal O\text {-}\mathrm {alg}$
. After removing the
$G^0$
-invariants in (5.7), we get two maps
$$ \begin{align*}\operatorname{\mathrm{Hom}}((\mathcal E\otimes M)_{\Delta}, B^{\times}) \xrightarrow{\cong} \operatorname{\mathrm{Rep}}_{G,\pi}^{\Gamma_1}(B) \to \operatorname{\mathrm{Hom}}(H_1(\Gamma_1, M), B^{\times}).\end{align*} $$
The first map comes from (4.13) and (5.3). The second map comes from (4.14) and (5.3). Hence, the composition is (4.15) with
$V = B^{\times }$
, so the last assertion follows from Lemma 4.8.
Lemma 5.7. Let
$\tau : G\hookrightarrow \mathbb A^n$
be a closed immersion of
$\mathcal O$
-group schemes and let
$\rho \in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi }(A)$
. Assume that
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi }(\mathcal O)$
is nonempty. Then
$\tau (\rho (\Gamma _1))$
is contained in a finitely generated
$\mathcal O[H_1(\Gamma _1, M)]$
-submodule of
$A^n = \mathbb A^n(A)$
.
Proof. Proposition 5.2 implies that it is enough to prove the statement, when
$A=\mathcal O[(\mathcal E\otimes M)_{\Delta }]$
and
$\rho = \Phi _{\mathrm {nat}}\rho _0$
, where
$\rho _0$
is any representation in
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi }(\mathcal O)$
and
$\Phi _{\mathrm {nat}}$
is the cocycle defined in Section 4.
Since
$\Gamma _2$
is of finite index in
$\Gamma _1$
, it is enough to show that
$\tau (\rho (\Gamma _2))$
is contained in a finitely generated
$\mathcal O[H_1(\Gamma _1, M)]$
-submodule of
$(\mathcal O[(\mathcal E\otimes M)_{\Delta }])^n$
. Since
$\Gamma _2$
acts trivially on M, we have a canonical isomorphism
$$ \begin{align*}H_1(\Gamma_2, M)\cong \Gamma_2^{\mathrm{ab}}\otimes M,\end{align*} $$
and it follows from (4.16) that the image of
$\mathcal O[\Gamma _2^{\mathrm {ab}}\otimes M]$
in
$\mathcal O[(\mathcal E\otimes M)_{\Delta }]$
is contained in
$\mathcal O[H_1(\Gamma _1, M)]$
. We thus may assume that
$G=\mathfrak D(M)$
and
$\Gamma _2=\Gamma _1$
. In this case,
$\mathcal O[H_1(\Gamma , M)]= \mathcal O[(\mathcal E\otimes M)_{\Delta }]$
, so there is nothing to prove.
6 Lafforgue’s G-pseudocharacters
We keep the notation of Section 5. We now recall Lafforgue’s notion of G-pseudocharacter in the form of [Reference Quast13, Definition 3.1]. The difference to Lafforgue’s original definition [Reference Lafforgue9, Section 11] is that we work over the base ring
$\mathcal O$
and allow G to be disconnected. The definition works for generalised reductive
$\mathcal O$
-group schemes as defined in [Reference Paškūnas and Quast12], but this generality is not needed here.
Definition 6.1. Let A be a commutative
$\mathcal O$
-algebra. A G-pseudocharacter
$\Theta $
of
$\Gamma $
over A is a sequence
$(\Theta _n)_{n \geq 1}$
of
$\mathcal O$
-algebra maps
$$ \begin{align*}\Theta_n : \mathcal O[G^n]^{G^0} \to \mathrm{Map}(\Gamma^n,A)\end{align*} $$
for
$n \geq 1$
, satisfying the following conditionsFootnote
1
:
-
(1) For each
$n,m \geq 1$
, each map
$\zeta : \{1, \dots , m\} \to \{1, \dots ,n\}$
,
$f \in \mathcal O[G^m]^{G^0}$
and
$\gamma _1, \dots , \gamma _n \in \Gamma $
, we have where
$$ \begin{align*}\Theta_n(f^{\zeta})(\gamma_1, \dots, \gamma_n) = \Theta_m(f)(\gamma_{\zeta(1)}, \dots, \gamma_{\zeta(m)})\end{align*} $$
$f^{\zeta }(g_1, \dots , g_n) = f(g_{\zeta (1)}, \dots , g_{\zeta (m)})$
.
-
(2) For each
$n \geq 1$
, for each
$\gamma _1, \dots , \gamma _{n+1} \in \Gamma $
and each
$f \in \mathcal O[G^n]^{G^0}$
, we have where
$$ \begin{align*}\Theta_{n+1}(\hat f)(\gamma_1, \dots, \gamma_{n+1}) = \Theta_n(f)(\gamma_1, \dots, \gamma_n\gamma_{n+1})\end{align*} $$
$\hat f(g_1, \dots , g_{n+1}) = f(g_1, \dots , g_ng_{n+1})$
.
We denote the set of G-pseudocharacters of
$\Gamma _1$
over A by
$\mathrm {PC}_G^{\Gamma _1}(A)$
. The functor
$A \mapsto \mathrm {PC}_G^{\Gamma _1}(A)$
is representable by an affine
$\mathcal O$
-scheme
$\mathrm {PC}_G^{\Gamma _1}$
[Reference Quast13, Theorem 3.20]. When
$\varphi : G \to H$
is a homomorphism of generalised tori over
$\mathcal O$
, the induced maps
$\varphi ^*_n : \mathcal O[H^n]^{H^0} \to \mathcal O[G^n]^{G^0}$
give rise to an H-pseudocharacter
$(\Theta _n \circ \varphi ^*_n)_{n \geq 1}$
. By analogy, with the notation for representations, we denote this H-pseudocharacter by
$\varphi \circ \Theta $
. Thus, we also have an induced map
$\mathrm {PC}_G^{\Gamma }(A) \to \mathrm {PC}_H^{\Gamma }(A)$
. It is easy to verify that specialisation along
$f : A \to B$
commutes with composition with
$\varphi $
, i.e.
$(\varphi \circ \Theta ) \otimes _A B = \varphi \circ (\Theta \otimes _A B)$
.
Recall that for every homomorphism
$\rho : \Gamma _1 \to G(A)$
, there is an associated G-pseudocharacter
$\Theta _{\rho } \in \mathrm {PC}^{\Gamma _1}_G(A)$
, which depends on
$\rho $
only up to
$G^0(A)$
-conjugation. For
$m \geq 1$
, the homomorphism
$(\Theta _{\rho })_m : \mathcal O(G^m)^{G^0} \to \mathrm {Map}(\Gamma _1^m, A)$
is defined by
There is a natural
$G^0(A)$
-equivariant map
$$ \begin{align} \operatorname{\mathrm{Rep}}^{\Gamma_1}_G(A) \to \mathrm{PC}^{\Gamma_1}_G(A), \quad\rho \mapsto \Theta_{\rho} , \end{align} $$
which induces maps of
$\mathcal O$
-schemes
$$ \begin{align} \operatorname{\mathrm{Rep}}^{\Gamma_1}_G \to \mathrm{PC}^{\Gamma_1}_G. \end{align} $$
Since
$G^0$
acts trivially on the target, (6.3) factors as
If
$\rho \in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(A)$
, then
$\Theta _{\rho }$
is admissible in the sense that the functorial image of
$\Theta _{\rho }$
under the map
$G \to \underline \Delta $
maps to the
$\underline \Delta $
-pseudocharacter attached to
$\pi $
in
$\mathrm {PC}^{\Gamma _1}_{\underline \Delta }(A)$
. We denote the set of admissible G-pseudocharacters by
$\mathrm {PC}^{\Gamma _1}_{G,\pi }(A)$
. The functor
$A \mapsto \mathrm {PC}^{\Gamma _1}_{G,\pi }(A)$
is representable by an affine
$\mathcal O$
-scheme
$\mathrm {PC}^{\Gamma _1}_{G,\pi }$
. We have natural maps
$$ \begin{align} \operatorname{\mathrm{Rep}}^{\Gamma_1}_{G, \pi} &\to \mathrm{PC}^{\Gamma_1}_{G, \pi} \end{align} $$
as above.
We will show, that (6.4) and (6.6) are isomorphisms. The result is specific to generalised toriFootnote 2 and might be of independent interest.
Proposition 6.2. The map
$$ \begin{align} \mathcal O(\mathrm{PC}^{\Gamma_1}_G) \to \mathcal O(\operatorname{\mathrm{Rep}}^{\Gamma_1}_G)^{G^0} \end{align} $$
corresponding to (6.4) is an isomorphism.
Proof. As
$G^0$
is linearly reductive the proof of [Reference Emerson and Morel8, Proposition 2.11 (i)] applies, we recall the argument here. Let
$\mathcal O(G^{\Gamma _1})$
be the
$\mathcal O$
-algebra which represents the functor
$$ \begin{align*}\mathcal O\text{-}\mathrm{alg} \to \text{Set}, ~A \mapsto \mathrm{Map}(\Gamma_1, G(A)).\end{align*} $$
We have a natural surjection
$\mathcal O(G^{\Gamma _1}) \twoheadrightarrow \mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_G)$
with kernel J. We also have a natural surjection
$\mathcal O(G^{\Gamma _1})^{G^0} \twoheadrightarrow \mathcal O(\mathrm {PC}^{\Gamma _1}_G)$
with kernel I. The structure of J and I is described explicitly in [Reference Emerson and Morel8, Proposition 2.5]; note that there G is assumed to be connected, but the description generalises easily to our situation. Namely, J is generated by the image of the
$G^0$
-equivariant maps
$$ \begin{align*}\varphi_{\gamma, \delta} : \mathcal O(G \times G^{\Gamma_1}) \to \mathcal O(G^{\Gamma_1})\end{align*} $$
for all
$\gamma , \delta \in \Gamma _1$
, where
$$ \begin{align*}\varphi_{\gamma, \delta}(f)((g_{\alpha})_{\alpha \in \Gamma_1}) = f(g_{\gamma\delta}, (g_{\alpha})_{\alpha \in \Gamma_1}) - f(g_{\gamma}g_{\delta}, (g_{\alpha})_{\alpha \in \Gamma_1}).\end{align*} $$
The ideal I is generated by the image of
$\varphi _{\gamma , \delta }(\mathcal O(G \times G^{\Gamma _1})^{G^0})$
for all
$\gamma , \delta \in \Gamma _1$
. Since taking
$G^0$
-invariants is exact, the natural map
$\mathcal O(G^{\Gamma _1})^{G^0}/I = \mathcal O(\mathrm {PC}^{\Gamma _1}_G) \to \mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_G)^{G^0} = \mathcal O(G^{\Gamma _1})^{G^0}/J^{G^0}$
is surjective. So it remains to show that
$J^{G^0} \subseteq I$
. Let
$h \in J^G$
and write
$h = \sum _{i=1}^n \varphi _{\gamma _i, \delta _i}(f_i),$
where
$\gamma _i, \delta _i \in \Gamma _1$
and
$f_i \in \mathcal O(G \times G^{\Gamma _1})$
. Denote the Reynolds operator on
$G^0$
-modules by E. It commutes with the
$G^0$
-equivariant maps
$\varphi _{\gamma _i, \delta _i}$
, so that we have
$$ \begin{align*}h = E(h) = \sum_{i=1}^n \varphi_{\gamma_i, \delta_i}(E(f_i)) \in I.\\[-46pt] \end{align*} $$
Lemma 6.3. If the conjugation action of
$G^0$
on all
$\mathcal O(G^m)$
is trivial, then the following hold:
-
(1) (6.3) is an isomorphism;
-
(2) If
$\Gamma _1$
is a topological group and A is a topological
$\mathcal O$
-algebra, then (6.2) induces a bijection between continuous representations and continuous G-pseudocharacters;
In particular, if
$G = G^0$
or
$G^0$
is trivial, then (6.3) is an isomorphism, and hence,
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{\underline \Delta } \cong \mathrm {PC}^{\Gamma _1}_{\underline \Delta }$
.
Proof. Choose
$\mathcal O$
-algebra generators
$f_1, \dots , f_r \in \mathcal O(G)$
and let
$\Theta \in \mathrm {PC}^{\Gamma _1}_G(A)$
. The functions
$\Theta _1(f_1), \dots , \Theta _1(f_r)$
define a unique map
$\rho : \Gamma _1 \to G(A)$
such that
${\Theta _1(f_1) = f_i \circ \rho }$
for all
$i=1, \dots , r$
. Let
$\mu : \mathcal O(G) \to \mathcal O(G) \otimes _{\mathcal O} \mathcal O(G)$
be the comultiplication map. We can write
$$ \begin{align*}\mu(f_i) = \sum_{j,k} a_{ijk} f_j \otimes f_k\end{align*} $$
for some
$a_{ijk} \in \mathcal O$
. By (2) of Definition 6.1, we have
$$ \begin{align*} f_i(\rho(\gamma_1\gamma_2)) &= \Theta_1(f_i)(\gamma_1\gamma_2) = \Theta_2(\mu(f_i))(\gamma_1, \gamma_2) \\ &= \sum_{j,k} a_{ijk} \Theta_1(f_j)(\gamma_1) \Theta_1(f_k)(\gamma_2) \\ &= \sum_{j,k} a_{ijk} f_j(\rho(\gamma_1)) f_k(\rho(\gamma_2)) \\ &= \mu(f_i)(\rho(\gamma_1), \rho(\gamma_2)) = f_i(\rho(\gamma_1) \rho(\gamma_2)), \end{align*} $$
so
$\rho $
is a homomorphism. For every
$m \geq 1$
, every function
$f \in \mathcal O(G^m)$
can be written as a linear combination of functions of the form
$(g_1, \dots , g_m) \mapsto f_i(g_j)$
. So by rule (1) of Definition 6.1, the function
$\Theta _m(f)$
is determined by
$\Theta _1(f_1), \dots , \Theta _1(f_r)$
. It follows that
$\Theta $
is the only G-pseudocharacter satisfying
$\Theta _1(f_i) = f_i \circ \rho $
for all i. By definition,
$\Theta = \Theta _{\rho }$
, so this shows surjectivity and injectivity of (6.3).
For the claim about continuity, we observe that continuity of
$\Theta _1(f_1), \dots , \Theta _1(f_r)$
is equivalent to continuity of
$\rho $
.
Corollary 6.4. The map
$$ \begin{align} \mathcal O(\mathrm{PC}^{\Gamma_1}_{G,\pi}) \to \mathcal O(\operatorname{\mathrm{Rep}}^{\Gamma_1}_{G,\pi})^{G^0} \end{align} $$
corresponding to (6.6) is an isomorphism.
Proof. Via the canonical projection
$\Pi : G \to \underline \Delta $
, we have a natural
$G^0$
-action on
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_{\underline \Delta })$
, which is trivial, since
$\underline \Delta ^0 = 1$
. So the natural
$G^0$
-equivariant map
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_{\underline \Delta }) \to \mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_G)$
induces a map
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_{\underline \Delta }) \to \mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_G)^{G^0}$
. The representation
$\pi : \Gamma _1 \to \Delta $
corresponds to a map
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_{\underline \Delta }) \to \mathcal O$
. The tensor product
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_G) \otimes _{\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_{\underline \Delta })} \mathcal O$
represents
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }$
. By virtue of Remark 2.3, we have a natural isomorphism
$$ \begin{align*} \mathcal O(\operatorname{\mathrm{Rep}}^{\Gamma_1}_G)^{G^0} \otimes_{\mathcal O(\operatorname{\mathrm{Rep}}^{\Gamma_1}_{\underline\Delta})} \mathcal O \xrightarrow{\cong} \mathcal O(\operatorname{\mathrm{Rep}}^{\Gamma_1}_{G,\pi})^{G^0}. \end{align*} $$
Moreover,
$\mathcal O(\mathrm {PC}^{\Gamma _1}_{G,\pi })$
is representable by
$$ \begin{align*}\mathcal O(\mathrm{PC}^{\Gamma_1}_G) \otimes_{\mathcal O(\mathrm{PC}^{\Gamma_1}_{\underline\Delta})} \mathcal O \cong \mathcal O(\mathrm{PC}^{\Gamma_1}_G) \otimes_{\mathcal O(\operatorname{\mathrm{Rep}}^{\Gamma_1}_{\underline\Delta})} \mathcal O\end{align*} $$
by Lemma 6.3. We conclude that (6.8) is obtained from (6.7) by applying
$- \otimes _{\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_{\underline \Delta })} \mathcal O$
. So by Proposition 6.2, (6.8) is an isomorphism as well.
Corollary 6.5. Assume there is a representation
$\rho _0 \in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(\mathcal O)$
. Then we have isomorphisms
$\mathcal O(\mathrm {PC}^{\Gamma _1}_{G,\pi }) \cong \mathcal O[H_1(\Gamma _1, M)] \cong \mathcal O[(\mathcal E\otimes M)_{\Delta }]^{G^0}$
via (5.7) and (6.8).
Lemma 6.6. Assume that
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(\mathcal O)$
is nonempty. Then the map
$\mathcal O(\mathrm {PC}^{\Gamma _1}_{G,\pi }) \to \mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi })$
corresponding to (6.5) has a section as
$\mathcal O$
-algebras. In particular, for all
$A \in \mathcal O\text {-}\mathrm {alg}$
and all
${\Theta \in \mathrm {PC}^{\Gamma _1}_{G,\pi }(A)}$
, there is a representation
$\rho \in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi }(A)$
, such that
$\Theta = \Theta _{\rho }$
.
Proof. Let
$\rho _0 \in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(\mathcal O)$
. In the diagram
the left solid square commutes since (5.4) is a
$G^0$
-equivariant isomorphism. The right solid square commutes since (5.7) is by Corollary 6.4 induced by
$\varphi _{\mathrm {nat}}$
. The section of the right vertical arrow is defined by mapping
$t_1, \dots , t_s$
to
$1$
and, since all horizontal maps are isomorphisms, defines a section
$\sigma $
of the left vertical arrow. Given a G-pseudocharacter
$\Theta \in \mathrm {PC}^{\Gamma _1}_{G,\pi }(A)$
, it corresponds to a homomorphism
$\mathcal O(\mathrm {PC}^{\Gamma _1}_{G,\pi }) \to A$
, and by restriction along
$\sigma $
, we get a representation
$\rho \in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(A)$
, which by restriction along (6.8) recovers
$\Theta $
, so
$\Theta = \Theta _{\rho }$
.
7 Profinite completion
So far, we have worked with representations and pseudocharacters of an abstract group
$\Gamma _1$
, and topology did not play a role. In this section, we transfer some of the results proved in Sections 5 and 6 to the profinite completion
$\widehat {\Gamma }_1$
of
$\Gamma _1$
and impose continuity conditions on the representations and pseudocharacters of
$\widehat {\Gamma }_1$
.
Let
$\mathcal O$
be the ring of integers in a finite extension of L with uniformiser
$\varpi $
and residue field k.
7.1 Deformations of representations
Let
$\overline {\rho }: \widehat {\Gamma }_1\rightarrow G(k)$
be a continuous representation, such that
$\overline {\rho }|_{\Gamma _1}\in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(k)$
.
Let
$D^{\square }_{\overline {\rho }}:\mathfrak A_{\mathcal O}\rightarrow \text { Set}$
be the functor such that
$D^{\square }_{\overline {\rho }}(A)$
is the set of continuous representations
$\rho _A: \widehat {\Gamma }_1\rightarrow G(A)$
such that
$\rho _A\equiv \overline {\rho }\ \pmod {\mathfrak m_A}$
. The functor
$D^{\square }_{\overline {\rho }}$
is pro-representable by
$R^{\square }_{\overline {\rho }}\in \widehat {\mathfrak A}_{\mathcal O}$
.
Lemma 7.1. There is a natural isomorphism in
$\widehat {\mathfrak A}_{\mathcal O}$
:
$$ \begin{align} R^{\square}_{\overline{\rho}}\cong \varprojlim_I \mathcal O(\operatorname{\mathrm{Rep}}^{\Gamma_1}_G)_{\mathfrak m}/I, \end{align} $$
where
$\mathfrak m$
is the maximal ideal of
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_G)$
corresponding to
$\overline {\rho }$
, and the limit is taken over all the ideals I, such that the quotient is finite.
Proof. If
$A\in \mathfrak A_{\mathcal O}$
and
$\rho \in D^{\square }_{\overline {\rho }}(A)$
, then
$\rho |_{\Gamma _1}\in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi }(A)$
, and hence, we obtain a natural homomorphism of
$\mathcal O$
-algebras
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_G)\rightarrow A$
. The condition
$\rho \equiv \overline {\rho }\ \pmod {\mathfrak m_A}$
implies that this map extends to the localisation
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_G)_{\mathfrak m}\rightarrow A$
. Since A is finite, the kernel of this map is one of the ideals I appearing in the inductive system in (7.1).
Conversely, if I is an ideal of
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_G)_{\mathfrak m}$
such that the quotient A is finite, then
$A\in \mathfrak A_{\mathcal O}$
. The map
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_G)\rightarrow k$
corresponding to
$\overline {\rho }$
factors through
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi })\rightarrow k$
as
$\overline {\rho }|_{\Gamma _1}\in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(k)$
by assumption. Since
$G/G^0$
is constant and A is a local ring, we have
$(G/G^0)(A)=(G/G^0)(k)$
, and hence, the map
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_G)\rightarrow A$
factors through
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi })\rightarrow A$
. By specialising the universal representation
$\Gamma _1\rightarrow G(\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi }))$
along this map, we obtain a representation
$\rho : \Gamma _1\rightarrow G(A)$
. Since the target is finite,
$\rho $
extends uniquely to a continuous representation
$\tilde {\rho }:\widehat {\Gamma }_1\rightarrow G(A)$
, which defines a point in
$D^{\square }_{\overline {\rho }}(A)$
.
Hence, the right-hand side of (7.1) pro-represents
$D^{\square }_{\overline {\rho }}$
and the assertion follows.
Lemma 7.2. Let
$\mathcal A$
be an abelian group and let
$\mathfrak m$
be a maximal ideal of the group ring
$\mathcal O[\mathcal A]$
with residue field k. Then there is an isomorphism in
$\widehat {\mathfrak A}_{\mathcal O}$
where N is the pro-p completion of
$\mathcal A$
, and the limit is taken over all the ideals I, such that the quotient is finite.
Proof. Let
$\overline {\psi }: \mathcal A \rightarrow k^{\times }$
be the character obtained by the composition
$\mathcal A\rightarrow \mathcal O[\mathcal A]/\mathfrak m=k$
and let
$\psi : \mathcal A\rightarrow \mathcal O^{\times }$
be any character lifting
$\overline {\psi }$
(for example, the Teichmüller lift). Then
$\mathfrak m$
is generated by
$\varpi $
and
$( \psi (a)^{-1} a - 1)$
for all
$a\in \mathcal A$
. The map
$\varphi : \mathcal O[\mathcal A]\rightarrow \mathcal O[\mathcal A]$
, which sends
$a\in \mathcal A$
to
$\psi (a) a$
, is an isomorphism of
$\mathcal O$
-algebras, which sends
$\mathfrak m$
to the maximal ideal corresponding to the trivial character. We thus may assume that
$\mathfrak m$
is generated by
$\varpi $
and the augmentation ideal of
$\mathcal O[\mathcal A]$
.
If P is a finite p-power order quotient of
$\mathcal A$
, then
$\mathcal O/\varpi ^n[P]$
is a finite local ring, and the surjection
$\mathcal O[\mathcal A]\twoheadrightarrow \mathcal O/\varpi ^n[P]$
maps
$\mathfrak m$
to the maximal ideal of
$\mathcal O/\varpi ^n[P]$
. Hence, the map factors through
$\mathcal O[\mathcal A]_{\mathfrak m}\twoheadrightarrow \mathcal O/\varpi ^n[P]$
. Conversely, if A is a quotient of
$\mathcal O[\mathcal A]_{\mathfrak m}$
, which is finite (as a set), then the image of
$\mathcal A$
is contained in
$1+\mathfrak m_A$
and hence is a finite group of p-power order. This implies the assertion.
Lemma 7.3. Let
$\mathcal A$
be an abelian group, let
$\mathfrak m$
be a maximal ideal of the group ring
$\mathcal O[\mathcal A]$
with residue field k and let N be the pro-p completion of
$\mathcal A$
. Then the following are equivalent:
-
(1) N is a finitely generated
$\mathbb Z_p$
-module; -
(2)
$\mathcal A/ p \mathcal A$
is a finite p-group; -
(3)
is noetherian.
is noetherian.If the equivalent conditions hold, then 
is naturally isomorphic to the
$\mathfrak m$
-adic completion of
$\mathcal O[\mathcal A]$
.
Proof. The equivalence of (1) and (2) follows from Nakayama’s lemma and the fact that
$N/pN$
is the pro-p completion of
$\mathcal A/p \mathcal A\cong \oplus _I \mathbb F_p$
for some set I.
If N is finitely generated as a
$\mathbb Z_p$
-module, then
$N\cong \mu \oplus \mathbb Z_p^r$
, where
$\mu $
is a finite p-group and
$r=\dim _{\mathbb {Q}_p} N\otimes _{\mathbb Z_p} \mathbb {Q}_p$
. Hence, 
and hence, (1) implies (3).
If N is not finitely generated as a
$\mathbb Z_p$
-module, then we may find a strictly increasing nested family of finitely generated
$\mathbb Z_p$
-submodules
$N_i \subset N$
. Since they are finitely generated, they are closed in N. The kernels of 
form a strictly increasing nested family of ideals in 
. Hence, (3) implies (1).
As in the proof of Lemma 7.2, we may assume that
$\mathfrak m$
is generated by
$\varpi $
and the augmentation ideal of
$\mathcal O[\mathcal A]$
. The map
$\mathcal A\rightarrow \mathfrak m$
,
$a\mapsto a-1$
induces an isomorphism of k-vector spaces
$k\otimes _{\mathbb Z} \mathcal A \cong \mathfrak m/(\varpi , \mathfrak m^2)$
. If (2) holds, then
$\dim _k \mathfrak m/(\varpi , \mathfrak m^2)$
is finite. Hence, the powers of
$\mathfrak m$
form a cofinal system in the inverse system in (7.2), and the last assertion follows from Lemma 7.2.
Lemma 7.4. If
$\Gamma _2^{\mathrm {ab}}/p \Gamma _2^{\mathrm {ab}}$
is a finite p-group and
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi }(\mathcal O)$
is nonempty, then
$R^{\square }_{\overline {\rho }}$
is naturally isomorphic to the
$\mathfrak m$
-adic completion of
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi })$
, where
$\mathfrak m$
is the maximal ideal corresponding to
$\overline {\rho }|_{\Gamma _1}$
.
Proof. Proposition 5.2 allows us to identify
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi })$
with
$\mathcal O[(\mathcal E\otimes M)_{\Delta }]$
. It follows from (4.10) that if
$\Gamma _2^{\mathrm {ab}}/p \Gamma _2^{\mathrm {ab}}$
is finite, then
$\mathcal E/p \mathcal E$
is also finite. Since M is a finite free
$\mathbb Z$
-module, the assertion follows from Lemma 7.1 and Lemma 7.3 applied to
$\mathcal A=(\mathcal E\otimes M)_{\Delta }$
.
Lemma 7.5. There is a natural isomorphism of local k-algebras between
$R^{\square }_{\overline {\rho }}/\varpi $
and the completed group algebra 
.
Proof. The ring
$R^{\square }_{\overline {\rho }}/\varpi $
represents the restriction of
$D^{\square }_{\overline {\rho }}$
to
$\mathfrak A_k$
. Lemma 7.1 implies that
$$ \begin{align} R^{\square}_{\overline{\rho}}/\varpi\cong \varprojlim_I \mathcal O(\operatorname{\mathrm{Rep}}^{\Gamma_1}_G)_{\mathfrak m}/(\varpi,I). \end{align} $$
Since
$\overline {\rho }|_{\Gamma _1}\in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi }(k)$
, we may apply Proposition 5.2 with the base ring
$\mathcal O$
equal to k to obtain
$$ \begin{align} \mathcal O(\operatorname{\mathrm{Rep}}^{\Gamma_1}_{G,\pi})/\varpi \cong k[(\mathcal E\otimes M)_{\Delta}]. \end{align} $$
Let
$\Gamma \twoheadrightarrow \widehat {\Gamma }_1$
be a surjection of profinite groups, such that for all
$A\in \mathfrak A_{\mathcal O}$
, every continuous representation
$\rho _A: \Gamma \rightarrow G(A)$
factors through
$\widehat {\Gamma }_1$
.
Let
$\operatorname {\mathrm {ad}} \overline {\rho }$
be the representation of
$\Gamma $
on
$\operatorname {\mathrm {Lie}} G_k$
obtained by composing
$\overline {\rho }$
with the adjoint representation. We write
$H^i_{\mathrm {cont}}(\Gamma , \operatorname {\mathrm {ad}} \overline {\rho })$
for the i-th continuous group cohomology and denote by
$h^i_{\mathrm {cont}}(\Gamma , \operatorname {\mathrm {ad}}\overline {\rho })$
its dimension as a k-vector space.
Proposition 7.6. If
$h^1_{\mathrm {cont}}(\Gamma , \operatorname {\mathrm {ad}}\overline {\rho })$
is finite, then
$((\mathcal E\otimes M)_{\Delta })^{\wedge , p}$
is a finitely generated
$\mathbb Z_p$
-module. Moreover, if
$h^2_{\mathrm {cont}}(\Gamma , \operatorname {\mathrm {ad}}\overline {\rho })$
is also finite and
$$ \begin{align} h^1_{\mathrm{cont}}(\Gamma, \operatorname{\mathrm{ad}}\overline{\rho}) -h^0_{\mathrm{cont}}(\Gamma, \operatorname{\mathrm{ad}}\overline{\rho})- h^2_{\mathrm{cont}}(\Gamma, \operatorname{\mathrm{ad}}\overline{\rho}) \ge \mathop{\mathrm{ rank}}\nolimits_{\mathbb Z_p} ((\mathcal E\otimes M)_{\Delta})^{\wedge, p} -\dim G_k, \end{align} $$
then
$R^{\square }_{\overline {\rho }}$
is complete intersection,
$\mathcal O$
-flat of relative dimension
$\mathop {\mathrm { rank}}\nolimits _{\mathbb Z_p} ((\mathcal E\otimes M)_{\Delta })^{\wedge , p}$
.
Proof. The assumption on
$\Gamma $
implies that we could have equivalently defined the deformation problem
$D^{\square }_{\overline {\rho }}$
with
$\Gamma $
instead of
$\widehat \Gamma _1$
.
By standard obstruction theory due to Mazur, we have a presentation
where
$r=\dim _k Z^1_{\mathrm {cont}}(\Gamma , \operatorname {\mathrm {ad}} \overline {\rho })$
and
$s=h^2_{\mathrm {cont}}(\Gamma , \operatorname {\mathrm {ad}}\overline {\rho })$
. The exact sequence
$$ \begin{align*}0\rightarrow (\operatorname{\mathrm{ad}} \overline{\rho})^{\Gamma}\rightarrow \operatorname{\mathrm{ad}} \overline{\rho} \rightarrow Z^1_{\mathrm{cont}}(\Gamma, \operatorname{\mathrm{ad}} \overline{\rho})\rightarrow H^1_{\mathrm{cont}}(\Gamma, \operatorname{\mathrm{ad}} \overline{\rho})\rightarrow 0\end{align*} $$
implies that
$r= h^1_{\mathrm {cont}}(\Gamma , \operatorname {\mathrm {ad}}\overline {\rho }) -h^0_{\mathrm {cont}}(\Gamma , \operatorname {\mathrm {ad}}\overline {\rho })+\dim G_k$
. Let 
. By considering (7.6) modulo
$\varpi $
and using Lemma 7.5, we deduce that
$\mathop {\mathrm { rank}}\nolimits _{\mathbb Z_p} N\ge r -s$
. Moreover, the assumption (7.5) implies that
$r-s \ge \mathop {\mathrm { rank}}\nolimits _{\mathbb Z_p} N$
. Hence,
$r-s = \mathop {\mathrm { rank}}\nolimits _{\mathbb Z_p} N$
, and
$\varpi , f_1, \ldots , f_s$
can be extended to a system of parameters of a regular ring 
. Thus,
$\varpi , f_1, \ldots , f_s$
are a part of a regular sequence, and hence,
$R^{\square }_{\overline {\rho }}$
is complete intersection, flat over
$\mathcal O$
of relative dimension of
$\mathop {\mathrm { rank}}\nolimits _{\mathbb Z_p} N$
.
Corollary 7.7. If the assumptions of Proposition 7.6 hold, then there is an isomorphism of local
$\mathcal O'$
-algebras:
where
$\mathcal O'$
is the ring of integers in a finite extension of L. In particular,
$\operatorname {\mathrm {Rep}}^{\widehat \Gamma _1}_{G,\pi }(\mathcal O')$
and
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(\mathcal O')$
are nonempty.
Proof. It follows from Proposition 7.6 that there is a homomorphism of
$\mathcal O$
-algebras
$x:R^{\square }_{\overline {\rho }}\rightarrow \overline {\mathbb {Q}}_p$
. Then
$\kappa (x)$
is a finite extension of L, and the image of x is contained in the ring of integers of
$\kappa (x)$
, which we denote by
$\mathcal O'$
. Let
$\rho _x: \widehat {\Gamma }_1\rightarrow G(\mathcal O')$
be the specialisation of the universal deformation along
$x: R^{\square }_{\overline {\rho }}\rightarrow \mathcal O'$
. Then
$\rho _x\in \operatorname {\mathrm {Rep}}^{\widehat \Gamma _1}_{G,\pi }(\mathcal O')$
and its restriction to
$\Gamma _1$
defines a point in
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_G(\mathcal O')$
.
Since
$\mathcal O'\otimes _{\mathcal O} R^{\square }_{\overline {\rho }}$
represents the functor
$D^{\square }_{\overline {\rho }_{k'}}: \mathfrak A_{\mathcal O'}\rightarrow \text {Set}$
, where
$\overline {\rho }_{k'}$
is the composition of
$\overline {\rho }$
with
$G(k)\hookrightarrow G(k')$
, where
$k'$
is the residue field of
$\mathcal O'$
, we may assume that
$\mathcal O'=\mathcal O$
.
Proposition 5.2 allows us to identify
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi })$
with
$\mathcal O[(\mathcal E\otimes M)_{\Delta }]$
. This identification depends on the lift
$\rho _x$
. The assertion follows from Lemmas 7.1 and 7.2.
7.2 Deformations of pseudocharacters
If A is a topological ring, then a G-pseudocharacter
$\Theta \in \mathrm {PC}_{G,\pi }^{\widehat {\Gamma }_1}(A)$
is continuous, if every
$\Theta _m$
takes values in the subset
$\mathcal C(\widehat {\Gamma }_1^m, A) \subseteq \mathrm {Map}(\widehat {\Gamma }_1^m, A)$
of continuous maps. We write
$\mathrm {cPC}^{\widehat {\Gamma }_1}_{G,\pi }(A) \subseteq \mathrm {PC}^{\widehat {\Gamma }_1}_{G,\pi }(A)$
for the subset of continuous G-pseudocharacters.
Lemma 7.8. Assume that
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(\mathcal O)$
is nonempty. Let A be a finite discrete
$\mathcal O$
-algebra. Then the natural map
$\mathrm {cPC}^{\widehat {\Gamma }_1}_{G,\pi }(A) \to \mathrm {PC}^{\Gamma _1}_{G,\pi }(A), \Theta \mapsto \Theta |_{\Gamma _1}$
is bijective.
Proof. As the image of
$\Gamma _1$
in
$\widehat {\Gamma }_1$
is dense, injectivity follows from [Reference Quast13, Lemma 3.2]. Let
$\Theta \in \mathrm {PC}^{\Gamma _1}_{G,\pi }(A)$
. By Lemma 6.6, there is a representation
$\rho \in \operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(A)$
, such that
$\Theta = \Theta _{\rho }$
. Since A is finite,
$\rho $
extends to a continuous representation
$\tilde \rho : \widehat \Gamma _1 \to G(A)$
, and we have
$\Theta _{\tilde \rho }|_{\Gamma _1} = \Theta $
.
Let
$\overline {\Theta }$
be the G-pseudocharacter of
$\widehat {\Gamma }_1$
associated to
$\overline {\rho }$
. Let
$D^{\mathrm {ps}}_{\overline {\Theta }}: \mathfrak A_{\mathcal O}\rightarrow \text {Set}$
be the deformation functor which sends A to the set of continuous G-pseudocharacters
$\Theta $
of
$\widehat {\Gamma }_1$
valued in A such that
$\Theta \otimes _A k= \overline {\Theta }.$
The functor
$D^{\mathrm {ps}}_{\overline {\Theta }}$
is pro-represented by
$R^{\mathrm {ps}}_{\overline {\Theta }}\in \widehat {\mathfrak A}_{\mathcal O}$
by [Reference Quast13, Theorem 5.4].
Proposition 7.9. Assume that
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(\mathcal O)$
is nonempty. Then there is an isomorphism in
$\widehat {\mathfrak A}_{\mathcal O}$
:
$$ \begin{align} R^{\mathrm{ps}}_{\overline{\Theta}} \cong \varprojlim_I \mathcal O(\mathrm{PC}^{\Gamma_1}_{G,\pi})_{\mathfrak m}/ I, \end{align} $$
where
$\mathfrak m$
is the maximal ideal of
$\mathcal O(\mathrm {PC}^{\Gamma _1}_{G,\pi })$
corresponding to
$\overline {\Theta }|_{\Gamma _1}$
, and the limit is taken over all ideals I such that the quotient is finite.
Proof. If
$A \in \mathfrak A_{\mathcal O}$
, then we have natural bijections
$$ \begin{align*} \operatorname{\mathrm{Hom}}_{\widehat{\mathfrak A}_{\mathcal O}}(R^{\mathrm{ps}}_{\overline{\Theta}}, A) \cong \{\Theta \in \mathrm{cPC}^{\widehat\Gamma_1}_{G, \pi}(A) \mid \Theta \otimes_A k = \overline{\Theta}\},\\ \operatorname{\mathrm{Hom}}_{\text{local }\mathcal O\text{-}\mathrm{alg}}(\mathcal O(\mathrm{PC}^{\Gamma_1}_{G,\pi})_{\mathfrak m}, A) \cong \{\Theta \in \mathrm{PC}^{\Gamma_1}_{G, \pi}(A) \mid \Theta \otimes_A k = \overline{\Theta}\}. \end{align*} $$
As
$\mathrm {cPC}^{\widehat {\Gamma }_1}_{G, \pi }(A) \cong \mathrm {PC}^{\Gamma _1}_{G, \pi }(A)$
by Lemma 7.8, the claim follows.
Lemma 7.10. Assume that
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(\mathcal O)$
is nonempty. Then
$R^{\mathrm {ps}}_{\overline {\Theta }}$
is isomorphic to the completed group algebra 
.
Proof. Since
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(\mathcal O)$
is nonempty, Corollary 6.5 allows us to identify
$\mathcal O(\mathrm {PC}^{\Gamma _1}_{G,\pi })$
with
$\mathcal O[H_1(\Gamma , M)]$
. The assertion follows from Proposition 7.9 and Lemma 7.2.
Lemma 7.11. If
$\Gamma _2^{\mathrm {ab}}/p \Gamma _2^{\mathrm {ab}}$
is a finite p-group and
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(\mathcal O)$
is nonempty, then
$R^{\mathrm {ps}}_{\overline {\Theta }}$
is noetherian and is naturally isomorphic to the
$\mathfrak m$
-adic completion of
$\mathcal O(\mathrm {PC}^{\Gamma _1}_{G,\pi })$
, where
$\mathfrak m$
is the maximal ideal of
$\mathcal O(\mathrm {PC}^{\Gamma _1}_{G,\pi })$
corresponding to
$\overline {\Theta }|_{\Gamma _1}$
.
Proof. The exact sequence of
$\Gamma _1$
-modules
$0\rightarrow I_{\Delta }\otimes M \rightarrow \mathbb Z[\Delta ]\otimes M \rightarrow M\rightarrow 0$
induces an exact sequence in homology:
$$ \begin{align*}H_1(\Gamma_2, M)\rightarrow H_1(\Gamma_1, M)\rightarrow (I_{\Delta}\otimes M)_{\Delta}.\end{align*} $$
Since the action of
$\Gamma _2$
on M is trivial, we have a canonical isomorphism
$H_1(\Gamma _2, M)\cong \Gamma _2^{\mathrm {ab}}\otimes M$
. Since
$I_{\Delta }$
and M are free
$\mathbb Z$
-modules of finite rank, we conclude that the assumption
$\Gamma _2^{\mathrm {ab}}/p \Gamma _2^{\mathrm {ab}}$
is finite implies that
$\mathcal A/p \mathcal A$
is finite, where
$\mathcal A=H_1(\Gamma _1, M)$
. The assertion follows from Lemmas 7.10 and 7.3.
7.3 Moduli space of representations
We study the following schemes.
Definition 7.12. For
$\mathcal G= \Gamma _1$
or
$\widehat {\Gamma }_1$
, let
$X^{\mathrm {gen}, \mathcal G}_{G, \overline {\Theta }}: R^{\mathrm {ps}}_{\overline {\Theta }}\text {-}\mathrm {alg}\rightarrow \text {Set}$
be the functor
where
$\Theta ^u\in D^{\mathrm {ps}}_{\overline {\Theta }}(R^{\mathrm {ps}}_{\overline {\Theta }})$
is the universal deformation of
$\overline {\Theta }$
, and we consider its restriction to
$\Gamma _1$
if
$\mathcal G=\Gamma _1$
.
Proposition 7.13. Assume that
$\operatorname {\mathrm {Rep}}^{\widehat {\Gamma }_1}_{G,\pi }(\mathcal O)$
is nonempty. The restriction to
$\Gamma _1$
induces an isomorphism
$$ \begin{align*}X^{\mathrm{gen},\widehat{\Gamma}_1}_{G,\overline{\Theta}} \overset{\cong}{\longrightarrow} X^{\mathrm{gen},\Gamma_1}_{G,\overline{\Theta}}.\end{align*} $$
In particular,
$X^{\mathrm {gen},\widehat {\Gamma }_1}_{G,\overline {\Theta }}$
is representable by the
$R^{\mathrm {ps}}_{\overline {\Theta }}$
-algebra isomorphic to
$$ \begin{align*}R^{\mathrm{ps}}_{\overline{\Theta}}\otimes_{\mathcal O(\mathrm{PC}^{\Gamma_1}_{G, \pi})} \mathcal O(\operatorname{\mathrm{Rep}}^{\Gamma_1}_{G, \pi})\cong R^{\mathrm{ps}}_{\overline{\Theta}}[t_1^{\pm 1}, \ldots, t_s^{\pm 1}],\end{align*} $$
where
$s= \mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M - \mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M_{\Delta }$
.
Proof. We let
$\mathcal G$
be either
$\Gamma _1$
or its profinite completion
$\widehat {\Gamma }_1$
in the proof. The functor
$X^{\mathrm {gen},\mathcal G}_{G,\overline {\Theta }}$
is representable by
$R^{\mathrm {ps}}_{\overline {\Theta }}\otimes _{\mathcal O(\mathrm {PC}^{\mathcal G}_{G, \pi })} \mathcal O(\operatorname {\mathrm {Rep}}^{\mathcal G}_{G, \pi })$
. Let
$\rho _0 \in \operatorname {\mathrm {Rep}}^{\widehat {\Gamma }_1}_{G,\pi }(\mathcal O)$
. Then
$\rho _0|_{\Gamma _1}$
is in
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G,\pi }(\mathcal O)$
, and using these representations, we may identify
$\operatorname {\mathrm {Rep}}^{\mathcal G}_{G,\pi }$
with the space of
$1$
-cocycles
$Z^1(\mathcal G, \mathfrak D(M)(-))$
by Proposition 5.2. Corollaries 4.11 and 6.5 imply that after choosing a basis of
$I_{\Delta }M$
as a
$\mathbb Z$
-module, we may identify
$$ \begin{align*}\mathcal O(\operatorname{\mathrm{Rep}}^{\mathcal G}_{G,\pi})\cong \mathcal O(\mathrm{PC}^{\mathcal G}_{G, \pi})[t_1^{\pm 1}, \ldots, t_s^{\pm 1}].\end{align*} $$
Thus, for both
$\mathcal G= \Gamma _1$
and
$\mathcal G=\widehat {\Gamma }_1$
, we have
$$ \begin{align} R^{\mathrm{ps}}_{\overline{\Theta}}\otimes_{\mathcal O(\mathrm{PC}^{\mathcal G}_{G, \pi})} \mathcal O(\operatorname{\mathrm{Rep}}^{\mathcal G}_{G, \pi})\cong R^{\mathrm{ps}}_{\overline{\Theta}}[t_1^{\pm 1}, \ldots, t_s^{\pm 1}], \end{align} $$
and under these isomorphisms, the restriction to
$\Gamma _1$
is just the identity map.
The following Lemma will allow us to relate the scheme
$X^{\mathrm {gen}, \widehat {\Gamma }_1}_{G, \overline {\Theta }}$
to the scheme
$X^{\mathrm {gen}}_{G, \bar {\rho }^{\mathrm {ss}}}$
introduced in [Reference Paškūnas and Quast12].
Lemma 7.14. Let
$\tau : G\hookrightarrow \mathbb A^n$
be a closed immersion of
$\mathcal O$
-schemes, let A be an
$R^{\mathrm {ps}}_{\overline {\Theta }}$
-algebra and let
$\rho \in X^{\mathrm {gen}, \widehat {\Gamma }_1}_{G, \overline {\Theta }}(A)$
. Assume that
$\operatorname {\mathrm {Rep}}^{\widehat \Gamma _1}_{G, \pi }(\mathcal O)$
is nonempty. Then
$\tau (\rho (\widehat {\Gamma }_1))$
is contained in a finitely generated
$R^{\mathrm {ps}}_{\overline {\Theta }}$
-submodule of
$A^n = \mathbb A^n(A)$
.
Proof. Since
$X^{\mathrm {gen}, \widehat {\Gamma }_1}_{G, \overline {\Theta }}$
is represented by
$R^{\mathrm {ps}}_{\overline {\Theta }}\otimes _{\mathcal O(\mathrm {PC}^{\widehat {\Gamma }_1}_{G, \pi })} \mathcal O(\operatorname {\mathrm {Rep}}^{\widehat \Gamma _1}_{G, \pi })$
, the assertion follows from Lemma 5.7 applied to
$\widehat {\Gamma }_1$
, and Corollary 6.5.
7.4 Irreducible components
Let N be a finitely generated
$\mathbb Z_p$
-module and let 
be the completed group algebra of N, where
$\mathcal O$
is the ring of integers in a finite extension L of
$\mathbb {Q}_p$
. Let
$\widehat {\mathfrak D}(N): \widehat {\mathfrak A}_{\mathcal O} \rightarrow \operatorname {\mathrm {Ab}}$
be a formal group scheme, given by
Multiplication in
$\widehat {\mathfrak D}(N)$
is induced by the map
Let
$\mu $
be the torsion subgroup of N. Then we have a non-canonical isomorphism
$N\cong \mu \oplus \mathbb Z_p^r$
, where
$r=\dim _{\mathbb {Q}_p} N\otimes _{\mathbb Z_p} \mathbb {Q}_p$
. This induces an isomorphism
We assume that L contains all the
$p^m$
-th roots of unity, where
$p^m$
is the order of
$\mu $
. Then the group
$\mathrm X(\mu )$
of characters
$\chi : \mu \rightarrow \mathcal O^{\times }$
also has order
$p^m$
. The following Lemma is an immediate consequence of (7.11):
Lemma 7.15. The following hold:
-
(1) the irreducible components of
are in canonical bijection with
$\mathrm X(\mu )$
, so that the irreducible component corresponding to
$\chi \in \mathrm X(\mu )$
is given by ; -
(2) every irreducible component of
contains an
$\mathcal O$
-rational point
$\psi \in \widehat {\mathfrak D}(N)(\mathcal O)$
; -
(3) a point
$\psi \in \widehat {\mathfrak D}(N)(\mathcal O)=\operatorname {\mathrm {Hom}}^{\mathrm {cont}}_{\operatorname {\mathrm {Group}}}(N, \mathcal O^{\times })$
lies on the irreducible component corresponding to
$\chi \in \mathrm X(\mu )$
if and only if
$\psi (x)=\chi (x)$
for all
$x\in \mu $
.
are in canonical bijection with 
;
contains an 
Let R be a complete local noetherian
$\mathcal O$
-algebra with residue field k. Let
${\mathcal X:\widehat {\mathfrak A}_{\mathcal O} \rightarrow \text {Set}}$
be the functor
$\mathcal X(A)= \operatorname {\mathrm {Hom}}_{\widehat {\mathfrak A}_{\mathcal O}}(R, A)$
. This functor is represented by a formal scheme
$\operatorname {\mathrm {Spf}} R$
. Let us suppose that we have a faithful and transitive action of
$\widehat {\mathfrak D}(N)$
on
$\mathcal X$
. Concretely, this means that for all
$A\in \widehat {\mathfrak A}_{\mathcal O}$
, we have a faithful and transitive action of the group
$\widehat {\mathfrak D}(N)(A)$
on the set
$\mathcal X(A)$
, which is functorial in A. It is enough to restrict to
$A\in \mathfrak A_{\mathcal O}$
as the general case follows by continuity. The action map
$\widehat {\mathfrak D}(N)\times \mathcal X \rightarrow \mathcal X$
gives us a morphism in
$\widehat {\mathfrak A}_{\mathcal O}$
:
Let us assume that
$\mathcal X(\mathcal O)$
is nonempty and choose
$x\in \mathcal X(\mathcal O)$
. Since the action map is faithful and transitive, for every
$A\in \widehat {\mathfrak A}_{\mathcal O}$
, the map
$$ \begin{align} \widehat{\mathfrak D}(N)(A) \rightarrow \mathcal X(A), \quad \psi\mapsto \psi\cdot x_A \end{align} $$
is bijective, where
$x_A$
is the image of x in
$\mathcal X(A)$
. This implies that the composition
is an isomorphism.
Example 7.16. If
$\mathcal X=\widehat {\mathfrak D}(N)$
and the action is given by left translations, then it follows from (7.9), (7.10) that
$\alpha _{\psi }$
corresponds to the character 
,
$n \mapsto \psi (n)n$
.
Lemma 7.17. Assume that
$\mathcal X(\mathcal O)$
is nonempty. Then the following hold:
-
(1) every irreducible component of
$\operatorname {\mathrm {Spec}} R$
has an
$\mathcal O$
-rational point
$x\in \mathcal X(\mathcal O)$
; -
(2) every
$x\in \mathcal X(\mathcal O)$
lies on a unique irreducible component of
$\operatorname {\mathrm {Spec}} R$
; -
(3) if
$x, y\in \mathcal X(\mathcal O)$
, then there exists a unique
$\psi \in \widehat {\mathfrak D}(N)(\mathcal O)$
such that
$\psi \cdot x = y$
; -
(4) let x, y and
$\psi $
be as in part
$(3)$
. Then x, y lie on the same irreducible component of
$\operatorname {\mathrm {Spec}} R$
if and only if
$\psi |_{\mu } =1$
.
Proof. If
$\mathcal X= \widehat {\mathfrak D}(N)$
with the action given by left translations, then the assertions follow readily from Lemma 7.15.
In the general case, we pick
$x\in \mathcal X(\mathcal O)$
. The isomorphism 
of
$\mathcal O$
-algebras allows us to reduce the question to the case above. For parts (3) and (4), we note that the bijection (7.13) is
$\widehat {\mathfrak D}(N)(A)$
-equivariant for the action by left translations on the source.
Lemma 7.18. Assume that
$\mathcal X(\mathcal O)$
is nonempty. Then the action of
$\widehat {\mathfrak D}(N)(\mathcal O)$
on
$\mathcal X(\mathcal O)$
induces a canonical action of
$\mathrm X(\mu )$
on the set of irreducible components of
$\operatorname {\mathrm {Spec}} R$
. This action is faithful and transitive.
Proof. Since
$\mu $
is a direct summand of N, the map
$\psi \mapsto \psi |_{\mu }$
induces a surjective group homomorphism
$\widehat {\mathfrak D}(N)(\mathcal O)\twoheadrightarrow \mathrm X(\mu )$
. Let K be the kernel of this map. Lemma 7.17 implies that there is a natural bijection between the set of irreducible components of
$\operatorname {\mathrm {Spec}} R$
and the set of K-orbits in
$\mathcal X(\mathcal O)$
. The action
$\widehat {\mathfrak D}(N)(\mathcal O)$
on the set of K-orbits factors through the action of
$\mathrm X(\mu )$
, which induces the sought-after action on the set of irreducible components. Since the action of
$\widehat {\mathfrak D}(N)(\mathcal O)$
on
$\mathcal X(\mathcal O)$
is faithful and transitive, the same applies to the action of
$\mathrm X(\mu )$
.
We will now get back to our particular example. Let
$\widehat {Z}^1: \mathfrak A_{\mathcal O}\rightarrow \operatorname {\mathrm {Ab}}$
be the functor such that
$\widehat {Z}^1(A)$
is the set of continuous
$1$
-cocycles
$\Phi :\widehat {\Gamma }_1\rightarrow \operatorname {\mathrm {Hom}}(M, 1+\mathfrak m_A)$
for the discrete topology on the target.
Lemma 7.19. The functor
$\widehat {Z}^1$
is pro-represented by 
.
Proof. Let
$G=\mathfrak D(M)\,{\rtimes}\, \underline {\Delta }$
,
$\overline {\rho }: \widehat {\Gamma }_1\rightarrow G(k), \gamma \mapsto (1, \pi (\gamma ))$
and let
$\rho _0: \widehat {\Gamma }_1\rightarrow G(\mathcal O), \gamma \mapsto (1, \pi (\gamma ))$
. The map
$\Phi \mapsto \Phi \rho _0$
induces a natural bijection between
$\widehat {Z}^1(A)$
and
$D^{\square }_{\overline {\rho }}(A)$
for all
$A\in \mathfrak A_{\mathcal O}$
; this can be shown by the same argument as in the proof of Proposition 5.2. Thus,
$\widehat {Z}^1$
is pro-represented by
$R^{\square }_{\overline {\rho }}$
, and the assertion follows from Proposition 5.2, Lemma 7.1 and Lemma 7.2.
It follows from Proposition 5.2 that if
$\Phi \in \widehat {Z}^1(A)$
and
$\rho \in D^{\square }_{\overline {\rho }}(A)$
, then
$\gamma \mapsto \Phi (\gamma ) \rho (\gamma )$
defines a representation
$\Phi \rho \in D^{\square }_{\overline {\rho }}(A)$
, and the map
$\widehat {Z}^1\times D^{\square }_{\overline {\rho }}\rightarrow D^{\square }_{\overline {\rho }}$
,
$(\Phi , \rho ) \mapsto \Phi \rho $
defines a faithful and transitive action of
$\widehat {Z}^1$
on
$D^{\square }_{\overline {\rho }}$
.
Proposition 7.20. Assume that
$\Gamma _2^{\mathrm {ab}}/p \Gamma _2^{\mathrm {ab}}$
is finite and let
$\mu $
be the torsion subgroup of
$((\mathcal E\otimes M)_{\Delta })^{\wedge , p}$
. Assume further that
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi }(\mathcal O)$
is nonempty and
$\mathcal O$
contains all the
$p^m$
-th roots of unity, where
$p^m$
is the order of
$\mu $
. Then there is a canonical action of the character group
$\mathrm X(\mu )$
on the set of irreducible components of
$\operatorname {\mathrm {Spec}} R^{\square }_{\overline {\rho }}$
,
$\operatorname {\mathrm {Spec}} R^{\mathrm {ps}}_{\overline {\Theta }}$
and
$X^{\mathrm {gen}}_{G, \overline {\Theta }}$
, respectively. Moreover, the following hold:
-
(1) this action is faithful and transitive;
-
(2) irreducible components of
$\operatorname {\mathrm {Spec}} R^{\square }_{\overline {\rho }}$
and
$\operatorname {\mathrm {Spec}} R^{\mathrm {ps}}_{\overline {\Theta }}$
are formally smooth over
$\mathcal O$
; -
(3) irreducible components of
$X^{\mathrm {gen}}_{G, \overline {\Theta }}$
are of the form where
$r=\mathop {\mathrm { rank}}\nolimits _{\mathbb Z_p} H_1(\Gamma _1, M)^{\wedge , p}$
and
$r+s=\mathop {\mathrm { rank}}\nolimits _{\mathbb Z_p} ((\mathcal E\otimes M)_{\Delta })^{\wedge , p}$
.
Proof. The assumption that
$\Gamma _2^{\mathrm {ab}}/p \Gamma _2^{\mathrm {ab}}$
is finite implies that
$((\mathcal E\otimes M)_{\Delta })^{\wedge , p}$
is a finitely generated
$\mathbb Z_p$
-module by Lemma 7.3, and hence,
$\mu $
is a finite p-group.
The assumption that
$\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi }(\mathcal O)$
is nonempty implies via Proposition 5.2 that
$\mathcal O(\operatorname {\mathrm {Rep}}^{\Gamma _1}_{G, \pi })\cong \mathcal O[(\mathcal E\otimes M)_{\Delta }]$
. It follows from Lemmas 7.1 and 7.2 that 
. In particular,
$D^{\square }_{\overline {\rho }}(\mathcal O)$
is nonempty. It follows from Lemma 7.18 that the action of
$\widehat {Z}^1(\mathcal O)$
on
$D^{\square }_{\overline {\rho }}(\mathcal O)$
induces a transitive and faithful action of
$\mathrm X(\mu )$
on the set of irreducible components of
$R^{\square }_{\overline {\rho }}$
.
Let us spell out Lemma 7.17 in our context. Given an irreducible component X of
$\operatorname {\mathrm {Spec}} R^{\square }_{\overline {\rho }}$
,
$X(\mathcal O)$
is nonempty, and we pick any
$\rho \in X(\mathcal O)$
; given
$\chi \in \mathrm X(\mu )$
, we pick any
$\Phi \in \widehat {Z}^1(\mathcal O)$
such that
$\Phi (\gamma )=\chi (\gamma )$
for all
$\gamma \in \mu $
. Then
$\chi \cdot X$
is the unique irreducible component of
$\operatorname {\mathrm {Spec}} R^{\square }_{\overline {\rho }}$
such that
$\Phi \rho \in (\chi \cdot X)(\mathcal O)$
.
It follows from Proposition 4.9 that
$$ \begin{align*}((\mathcal E\otimes M)_{\Delta})^{\wedge, p}\cong H_1(\Gamma_1, M)^{\wedge, p}\oplus \mathbb Z_p^s\end{align*} $$
for some
$s\ge 0$
. Hence, 
and
$\mu $
is the torsion subgroup of
$H_1(\Gamma _1, M)^{\wedge , p}$
. Thus, the map
$\operatorname {\mathrm {Spec}} R^{\square }_{\overline {\rho }} \rightarrow \operatorname {\mathrm {Spec}} R^{\mathrm {ps}}_{\overline {\Theta }}$
is
$\mathrm X(\mu )$
-equivariant and induces an
$\mathrm X(\mu )$
-equivariant bijection between the irreducible components.
Since
$A^{\mathrm {gen}}_{G, \overline {\Theta }}\cong R^{\mathrm {ps}}_{\overline {\Theta }}[t_1^{\pm 1}, \ldots , t_s^{\pm 1}]$
, by Proposition 7.13, the map
$X^{\mathrm {gen}}_{G, \overline {\Theta }}\rightarrow \operatorname {\mathrm {Spec}} R^{\mathrm {ps}}_{\overline {\Theta }}$
is
$\mathrm X(\mu )$
-equivariant and induces an
$\mathrm X(\mu )$
-equivariant bijection between the sets of irreducible components.
The isomorphism 
allows us to consider
$R^{\mathrm {ps}}_{\overline {\Theta }}$
as an
$\mathcal O[\mu ]$
-algebra. This is non-canonical: it amounts to distinguishing one irreducible component of
$\operatorname {\mathrm {Spec}} R^{\mathrm {ps}}_{\overline {\Theta }}$
. Once we do this, the other irreducible components are given by
$R^{\mathrm {ps}}_{\overline {\Theta }}\otimes _{\mathcal O[\mu ],\chi } \mathcal O$
for
$\chi \in \mathrm X(\mu )$
, and the special component corresponds to the trivial character. These are isomorphic to 
and hence are formally smooth.
Similarly, irreducible components of
$\operatorname {\mathrm {Spec}} R^{\square }_{\overline {\rho }}$
and
$X^{\mathrm {gen}}_{G, \overline {\Theta }}$
are given by
and
respectively.
8 Rank calculations
Let E be a finite Galois extension of F, let 
and let M be a free
$\mathbb Z$
-module of finite rank with an action of
$\Delta $
. In this section, we compute the
$\mathbb Z_p$
-rank and the torsion subgroup of the pro-p completion of
$(E^{\times }\otimes M)^{\Delta }$
and related modules. These calculations are used in the next section.
If
$\mathcal A$
is an abelian group, we denote its pro-p completion by
$\mathcal A^{\wedge , p}$
. If N is a
$\mathbb Z_p$
-module, we define 
.
Let
$\Gamma _E^{\mathrm {ab}}$
be the maximal abelian pro-finite quotient of
$\Gamma _E$
and let
$\Gamma _E^{\mathrm {ab},p}$
be the maximal abelian pro-p quotient of
$\Gamma _E$
. The Artin map
$\operatorname {\mathrm {Art}}_E: E^{\times }\rightarrow \Gamma _E^{\mathrm {ab}}$
induces an isomorphism between the profinite completion of
$E^{\times }$
and
$\Gamma _E^{\mathrm {ab}}$
. Thus,
$(E^{\times }\otimes M)^{\wedge , p}\cong \Gamma _E^{\mathrm {ab},p}\otimes M$
.
Lemma 8.1. Let N be a finitely generated
$\mathbb Z[\Delta ]$
(resp.
$\mathbb Z_p[\Delta ]$
) module. Then the Tate cohomology groups
$\widehat {H}^i( \Delta , N)$
are finite for all
$i\in \mathbb Z$
.
Proof. If N is finitely generated over
$\mathbb Z$
, then the statement is proved in [Reference Cassels and Fröhlich5, Corollary 2, p. 105]. The same argument carries over if N is finitely generated over
$\mathbb Z_p$
: the cohomology groups are finitely generated
$\mathbb Z_p$
-modules since the complex computing cohomology consists of finitely generated
$\mathbb Z_p$
-modules. Moreover, they are annihilated by the order of
$\Delta $
.
Lemma 8.2.
$\mathop {\mathrm { rank}}\nolimits _{\mathbb Z_p} (M^{\wedge , p} \otimes I_{\Delta })_{\Delta } = \mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M - \mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M_{\Delta }.$
Proof. The long exact sequence in homology attached to
$$ \begin{align*}0 \rightarrow M^{\wedge, p}\otimes I_{\Delta}\rightarrow M^{\wedge, p}\otimes \mathbb Z[\Delta]\rightarrow M^{\wedge, p}\rightarrow 0\end{align*} $$
yields an exact sequence
$$ \begin{align*}H_1(\Delta, M^{\wedge, p})\rightarrow (M^{\wedge, p}\otimes I_{\Delta})_{\Delta} \rightarrow M^{\wedge, p} \rightarrow (M^{\wedge, p})_{\Delta}\rightarrow 0,\end{align*} $$
as
$M \otimes \mathbb Z[\Delta ]$
is induced by the projection formula. Since we may swap coinvariants with completions and
$M_{\Delta }$
is finitely generated, we have
$$ \begin{align*}\mathop{\mathrm{ rank}}\nolimits_{\mathbb Z_p} (M^{\wedge, p})_{\Delta} = \mathop{\mathrm{ rank}}\nolimits_{\mathbb Z_p} (M_{\Delta})^{\wedge, p}=\mathop{\mathrm{ rank}}\nolimits_{\mathbb Z} M_{\Delta}.\end{align*} $$
Since
$M^{\wedge , p}$
is finitely generated over
$\mathbb Z_p$
and
$\Delta $
is finite, the group is
$H_1(\Delta , M^{\wedge , p})$
is finite by Lemma 8.1. This implies the assertion.
Lemma 8.3.
$\mathop {\mathrm { rank}}\nolimits _{\mathbb Z_p} (\Gamma _E^{\mathrm {ab},p}\otimes M)_{\Delta }=\mathop {\mathrm { rank}}\nolimits _{\mathbb Z_p} ((E^{\times }\otimes M)^{\Delta })^{\wedge ,p}.$
Proof. Since M is a free
$\mathbb Z$
-module, we have an exact sequence of
$\Delta $
-modules
$$ \begin{align*}0\rightarrow (1+\mathfrak p_E)\otimes M \rightarrow E^{\times}\otimes M \rightarrow (E^{\times}/(1+\mathfrak p_E))\otimes M\rightarrow 0.\end{align*} $$
Since
$1+\mathfrak p_E$
is a finitely generated
$\mathbb Z_p$
-module and
$E^{\times }/(1+\mathfrak p_E)$
is a finitely generated
$\mathbb Z$
-module, we deduce that
$\widehat {H}^i(\Delta , E^{\times }\otimes M)$
are finite for all
$i\in \mathbb Z$
. From the exact sequence
$$ \begin{align*}0\rightarrow \widehat{H}^{-1}(\Delta, E^{\times}\otimes M)\rightarrow (E^{\times}\otimes M)_{\Delta} \rightarrow (E^{\times}\otimes M)^{\Delta}\rightarrow \widehat{H}^0(\Delta, E^{\times}\otimes M)\rightarrow 0,\end{align*} $$
we deduce that the pro-p completions of
$(E^{\times }\otimes M)_{\Delta }$
and of
$(E^{\times }\otimes M)^{\Delta }$
have the same
$\mathbb Z_p$
-rank. It follows from the universal property of pro-p completion that it commutes with taking
$\Delta $
-coinvariants. Hence, the pro-p completion of
$(E^{\times }\otimes M)_{\Delta }$
is isomorphic to
$((E^{\times })^{\wedge , p}\otimes M)_{\Delta }\cong (\Gamma _E^{\mathrm {ab},p}\otimes M)_{\Delta }$
.
Lemma 8.4.
$\mathop {\mathrm { rank}}\nolimits _{\mathbb Z_p} (\Gamma _E^{\mathrm {ab},p}\otimes M)_{\Delta } = \mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M \cdot [F:\mathbb Q_p] + \mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M_{\Delta }$
.
Proof. We consider the long exact sequence in homology
$$ \begin{align*}H_1(\Delta, M^{\wedge, p}) \to ((\mathcal O_E^{\times})^{\wedge, p} \otimes M)_{\Delta} \to ((E^{\times})^{\wedge, p} \otimes M)_{\Delta} \to (M^{\wedge, p})_{\Delta} \to 0.\end{align*} $$
Since completion commutes with taking
$\Delta $
-coinvariants we have
$$ \begin{align*}\mathop{\mathrm{ rank}}\nolimits_{\mathbb Z_p} (M^{\wedge, p})_{\Delta}=\mathop{\mathrm{ rank}}\nolimits_{\mathbb Z} M_{\Delta}.\end{align*} $$
Since
$H_1(\Delta , M^{\wedge , p})$
is finite by Lemma 8.1, we are left to compute the
$\mathbb Z_p$
-rank of
$((\mathcal O_E^{\times })^{\wedge , p} \otimes M)_{\Delta }$
. We note that
$(\mathcal O_E^{\times })^{\wedge , p}$
is equal to
$1+\mathfrak p_E$
, and another application of Lemma 8.1 shows that the rank does not change if we replace
$(\mathcal O_E^{\times })^{\wedge , p}$
with any open
$\Delta $
-stable subgroup V of
$1+\mathfrak p_E$
.
We choose V to be the image of a p-adic exponential function defined on
$\mathfrak p_E^n$
for some large enough
$n \geq 1$
. We then have an isomorphism
$\mathfrak p_E^n \otimes M \cong V \otimes M$
of
$\Delta $
-modules. Since
$\mathfrak p_E^n \cong \mathcal O_E$
is isomorphic to
$\mathcal O_F[\Delta ]$
as
$\mathbb Z_p[\Delta ]$
-module (see the proof of [Reference Serre14, Section 1.4]),
$\mathcal O_E \otimes M$
is free and thus
$(\mathcal O_E \otimes M)_{\Delta } \cong \mathcal O_F \otimes M$
. Thus,
$\mathop {\mathrm { rank}}\nolimits _{\mathbb Z_p} ((\mathcal O_E^{\times })^{\wedge , p} \otimes M)_{\Delta }= [F:\mathbb {Q}_p] \cdot \mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M$
, and the assertion follows.
Lemma 8.5. The torsion subgroup of
$((E^{\times }\otimes M)^{\Delta })^{\wedge , p}$
is equal to
$(\mu _{p^{\infty }}(E)\otimes M)^{\Delta }$
.
Proof. The image of
$(E^{\times }\otimes M)^{\Delta }\rightarrow (E^{\times }/\mathcal O_E^{\times })\otimes M$
is a free
$\mathbb Z$
-module of finite rank, which we denote by s, as the target is a free
$\mathbb Z$
-module of finite rank. The kernel of this map is equal to
$(\mathcal O_E^{\times }\otimes M)^{\Delta }$
. The Teichmüller lift gives an isomorphism of
$\Delta $
-modules
$\mathcal O_{E}^{\times } \cong (1+\mathfrak p_E) \oplus k_E^{\times }$
, and hence, we have an isomorphism of abelian groups
$$ \begin{align} (E^{\times}\otimes M)^{\Delta}\cong ((1+\mathfrak p_E)\otimes M)^{\Delta} \oplus(k_E^{\times}\otimes M)^{\Delta} \oplus \mathbb Z^s. \end{align} $$
Since the order of
$k_E^{\times }$
is prime to p and
$((1+\mathfrak p_E)\otimes M)^{\Delta }$
is closed in
$(1+\mathfrak p_E)\otimes M$
and hence p-adically complete, we conclude that the torsion subgroup in the pro-p completion of
$(E^{\times }\otimes M)^{\Delta }$
coincides with the torsion subgroup in
$((1+\mathfrak p_E)\otimes M)^{\Delta }$
, which is equal to
$(\mu _{p^{\infty }}(E)\otimes M)^{\Delta }$
.
Lemma 8.6. If
$\mathcal A$
is a finitely generated
$\mathbb Z[\Delta ]$
-module, then
$(\mathcal A^{\wedge , p})^{\Delta }\cong (\mathcal A^{\Delta })^{\wedge , p}$
.
Proof. We have
$\mathcal A^{\wedge , p}\cong \mathcal A\otimes \mathbb Z_p$
and
$(\mathcal A^{\Delta })^{\wedge , p}\cong \mathcal A^{\Delta }\otimes \mathbb Z_p$
by [Reference Project15, Tag 00MA]. We may express
$\mathcal A^{\Delta }$
as the kernel of
$$ \begin{align*}\mathcal A\rightarrow \bigoplus_{\delta\in \Delta} \mathcal A, \quad a\mapsto (\delta a -a)_{\delta\in \Delta}.\end{align*} $$
Since
$\mathbb Z_p$
is a flat
$\mathbb Z$
-module, we conclude that
$\mathcal A^{\Delta }\otimes \mathbb Z_p \cong (\mathcal A\otimes \mathbb Z_p)^{\Delta }.$
Lemma 8.7.
$((E^{\times }\otimes M)^{\Delta })^{\wedge , p}\cong (\Gamma _E^{\mathrm {ab},p}\otimes M)^{\Delta }$
.
Proof. Let
$n_0$
be an integer such that
$\exp : \mathfrak p_E^n \rightarrow 1+\mathfrak p_E^n$
converges for all
$n\ge n_0$
and let 
. Then
$V_n$
for
$n\ge n_0$
form a basis of open neighbourhoods of
$1$
in
$1+\mathfrak p_E$
. Since
$\mathfrak p_E^n\cong \mathcal O_E \cong \mathcal O_F[\Delta ]$
as
$\Delta $
-modules and
$\exp $
is
$\Delta $
-equivariant, we have an isomorphism
$V_n\otimes M \cong \operatorname {\mathrm {Ind}}^{\Delta }_{1} (\mathcal O_F \otimes M)$
, and hence,
$H^1(\Delta , V_n\otimes M)=0$
. Thus, for all
$n\ge n_0$
, we obtain an exact sequence
$$ \begin{align} 0\rightarrow (V_n\otimes M)^{\Delta} \rightarrow (E^{\times}\otimes M)^{\Delta} \rightarrow ((E^{\times}/ V_n)\otimes M)^{\Delta}\rightarrow 0. \end{align} $$
The completion and
$\varprojlim _n$
are both limits and hence commute with each other. Lemma 8.6 and (8.2) imply that
$$ \begin{align} ((E^{\times}\otimes M)^{\Delta})^{\wedge, p}\cong \varprojlim_n ((E^{\times}/ V_n)\otimes M)^{\Delta})^{\wedge, p}\cong \varprojlim_n ((E^{\times}/ V_n)\otimes M)^{\wedge, p})^{\Delta}. \end{align} $$
The isomorphism
$(E^{\times }\otimes M)^{\wedge , p}\cong \Gamma _E^{\mathrm {ab},p}\otimes M$
induces an isomorphism
$$ \begin{align} ((E^\times/V_n)\otimes M)^{\wedge, p}\cong (\Gamma_E^{\mathrm{ab},p}/\operatorname{\mathrm{Art}}_E(V_n))\otimes M. \end{align} $$
Since
$H^1(\Delta , V_n \otimes M)=0$
, we have an exact sequence
$$ \begin{align} 0\rightarrow (V_n\otimes M)^{\Delta} \rightarrow (\Gamma_E^{\mathrm{ab},p}\otimes M)^{\Delta} \rightarrow ((\Gamma_E^{\mathrm{ab},p}/ \operatorname{\mathrm{Art}}_E(V_n))\otimes M)^{\Delta}\rightarrow 0. \end{align} $$
We thus have
$$ \begin{align*}(\Gamma^{\mathrm{ab},p}_E \otimes M)^{\Delta}\cong \varprojlim_n ((\Gamma_E^{\mathrm{ab},p}/ \operatorname{\mathrm{Art}}_E(V_n))\otimes M)^{\Delta},\end{align*} $$
Corollary 8.8. There is an isomorphism of
$\mathbb Z_p$
-modules:
$$ \begin{align*}(\Gamma_E^{\mathrm{ab},p}\otimes M)^{\Delta}\cong (\mu_{p^{\infty}}(E)\otimes M)^{\Delta} \times \mathbb Z_p^r,\end{align*} $$
where
$r=\mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M \cdot [F:\mathbb {Q}_p] + \mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M_{\Delta }$
.
Proposition 8.9. Let
$0 \rightarrow E^{\times }\rightarrow \mathcal E\rightarrow I_{\Delta }\rightarrow 0$
be any extension of
$\mathbb Z[\Delta ]$
-modules.
$$ \begin{align} \mathop{\mathrm{ rank}}\nolimits_{\mathbb Z_p} ((\mathcal E\otimes M)_{\Delta})^{\wedge,p}= ([F:\mathbb {Q}_p]+1) \mathop{\mathrm{ rank}}\nolimits_{\mathbb{Z}} M. \end{align} $$
Proof. Since
$I_{\Delta }\otimes M$
is a free
$\mathbb Z$
-module, the surjection
$\mathcal E\otimes M\twoheadrightarrow I_{\Delta }\otimes M$
has a section, and hence, we have an exact sequence of
$\mathbb Z_p[\Delta ]$
-modules
$$ \begin{align*}0\rightarrow (E^{\times}\otimes M)^{\wedge, p}\rightarrow (\mathcal E\otimes M)^{\wedge,p} \rightarrow (I_{\Delta}\otimes M)^{\wedge,p}\rightarrow 0.\end{align*} $$
Since M is a free
$\mathbb Z$
-module of finite rank and
$(E^{\times })^{\wedge , p}\cong \Gamma ^{\mathrm {ab}, p}_E$
, we have
$$ \begin{align*}(E^{\times}\otimes M)^{\wedge, p}\cong (E^{\times})^{\wedge, p}\otimes M\cong \Gamma^{\mathrm{ab},p}_E\otimes M.\end{align*} $$
Since
$I_{\Delta }$
is a free
$\mathbb Z$
-module
$(I_{\Delta }\otimes M)^{\wedge , p}\cong I_{\Delta }\otimes M^{\wedge , p}$
, so that we obtain an exact sequence of
$\mathbb Z_p[\Delta ]$
-modules:
$$ \begin{align*}0\rightarrow \Gamma_E^{\mathrm{ab},p}\otimes M \rightarrow (\mathcal E\otimes M)^{\wedge, p}\rightarrow I_{\Delta}\otimes M^{\wedge, p}\rightarrow 0.\end{align*} $$
Taking
$\Delta $
-coinvariants and observing that
$H_1(\Delta , I_{\Delta }\otimes M^{\wedge , p})\cong H_2(\Delta , M^{\wedge , p})$
, we obtain an exact sequence:
$$ \begin{align} H_2(\Delta, M^{\wedge, p})\rightarrow (\Gamma_E^{\mathrm{ab}, p} \otimes M)_{\Delta}\rightarrow ((\mathcal E\otimes M)^{\wedge, p})_{\Delta} \rightarrow (I_{\Delta}\otimes M^{\wedge, p})_{\Delta}\rightarrow 0. \end{align} $$
Since
$H_2(\Delta , M^{\wedge , p})$
is a torsion module by Lemma 8.1, we deduce that
$$ \begin{align*}\mathop{\mathrm{ rank}}\nolimits_{\mathbb Z_p} ((\mathcal E\otimes M)^{\wedge, p})_{\Delta} = \mathop{\mathrm{ rank}}\nolimits_{\mathbb Z_p} (\Gamma_E^{\mathrm{ab}, p} \otimes M)_{\Delta} + \mathop{\mathrm{ rank}}\nolimits_{\mathbb Z_p} (I_{\Delta}\otimes M^{\wedge, p})_{\Delta}.\end{align*} $$
9 Galois deformations
We will apply the machinery developed in the previous sections in an arithmetic situation. Let F be a finite extension of
$\mathbb {Q}_p$
. We fix an algebraic closure
$\overline F$
and let 
. Let
$\overline {\rho }: \Gamma _F \rightarrow G(k)$
be a continuous representation, where G is a generalised torus over
$\mathcal O$
, and let
$\overline {\Theta }$
be the G-pseudocharacter associated to
$\overline {\rho }$
. Let
$\Gamma _E$
be the kernel of the composition
$\Gamma _F \overset {\overline {\rho }}{\longrightarrow } G(k)\rightarrow (G/G^0)(k)$
and let 
. Let
$\pi : \Gamma _F \rightarrow \Delta $
and
$\Pi : G\rightarrow G/G^0$
be the projection maps.
We let
$D^{\mathrm {ps}}_{\overline {\Theta }}: \mathfrak A_{\mathcal O} \rightarrow \text {Set}$
be the functor
$$ \begin{align*}D^{\mathrm{ps}}_{\overline{\Theta}}(A)=\{\Theta\in \mathrm{cPC}^{\Gamma_F}_{G,\pi}(A): \Theta\otimes_A k= \overline{\Theta}\}\end{align*} $$
and let
$R^{\mathrm {ps}}_{\overline {\Theta }}\in \widehat {\mathfrak A}_{\mathcal O}$
be the ring pro-representing
$D^{\mathrm {ps}}_{\overline {\Theta }}$
and let
$\Theta ^u$
be the universal deformation of
$\overline {\Theta }$
.
We let
$X^{\mathrm {gen}}_{G, \overline {\Theta }}: R^{\mathrm {ps}}_{\overline {\Theta }}\text {-}\mathrm {alg}\rightarrow \text {Set}$
be the functor
$$ \begin{align*}X^{\mathrm{gen}}_{G, \overline{\Theta}}(A)=\{ \rho\in \operatorname{\mathrm{Rep}}^{\Gamma_F}_{G, \pi}(A): \Theta_{\rho}=\Theta^u\otimes_{R^{\mathrm{ps}}_{\overline{\Theta}}} A\}.\end{align*} $$
Let
$A^{\mathrm {gen}}_{G, \overline {\Theta }}$
be the
$R^{\mathrm {ps}}_{\overline {\Theta }}$
-algebra representing
$X^{\mathrm {gen}}_{G, \overline {\Theta }}$
.
We let
$D^{\square }_{\overline {\rho }}: \mathfrak A_{\mathcal O} \rightarrow \text {Set}$
be the functor such that
$D^{\square }_{\overline {\rho }}(A)$
is the set of continuous representations
$\rho : \Gamma _F\rightarrow G(A)$
such that
$\rho \equiv \overline {\rho } \mod {\mathfrak m_A}$
. Let
$R^{\square }_{\overline {\rho }}\in \widehat {\mathfrak A}_{\mathcal O}$
be the the ring pro-representing
$D^{\square }_{\overline {\rho }}$
.
The Weil group
$W_{E/F}$
fits into a short exact sequence
$$ \begin{align*}0\rightarrow E^{\times} \rightarrow W_{E/F}\rightarrow \Delta\rightarrow 0,\end{align*} $$
corresponding to the fundamental class
$[u_{E/F}]\in H^2(\Delta , E^{\times })$
by [Reference Tate16, (1.2)]. Let
$\widehat {W}_{E/F}$
be the profinite completion of
$W_{E/F}$
. The Artin map
$\operatorname {\mathrm {Art}}_E: E^{\times }\rightarrow \Gamma _E^{\mathrm {ab}}$
induces an isomorphism between the profinite completion of
$E^{\times }$
and
$\Gamma _E^{\mathrm {ab}}$
. This yields a natural isomorphism
$$ \begin{align} \widehat{W}_{E/F} \cong \Gamma_F/ [\Gamma_E, \Gamma_E], \end{align} $$
where
$[\Gamma _E, \Gamma _E]$
is the closure of the subgroup generated by the commutators in
$\Gamma _E$
.
Lemma 9.1. Every continuous representation
$\rho : \Gamma _F \rightarrow G(A)$
satisfying
$\Pi \circ \rho = \pi $
, where A is a finite discrete
$\mathcal O$
-algebra, factors through the quotient
$\Gamma _F\twoheadrightarrow \widehat {W}_{E/F}$
.
Proof. Since
$\rho (\Gamma _E) \subseteq G^0(A)$
and
$G^0(A)$
is commutative and Hausdorff, the assertion follows from (9.1).
Lemma 9.2. Let
$A\in R^{\mathrm {ps}}_{\overline {\Theta }}\text {-}\mathrm {alg}$
. Then every
$\rho \in X^{\mathrm {gen}}_{G, \overline {\Theta }}(A)$
factors through the quotient
$\Gamma _F\twoheadrightarrow \widehat {W}_{E/F}$
.
Proof. The restriction of
$\overline {\rho }$
to
$\Gamma _E$
takes values in
$G^0(k)$
. Let
$\overline {\Psi }$
be the
$G^0$
-pseudocharacter of
$\overline {\rho }|_{\Gamma _E}$
and let
$\Psi ^u\in D^{\mathrm {ps}}_{\overline {\Psi }}(R^{\mathrm {ps}}_{\overline {\Psi }})$
be the universal deformation of
$\overline {\Psi }$
. Since
$G^0$
is commutative, Lemma 6.3 implies that there is a continuous group homomorphism
$\psi ^u: \Gamma _E \rightarrow G^0(R^{\mathrm {ps}}_{\overline {\Psi }})$
such that
$\Psi ^u = \Theta _{\psi ^u}$
. In particular,
$$ \begin{align} \psi^u(\gamma)=1, \quad \forall \gamma \in [\Gamma_E, \Gamma_E]. \end{align} $$
Since
$\Theta ^u|_{\Gamma _E}$
is a deformation of
$\overline {\Psi }$
to
$R^{\mathrm {ps}}_{\overline {\Theta }}$
, there is a homomorphism
$R^{\mathrm {ps}}_{\overline {\Psi }}\rightarrow R^{\mathrm {ps}}_{\overline {\Theta }}$
such that
$$ \begin{align} \Theta^u|_{\Gamma_E} = \Psi^u\otimes_{R^{\mathrm{ps}}_{\overline{\Psi}}} R^{\mathrm{ps}}_{\overline{\Theta}}. \end{align} $$
If
$\rho \in X^{\mathrm {gen}}_{G, \overline {\Theta }}(A)$
, then
$\Theta _{\rho }= \Theta ^u\otimes _{R^{\mathrm {ps}}_{\overline {\Theta }}} A$
, and it follows from (9.3) that
$\rho |_{\Gamma _E}= \psi ^u \otimes _{R^{\mathrm {ps}}_{\overline {\Psi }}} A$
. The assertion follows from (9.2).
Theorem 9.3. There is a finite extension
$L'$
of L with the ring of integers
$\mathcal O'$
, such that the following hold:
-
(1)
$G^0_{\mathcal O'}$
is split and
$(G/G^0)_{\mathcal O'}$
is a constant group scheme; -
(2)
$\operatorname {\mathrm {Rep}}^{\Gamma _F}_{G, \pi }(\mathcal O')$
is nonempty.
Moreover, if
$(1)$
and
$(2)$
hold, then there are isomorphisms of
$\mathcal O'$
-algebras:
-
(3)
; -
(4)
$A^{\mathrm {gen}}_{G, \overline {\Theta }}\otimes _{\mathcal O} \mathcal O'\cong R^{\mathrm {ps}}_{\overline {\Theta }}\otimes _{\mathcal O} \mathcal O'[t_1^{\pm 1}, \ldots , t_s^{\pm 1}]$
; -
(5)
,
;
,where M is the character lattice of
$G^0_{\mathcal O'}$
,
$r= \mathop {\mathrm { rank}}\nolimits _{\mathbb {Z}} M \cdot [F:\mathbb {Q}_p] +\mathop {\mathrm { rank}}\nolimits _{{\mathbb {Z}}} M_{\Delta }$
,
$s=\mathop {\mathrm { rank}}\nolimits _{\mathbb {Z}} M -\mathop {\mathrm { rank}}\nolimits _{\mathbb {Z}} M_{\Delta }$
.
Proof. If
$L'$
is a finite extension of L with the ring of integers
$\mathcal O'$
and residue field
$k'$
, then the functor
$D^{\mathrm {ps}}_{\overline {\Theta }_{k'}}: \mathfrak A_{\mathcal O'}\rightarrow \text {Set}$
is pro-representable by
$R^{\mathrm {ps}}_{\overline {\Theta }}\otimes _{\mathcal O} \mathcal O'$
, and analogous statements hold for
$X^{\mathrm {gen}}_{G, \overline {\Theta }_{k'}}$
and
$D^{\square }_{\overline {\rho }_{k'}}$
. Thus, it is enough to prove the statement for
$\mathcal O'=\mathcal O$
, after replacing L by a finite extension.
Since G is a generalised torus, after replacing L by a finite unramified extension, we may assume that
$G^0$
is split and
$G/G^0$
is a constant group scheme. We may assume that
$G/G^0=\underline {\Delta }$
, as replacing G by the preimage of
$\underline {\Delta }$
does not change the functors under consideration. The character lattice M of
$G^0$
does not change if we further replace L by a finite extension. As explained at the beginning of Section 5, we have an action of
$\Delta $
on M.
We will apply the results of previous sections with
$\Gamma _1=W_{E/F}$
and
$\Gamma _2=E^{\times }$
. It is a fundamental result of Langlands proved in [Reference Langlands11] (see also a nice exposition by Birkbeck [Reference Birkbeck2, Proposition 2.0.3]) that there are natural isomorphisms:
$$ \begin{align} H_1(W_{E/F}, M) \cong H_1(E^{\times}, M)^{\Delta} \cong (E^{\times}\otimes M)^{\Delta}. \end{align} $$
For each
$c\in \Delta $
, we choose a coset representative
$\bar {c}\in W_{E/F}$
. We have constructed a
$\Delta $
-action on 
, which depends on this choice, such that we have an extension of
$\mathbb Z[\Delta ]$
-modules
$$ \begin{align*}0\rightarrow E^{\times} \rightarrow \mathcal E \rightarrow I_{\Delta}\rightarrow 0,\end{align*} $$
and the image of the extension class in
$H^2(\Delta , E^{\times })$
under the isomorphisms
$$ \begin{align*}\operatorname{\mathrm{Ext}}^1_{\mathbb Z[\Delta]}(I_{\Delta}, E^{\times})\cong \operatorname{\mathrm{Ext}}^2_{\mathbb Z[\Delta]}(\mathbb Z, E^{\times})\cong H^2(\Delta, E^{\times})\end{align*} $$
is equal to
$[u_{E/F}]$
.
Let N be the pro-p completion of
$(\mathcal E\otimes M)_{\Delta }$
. It follows from Proposition 8.9 and Euler–Poincaré characteristic formula that
$$ \begin{align} h^1_{\mathrm{cont}}(\Gamma_F, \operatorname{\mathrm{ad}}\overline{\rho})-h^0_{\mathrm{cont}}(\Gamma_F, \operatorname{\mathrm{ad}} \overline{\rho})- h^2_{\mathrm{cont}}(\Gamma_F, \operatorname{\mathrm{ad}} \overline{\rho}) =\mathop{\mathrm{ rank}}\nolimits_{\mathbb Z_p} N -\dim G_k. \end{align} $$
Lemma 9.1 and (9.5) ensure that the assumptions of Proposition 7.6 are satisfied, and hence, after replacing L by a finite extension, we may ensure that
$\operatorname {\mathrm {Rep}}^{\widehat {W}_{E/F}}_{G, \pi }(\mathcal O)$
is nonempty by Corollary 7.7. In particular,
$\operatorname {\mathrm {Rep}}^{\Gamma _F}_{G,\pi }(\mathcal O)$
and
$\operatorname {\mathrm {Rep}}^{W_{E/F}}_{G, \pi }(\mathcal O)$
are also nonempty.
Lemma 6.6 implies that there is
$\rho \in X^{\mathrm {gen}}_{G, \bar {\rho }^{\mathrm {ss}}}(R^{\mathrm {ps}}_{\overline {\Theta }})$
such that
$\Theta ^u = \Theta _{\rho }$
. Since
$\rho $
factors through the quotient
$\Gamma _F \twoheadrightarrow \widehat {W}_{E/F}$
by Lemma 9.2, we conclude that
$\Theta ^u$
is obtained from a G-pseudocharacter of
$\widehat {W}_{E/F}$
via inflation to
$\Gamma _F$
. This together with Lemmas 9.1 and 9.2 implies that in the definitions of the functors
$D^{\mathrm {ps}}_{\overline {\Theta }}$
,
$D^{\square }_{\overline {\rho }}$
and
$X^{\mathrm {gen}}_{G, \overline {\Theta }}$
, we can replace
$\Gamma _F$
with
$\widehat {W}_{E/F}$
without changing the functors themselves.
Lemma 7.10 and (9.4) imply that
$R^{\mathrm {ps}}_{\overline {\Theta }}$
is isomorphic to 
, where N is the pro-p completion of
$(E^{\times }\otimes M)^{\Delta }$
. It follows from Lemma 8.7 that
$N\cong (\Gamma ^{\mathrm {ab}, p}_E\otimes M)^{\Delta }$
, which is a finitely generated
$\mathbb Z_p$
-module of rank r and torsion subgroup isomorphic to
$(\mu _{p^{\infty }}(E)\otimes M)^{\Delta }$
by Corollary 8.8. This yields the isomorphisms in part (3). Part (4) follows from Proposition 7.13.
The map
$\rho \mapsto \Theta _{\rho }$
induces a map of local
$\mathcal O$
-algebras
$R^{\mathrm {ps}}_{\overline {\Theta }}\rightarrow R^{\square }_{\overline {\rho }}$
and hence a map of
$R^{\mathrm {ps}}_{\overline {\Theta }}$
-algebras
$A^{\mathrm {gen}}_{G, \overline {\Theta }}\rightarrow R^{\square }_{\overline {\rho }}$
. Since
$E^{\times }/ (E^{\times })^p$
is finite, Lemma 7.4 implies that
$R^{\square }_{\overline {\rho }}$
is the completion of
$\mathcal O(\operatorname {\mathrm {Rep}}^{W_{E/F}}_{G,\pi })$
with respect to the maximal ideal corresponding to
$\overline {\rho }$
. By considering the composition
$\mathcal O(\operatorname {\mathrm {Rep}}^{W_{E/F}}_{G,\pi }) \rightarrow A^{\mathrm {gen}}_{G, \overline {\Theta }}\rightarrow R^{\square }_{\overline {\rho }}$
, we deduce that it induces a natural isomorphism between
$R^{\square }_{\overline {\rho }}$
, and the completion of
$A^{\mathrm {gen}}_{G, \overline {\Theta }}$
with respect to the maximal ideal corresponding to
$\overline {\rho }$
. Part (5) then follows from parts (3) and (4).
We will now deduce some corollaries, which hold without extending the scalars.
Corollary 9.4. The map
$R^{\mathrm {ps}}_{\overline {\Theta }} \rightarrow R^{\square }_{\overline {\rho }}$
is formally smooth. In particular, it is flat and induces a bijection between the sets of irreducible components.
Proof. Since the map
$R^{\mathrm {ps}}_{\overline {\Theta }}\rightarrow \mathcal O'\otimes _{\mathcal O}R^{\mathrm {ps}}_{\overline {\Theta }}$
is faithfully flat, the assertion follows from [Reference Project15, Tag 06CM] and part (5) of Theorem 9.3.
Corollary 9.5. The map
$R^{\mathrm {ps}}_{\overline {\Theta }}\rightarrow A^{\mathrm {gen}}_{G, \overline {\Theta }}$
is smooth and induces a bijection between the sets of irreducible components. Moreover,
$A^{\mathrm {gen}}_{G, \overline {\Theta }}$
is flat over
$\mathcal O$
of relative dimension
$\dim G_k \cdot ([F:\mathbb {Q}_p]+1)$
.
Proof. The map
$R^{\mathrm {ps}}_{\overline {\Theta }}\rightarrow A^{\mathrm {gen}}_{G, \overline {\Theta }}$
is formally smooth by the same argument as in Corollary 9.4 using part (4) of Theorem 9.3. Since it is of finite type, it is smooth by [Reference Project15, Tag 00TN]. The map 
is flat and of finite presentation and hence open by [Reference Project15, Tag 00I1]. By [Reference Project15, Tag 004Z], it is enough to show the fibres of
$X^{\mathrm {gen}}_{G, \overline {\Theta }} \rightarrow X^{\mathrm {ps}}_{\overline {\Theta }}$
are irreducible.
To ease the notation, we let
$R=R^{\mathrm {ps}}_{\overline {\Theta }}$
,
$A=A^{\mathrm {gen}}_{G, \overline {\Theta }}$
,
$R'=R\otimes _{\mathcal O} \mathcal O'$
and
$A'=A\otimes _{\mathcal O} \mathcal O'$
. Let
$x:\operatorname {\mathrm {Spec}} \kappa \rightarrow \operatorname {\mathrm {Spec}} R$
be a geometric point and let
$x': \operatorname {\mathrm {Spec}} \kappa \rightarrow \operatorname {\mathrm {Spec}} R'$
be a point above x. The map
$$ \begin{align*}A\otimes_{R, x} \kappa \rightarrow A'\otimes_{R', x'}\kappa\end{align*} $$
is an isomorphism. Part (2) of Theorem 9.3 implies that
$$ \begin{align*}A'\otimes_{R', x'}\kappa\cong \kappa[t_1^{\pm 1}, \ldots, t_s^{\pm 1} ].\end{align*} $$
Thus, the fibres of
$X^{\mathrm {gen}}_{G, \overline {\Theta }} \rightarrow X^{\mathrm {ps}}_{\overline {\Theta }}$
are irreducible, and the result follows.
The last assertion follows from the fact that
$\mathcal O'$
is finite and free over
$\mathcal O$
, parts (3) and (4) of Theorem 9.3 and
$\dim G_k = \mathop {\mathrm { rank}}\nolimits _{\mathbb Z} M$
.
Lemma 9.6. Let
$A\in R^{\mathrm {ps}}_{\overline {\Theta }}\text {-}\mathrm {alg}$
, let
$\rho \in X^{\mathrm {gen}}_{G,\overline {\Theta }}(A)$
and let
$\tau : G\hookrightarrow \mathbb A^n$
be a closed immersion of
$\mathcal O$
-schemes. Then
$\tau (\rho (\Gamma _F))$
is contained in a finitely generated
$R^{\mathrm {ps}}_{\overline {\Theta }}$
-submodule of
$A^n=\mathbb A^n(A)$
.
Proof. It is enough to verify the assertion after extending the scalars to
$\mathcal O'$
given by Theorem 9.3. This follows from Lemma 9.2 and Lemma 9.6.
Remark 9.7. Lemma 9.6 implies that the scheme
$X^{\mathrm {gen}}_{G, \overline {\Theta }}$
coincides with the scheme
$X^{\mathrm {gen}}_{G, \bar {\rho }^{\mathrm {ss}}}$
defined in [Reference Paškūnas and Quast12] for a generalised reductive group G; see [Reference Paškūnas and Quast12, Proposition 7.3].
Corollary 9.8. Let
$p^m$
be the order of
$(\mu _{p^{\infty }}(E)\otimes M)^{\Delta }$
. Assume that
$\mathcal O$
contains all the
$p^m$
-th roots of unity and (1) and (2) in Theorem 9.3 hold with
$\mathcal O'=\mathcal O$
. Then there is a canonical action of the character group
$\mathrm X((\mu _{p^{\infty }}(E)\otimes M)^{\Delta })$
on the set of irreducible components of
$\operatorname {\mathrm {Spec}} R^{\mathrm {ps}}_{\overline {\Theta }}$
,
$\operatorname {\mathrm {Spec}} R^{\square }_{\overline {\rho }}$
and
$X^{\mathrm {gen}}_{G, \overline {\Theta }}$
, respectively. Moreover, the following hold:
-
(1) this action is faithful and transitive;
-
(2) irreducible components of
$\operatorname {\mathrm {Spec}} R^{\square }_{\overline {\rho }}$
and
$\operatorname {\mathrm {Spec}} R^{\mathrm {ps}}_{\overline {\Theta }}$
are formally smooth over
$\mathcal O$
; -
(3) irreducible components of
$X^{\mathrm {gen}}_{G, \overline {\Theta }}$
are of the form where r and s are as in Theorem 9.3.
Proof. The assertion follows from Proposition 7.20.
Corollary 9.9. Let
$\varphi : G \to H$
be a surjection of generalised tori over
$\mathcal O$
and let
$\overline {\rho } : \Gamma _F \to G(k)$
be a continuous representation. Then the map
$(R^{\square }_{\varphi \circ \overline {\rho }}/\varpi )^{\mathrm {red}}\rightarrow (R^{\square }_{\overline {\rho }}/\varpi )^{\mathrm {red}}$
is flat, where
$\mathrm {red}$
indicates reduced rings, and the fibre at the closed point has dimension
$([F:\mathbb {Q}_p]+1)(\dim G_k - \dim H_k)$
.
Proof. If
$\mathcal P$
is a finitely generated
$\mathbb Z_p$
-module and 
is the completed group algebra of
$\mathcal P$
, then 
, where
$\mathcal P^{\mathrm {tf}}$
is the maximal torsion-free quotient of
$\mathcal P$
. In particular, 
, where
$d= \dim _{\mathbb {Q}_p} (\mathcal P\otimes _{\mathbb Z_p} \mathbb {Q}_p)$
.
Let
$f:\mathcal P_1\rightarrow \mathcal P_2$
be a homomorphism of finitely generated
$\mathbb Z_p$
-modules and let 
. If
$\ker f$
is torsion, then
$\mathcal P_1^{\mathrm {tf}}$
is a submodule of
$\mathcal P_2^{\mathrm {tf}}$
and the fibre
$k\otimes _{R_1^{\mathrm {red}}} R_2^{\mathrm {red}}$
is isomorphic to 
. Since the dimension of the fibre is equal to
$d_2-d_1$
, where
$d_i= \dim _{\mathbb {Q}_p} (\mathcal P_i\otimes _{\mathbb Z_p} \mathbb {Q}_p)$
, we deduce from [Reference Project15, Tag 00R4] that
$R_1^{\mathrm {red}}\rightarrow R_2^{\mathrm {red}}$
is flat.
Let M be the character lattice of
$G^0$
and let N be the character lattice of
$H^0$
. Then
$N\subseteq M$
, and it follows from (8.7) and Lemma 8.1 that the kernel of the map
$((\mathcal E \otimes N)^{\wedge , p})_{\Delta }\rightarrow ((\mathcal E \otimes M)^{\wedge , p})_{\Delta }$
is torsion. Since 
by Lemma 7.5 and we may swap pro-p completion with
$\Delta $
-coinvariants, we obtain the assertion by letting
$\mathcal P_1=((\mathcal E \otimes N)_{\Delta })^{\wedge , p}$
and
$\mathcal P_2=((\mathcal E \otimes M)_{\Delta })^{\wedge , p}$
. The assertion about the dimension of the fibre follows from Proposition 8.9.
Acknowledgements
VP would like to thank Timo Richarz for a helpful discussion concerning the Langlands correspondence for tori. The authors would like to thank Toby Gee and James Newton for their comments on a draft version of this paper.
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.
Competing interest
The authors have no competing interest to declare.
Funding statement
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.
Ethical standards
The research meets all ethical guidelines, including adherence to the legal requirements of the study country.
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