On patched completed homology and a conjecture of Venkatesh

Douglas Molin

Abstract

Let $F$ be a CM field and $\Pi$ a regular algebraic cuspidal cohomological representation of $\mathbf{G}=\operatorname{PGL}_{2}/F$ . A conjecture of Venkatesh describes the structure of the contribution of $\Pi$ to the homology of the locally symmetric spaces associated to $\mathbf{G}$ . We investigate this conjecture in the setting of $p$ -adic homology with $p$ a totally split prime. Along the way, we elaborate on the relations between Venkatesh’s conjecture and completed homology, the Taylor-Wiles method and the $p$ -adic local Langlands correspondence. Our main result is a ‘big $R=T$ ’ theorem in characteristic 0, from which we deduce a variant of the $p$ -adic realisation of Venkatesh’s conjecture, conditional on various natural conjectures and technical assumptions.

1 Introduction

Let $\mathbf{G}=\operatorname{PGL}_{2}/F$ where $F$ is a CM field and suppose $\Pi$ is a regular algebraic cuspidal automorphic representation of $\mathbf{G}(\mathbb{A}_{F})$ which is cohomological with respect to an algebraic representation of some weight $\lambda$ . Then $\Pi$ contributes to the homology of the locally symmetric spaces associated to $\mathbf{G}$ in the following sense.

Let $l_{0}=[F:\mathbb{Q}]/2$ and fix a compact open subgroup $U\subset\mathbf{G}(\mathbb{A}_{F}^{\infty})$ . There is an associated locally symmetric space $X_{U}$ which – if $U$ is small enough – is a smooth manifold of dimension $3l_{0}$ , and we can consider its homology $H_{\ast}(X_{U},\mathcal{L}_{\lambda})$ with coefficients in a $p$ -adic local system $\mathcal{L}_{\lambda}$ associated to the weight $\lambda$ of $\Pi$ . The homology is a finite-dimensional $\mathbb{Q}_{p}$ -vector space with an action of Hecke operators, and we define its ‘ $\Pi$ -part’ as the eigenspace

H_{\ast}(X_{U},\mathcal{L}_{\lambda})_{\Pi}\subset H_{\ast}(X_{U},\mathcal{L}_{\lambda})

of homology classes with the same system of Hecke eigenvalues as $\Pi$ . Let us assume for the purpose of this introduction that all eigenvalues of $\Pi$ are rational and that $\Pi$ is unramified at all places above $p$ . Using $(\mathfrak{g},K)$ -cohomology, one can compute the dimensions of the graded pieces of the $\Pi$ -part and see that, if $d=\dim(\Pi^{\infty})^{U}$ , then

\dim_{\mathbb{Q}_{p}}H_{l_{0}+i}(X_{U},\mathcal{L}_{\lambda})_{\Pi}=\begin{cases}d\cdot\binom{l_{0}}{i}&\text{ if }i=0,\dots,l_{0}\\ 0&\text{ otherwise.}\end{cases}

To give an arithmetic explanation of this ‘spreading out’ in multiple degrees of Hecke eigenspaces, Venkatesh conjectures the existence of a vector space $V_{\Pi}$ and a natural action of the exterior algebra $\wedge^{\ast}V_{\Pi}$ on $H_{\ast}(X_{U},\mathcal{L}_{\lambda})_{\Pi}$ such that the homology is a free graded module of rank $d$ over $\wedge^{\ast}V_{\Pi}$ . In fact, Venkatesh’s conjecture is a more refined statement at the level of only conjecturally existing motives, but in the case of $p$ -adic homology one can make a concrete prediction. Letting $S$ denote the set of places of $F$ where $\Pi$ ramifies, one can associate a Galois representation $\rho:G_{F,S}\to\operatorname{GL}_{2}(\mathbb{Q}_{p})$ to $\Pi$ using the construction in [19] or [34]. Here, $G_{F,S}$ is the Galois group of the maximal extension of $F$ unramified outside of $S$ . One expects $V_{\Pi}$ to be a Galois cohomology group defined in terms of $\rho$ , namely

V_{\Pi}=H^{1}_{f}(G_{F,S},\operatorname{ad}^{0}\rho(1)),

the dual of the adjoint Bloch-Kato Selmer group (see Section 4). The Bloch-Kato conjecture predicts that the dimension of this vector space equals the order of vanishing at $s=1$ of the adjoint $L$ -function $L(s,\operatorname{ad}^{0}\Pi)$ , which is known to be $l_{0}=[F:\mathbb{Q}]/2$ . In this way, one hopes to obtain an arithmetic proof of the dimension formula $(\star)$ .

In this article, we prove a version of the $p$ -adic conjecture under various conjectures and technical assumptions on $\rho$ , as well as the assumptions that $p\geq 5$ and $p$ is totally split in $F$ . Before stating our main result, let us describe in broad terms the strategy of proof. The main idea is to place the $\Pi$ -part in a $p$ -adic family by relating it to completed homology. This is a $p$ -adic representation $\widetilde{H}$ of $G=\prod_{v\mid p}\mathbf{G}(F_{v})$ and it has an action of a ‘big’ Hecke algebra $T$ . The $\Pi$ -part is related to completed homology by a spectral sequence. Using the Taylor-Wiles method for completed homology developed by Gee-Newton [16], we prove a ‘big $R=T$ ’ theorem in characteristic $0$ which identifies the spectrum of $T$ with the unrestricted deformation space of the Galois representation $\rho$ associated to $\Pi$ .

The proof of the ‘ $R=T$ ’ theorem depends crucially on the condition that $p$ is totally split in $F$ and a type of local-global compatibility assumption at the places $v\mid p$ . Under these assumptions, we are able to utilise the $p$ -adic local Langlands correspondence for $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ as described in [32]. Paškūnas’ theory provides an equivalence between a category of $G$ -representations in which we find $\widetilde{H}$ and a certain category of modules over a local deformation ring $R_{\operatorname{loc}}$ . This is the $\mathbb{Q}_{p}$ -algebra representing deformations of the tuple $(\rho_{v})_{v\mid p}$ , where $\rho_{v}$ denotes the restriction of $\rho$ to a decomposition group at $v$ . Under this equivalence, the image of $\widetilde{H}$ is finitely generated, and thus we are able to carry out the depth estimates in the Taylor-Wiles method with this object. We learned of this strategy in [31].

Let us return to the $\Pi$ -part $H_{\ast}(X_{U},\mathcal{L}_{\lambda})_{\Pi}$ . It is a finitely generated module over the big Hecke algebra $R=T$ , and we think of it as a coherent sheaf on the deformation space of $\rho$ . Its support is a closed subscheme consisting of deformations of $\rho$ satisfying local conditions coming from $p$ -adic Hodge theory. This subscheme can tautologically be described as the intersection between a space of deformations of $\rho$ with the space of deformations of the local factors $\rho_{v}$ satisfying the aforementioned conditions, where the intersection takes place within the space of all deformations of $(\rho_{v})_{v\mid p}$ . Under the Bloch-Kato conjecture, this intersection consists of a single point. In this way, one is able to view the $\Pi$ -part as a module over the $\operatorname{Tor}$ -algebra representing the derived intersection, and the goal is to prove that it is free and moreover to identify the $\operatorname{Tor}$ -algebra with the exterior algebra $\wedge^{\ast}V_{\Pi}$ mentioned above. Thus, the non-transverseness of the intersection of deformation spaces explains the spreading out into multiple degrees.

For the purpose of stating a simplified version of our main result, let $R_{\operatorname{loc}}$ be as above, i.e. the deformation ring parametrising deformations of the local representations $\rho_{v}$ for $v\mid p$ . There exists a quotient $R_{\operatorname{loc}}\twoheadrightarrow R_{\operatorname{loc}}(\lambda)$ corresponding to deformations satisfying the $p$ -adic Hodge theoretic conditions determined by the weight $\lambda$ . Finally, let $R_{\operatorname{gl}}$ denote the representing ring of deformations of $\rho$ unramified outside of $S$ . It turns out that $R_{\operatorname{gl}}$ represents a closed subscheme of the formal spectrum of $R_{\operatorname{loc}}$ , and the space of global deformations satisfying the local conditions induced by $\lambda$ is represented by the completed tensor product $R_{\operatorname{gl}}\otimes_{R_{\operatorname{loc}}}R_{\operatorname{loc}}(\lambda)$ .

We now present our main result (Theorem 7.12) without spelling out all of the assumptions.

Theorem 1.1.

Suppose $p\geq 5$ is totally split in $F$ and that $\rho$ has no global deformations satisfying the local conditions induced by $\lambda$ . Then, under various additional assumptions on $\rho$ , the graded $L$ -vector space $H_{\ast}(X_{U},\mathcal{L}_{\lambda})_{\Pi}$ has a canonical structure of free module of rank $d$ over the $\operatorname{Tor}$ -algebra

\operatorname{Tor}_{\ast}^{R_{\operatorname{loc}}}\big{(}R_{\operatorname{gl}},R_{\operatorname{loc}}(\lambda)\big{)}.

If, in addition, $R_{\operatorname{gl}}$ is formally smooth, there is a canonical isomorphism of graded-commutative rings

\operatorname{Tor}_{\ast}^{R_{\operatorname{loc}}}\big{(}R_{\operatorname{gl}},R_{\operatorname{loc}}(\lambda)\big{)}\cong\wedge^{\ast}_{L}H^{1}_{f}\big{(}G_{F,S},\operatorname{ad}^{0}\rho(1)\big{)}.

The assumption that $\rho$ has no global deformations satisfying the conditions induced by $\lambda$ is predicted to hold by the Bloch-Kato conjecture and is equivalent to the statement $\dim V_{\Pi}=l_{0}$ . Proving the final part of the theorem amounts to leveraging the smoothness assumption to compute the $\operatorname{Tor}$ -groups of the intersection $R_{\operatorname{gl}}\otimes_{R_{\operatorname{loc}}}R_{\operatorname{loc}}(\lambda)$ . This can be done by studying a natural short exact sequence of Galois cohomology groups coming from the Poitou-Tate sequence, by which the statement follows from an explicit computation with Koszul complexes.

A similar theorem for $\operatorname{GL}_{n}/\mathbb{Q}$ , $n\geq 2$ , has been proved using the theory of eigenvarieties and overconvergent cohomology by Hansen-Thorne [18, Theorem 1.1], and our application of the Poitou-Tate sequence is based on an analogous argument therein. Notably, our methods require no ‘small slope’ (or even finite slope) assumption on $\Pi$ since we do not use overconvergent cohomology. However, with our methods we can at present only consider the case $n=2$ and $p$ totally split.

Let us briefly outline the content of this article. In section 2, we recall some facts about the homology of locally symmetric spaces and their Hecke operators. In section 3, we discuss some notions from homological and commutative algebra which are used in later sections. Section 4 contains a brief discussion of Galois cohomology and Selmer groups, and Section 5 is devoted to deformation rings. In section 6, we turn to the representation-theoretic part of the story and introduce completed homology. The final section is devoted to the proof of our main result.

This article forms the basis of the author’s ‘mittseminarium’ given in February of 2024 at the University of Gothenburg.

2 Setup

Let $p\geq 5$ be a prime and $L/\mathbb{Q}_{p}$ a finite extension with ring of integers $\mathcal{O}$ , uniformiser $\varpi$ and residue field $k=\mathcal{O}/\varpi$ . At times, we tacitly assume $L$ is ‘large enough’ (e.g. to contain Hecke eigenvalues). We fix an isomorphism $\iota:\overline{\mathbb{Q}}_{p}\simeq\mathbb{C}$ and a CM field $F$ in which $p$ is totally split.

Unless otherwise stated, completed tensor products are taken over $\mathcal{O}$ . When $R$ is an $\mathcal{O}$ -algebra and $\mathfrak{p}\subset R$ is an ideal not containing $\varpi$ , we use the same symbol to denote the ideal $\mathfrak{p}$ and the ideal it generates inside $R[1/\varpi]$ . We denote by $\mathbb{A}^{\infty}_{F}$ the ring of finite adèles of $F$ .

2.1 Arithmetic locally symmetric spaces

We begin by recalling the construction of the locally symmetric spaces associated to $\mathbf{G}=\operatorname{PGL}_{2}/F$ . A complete reference is [22, §6.1-2]. We introduce the following notation:

•

$\mathbf{G}=\operatorname{PGL}_{2}/F$ ,
•

$G_{\infty}=\mathbf{G}(F\otimes_{\mathbb{Q}}\mathbb{R})$ ,
•

$K_{\infty}\subset G_{\infty}$ a maximal compact connected subgroup,
•

$D_{\infty}=G_{\infty}/K_{\infty}$ ,
•

$l_{0}=\operatorname{rank}G_{\infty}-\operatorname{rank}K_{\infty}=[F:\mathbb{Q}]/2$ ,
•

$q_{0}=\frac{1}{2}(\dim D_{\infty}-l_{0})$ .

The integer $l_{0}$ is called the defect of $\mathbf{G}$ , and turns up in many different settings. In fact, we have $\dim D_{\infty}=3l_{0}$ and hence $q_{0}=l_{0}$ . For groups other than $\operatorname{PGL}_{2}/F$ , one usually has $q_{0}\neq l_{0}$ , and we have opted to maintain the distinction in what follows.

Definition 2.1.

Let $U\subset\mathbf{G}(\mathbb{A}^{\infty}_{F})$ be an open compact subgroup. The locally symmetric space associated to $U$ is the double quotient

X_{U}=\mathbf{G}(F)\backslash(D_{\infty}\times\mathbf{G}(\mathbb{A}^{\infty}_{F})/U),

where the action of $\mathbf{G}(F)$ is the diagonal action.

The space $X_{U}$ decomposes as a finite disjoint union of subspaces of the form $\Gamma_{i}\backslash D_{\infty}$ where $\Gamma_{i}=\mathbf{G}(F)\cap g_{i}Ug_{i}^{-1}$ for some $g\in\mathbf{G}(\mathbb{A}_{F}^{\infty})$ .

Definition 2.2.

A good subgroup $U\subset\mathbf{G}(\mathbb{A}^{\infty}_{F})$ is an open compact subgroup of the form $\prod_{v}U_{v}$ such that:

(i)

For every $v$ , $U_{v}\subseteq\operatorname{PGL}_{2}(\mathcal{O}_{F_{v}})$ .
(ii)

For every $g\in\mathbf{G}(\mathbb{A}^{\infty}_{F})$ and every $h\in gUg^{-1}\cap\mathbf{G}(F),$ the eigenvalues of $h$ generate a torsion-free subgroup of $\overline{F}$ (i.e., the subgroup $gUg^{-1}$ is ‘neat’).

Proposition 2.3.

[22, Lemma 6.1] Let $U\subset\mathbf{G}(\mathbb{A}^{\infty}_{F})$ be a good subgroup. Then $X_{U}$ is a smooth manifold of dimension $3l_{0}$ and homotopy equivalent to the geometric realisation of a finite simplicial complex. Moreover, if $U^{\prime}\subset U$ is a normal compact open subgroup, then $U^{\prime}$ is also good and $X_{U^{\prime}}\to X_{U}$ is a Galois cover of smooth manifolds with Galois group $U/U^{\prime}$ .

2.2 Homology of arithmetic locally symmetric spaces

Throughout this section, $U=U_{p}U^{p}\subset\mathbf{G}(\mathbb{A}^{\infty}_{F})$ is a good subgroup. Here, $U_{p}=\prod_{v\mid p}U_{v}$ and $U^{p}=\prod_{v\nmid p}U_{v}$ . We fix a discrete left $\mathbb{Z}[U_{p}]$ -module $M$ , viewed as a $\mathbb{Z}[U]$ -module by letting $U$ act via the projection $U\to U_{p}$ .

Definition 2.4.

The local system associated to $M$ is the sheaf $\mathcal{L}_{M}$ of continuous sections of the map

\mathbf{G}(F)\backslash(D_{\infty}\times\mathbf{G}(\mathbb{A}^{\infty}_{F})/U)\times M/U\to\mathbf{G}(F)\backslash(D_{\infty}\times\mathbf{G}(\mathbb{A}^{\infty}_{F})/U),

We denote by $H_{\ast}(X_{U},\mathcal{L}_{M})$ the homology of $X_{U}$ with local coefficients $\mathcal{L}_{M}$ .

There are two complexes commonly used to compute the homology with local coefficients displayed above. The first is the adèlic complex, which is useful for defining the Hecke action in a natural way. The drawback of the adèlic complex is its large size, and when finiteness properties are required one uses a Borel-Serre complex instead.

Definition 2.5.

The adélic complex of $X_{U}$ with coefficients in $M$ is the chain complex

C^{\operatorname{ad}}_{\bullet}(U,M)=\operatorname{Sing}_{\bullet}\big{(}G_{\infty}/K_{\infty}\times\mathbf{G}(\mathbb{A}^{\infty}_{F})\big{)}\otimes_{\mathbb{Z}[\mathbf{G}(F)\times U]}M,

where $\operatorname{Sing}_{\bullet}\big{(}G_{\infty}/K_{\infty}\times\mathbf{G}(\mathbb{A}^{\infty}_{F})\big{)}$ denotes the complex of singular chains with $\mathbb{Z}$ -coefficients, viewed as a complex of right $\mathbb{Z}[\mathbf{G}(F)\times U]$ -modules.

To define our Borel-Serre complex, we consider the Borel-Serre bordification $D_{\infty}\subset D_{\infty}^{\operatorname{BS}}$ (see [6, §7.1, Proposition 7.6]) and the principal $U_{p}$ -bundle

\mathbf{G}(F)\backslash(D_{\infty}^{\operatorname{BS}}\times\mathbf{G}(\mathbb{A}^{\infty}_{F})/\mathbf{1}_{p}U^{p})\twoheadrightarrow\mathbf{G}(F)\backslash(D_{\infty}^{\operatorname{BS}}\times\mathbf{G}(\mathbb{A}^{\infty}_{F})/U_{p}U^{p})=X^{\operatorname{BS}}_{U_{p}U^{p}},

where $\mathbf{G}(\mathbb{A}^{\infty}_{F})$ is equipped with the discrete topology.

Fix a finite triangulation of $X^{\operatorname{BS}}_{U_{p}U^{p}}$ , and consider its associated complex of simplicial chains with $\mathbb{Z}$ -coefficients. Pulling back the triangulation via the map above we obtain – possibly after finite refinement – a $K_{p}$ -equivariant triangulation on the ‘infinite $p$ -level’ space $\mathbf{G}(F)\backslash(D_{\infty}^{\operatorname{BS}}\times\mathbf{G}(\mathbb{A}^{\infty}_{F})/\mathbf{1}_{p}U^{p})$ . The associated complex of simplicial chains is then a bounded complex of finitely generated and free $\mathbb{Z}[U_{p}]$ -modules which we denote by $C^{\operatorname{BS}}_{\bullet}(U)$ .

Definition 2.6.

The Borel-Serre complex with $M$ -coefficients is the complex

C^{\operatorname{BS}}_{\bullet}(U,M)=C^{\operatorname{BS}}_{\bullet}(U)\otimes_{\mathbb{Z}[U_{p}]}M.

Proposition 2.7.

The complexes $C^{\operatorname{ad}}_{\bullet}(U,M)$ and $C^{\operatorname{BS}}_{\bullet}(U,M)$ are chain homotopic as $\mathbb{Z}[U_{p}]$ -complexes, and

H_{\ast}(C^{\operatorname{ad}}_{\bullet}(U,M))\cong H_{\ast}(C^{\operatorname{BS}}_{\bullet}(U,M))\cong H_{\ast}(X_{U},\mathcal{L}_{M}).

In light of this result, we fix once and for all a chain homotopy equivalence $C^{\operatorname{ad}}_{\bullet}(U,\mathbb{Z})\stackrel{{\scriptstyle\sim}}{{\to}}C^{\operatorname{BS}}_{\bullet}(U)$ , and we define

H_{\ast}(X_{U},M):=H_{\ast}(X_{U},\mathcal{L}_{M}),

2.3 Hecke operators

Let $S$ be a finite set of finite places of $F$ containing the set $S_{p}$ of places above $p$ . We will only consider Hecke operators away from $p$ .

Definition 2.8.

The abstract Hecke algebra is the countably generated free $\mathcal{O}$ -algebra

\mathbb{T}^{S}=\mathcal{O}[T_{v}\mid v\notin S].

The variable $T_{v}$ corresponds to the double coset operator given above $v$ by

[K_{v}\begin{pmatrix}\varpi_{v}&0\\ 0&1\end{pmatrix}K_{v}],

where $\varpi_{v}$ denotes an arbitrary choice of uniformiser of $\mathcal{O}_{F_{v}}$ and $K_{v}=\operatorname{PGL}_{2}(\mathcal{O}_{F_{v}})$ . Given an automorphic representation $\Pi$ (and a choice of $\iota:\overline{\mathbb{Q}}_{p}\cong\mathbb{C}$ ), we have a map

\mathbb{T}^{S}\to\operatorname{End}_{\mathbb{C}}(\otimes^{\prime}_{v\notin S}\Pi_{v}^{K_{v}})

with kernel a maximal ideal.

Definition 2.9.

Let $\Pi$ be an automorphic representation and suppose $S$ contains the ramified places of $\Pi$ . Let $K_{v}=\operatorname{PGL}_{2}(\mathcal{O}_{F_{v}})$ and

\mathfrak{N}_{\Pi,\iota}=\ker\big{(}\mathbb{T}^{S}\to\operatorname{End}_{\mathbb{C}}(\otimes^{\prime}_{v\notin S}\Pi_{v}^{K_{v}})\big{)}.

The $\Pi$ -part of the homology $H_{\ast}(X_{U},M)$ is defined as the localisation

H_{\ast}(X_{U},M)_{\mathfrak{N}_{\Pi,\iota}}.

The dimensions of the graded pieces are given by the following formula.

Theorem 2.10.

Let $\mathfrak{p}\subset\mathbb{T}^{S}[1/\varpi]$ be the maximal ideal associated to the cuspidal $\operatorname{PGL}_{2}/F$ -representation $\Pi$ contributing to homology with coefficients in $\sigma$ , and set $d=\dim_{\mathbb{C}}(\Pi^{\infty})^{U^{p}}$ . Then

\dim_{L}H_{i}(X_{KU^{p}},\sigma)_{\mathfrak{p}}=\begin{cases}d\cdot\binom{l_{0}}{i}&\text{ if }i\in[q_{0},q_{0}+l_{0}],\\ 0&\text{ otherwise.}\end{cases}

Proof.

The result follows from the proof of [18, Proposition 4.2]. ∎

We will consider the notion above when $M$ is an $\mathcal{O}[U_{p}]$ -module which is finitely generated over $\mathcal{O}$ . In particular, if $s\geq 1$ and $M=\mathcal{O}/\varpi^{s}$ with the trivial group action, we have for any open normal subgroup $U_{p}\subset K$ a homomorphism

\mathbb{T}^{S}\to\operatorname{End}_{D((\mathcal{O}/\varpi^{s})[K/U_{p}])}(C^{\operatorname{ad}}_{\bullet}(U_{p}U^{p},\mathcal{O}/\varpi^{s})),

where $D(-)$ denotes the unbounded derived category. We define $\mathbb{T}^{S}(U_{p}U^{p},\mathcal{O}/\varpi^{s})$ as the image of the homomorphism above. Varying $s$ and $U_{p}$ , one obtains a projective system.

Definition 2.11.

The big Hecke algebra is defined as the projective limit

\mathbb{T}^{S}(U^{p})=\varprojlim_{s,U_{p}}\mathbb{T}^{S}(U_{p}U^{p},\mathcal{O}/\varpi^{s}).

The big Hecke algebra is a semi-local profinite $\mathcal{O}$ -algebra, such that for any maximal ideal $\mathfrak{m}\subset\mathbb{T}^{S}(U^{p})$ the localisation $\mathbb{T}^{S}(U^{p})_{\mathfrak{m}}$ is a local $\mathfrak{m}$ -adically complete $\mathcal{O}$ -algebra with residue field a finite extension of $k$ ([16, Lemma 2.1.14]).

3 Commutative and homological algebra

In this section we collect some general lemmas from commutative and homological algebra that will be used in later sections.

3.1 Projective limits

Lemma 3.1.

Let $A$ be a ring, $(M_{\alpha})$ a projective system of finite length $A$ -modules and $N$ a finitely presented $A$ -module. Then there is a canonical isomorphism

(\varprojlim M_{\alpha})\otimes_{A}N\to\varprojlim M_{\alpha}\otimes_{A}N.

Proof.

See e.g. [1, Lemma 2.3.4]. ∎

Lemma 3.2.

([29, Proposition IV.2.7]) Let $A$ be a topological ring. Then the functor

\varprojlim:\operatorname{Pro}-\operatorname{Mod}^{\operatorname{cpt}}(A)\to\operatorname{Mod}^{\operatorname{cpt}}(A)

is exact, where $\operatorname{Pro}-\operatorname{Mod}^{\operatorname{cpt}}(A)$ is the category of projective systems of compact $A$ -modules.

3.2 Depth and regular sequences

Throughout this section, let $(A,\mathfrak{m})$ be a Noetherian local ring, $I\subset A$ a proper ideal and $M$ a finitely generated $A$ -module. We recall some basic facts about regular sequences and the depth of modules, following [14, §18].

Definition 3.3.

An $M$ -regular $I$ -sequence is a sequence $\underline{a}=a_{1},\dots,a_{r}\in I$ such that for $i=1,\dots,r$ , multiplication by $a_{i}$ is injective on $M/(a_{1},\dots,a_{i-1})$ .

We refer to $A$ -regular $\mathfrak{m}$ -sequences simply as regular sequences.

Definition 3.4.

The $I$ -depth of $M$ is denoted $\operatorname{dp}_{I}(M)$ and is the supremum of the lengths of $M$ -regular $I$ -sequences.

When $I=\mathfrak{m}$ , we denote the $\mathfrak{m}$ -depth of $M$ by $\operatorname{dp}_{A}(M)$ . It is clear from the definition that

\operatorname{dp}_{I}M\leq\operatorname{dp}_{A}M.

The property of being $M$ -regular can be formulated in a homological way using Koszul complexes.

Definition 3.5.

Let $\underline{a}=a_{1},\dots,a_{r}$ be an arbitrary sequence of elements of $A$ . The Koszul complex of $\underline{a}$ with coefficients in $A$ is the complex $K_{\bullet}^{A}(\underline{a})$ of $A$ -modules which is non-zero only in degrees $[0,r]$ where it is given by

	$\displaystyle K_{n}^{A}(\underline{a})$	$\displaystyle=\bigoplus_{1\leq j_{1}<\dots<j_{n}\leq r}A$
	$\displaystyle d(b_{1},\dots,b_{n})$	$\displaystyle=\sum_{i=1}^{r}(-1)^{i-1}a_{i}(b_{1},\dots,b_{i-1},b_{i+1},\dots,b_{n}).$

The Koszul complex is exact precisely when the sequence $\underline{a}$ is regular on $A$ , in which case $K_{n}^{A}(\underline{a})$ is a free resolution of $A/(\underline{a})$ . More generally, one has the following homological characterisation of regular sequences.

Proposition 3.6.

Let $\underline{a}$ be an arbitrary sequence in $A$ . Then $\underline{a}$ is $M$ -regular if and only if, for every $i\geq 1$ ,

H_{i}(K_{\bullet}^{A}(\underline{a})\otimes_{A}M)=0.

Note that if $\underline{a}$ is $A$ -regular, then $H_{\ast}(K_{\bullet}^{A}(\underline{a})\otimes_{A}M)\cong\operatorname{Tor}_{\ast}(A/(\underline{a}),M)$ . Restated slightly, we have:

Corollary 3.7.

Suppose $I\subset A$ can be generated by an $A$ -regular sequence. Then $I$ can be generated by an $M$ -regular sequence if and only if

M\otimes^{\mathbf{L}}_{A}A/I\cong M/IM,

where we interpret $M/IM$ as a complex concentrated in degree $0$ .

Lemma 3.8.

Let $M,N$ be finitely generated modules over $A$ . Suppose $I\subset\operatorname{Ann}N$ is an ideal generated by an $A$ -regular and $M$ -regular sequence. Then we have a natural isomorphism of complexes

M\otimes^{\mathbf{L}}_{A}N\cong M/IM\otimes^{\mathbf{L}}_{A/I}N

Proof.

Since $N\cong N/IN$ , we have

\begin{split}M\otimes^{\mathbf{L}}_{A}N&\cong M\otimes^{\mathbf{L}}_{A}A/I\otimes^{\mathbf{L}}_{A/I}N\\ &\cong M/IM\otimes^{\mathbf{L}}_{A/I}N,\end{split}

using the assumptions and the homological criterion. ∎

The depth can also be defined in homological terms using the following result.

Proposition 3.9.

([14, Proposition 18.4])

\operatorname{dp}_{I}(M)=\min\{i\in\mathbb{N}\mid\operatorname{Ext}^{i}_{A}(A/I,M)\neq 0\}.

Lemma 3.10.

Suppose $I\subset A$ can be generated by a regular sequence. Then

\operatorname{dp}_{I}M+\operatorname{dp}^{\ast}_{I}M=\operatorname{dp}_{I}A,

where $\operatorname{dp}^{\ast}_{I}M=\max\{i\mid\operatorname{Tor}_{i}^{A}(A/I,M)\neq 0\}$ .

Proof.

Let $r=\operatorname{dp}_{I}A$ and $\underline{a}=a_{1},\dots,a_{r}$ a regular sequence generating $I$ . By [14, Proposition 17.15], there is a natural isomorphism of chain complexes

K_{\bullet}^{A}(\underline{a})\otimes_{A}M\cong\operatorname{Hom}_{A}(K_{r-\bullet}^{A}(\underline{a}),M)

and hence for every $i$ an isomorphism

\operatorname{Tor}_{i}^{A}(A/I,M)\cong\operatorname{Ext}^{r-i}_{A}(A/I,M).

The statement follows. ∎

Lemma 3.11.

Let $\widehat{A}$ and $\widehat{M}$ denote the $\mathfrak{m}$ -adic completions of $A$ and $M$ . Then

\operatorname{dp}_{A}M=\operatorname{dp}_{\widehat{A}}\widehat{M}.

Proof.

Since $A$ is Noetherian, $\widehat{A}$ is a faithfully flat $A$ -module [25, Theorem 1.3.16(b)] and the maximal ideal of $\widehat{A}$ is $\mathfrak{m}\widehat{A}$ . By flat base change, we have an isomorphism of $A$ -modules (see [39, Proposition 3.3.10])

\operatorname{Ext}_{\widehat{A}}^{i}(\widehat{A}/\mathfrak{m}\widehat{A},\widehat{M})\cong\operatorname{Ext}_{A}^{i}(A/\mathfrak{m},M)\otimes_{A}\widehat{A}.

Since $\widehat{A}$ is faithfully flat, the right hand side equals $0$ if and only if $\operatorname{Ext}^{i}_{A}(A/\mathfrak{m},M)$ does, and the statement thus follows from Proposition 3.9. ∎

3.3 Cohen-Macaulay modules

We keep the notation from the previous section. One always has the inequalities

\operatorname{dp}_{A}M\leq\dim_{A}M\leq\dim A.

Definition 3.12.

We say $M$ is Cohen-Macaulay if $\operatorname{dp}_{A}M=\dim_{A}M$ , and maximal Cohen-Macaulay if $\operatorname{dp}_{A}M=\dim A$ .

Lemma 3.13.

([36, Tag 00NT]) Suppose $A$ is regular and that $M$ is maximal Cohen-Macaulay over $A$ . Then $M$ is free.

Lemma 3.14.

Let $A$ be a complete local Noetherian $\mathcal{O}$ -algebra and $M$ a finitely generated maximal Cohen-Macaulay $A$ -module. Suppose $\mathfrak{p}\subset\operatorname{Spec}A[1/\varpi]$ is a regular closed point. Then $\widehat{M}_{\mathfrak{p}}$ is free over $\widehat{A}_{\mathfrak{p}}$ .

Proof.

We prove that $\widehat{M}_{\mathfrak{p}}$ is maximal Cohen-Macaulay over $\widehat{A}_{\mathfrak{p}}$ and use Proposition 3.13. It suffices to prove $\dim A_{\mathfrak{p}}\leq\operatorname{dp}_{A_{\mathfrak{p}}}M_{\mathfrak{p}}$ . By the Cohen structure theorem, $A$ is a quotient of a ring of formal power series over $\mathcal{O}$ and hence the quotient $A/\mathfrak{p}$ is a finite extension of $\mathcal{O}$ and moreover,

\dim A_{\mathfrak{p}}\leq\dim A-1=\operatorname{dp}_{A}M-1,

where we in the second equality have used the maximal Cohen-Macaulay property of $M$ . Moving on, since $\mathfrak{m}=\mathfrak{p}+(\varpi)$ , an application of [14, Lemma 18.3] gives the inequality

\operatorname{dp}_{A}M-1=\operatorname{dp}_{\mathfrak{p}+(\varpi)}M-1\leq\operatorname{dp}_{\mathfrak{p}}M.

We have

\operatorname{dp}_{\mathfrak{p}}M\leq\operatorname{dp}_{A_{\mathfrak{p}}}M_{\mathfrak{p}}

since the $A_{\mathfrak{p}}$ -depth can be calculated in terms of the group

\operatorname{Ext}^{\ast}_{A_{\mathfrak{p}}}(A_{\mathfrak{p}}/\mathfrak{p},M_{\mathfrak{p}})\cong\operatorname{Ext}^{\ast}_{A}(A/\mathfrak{p},M)\otimes_{A}A_{\mathfrak{p}}.

Thus we have established $\dim A_{\mathfrak{p}}\leq\operatorname{dp}_{A_{\mathfrak{p}}}M_{\mathfrak{p}}$ , and the theorem follows. ∎

4 Galois cohomology

In this section, we discuss Galois cohomology and Bloch-Kato Selmer groups. For a complete reference, see [28], and for an introduction see [4].

Suppose $\rho:G_{F,S}\to\operatorname{GL}_{2}(L)$ where $S\supseteq S_{p}$ is a finite set of finite places, and that $\rho$ is de Rham at all places above $p$ . First, we introduce the local Bloch-Kato Selmer groups. Let $v\in S$ . If $v\mid p$ , then

H^{1}_{f}(G_{F_{v}},\operatorname{ad}^{0}\rho)=\ker\big{(}H^{1}(G_{F_{v}},\operatorname{ad}^{0}\rho)\to H^{1}(G_{F_{v}},\operatorname{ad}^{0}\rho\otimes_{\mathbb{Q}_{p}}B_{\operatorname{cris}})\big{)},

where the map on the right is the natural one induced by the tensor product and $B_{\operatorname{cris}}$ is Fontaine’s crystalline period ring. If $v\nmid p$ , let $I_{F_{v}}\subset G_{F_{v}}$ denote the inertia subgroup and set

H^{1}_{f}(G_{F_{v}},\operatorname{ad}^{0}\rho)=\ker\big{(}H^{1}(G_{F,S},\operatorname{ad}^{0}\rho)\to H^{1}(I_{F_{v}},\operatorname{ad}^{0}\rho)\big{)}.

where the map on the right is the one induced by the inclusion $I_{F_{v}}\subset G_{F_{v}}$ .

For every place $v$ of $F$ there is a restriction map

\operatorname{res}_{v}:H^{1}(G_{F,S},\operatorname{ad}^{0}\rho)\to H^{1}(G_{F_{v}},\operatorname{ad}^{0}\rho),

and we define the global Bloch-Kato Selmer group as

H^{1}_{f}(G_{F,S},\operatorname{ad}^{0}\rho)=\{c\in H^{1}(G_{F,S},\operatorname{ad}^{0}\rho)\mid\forall v\in S:\operatorname{res}_{v}(c)\in H^{1}_{f}(G_{F_{v}},\operatorname{ad}^{0}\rho)\},

or equivalently

\ker\bigg{(}H^{1}(G_{F,S},\operatorname{ad}^{0}\rho)\to\prod_{v\in S}\frac{H^{1}(G_{F_{v}},\operatorname{ad}^{0}\rho)}{H^{1}_{f}(G_{F_{v}},\operatorname{ad}^{0}\rho)}\bigg{)}.

We use lowercase $h$ to denote dimension, e.g.

h^{1}_{f}(G_{F,S},V)=\dim_{L}H^{1}_{f}(G_{F,S},V).

If $v\in S\setminus S_{p}$ and the Weil-Deligne representation $\operatorname{WD}(\rho_{v})$ associated to $\rho_{v}$ is generic we have $H^{0}(G_{F_{v}},\operatorname{ad}^{0}\rho(1))=0$ by [2, Lemma 1.1.5]. Thus, by local Tate duality and the formula for the Euler-Poincaré characteristic,

h^{0}(G_{F_{v}},\operatorname{ad}^{0}\rho)=h^{1}(G_{F_{v}},\operatorname{ad}^{0}\rho).

Proposition 4.1.

Suppose $\rho:G_{F,S}\to\operatorname{GL}_{2}(L)$ is irreducible, where $F$ is a totally complex field in which $p$ is totally split, that $\rho$ is de Rham with distinct Hodge-Tate weights at all places above $p$ , and moreover that $\operatorname{WD}(\rho_{v})$ is generic for all $v\in S\setminus S_{p}$ . Then if $l_{0}=[F:\mathbb{Q}]/2$ we have

h^{1}_{f}(G_{F,S},\operatorname{ad}^{0}\rho)=h^{1}_{f}(G_{F,S},\operatorname{ad}^{0}\rho(1))-l_{0}.

Proof.

By the Greenberg-Wiles formula, and the irreducibility of $\rho$ , we have

\begin{split}&h^{1}_{f}(G_{F,S},\operatorname{ad}^{0}\rho)-h^{1}_{f}(G_{F,S},\operatorname{ad}^{0}\rho(1))=\\ \sum_{v\in S}\big{(}&h^{1}_{f}(G_{F_{v}},\operatorname{ad}^{0}\rho)-h^{0}(G_{F_{v}},\operatorname{ad}^{0}\rho)\big{)}-\sum_{v\mid\infty}h^{0}(G_{F_{v}},\operatorname{ad}^{0}\rho).\\ \end{split}

Since $\operatorname{WD}(\rho_{v})$ is generic for every $v\in S\setminus S_{p}$ , the corresponding terms in the sum vanish. Furthermore, since $F$ is totally complex, the expression simplifies to

\sum_{v\mid p}\big{(}h^{1}_{f}(G_{F_{v}},\operatorname{ad}^{0}\rho)-h^{0}(G_{F_{v}},\operatorname{ad}^{0}\rho)\big{)}-\sum_{v\mid\infty}\dim\operatorname{ad}^{0}\rho.

Now, $p$ is totally split in $F$ and the sum over $v\mid p$ counts the total multiplicities of the negative Hodge-Tate weights of the $\operatorname{ad}^{0}\rho|_{G_{F_{v}}}$ , (see [5, Corollary 3.8.4]). Since the Hodge-Tate weights of $\rho_{v}$ are assumed to be distinct, the total expression therefore equals $-[F:\mathbb{Q}]/2=-l_{0}$ . ∎

Theorem 4.2.

Suppose $\rho:G_{F,S}\to\operatorname{GL}_{2}(L)$ is de Rham at all places above $p$ . Then there is an exact sequence of $L$ -vector spaces

	$\displaystyle 0$	$\displaystyle\to H^{1}_{f}\big{(}G_{F,S},\operatorname{ad}^{0}\rho\big{)}\to H^{1}\big{(}G_{F,S},\operatorname{ad}^{0}\rho\big{)}\to\prod_{v\in S}\frac{H^{1}\big{(}G_{F_{v}},\operatorname{ad}^{0}\rho\big{)}}{H^{1}_{f}\big{(}G_{F_{v}},\operatorname{ad}^{0}\rho\big{)}}$
		$\displaystyle\to H^{1}_{f}\big{(}G_{F,S},\operatorname{ad}^{0}\rho(1)\big{)}^{\vee}\to H^{2}(G_{F,S},\operatorname{ad}^{0}\rho).$

Moreover, if the Weil-Deligne representation $\operatorname{WD}(\rho_{v})$ associated to $\rho_{v}$ is generic for every $v\in S\setminus S_{p}$ , the corresponding factors in the third term are $0$ .

Proof.

On [38, p.119], the analogous exact sequence with finite coefficients is derived from the Poitou-Tate sequence. We argue in the same way using the Poitou-Tate sequence for cohomology in characteristic $0$ , which is readily obtained from [38, Proposition 10] by identifying, for any $L$ -vector space $V$ with a continuous action of $G_{F,S}$ ,

H^{1}(G_{F,S},V)\cong(\varprojlim_{s}H^{1}(G_{F,S},\Theta/\varpi^{s}))\otimes_{\mathcal{O}}L,

where $\Theta$ is an arbitrary choice of $G_{F,S}$ -stable lattice in $V$ .

To prove the second part, recall that the genericity assumption at a place $v\in S\setminus S_{p}$ implies the equality

h^{0}(G_{F_{v}},\operatorname{ad}^{0}\rho)=h^{1}(G_{F_{v}},\operatorname{ad}^{0}\rho).

Thus, it suffices to prove that the left-hand side equals $h^{1}_{\operatorname{ur}}(G_{F_{v}},\operatorname{ad}^{0}\rho)$ . We follow [4, Proposition 2.3(a)]. Let $V=\operatorname{ad}^{0}\rho$ . The inflation-restriction sequence yields an isomorphism

H^{1}(G_{F_{v}}/I_{v},V^{I_{v}})\cong H^{1}_{f}(G_{F_{v}},V),

and since $G_{F_{v}}/I_{v}\cong\widehat{\mathbb{Z}}$ has cohomological dimension $1$ ([17, Proposition 6.1.9]), the Euler-Poincaré characteristic formula implies

h^{1}(G_{F_{v}}/I_{v},V^{I_{v}})=h^{0}(G_{F_{v}}/I_{v},V^{I_{v}})=h^{0}(G_{F_{v}},V).

Hence $h^{1}_{f}(G_{F_{v}},V)=h^{0}(G_{F_{v}},\operatorname{ad}^{0}\rho)$ , as claimed. ∎

5 Galois deformation theory

In this section, we discuss the deformation theory of Galois representations, giving definitions and citing results from the literature that we shall need in the sequel.

5.1 Deformations of Galois representations

Let $\mathfrak{A}_{\mathcal{O}}$ be the category of discrete Artinian local $\mathcal{O}$ -algebras with residue field $k$ , and $\widehat{\mathfrak{A}}_{\mathcal{O}}$ the category of complete Noetherian local $\mathcal{O}$ -algebras with residue field $k$ , with morphisms the continuous $\mathcal{O}$ -algebra homomorphisms. The latter is equivalent to a full subcategory of the category $\operatorname{Pro}-\mathfrak{A}_{\mathcal{O}}$ , and contains $\mathfrak{A}_{\mathcal{O}}$ as a full subcategory.

Let $S$ be a finite set of finite places of $F$ and let $\Gamma$ be one of the groups $G_{F,S}$ and $G_{F_{v}}$ . Suppose $\bar{\rho}:\Gamma\to\operatorname{GL}_{2}(k)$ is a continuous representation and fix a continuous character $\chi:\Gamma\to\mathcal{O}^{\times}$ such that $\chi\equiv\det\bar{\rho}\ \operatorname{mod}\varpi.$ For any $A\in\mathfrak{A}_{\mathcal{O}}$ , let $\operatorname{red_{A}}$ be the reduction map $\operatorname{GL}_{2}(A)\to\operatorname{GL}_{2}(k)$ , and $\chi_{A}$ the composition $\Gamma\stackrel{{\scriptstyle\chi}}{{\to}}\mathcal{O}^{\times}\to A^{\times}$ .

Definition 5.1.

Keeping the above notation, we make the following definitions.

(i)

A framed deformation of $\bar{\rho}$ to $A$ with determinant $\chi$ is a continuous homomorphism $\rho_{A}:\Gamma\to\operatorname{GL}_{2}(A)$ fitting into a diagram

and satisfying $\det\rho_{A}=\chi_{A}$ .
(ii)

Two framed deformations $\rho_{A},\rho_{A}^{\prime}$ are strictly equivalent if there is an element $\gamma\in\ker(\operatorname{red}_{A})$ such that

$\gamma\rho_{A}\gamma^{-1}=\rho_{A}^{\prime}.$
(iii)

A deformation of $\bar{\rho}$ to $A$ is an equivalence class of framed deformations to $A$ under the relation of strict equivalence.

All in all, we obtain functors

D_{\bar{\rho}}^{\chi},D_{\bar{\rho}_{v}}^{\chi},D^{\chi,\square}_{\bar{\rho}_{v}}:\mathfrak{A}_{\mathcal{O}}\to\mathbf{Set},

where for example $D_{\bar{\rho}}^{\chi}(A)$ is the set of deformations of $\bar{\rho}$ to $A$ of determinant $\chi$ . The superscript $\square$ means ‘framed’. These functors extend in a canonical way to $\widehat{A}_{\mathcal{O}}$ , and from now on we consider all functors of deformations as defined on the category $\widehat{A}_{\mathcal{O}}$ .

Deformations of a continuous representation $\rho:\Gamma\to\operatorname{GL}_{2}(L)$ in characteristic $0$ is defined in the same way. Let $\widehat{\mathfrak{A}}_{L}$ be the category of complete Noetherian local $L$ -algebras with residue field $L$ . As before, we define functors

D_{\rho}^{\chi},D_{\rho_{v}}^{\chi},D^{\chi,\square}_{\rho_{v}}:\widehat{\mathfrak{A}}_{L}\to\mathbf{Set}.

5.2 Deformations of pseudorepresentations

The notion of (non-framed) deformations is sometimes too restrictive in the sense that the corresponding functors can fail to be representable. To circumvent this issue, one introduces pseudorepresentations, which are roughly speaking functions that behave like traces of representations. Pseudodeformations can be defined in great generality, but for our purposes the following narrow definition will suffice.

Definition 5.2.

Let $A$ be a topological ring such that $2\in A^{\times}$ . An $A$ -valued pseudorepresentation (of dimension $2$ ) of $G_{F_{v}}$ is a continuous function $\tau_{v}:G_{F_{v}}\to A$ such that:

(i)

$\tau(\operatorname{id})=2$ .
(ii)

For every $g_{1},g_{2}\in G,\ \tau(g_{1}g_{2})=\tau(g_{2}g_{1}).$

(iii)

For every $g_{1},g_{2},g_{3}\in G$ ,

\begin{split}\tau(g_{1}g_{2}g_{3})&=\tau(g_{1}g_{2})\tau(g_{3})+\tau(g_{1}g_{3})\tau(g_{2})+\tau(g_{2}g_{3})\tau(g_{1})\\ &-\tau(g_{1}g_{3}g_{2})-\tau(g_{1})\tau(g_{2})\tau(g_{3}).\end{split}

Definition 5.3.

Let $\tau_{v}$ be an $A$ -valued pseudorepresentation of $G_{F_{v}}$ . The determinant of $\tau_{v}$ is the function $\det\tau_{v}:G_{F_{v}}\to A$ defined by

\det\tau_{v}(g)=\frac{1}{2}\big{(}\tau_{v}(g)^{2}-\tau_{v}(g^{2})\big{)}

We will consider $A$ -valued pseudorepresentations where $A\in\widehat{\mathfrak{A}}_{\mathcal{O}}$ is equipped with its profinite topology. Just as for representations, if $\overline{\tau}_{v}:G_{F_{v}}\to k$ is a pseudorepresentation we define a functor

D_{\overline{\tau}_{v}}^{\operatorname{ps},\chi_{v}}:\widehat{\mathfrak{A}}_{\mathcal{O}}\to\mathbf{Set}

taking an artinian $\mathcal{O}$ -algebra $A$ to the set of $A$ -valued pseudorepresentations with determinant $\chi_{v}$ lifting $\overline{\tau}_{v}$ .

5.3 Deformation problems

In this brief section, we recall the definition of a deformation problem, following [22, §4].

Definition 5.4.

A local deformation problem for $\bar{\rho}_{v}$ is a representable closed subfunctor $D_{v}\subseteq D^{\chi,\square}_{\bar{\rho}_{v}}$ such that $D_{v}(A)$ is stable under conjugation by elements of $\ker\operatorname{red}_{A}$ .

Definition 5.5.

A global deformation problem is a tuple $\mathcal{S}=(\bar{\rho},\chi,S,\{D_{v}\}_{v\in S})$ where

•

$\bar{\rho}:G_{F}\to\operatorname{GL}_{2}(k)$ is an absolutely irreducible representation,
•

$\chi:G_{F}\to\mathcal{O}^{\times}$ is a continuous character such that $\det\bar{\rho}=\overline{\chi}$ ,
•

$S$ is a finite set of finite places of $F$ such that $\bar{\rho}$ and $\chi$ are unramified outside $S$ (i.e. factor through $G_{F}\twoheadrightarrow G_{F,S}$ ),
•

$D_{v}$ is a local deformation problem for every $v\in S$ .

Definition 5.6.

Let $\mathcal{S}=(\bar{\rho},\chi,S,\{D_{v}\}_{v\in S})$ be a global deformation problem. A deformation $[\rho_{A}]\in D_{\bar{\rho}}^{\chi}(A)$ is of type $\mathcal{S}$ if $\rho_{A}$ is unramified outside $S$ , satisfies $\det\rho_{A}=\chi$ and for every $v\in S$ ,

\rho_{A}|_{G_{F_{v}}}\in D_{v}(A).

Implicit in this definition is the fact that the conditions are independent of the choice of representative $\rho_{A}$ of the strict equivalence class $[\rho_{A}]$ . We define a functor

D_{\mathcal{S}}:\widehat{\mathfrak{A}}_{\mathcal{O}}\to\mathbf{Set},

by letting $D_{\mathcal{S}}(A)\subseteq D_{\bar{\rho}}^{\chi}(A)$ be the set of deformations of type $\mathcal{S}$ .

5.4 Universal deformation rings

When deformation functors are representable, many questions about deformations translate to questions about the representing objects, which are complete Noetherian $\mathcal{O}$ - or $L$ -algebras. As mentioned earlier, one reason to consider pseudorepresentations is that the corresponding functors are always representable. The same is true for framed deformations, as can be proved using Schlessinger’s criterion as in [27].

Theorem 5.7.

For any place $v$ , the functors $D^{\chi,\square}_{\bar{\rho}_{v}}$ and $D_{\overline{\tau}_{v}}^{\operatorname{ps},\chi_{v}}$ are representable. We denote the representing objects by $R_{v}^{\square}$ and $R^{\operatorname{ps}}_{v}$ , respectively.

In contrast, functors of deformations are not representable in general, but the following class of representations have representable functors of deformations.

Definition 5.8.

A representation is Schur if it has only scalar endomorphisms.

Absolutely irreducible representations are always Schur. Recall that the definition of a global deformation problem includes the hypothesis that the residual representation is absolutely irreducible. The following results are deduced in the same way as Theorem 5.7.

Theorem 5.9.

Suppose $\rho$ (resp. $\rho_{v}$ , $\bar{\rho},$ $\bar{\rho}_{v}$ ) is Schur. Then $D_{\rho}^{\chi}$ (resp. $D_{\rho_{v}}^{\chi},$ $D_{\bar{\rho}}^{\chi}$ , $D_{\bar{\rho}_{v}}^{\chi}$ ) is represented by an object $R_{\rho}$ (resp. $R_{\rho_{v}}$ , $R$ , $R_{v}$ ) in $\widehat{\mathfrak{A}}_{\mathcal{O}}$ .

Theorem 5.10.

Let $\mathcal{S}$ be a global deformation problem. Then $D_{\mathcal{S}}$ is represented by an object $R_{\mathcal{S}}$ in $\widehat{\mathfrak{A}}_{\mathcal{O}}$ .

There are natural transformations

D^{\chi,\square}_{\rho_{v}}\to D_{\rho_{v}}^{\chi}\stackrel{{\scriptstyle\operatorname{tr}}}{{\to}}D_{\overline{\tau}_{v}}^{\operatorname{ps},\chi_{v}},

and thus $R_{v}^{\square}$ is a $R^{\operatorname{ps}}_{v}$ -algebra. Moreover, if $R_{v}$ exists then we have homomorphisms

R^{\operatorname{ps}}_{v}\to R_{v}\to R_{v}^{\square},

and $R_{v}^{\square}$ is formally smooth of relative dimension $3$ over $R_{v}$ .

Suppose $\mathcal{S}$ is a deformation problem, with local conditions $D_{v}$ represented by $\overline{R}_{v}$ . By definition, $\overline{R}_{v}$ is a quotient of $R_{v}^{\square}$ . We define

	$\displaystyle R_{\mathcal{S}}^{\operatorname{loc}}$	$\displaystyle=\widehat{\bigotimes}_{v\in S}\overline{R}_{v},$
	$\displaystyle R^{\operatorname{ps}}_{p}$	$\displaystyle=\widehat{\bigotimes}_{v\mid p}R^{\operatorname{ps}}_{v}.$

These rings can also be interpreted as representing objects for certain functors, as the first item of the following lemma shows.

Lemma 5.11.

Let $T$ be a finite set, and suppose we have representable functors $D_{v}:\widehat{\mathfrak{A}}_{\mathcal{O}}\to\mathbf{Set}$ for every $v\in T$ . Denote by $R_{v}$ the representing object of $D_{v}$ , and consider the functor $D_{T}=\prod_{v\in T}D_{v}:\widehat{\mathfrak{A}}_{\mathcal{O}}\to\mathbf{Set}$ . Then we have the following:

(i)

$D_{T}$ is represented by $R_{T}=\widehat{\bigotimes}_{v\in T}R_{v}$ .
(ii)

If $\mathfrak{p}_{T}\subset R_{T}[1/\varpi]$ is a closed point generated by the joint image of closed points $\mathfrak{p}_{v}\subset R_{v}[1/\varpi]$ , we have a canonical isomorphism of $L$ -algebras

$(R_{T})_{\mathfrak{p}_{T}}^{\wedge}\cong\widehat{\bigotimes}_{v\in T,L}(R_{v})_{\mathfrak{p}_{v}}^{\wedge}.$

Proof.

(i) Let $\mathfrak{m}_{v}$ be the maximal ideal of $R_{v}$ , and $A=(A_{i})\in\widehat{\mathfrak{A}}_{\mathcal{O}}$ . We have

\begin{split}(\prod_{v\in T}D_{v})(A)&\cong\prod_{v\in T}\varprojlim_{i}\operatorname{Hom}_{\mathcal{O}}(R_{v},A_{i})\\ &=\prod_{v\in T}\varprojlim_{i}\varinjlim_{r\geq 1}\operatorname{Hom}_{\mathcal{O}}(R_{v}/\mathfrak{m}_{v}^{r},A_{i})\\ &=\varprojlim_{i}\varinjlim_{r\geq 1}\prod_{v\in T}\operatorname{Hom}_{\mathcal{O}}(R_{v}/\mathfrak{m}_{v}^{r},A_{i})\\ &=\varprojlim_{i}\varinjlim_{r\geq 1}\operatorname{Hom}_{\mathcal{O}}(\bigotimes_{v\in T}R_{v}/\mathfrak{m}_{v}^{r},A)\\ &=\operatorname{Hom}_{\mathcal{O}}(\widehat{\bigotimes}_{v\in T}R_{v},A).\end{split}

Here, we use that $A$ is artinian which implies that $\varprojlim$ commutes with $\operatorname{Hom}$ in the above way, and moreover that $\varprojlim$ and $\varinjlim$ commute with finite products.

(ii) Following [24, §2.3], the ring $(R_{T})_{\mathfrak{p}_{T}}^{\wedge}$ represents a functor

D_{(\mathfrak{p}_{T})}:\widehat{\mathfrak{A}}_{L}\to\mathbf{Set},

constructed as a filtered colimit. Since filtered colimits commute with finite limits, it is not hard to see that

D_{(\mathfrak{p}_{T})}\cong\prod_{v\mid p}D_{(\mathfrak{p}_{v})},

and now the argument from (i) carried out in the category $\widehat{\mathfrak{A}}_{L}$ proves (ii). ∎

Theorem 5.12.

[2, Proposition 1.2.2] Let $v\in S\setminus S_{p}$ and $\mathfrak{p}_{v}\subset R_{v}^{\square}[1/\varpi]$ be the point corresponding to a representation $\rho_{v}:G_{F_{v}}\to\operatorname{GL}_{2}(L)$ . Then the Weil-Deligne representation $\operatorname{WD}(\rho_{v})$ is generic if and only if $\mathfrak{p}_{v}$ is a regular point of $\operatorname{Spec}R_{v}^{\square}$ .

The dimensions of local deformation rings have been computed by Shotton [35]. Note that we are considering fixed-determinant deformations.

Theorem 5.13.

[35, Theorem 2.5] Let $v\in S\setminus S_{p}$ . Then the local framed (fixed-determinant) deformation ring $R_{v}^{\square}$ is an equidimensional reduced complete intersection ring, flat of relative dimension $3$ over $\mathcal{O}$ .

Having discussed the local framed deformation rings at places of $S\setminus S_{p}$ , we move on to the subtler story of the places above $p$ . At these places, our local deformation problems will encode conditions from $p$ -adic Hodge theory. We denote the cyclotomic character by $\varepsilon$ .

Definition 5.14.

Let $v\in S_{p}$ . A $v$ -adic Hodge type is a triple $(\mathbf{w}_{v},\tau_{v},\chi_{v})$ where

•

$\mathbf{w}_{v}=(a_{v},b_{v})\in\mathbb{Z}^{2}$ such that $b>a$ .
•

$\tau_{v}:I_{F_{v}}\to\operatorname{GL}_{2}(L)$ is a representation with open kernel.
•

$\chi_{v}:G_{F_{v}}\to\mathcal{O}^{\times}$ is a continuous character such that $\chi_{v}|_{I_{F_{v}}}=\varepsilon^{a_{v}+b_{v}}\det\tau$ .

Definition 5.15.

Let ${(\mathbf{w}_{v},\tau_{v},\chi_{v})}$ be a $v$ -adic Hodge type. A continuous representation

\rho_{v}:G_{F_{v}}\to\operatorname{GL}_{2}(L)

is of type ${(\mathbf{w}_{v},\tau_{v},\chi_{v})}$ if it is potentially semi-stable with Hodge-Tate weights equal to $\mathbf{w}_{v}$ , determinant equal to $\chi_{v}$ and if $\operatorname{WD}(\rho_{v})|_{I_{F_{v}}}\cong\tau_{v}$ , where $\operatorname{WD}(\rho_{v})$ is the Weil-Deligne representation associated to $\rho_{v}$ .

The representations of fixed $v$ -adic Hodge type form a Zariski closed subset of the generic fiber of the unrestricted deformation ring, as the following theorem shows.

Theorem 5.16.

[23, Theorem 2.7.6], [2, Theorem D] Let $v\in S_{p}$ and suppose $\bar{\rho}_{v}:G_{F_{v}}\to\operatorname{GL}_{2}(k)$ is Schur with deformation ring $R_{v}$ . For any $v$ -adic Hodge type ${(\mathbf{w}_{v},\tau_{v},\chi_{v})}$ with $\det\bar{\rho}_{v}=\overline{\chi}_{v}$ , there exists a reduced, $\mathcal{O}$ -flat quotient $R_{v}\twoheadrightarrow R_{v}(\mathbf{w}_{v},\tau_{v})$ such that for any closed point $\mathfrak{p}_{v}\subset R_{v}[1/\varpi]$ , the corresponding representation is of type ${(\mathbf{w}_{v},\tau_{v},\chi_{v})}$ if and only if $\mathfrak{p}_{v}\in\operatorname{Spec}R_{v}(\mathbf{w}_{v},\tau_{v})[1/\varpi]$ . Moreover, if $\mathfrak{p}_{v}$ corresponds to a representation $\rho_{v}$ such that $\operatorname{WD}(\rho_{v})$ is generic, then $\mathfrak{p}_{v}$ is a regular point of $\operatorname{Spec}R_{v}(\mathbf{w}_{v},\tau_{v})[1/\varpi]$

The relation between deformation rings in characteristic $0$ and $p$ is explicated in [24, Lemma 2.3.3, Proposition 2.3.5], from which one readily deduces the following two results.

Theorem 5.17.

Let $\bar{\rho}_{v}:G_{F_{v}}\to\operatorname{GL}_{2}(k)$ be a representation with framed deformation ring $R_{\bar{\rho}_{v}}^{\square}$ . Suppose $\mathfrak{p}_{v}\subset R_{\bar{\rho}_{v}}^{\square}$ is a closed point corresponding to $\rho_{v}:G_{F_{v}}\to\operatorname{GL}_{2}(L)$ . Then the localisation and completion $(R_{\bar{\rho}_{v}}^{\square})_{\mathfrak{p}_{v}}^{\wedge}$ represents $D_{\rho_{v}}^{\square}$ . If $\bar{\rho}_{v}$ is Schur, the analogous statement holds for the deformation rings.

Theorem 5.18.

Let $\mathcal{S}=(\bar{\rho},\chi,S,\{D^{\chi,\square}_{\bar{\rho}_{v}}\}_{v\in S})$ be the global deformation problem of deformations unramified outside of $S$ , represented by $R_{\mathcal{S}}$ . Suppose $\mathfrak{p}\subset R_{\mathcal{S}}[1/\varpi]$ is a closed point corresponding to a representation $\rho:G_{F,S}\to\operatorname{GL}_{2}(L)$ . Then the localisation and completion $R_{\mathcal{S},\rho}:={(R_{\mathcal{S}})}_{\mathfrak{p}}^{\wedge}$ represents the functor $D_{\mathcal{S},\rho}:\widehat{\mathfrak{A}}_{L}\to\mathbf{Set}$ for which $D_{\mathcal{S},\rho}(A)$ is the set of deformations of $\rho$ to $A$ of determinant $\chi$ which are unramified outside of $\mathcal{S}$ .

When dealing with irreducible representations, passage from pseudodeformation rings to deformation rings is enabled by the following result.

Theorem 5.19.

Let $\rho_{v}:G_{F_{v}}\to\operatorname{GL}_{2}(L)$ be an absolutely irreducible characteristic 0 representation of $G_{F_{v}}$ and $\mathfrak{p}_{v}\subset R^{\operatorname{ps},\chi_{v}}_{\operatorname{tr}\bar{\rho}_{v}}[1/\varpi]$ the closed point corresponding to $\operatorname{tr}\rho_{v}$ . Then

(R^{\operatorname{ps},\chi_{v}}_{\operatorname{tr}\bar{\rho}_{v}})_{\mathfrak{p}_{v}}^{\wedge}\cong R_{\rho_{v}},

the unrestricted deformation ring of $\rho_{v}$ .

Proof.

The ring $(R^{\operatorname{ps},\chi_{v}}_{\operatorname{tr}\bar{\rho}_{v}})_{\mathfrak{p}_{v}}^{\wedge}$ is isomorphic to $R^{\operatorname{ps},\chi_{v}}_{\operatorname{tr}\rho_{v}}$ by [23, Lemma 2.3.3, Proposition 2.3.5]. The latter is canonically isomorphic to $R_{\rho_{v}}$ since $\rho_{v}$ is absolutely irreducible (a consequence of the main theorem of [30]). ∎

The Galois cohomology groups discussed in Section are naturally isomorphic to tangent spaces of deformation rings, and local conditions on the cohomology classes corresponds to conditions on the restrictions $\rho|_{G_{F_{v}}}$ . For a ring $R$ , let $\operatorname{T}[R]$ denote its Zariski tangent space, i.e. $\operatorname{T}(R)=(\mathfrak{m}_{R}/\mathfrak{m}_{R}^{2})^{\vee}$ .

Theorem 5.20.

We have the following natural isomorphisms of $L$ -vector spaces (when the deformation rings exist):

	$\displaystyle H^{1}\big{(}G_{F,S},\operatorname{ad}^{0}\rho\big{)}$	$\displaystyle\cong\operatorname{T}[R_{\mathcal{S},\rho}]$
	$\displaystyle H^{1}\big{(}G_{F_{v}},\operatorname{ad}^{0}\rho\big{)}$	$\displaystyle\cong\operatorname{T}[R_{\rho_{v}}]$
	$\displaystyle H^{1}_{f}\big{(}G_{F_{v}},\operatorname{ad}^{0}\rho\big{)}$	$\displaystyle\cong\operatorname{T}[R_{\rho_{v}}(\sigma_{v})]$

Proof.

The first two follow from Theorems 5.17, 5.18 and a standard argument (see e.g. [1, Theorem 5.1.4]). The third item follows from the main theorem of [26] and [2, Proposition 1.3.12]. ∎

6 Representations and completed homology

In this section, we discuss the representation-theoretic input in our main result.

6.1 Iwasawa modules and categories of smooth representations

In this subsection, we recall some general facts about modules over Iwasawa algebras, following [15].

Definition 6.1.

The Iwasawa algebra (with $\mathcal{O}$ -coefficients) of a compact $p$ -adic analytic group $K$ is the profinite (possibly non-commutative) ring

\mathcal{O}[[K]]=\varprojlim_{\begin{subarray}{c}U_{p}\trianglelefteq K\end{subarray}}\mathcal{O}[K/U_{p}].

The inversion map $g\mapsto g^{-1}$ in $K$ induces an equivalence of categories between left- and right $\mathcal{O}[[K]]$ -modules.

Suppose $K$ is an open compact subgroup of a possibly non-compact $p$ -adic analytic group $G$ . The Iwasawa algebra $\mathcal{O}[[K]]$ contains the group algebra $\mathcal{O}[K]$ as a subring, which in turn sits inside the full group algebra $\mathcal{O}[G]$ . Of special interest to us are modules with an action of both $\mathcal{O}[[K]]$ and $\mathcal{O}[G]$ such that the restrictions to $\mathcal{O}[K]$ coincide.

Definition 6.2.

Let $G$ be a $p$ -adic analytic group, $K\subset G$ a compact subgroup and $\zeta:Z(G)\to\mathcal{O}^{\times}$ a central character. The category of profinite augmented $\mathcal{O}[G]$ -modules is the category $\operatorname{Mod}^{\operatorname{pfa}}_{G,\zeta}(\mathcal{O})$ with objects the profinite topological $\mathcal{O}[[K]]$ -modules admitting a neighbourhood basis of the identity given by $\mathcal{O}[[K]]$ -submodules, with a compatible action of $G$ and with central character $\zeta$ . The morphisms are $\mathcal{O}[G]$ -linear continuous maps.

The category $\operatorname{Mod}^{\operatorname{pfa}}_{G,\zeta}(\mathcal{O})$ abelian and independent of the choice of $K$ . Pontryagin duality defines an anti-equivalence

	$\displaystyle\operatorname{Mod}^{\operatorname{sm}}_{G,\zeta}(\mathcal{O})$	$\displaystyle\to\operatorname{Mod}^{\operatorname{pfa}}_{G,\zeta}(\mathcal{O})$
	$\displaystyle V$	$\displaystyle\mapsto V^{\vee}:=\operatorname{Hom}_{\mathcal{O}}(V,L/\mathcal{O})$

where $\operatorname{Mod}^{\operatorname{sm}}_{G,\zeta}(\mathcal{O})$ is the category of smooth $G$ -representations over $\mathcal{O}$ with central character $\zeta$ . The inverse of this functor is also given by $(\cdot)^{\vee}$ . Here, a $G$ -representation $V$ is called smooth if

V=\cup_{K,s}V^{K}[\varpi^{s}].

A stronger condition than smoothness is (local) admissibility. A smooth representation $V$ is said to be admissible if each term $V^{K}[\varpi^{s}]$ in the union above is finitely generated over $\mathcal{O}$ . We say that $V$ is locally admissible if, for every $v\in V$ , the subrepresentation of $V$ generated by $v$ is admissible.

The full subcategory of $\operatorname{Mod}^{\operatorname{sm}}_{G,\zeta}(\mathcal{O})$ consisting of locally admissible (resp. admissible) representations is denoted $\operatorname{Mod}^{\operatorname{ladm}}_{G,\zeta}(\mathcal{O})$ (resp. $\operatorname{Mod}^{\operatorname{adm}}_{G,\zeta}(\mathcal{O})$ ). These categories are also abelian, and the admissible representations are dual to the profinite augmented modules which are finitely generated as $\mathcal{O}[[K]]$ -modules. We define

\displaystyle\mathfrak{C}_{G,\zeta}(\mathcal{O}):=(\operatorname{Mod}^{\operatorname{ladm}}_{G,\zeta})^{\vee},

so that the categories

\operatorname{Mod}^{\operatorname{adm}}_{G,\zeta}(\mathcal{O})\subset\operatorname{Mod}^{\operatorname{ladm}}_{G,\zeta}(\mathcal{O})\subset\operatorname{Mod}^{\operatorname{sm}}_{G,\zeta}(\mathcal{O})

are anti-equivalent under Pontryagin duality to

\operatorname{Mod}_{G,\zeta}^{\operatorname{fga}}(\mathcal{O})\subset\mathfrak{C}_{G,\zeta}(\mathcal{O})\subset\operatorname{Mod}^{\operatorname{pfa}}_{G,\zeta}(\mathcal{O}).

6.2 Representations of $\operatorname{GL}_{2}(\mathbb{Q}_{p})$

In this subsection, we cite some facts about $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ -representations from [32]. For now, we let $G=\operatorname{GL}_{2}(\mathbb{Q}_{p})$ , $B\subset G$ the subgroup of upper triangular matrices and $K=\operatorname{GL}_{2}(\mathbb{Z}_{p})$ .

The category $\operatorname{Mod}^{\operatorname{ladm}}_{G,\zeta}(\mathcal{O})$ is canonically isomorphic to a direct product of subcategories

\operatorname{Mod}^{\operatorname{ladm}}_{G,\zeta}(\mathcal{O})\cong\prod_{\mathfrak{B}}\operatorname{Mod}^{\operatorname{ladm}}_{G,\zeta}(\mathcal{O})_{\mathfrak{B}}

([32, Proposition 5.34]). Here, $\operatorname{Mod}^{\operatorname{ladm}}_{G,\zeta}(\mathcal{O})_{\mathfrak{B}}$ denotes the full subcategory consisting of representations with the property that all irreducible subquotients are isomorphic to one of a finite number of representations lying inside the block $\mathfrak{B}$ , which is a finite set of $k$ -representations. Pontryagin duality preserves this decomposition, and thus

\mathfrak{C}_{G,\zeta}(\mathcal{O})\cong\prod_{\mathfrak{B}}\mathfrak{C}_{G,\zeta}(\mathcal{O})_{\mathfrak{B}}.

Since $k$ is not algebraically closed, there might irreducible representations which are not absolutely irreducible. After a finite base change, say from $k$ to $k^{\prime}$ , such a representation decomposes into a direct sum of absolutely irreducible $k^{\prime}$ -representations. Therefore, we will restrict our attention to blocks $\mathfrak{B}$ containing an absolutely irreducible representation and tacitly assume $k$ to be large enough throughout the rest of the paper.

Proposition 6.3.

([32, Proposition 5.42]) Suppose $p\geq 5$ . The blocks $\mathfrak{B}$ containing an absolutely irreducible representation are as follows:

(i)

$\mathfrak{B}=\{\pi\}$ , where $\pi$ is supersingular.
(ii)

$\mathfrak{B}=\{\operatorname{Ind}_{B}^{G}(\chi_{1}\otimes\chi_{2}\bar{\varepsilon}^{-1}),\operatorname{Ind}_{B}^{G}(\chi_{2}\otimes\chi_{1}\bar{\varepsilon}^{-1})\}$ with $\chi_{1}/\chi_{2}\neq 1,\bar{\varepsilon}^{\pm 1}$ .
(iii)

$\mathfrak{B}=\{\operatorname{Ind}_{B}^{G}(\chi\otimes\chi\bar{\varepsilon}^{-1})\}$ .
(iv)

$\mathfrak{B}=\{\eta,\operatorname{Sp}\otimes\eta,(\operatorname{Ind}_{B}^{G}\alpha)\otimes\eta\}$ , where $\eta=\eta_{0}\circ\det:G\to k^{\times}$ is a smooth character.

We will consider $\operatorname{PGL}_{2}$ -representations, meaning we will have $\zeta$ equal to the trivial central character. As a consequence, case (iii) above will not feature later on.

For every $\pi\in\mathfrak{B}$ , we let $P_{\pi}\twoheadrightarrow\pi^{\vee}$ be a projective envelope and define

	$\displaystyle P_{\mathfrak{B}}$	$\displaystyle=\bigoplus_{\pi\in\mathfrak{B}}P_{\pi}\twoheadrightarrow\bigoplus_{\pi\in\mathfrak{B}}\pi^{\vee},$
	$\displaystyle E_{\mathfrak{B}}$	$\displaystyle=\operatorname{End}_{\mathfrak{C}_{\mathfrak{B}}(\mathcal{O})}(P_{\mathfrak{B}}).$

Then $P_{\mathfrak{B}}$ is a projective envelope of $\bigoplus_{\pi\in\mathfrak{B}}\pi^{\vee}$ and is moreover a projective generator in the category $\mathfrak{C}_{G,\zeta}(\mathcal{O})_{\mathfrak{B}}$ . The ring $E_{\mathfrak{B}}$ is compact with respect to a natural topology and there is an equivalence of abelian categories

	$\displaystyle\mathfrak{C}_{G,\zeta}(\mathcal{O})_{\mathfrak{B}}$	$\displaystyle\to\operatorname{RMod}^{\operatorname{cpt}}(E_{\mathfrak{B}})$
	$\displaystyle V$	$\displaystyle\mapsto\operatorname{Hom}_{\mathfrak{C}_{\mathfrak{B}}(\mathcal{O})}(P_{\mathfrak{B}},V).$

The inverse of the functor $\operatorname{Hom}_{\mathfrak{C}_{\mathfrak{B}}(\mathcal{O})}(P_{\mathfrak{B}},-)$ is given by the completed tensor product, so that for any $V\in\mathfrak{C}_{\mathfrak{B}}(\mathcal{O})$ , there is a canonical isomorphism

V\cong\operatorname{Hom}_{\mathfrak{C}_{\mathfrak{B}}(\mathcal{O})}(P_{\mathfrak{B}},V)\widehat{\otimes}_{E_{\mathfrak{B}}}P_{\mathfrak{B}}.

So far we have only mentioned the ‘automorphic side’ of the $p$ -adic local Langlands correspondence. Let us now describe the ‘Galois side’ of the picture and relate $E_{\mathfrak{B}}$ to a pseudodeformation ring. To each block $\mathfrak{B}$ above, we associate a semisimple 2-dimensional $k$ -representation $\bar{\rho}_{\mathfrak{B}}$ of $G_{\mathbb{Q}_{p}}$ by the following recipe:

(i)

$\bar{\rho}_{\mathfrak{B}}=\mathbf{V}(\pi)$ where $\mathbf{V}$ is Colmez’ Montreal functor (see [32, 5.7]).
(ii)

$\bar{\rho}_{\mathfrak{B}}=\chi_{1}\oplus\chi_{2}$ (viewing $\chi_{1},\chi_{2}$ as Galois representations via local class field theory).
(iii)

$\bar{\rho}_{\mathfrak{B}}=\chi\oplus\chi$ .
(iv)

$\bar{\rho}_{\mathfrak{B}}=\eta_{0}\oplus\eta_{0}\bar{\varepsilon}$ .

In each case, we have $\det\bar{\rho}_{\mathfrak{B}}=\zeta\bar{\varepsilon}$ .

Theorem 6.4.

Let $Z_{\mathfrak{B}}=Z(E_{\mathfrak{B}})$ denote the center of $E_{\mathfrak{B}}$ . Then we have the following:

(a)

$E_{\mathfrak{B}}$ is a finitely generated module over $Z_{\mathfrak{B}}$ .
(b)

There is a canonical isomorphism of $\mathcal{O}$ -algebras

$Z_{\mathfrak{B}}\cong R^{\operatorname{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}$

where $R^{\operatorname{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}$ is the pseudodeformation ring of $2$ -dimensional pseudorepresentations of $G_{\mathbb{Q}_{p}}$ with determinant $\zeta\varepsilon$ lifting $\operatorname{tr}\bar{\rho}_{\mathfrak{B}}$ .
(c)

Let $\mathfrak{B}$ be one of the blocks (i), (ii) or (iv), and suppose $\mathfrak{p}_{v}\subset R^{\operatorname{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}[1/\varpi]$ is a closed point corresponding to an irreducible $G_{\mathbb{Q}_{p}}$ -representation $\rho_{v}$ , so that $(R^{\operatorname{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}})_{\mathfrak{p}_{v}}^{\wedge}\cong R_{\rho_{v}}$ . Then

$(E_{\mathfrak{B}})_{\mathfrak{p}_{v}}^{\wedge}\cong M_{|\mathfrak{B}|}(R_{\rho_{v}}),$

the ring of $|\mathfrak{B}|$ -by- $|\mathfrak{B}|$ matrices with coefficients in $R_{\rho_{v}}$ , the unrestricted deformation ring of $\rho_{v}$ .

Proof.

(a,b) See [32, Theorem 1.5, Corollary 8.11, Corollary 9.25, Lemma 10.90].
(c) In the block (i), the natural map $Z_{\mathfrak{B}}\to E_{\mathfrak{B}}$ is an isomorphism ([32, Proposition 6.3]). The case (ii) is dealt with in [32, Corollary B.27], and we outline the proof here. We have a presentation

E_{\mathfrak{B}}=(\operatorname{Hom}_{\mathfrak{C}_{G,\zeta}(\mathcal{O})}(P_{i},P_{j}))_{1\leq i,j\leq 2}\cong\begin{pmatrix}Z_{\mathfrak{B}}\mathbf{1}&Z_{\mathfrak{B}}b_{1}\\ Z_{\mathfrak{B}}b_{2}&Z_{\mathfrak{B}}\mathbf{1}\end{pmatrix}

where $b_{1}\circ b_{2}=b_{2}\circ b_{1}=cZ_{\mathfrak{B}}$ for an element $c\in Z_{\mathfrak{B}}$ with the property that a point $\mathfrak{p}_{v}\subset R^{\operatorname{ps},\zeta\varepsilon}_{\operatorname{tr}\bar{\rho}_{\mathfrak{B}}}[1/\varpi]$ defines an irreducible representation if and only if $c\notin\mathfrak{p}_{v}$ (put differently, the reducibility ideal is principal, generated by $c$ ). Consequently the ring $(E_{\mathfrak{B}})_{\mathfrak{p}_{v}}^{\wedge}\cong(E_{\mathfrak{B}}[1/c])_{\mathfrak{p}_{v}}^{\wedge}$ is a $2$ -by- $2$ matrix algebra over $(Z_{\mathfrak{B}})_{\mathfrak{p}_{v}}^{\wedge}\cong R_{\rho_{v}}$ .

The block (iv) is dealt with similarly. We have a presentation of $E_{\mathfrak{B}}$ as a $3$ -by- $3$ matrix of homomorphisms $P_{i}\to P_{j}$ ([32, p.134]). Here, the relevant relations are generated by two elements $c_{0},c_{1}\in Z_{\mathfrak{B}}$ , and as before, the point $\mathfrak{p}_{v}$ defines an irreducible representation if $(c_{0},c_{1})\not\subset\mathfrak{p}_{v}$ . Since $c_{i}^{-1}(c_{0},c_{1})=Z_{\mathfrak{B}}[1/c_{i}]$ , the same argument as in the previous case proves the claim. ∎

6.3 Representations of $\prod{\operatorname{PGL}_{2}(\mathbb{Q}_{p})}$

Let us now turn to our case of interest, namely

	$\displaystyle G$	$\displaystyle=\prod_{v\mid p}\mathbf{G}(F_{v})\cong\prod_{v\mid p}\operatorname{PGL}_{2}(\mathbb{Q}_{p}),$
	$\displaystyle K$	$\displaystyle=\prod_{v\mid p}K_{v}\cong\prod_{v\mid p}\operatorname{PGL}_{2}(\mathbb{Z}_{p}).$

Our category of interest is $\mathfrak{C}_{G}(\mathcal{O})$ , the Pontryagin dual of the category of locally admissible $G$ -representations, and we make the identification

\mathfrak{C}_{G}(\mathcal{O})\cong\big{(}\operatorname{Mod}^{\operatorname{ladm}}_{\prod_{v\mid p}\operatorname{GL}_{2}(\mathbb{Q}_{p}),\mathbf{1}}(\mathcal{O})\big{)}^{\vee}.

The existence of a block decomposition relies only on general facts about locally finite categories and still hold for $\mathfrak{C}_{G}(\mathcal{O})$ . That is, there exists a set of blocks $\mathfrak{B}$ , projective generators $P_{\mathfrak{B}}$ and a decomposition

\mathfrak{C}_{G}(\mathcal{O})\cong\prod_{\mathfrak{B}}\mathfrak{C}_{\mathfrak{B}}(\mathcal{O})\cong\prod_{\mathfrak{B}}\operatorname{RMod}^{\operatorname{cpt}}(E_{\mathfrak{B}})

where $E_{\mathfrak{B}}$ is the endomorphism ring of $P_{\mathfrak{B}}$ .

Lemma 6.5.

Let $H\in\mathfrak{C}_{\mathfrak{B}}(\mathcal{O})$ and suppose $H$ is finitely generated as an $\mathcal{O}[[K]]$ -module. Then $\operatorname{Hom}_{\mathfrak{C}_{G}(\mathcal{O})}(P_{\mathfrak{B}},H)$ is a finitely generated $E_{\mathfrak{B}}$ -module.

Proof.

Recall that $H$ being finitely generated over $\mathcal{O}[[K]]$ is equivalent to $H^{\vee}$ being admissible. Since $E_{\mathfrak{B}}=\operatorname{End}_{\mathfrak{C}_{G}(\mathcal{O})}(P_{\mathfrak{B}})$ , it suffices to show that for some $r$ , there is a surjection

P_{\mathfrak{B}}^{\oplus r}\twoheadrightarrow H.

Indeed, applying $\operatorname{Hom}_{\mathfrak{C}_{G}(\mathcal{O})}(P_{\mathfrak{B}},-)$ to this diagram then proves the lemma.

We claim that the cosocle $\operatorname{cosoc}H$ is a finite direct sum $\bigoplus_{i}\pi_{i}^{\vee}$ of irreducible objects. Indeed, every $\pi_{i}\hookrightarrow\oplus_{i}\pi_{i}=\operatorname{soc}H^{\vee}$ has non-zero $K_{0}$ -invariants for any pro- $p$ group $K_{0}$ . Since the dual $H^{\vee}$ is admissible, we see that $\operatorname{soc}H^{\vee}$ is a finite direct sum of irreducibles. The same then holds for $\operatorname{cosoc}H^{\vee}=(\operatorname{soc}H^{\vee})^{\vee}.$ Now, choose a surjection

P_{\mathfrak{B}}^{\oplus r}\twoheadrightarrow\operatorname{cosoc}H

for some $r$ . By the projectivity of $P_{\mathfrak{B}}$ , this map factors as

P_{\mathfrak{B}}^{\oplus r}\to H\to\operatorname{cosoc}H,

and the admissibility of $V$ implies that the second arrow is a superfluous surjection ([11, Lemma 4.6]). Thus the first map is also surjective, which completes the proof. ∎

Pan [31] has extended the results of the previous section to our setting.

Proposition 6.6.

Let $(\mathfrak{B}_{v})_{v\mid p}$ be a tuple of blocks of $\mathfrak{C}_{\operatorname{GL}_{2}(\mathbb{Q}_{p}),\mathbf{1}}(\mathcal{O})$ , each containing an absolutely irreducible representation, and define

\mathfrak{B}:=\otimes_{v\mid p}\mathfrak{B}_{v}:=\{\otimes_{v\mid p}\pi_{v}:\pi_{v}\in\mathfrak{B}_{v}\}.

Then $\mathfrak{B}$ is a block of $\mathfrak{C}_{G}(\mathcal{O})$ , and moreover:

(a)

$P_{\mathfrak{B}}:=\widehat{\bigotimes}_{v\mid p}P_{\mathfrak{B}_{v}}$ is a projective envelope of $\widehat{\bigotimes}_{v\mid p}\pi_{v}^{\vee}$ .
(b)

$E_{\mathfrak{B}}=\operatorname{End}_{\mathfrak{C}_{G}(\mathcal{O})}(P_{\mathfrak{B}})\cong\widehat{\bigotimes}_{v\mid p}E_{\mathfrak{B}_{v}}$
(c)

$E_{\mathfrak{B}}$ is a finitely generated module over $R^{\operatorname{ps}}_{p}$ .
(d)

Let $\mathfrak{p}_{v}$ be the ideal of $R^{\operatorname{ps}}_{v}=R^{\operatorname{ps}}_{\operatorname{tr}\bar{\rho}_{v}}$ corresponding to the trace $\operatorname{tr}\rho_{v}$ of a irreducible representation $\rho_{v}$ , and let $\mathfrak{p}$ be the ideal of $R^{\operatorname{ps}}_{p}=\widehat{\bigotimes}_{v\mid p}R^{\operatorname{ps}}_{v}$ generated by the joint image of the $\mathfrak{p}_{v}$ ’s. Then

$(E_{\mathfrak{B}})_{\mathfrak{p}}^{\wedge}\cong M_{|\mathfrak{B}|}((R^{\operatorname{ps}}_{p})_{\mathfrak{p}}^{\wedge})$

as $L$ -algebras (possibly non-commutative).

Proof.

For (i-iii), see [31, §3.4]. For (iv), recall that for $v\mid p$ , $P_{\mathfrak{B}_{v}}$ is a direct sum $\bigoplus_{i_{v}=1}^{|\mathfrak{B}_{v}|}P_{\mathfrak{B}_{v},i_{v}}$ and we have a matrix presentation of $E_{\mathfrak{B}}$ over $R^{\operatorname{ps}}_{p}$ given by

\begin{split}E_{\mathfrak{B}}&=\operatorname{End}_{\mathfrak{C}_{G}(\mathcal{O})}(\widehat{\bigotimes}_{v\mid p}P_{\mathfrak{B}_{v}})\\ &=\operatorname{End}_{\mathfrak{C}_{G}(\mathcal{O})}\big{(}\widehat{\bigotimes_{v\mid p}}\bigoplus_{i_{v}=1}^{|\mathfrak{B}_{v}|}P_{\mathfrak{B}_{v},i_{v}}\big{)}\\ &=\big{(}\widehat{\bigotimes}_{v\mid p}\operatorname{Hom}_{\mathfrak{C}_{\operatorname{GL}_{2}(\mathbb{Q}_{p}),\mathbf{1}}(\mathcal{O})}(P_{\mathfrak{B}_{v},i_{v}},P_{\mathfrak{B}_{v},j_{v}})\big{)}_{(i_{v}),(j_{v})}\end{split}

where $(i_{v})$ and $(j_{v})$ run over sequences of length $|\mathfrak{B}_{v}|$ such that $1\leq i_{v},j_{v}\leq|\mathfrak{B}_{v}|$ . This defines a matrix presentation of $E_{\mathfrak{B}}$ over $R^{\operatorname{ps}}_{p}$ as in the proof of Theorem 6.4(iv), and there exists an element $c\in R^{\operatorname{ps}}_{p}$ such that $E_{\mathfrak{B}}[1/c]$ is a matrix algebra. Indeed, we may take $c$ equal to the product of $c_{v}\in R^{\operatorname{ps}}_{v}$ chosen such that $E_{\mathfrak{B}_{v}}[1/c_{v}]$ is a matrix algebra, as in the aforementioned proof. ∎

6.4 Functors of twisted coinvariants

In this section, we introduce the $\mathbb{Z}[U_{p}]$ -modules whose associated local systems we will use as coefficients in homology and their associated functors of twisted coinvariants. They are indexed by $v$ -adic Hodge types (Definition 5.14).

Theorem 6.7.

([7, A.1.5.]) Let $\tau_{v}:I_{F_{v}}\to\operatorname{GL}_{2}(L)$ be a representation with open kernel. There exists a unique smooth irreducible $K_{v}$ -representation $\sigma(\tau_{v})$ on an $L$ -vector space characterised by the property

\operatorname{Hom}_{K_{v}}(\pi_{v},\sigma(\tau_{v}))\neq 0\iff\operatorname{LL}(\pi_{v})|_{I_{F_{v}}}\cong\tau_{v},

when $\pi_{v}$ ranges over all smooth absolutely irreducible infinite-dimensional $G_{v}$ -representation over $L$ and $\operatorname{LL}(\pi_{v})$ is the Weil-Deligne representation associated to $\pi_{v}$ by the classical local Langlands correspondence (normalised as in [7]).

Definition 6.8.

Let ${(\mathbf{w}_{v},\tau_{v},\chi_{v})}$ be a $v$ -adic Hodge type. The $K_{v}$ -representation associated to ${(\mathbf{w}_{v},\tau_{v},\chi_{v})}$ is the representation

\sigma(\mathbf{w}_{v},\tau_{v})=\sigma(\tau_{v})\otimes(\operatorname{Sym}^{b_{v}-a_{v}-1}L^{2})\otimes(\det)^{a_{v}}

We write $\sigma(\mathbf{w}_{v})=(\operatorname{Sym}^{b_{v}-a_{v}-1}L^{2})\otimes(\det)^{a_{v}}$ so that $\sigma(\mathbf{w}_{v},\tau_{v})=\sigma(\tau_{v})\otimes\sigma(\mathbf{w}_{v})$ . Since $K_{v}$ is compact, there is a $K_{v}$ -stable $\mathcal{O}$ -lattice

\sigma^{\circ}(\mathbf{w}_{v},\tau_{v})={\sigma^{\circ}}(\tau_{v})\otimes{\sigma^{\circ}}(\mathbf{w}_{v})\subset\sigma(\mathbf{w}_{v},\tau_{v}).

Given a tuple $(\mathbf{w},\tau,\chi)={(\mathbf{w}_{v},\tau_{v},\chi_{v})}_{v\in S_{p}}$ of $v$ -adic Hodge types, we obtain a $K=\prod_{v\mid p}K_{v}$ -representation upon forming the tensor product over $v\in S_{p}$ ,

\sigma(\mathbf{w},\tau)=\bigotimes_{v\mid p}\sigma(\mathbf{w}_{v},\tau_{v}),

containing the $K$ -stable $\mathcal{O}$ -lattice

\sigma^{\circ}(\mathbf{w},\tau)=\bigotimes_{v\mid p}\sigma^{\circ}(\mathbf{w}_{v},\tau_{v})\subset\sigma(\mathbf{w},\tau),

which is a finitely generated $\mathcal{O}[[K]]$ -module. Given such a ${\sigma^{\circ}}=\sigma^{\circ}(\mathbf{w},\tau)$ and a compact $\mathcal{O}[[K]]$ -module $N$ , we have a natural isomorphism

N\hat{\otimes}_{\mathcal{O}[[K]]}{\sigma^{\circ}}\cong N\otimes_{\mathcal{O}[[K]]}{\sigma^{\circ}},

and $-\otimes_{\mathcal{O}[[K]]}{\sigma^{\circ}}$ defines a right exact functor $\operatorname{RMod}^{\operatorname{cpt}}(\mathcal{O}[[K]])\to\operatorname{Mod}^{\operatorname{cpt}}(\mathcal{O})$ ([8, Lemma 2.1]). If $N$ is of the form $N=\widehat{\bigotimes}_{v\mid p}N_{v},$ we have an isomorphism

N\otimes_{\mathcal{O}[[K]]}{\sigma^{\circ}}\cong\widehat{\bigotimes_{v\mid p}}(N_{v}\otimes_{\mathcal{O}[[K_{v}]]}\sigma^{\circ}_{v}),

where $\sigma^{\circ}_{v}=\sigma^{\circ}(\mathbf{w}_{v},\tau_{v})$ . We will use the notation

	$\displaystyle N({\sigma^{\circ}})$	$\displaystyle:=N\otimes_{\mathcal{O}[[K]]}{\sigma^{\circ}},$
	$\displaystyle N(\sigma)$	$\displaystyle:=N\otimes_{\mathcal{O}[[K]]}\sigma.$

Note that $N(\sigma)=N({\sigma^{\circ}})[1/\varpi]$ . On projective objects, this functor has the following alternate description which occurs in the literature.

Proposition 6.9.

[16, Remark 5.1.7] Suppose $P$ is projective in $\operatorname{RMod}^{\operatorname{cpt}}(\mathcal{O}[[K]])$ , and ${\sigma^{\circ}}=\sigma^{\circ}(\mathbf{w},\tau)$ . There is a natural isomorphism

P({\sigma^{\circ}})\cong\operatorname{Hom}_{\mathcal{O}[[K]]}^{\operatorname{cts}}(P,({\sigma^{\circ}})^{\operatorname{d}})^{\operatorname{d}}

where $(-)^{d}=\operatorname{Hom}_{\mathcal{O}}(-,\mathcal{O})$ with the topology of pointwise convergence.

6.5 Description of $P$

Let $\mathfrak{B}=\otimes_{v\mid p}\mathfrak{B}_{v}$ be a block of $G$ of the form considered in Theorem 6.6, and $\sigma=\sigma(\mathbf{w},\tau)$ . In this section, we will consider the module of twisted coinvariants $P_{\mathfrak{B}}(\sigma)=P_{\mathfrak{B}}\otimes_{\mathcal{O}[[K]]}\sigma$ of the projective generator $P_{\mathfrak{B}}$ of the category $\mathfrak{C}_{\mathfrak{B}}(\mathcal{O})$ , or rather its localisation and completion at a characteristic 0 point.

Let $R_{p,\rho}=(R^{\operatorname{ps}}_{p})_{\mathfrak{p}}^{\wedge}$ . The localisation simplifies things greatly, since $(E_{\mathfrak{B}})_{\mathfrak{p}}^{\wedge}$ is isomorphic to a matrix algebra over $R_{p,\rho}$ when $\mathfrak{p}$ is the ideal corresponding to a tuple $(\rho_{v})$ of irreducible representations (Proposition 6.6(d)), and thus $P_{\mathfrak{B}}(\sigma)_{\mathfrak{p}}^{\wedge}$ is a direct sum of $|\mathfrak{B}|$ pairwise isomorphic $R_{p,\rho}$ -modules. For this reason, it will suffice to consider a single summand of $P_{\mathfrak{B}}$ .

For every $v\mid p$ , let

\pi_{v}=\operatorname{Ind}_{P}^{G}(\chi_{v,1}\otimes\chi_{v,2}\varepsilon^{-1})\in\mathfrak{B}_{v}

where $\chi_{v,1}/\chi_{v,2}\neq 1,\varepsilon$ (but allowing $\chi_{v,1}\varepsilon=\chi_{v,2}$ ) and let $P=\widehat{\bigotimes}_{v\mid p}P_{v}\twoheadrightarrow\widehat{\bigotimes}_{v\mid p}\pi_{v}^{\vee}$ be the projective envelope as in Proposition 6.6(a). Then, as explained above,

P_{\mathfrak{B}}(\sigma)_{\mathfrak{p}}^{\wedge}\cong\big{(}P(\sigma)_{\mathfrak{p}}^{\wedge}\big{)}^{\oplus|\mathfrak{B}|}

where we view the elements of this module as vectors and the matrix algebra $(E_{\mathfrak{B}})_{\mathfrak{p}}^{\wedge}$ acts by matrix multiplication. There is a natural forgetful functor

\mathfrak{C}_{G}(\mathcal{O})\to\operatorname{RMod}^{\operatorname{cpt}}(\mathcal{O}[[K]]).

Proposition 6.10.

$P$ is projective in the category of compact right $\mathcal{O}[[K]]$ -modules.

Proof.

We mimic the proof of [16, Lemma B.8]. We have $P=\widehat{\otimes}_{v\mid p}P_{v}$ , a tensor product of modules over

\widehat{\bigotimes}_{v\mid p}\mathcal{O}[[K_{v}]]\simeq\widehat{\bigotimes}_{v\in S_{p}}\mathcal{O}[[\operatorname{PGL}_{2}(\mathbb{Z}_{p})]].

We proceed by induction on the size of the set $S_{p}$ . By [33, Corollary 5.3], $P_{v}$ is projective in $\operatorname{RMod}^{\operatorname{cpt}}(\mathcal{O}[[K_{v}]])$ . For the induction step, let $w\in S_{p}$ and define

	$\displaystyle K^{w}$	$\displaystyle=\prod_{v\in S_{p}\setminus\{w\}}K_{v},$
	$\displaystyle P^{w}$	$\displaystyle=\widehat{\otimes}_{v\in S_{p}\setminus\{w\}}P_{v},$

so that $K=K_{w}\times K^{w}$ and $P=P_{w}\widehat{\otimes}P^{w}$ . By the induction hypothesis, $P_{w}$ and $P^{w}$ are projective over $\mathcal{O}[[K_{w}]]$ and $\mathcal{O}[[K^{w}]]$ , respectively. The universal property of the completed tensor product implies that for any compact $\mathcal{O}[[K]]$ -module $N$ ,

\operatorname{Hom}^{\operatorname{cts}}_{\mathcal{O}[[K_{w}\times K^{w}]]}(P_{w}\widehat{\otimes}_{\mathcal{O}[[K]]}P^{w},N)\cong\operatorname{Hom}^{\operatorname{cts}}_{\mathcal{O}[[K_{w}]]}(P_{w},\operatorname{Hom}^{\operatorname{cts}}_{\mathcal{O}[[K^{w}]]}(P^{w},N)).

Hence, the projectivity of $P$ follows from that of $P_{w}$ and $P^{w}$ . ∎

Given $\sigma=\sigma(\mathbf{w},\tau)$ we write $R_{v}(\sigma_{v})$ for the ring $R_{v}(\mathbf{w}_{v},\tau_{v})$ introduced in Theorem 5.16, and $R_{p}({\sigma^{\circ}})=\widehat{\bigotimes}_{v\mid p}R_{v}(\sigma_{v})$ .

Theorem 6.11.

Let $\sigma=\sigma(\mathbf{w},\tau)$ . Then the action of $R^{\operatorname{ps}}_{p}$ on $P({\sigma^{\circ}})$ factors through $R_{p}(\sigma)$ and $P(\sigma)$ is locally free of rank $1$ over the regular locus of $\operatorname{Spec}R_{p}(\sigma)[1/\varpi]$ .

Proof.

We have

P({\sigma^{\circ}})\cong\widehat{\bigotimes}_{v\mid p}P_{v}(\sigma^{\circ}_{v})

The modules $P_{v}(\sigma^{\circ}_{v})$ are $\mathcal{O}$ -flat by [33, Lemma 2.10] and maximal Cohen-Macaulay over $R_{v}(\sigma_{v})$ by [33, Corollary 6.4, 6.5] (here, we use Proposition 6.9). For every $v\mid p$ , we fix a maximal regular sequence on $P_{v}(\sigma^{\circ}_{v})$ of length $\dim R_{v}(\sigma_{v})$ containing $\varpi$ (by extending $\varpi$ to a maximal regular sequence). For a flat compact $\mathcal{O}$ -module $M$ , the functor $-\widehat{\otimes}_{\mathcal{O}}M$ is exact and thus the union of the regular sequences form a regular sequence of length

1+\sum_{v\mid p}(\dim R_{v}(\sigma_{v})-1)=\dim R_{p}(\sigma),

and thus $\operatorname{dp}_{R_{p}(\sigma)}P({\sigma^{\circ}})=\dim R_{p}(\sigma)$ . Thus, $P({\sigma^{\circ}})$ is maximal Cohen-Macaulay over $R_{p}(\sigma)$ . Therefore, if $\mathfrak{p}\subset R_{p}(\sigma)[1/\varpi]$ is a regular closed point, $(P({\sigma^{\circ}}))_{\mathfrak{p}}$ is free by Lemma 3.13. Finally, to compute the rank, let $k(\mathfrak{p})=\otimes_{v\mid p}k(\mathfrak{p}_{v})\cong L$ be the residue field at $\mathfrak{p}$ . By [33, Prop. 2.22, 4.14], each $P_{v}(\sigma_{v}^{\circ})_{\mathfrak{p}_{v}}$ is of rank $1$ over $R_{v}(\sigma_{v})_{\mathfrak{p}_{v}}$ , and hence

\operatorname{rk}_{R_{p}(\sigma)_{\mathfrak{p}}}\big{(}P({\sigma^{\circ}})_{\mathfrak{p}}\big{)}=\dim_{k(\mathfrak{p})}\big{(}k(\mathfrak{p})\otimes_{R_{p}(\sigma)_{\mathfrak{p}}}(P({\sigma^{\circ}}))_{\mathfrak{p}}\big{)}=1.

∎

Corollary 6.12.

Let $\mathfrak{p}\subset R^{\operatorname{ps}}_{p}[1/\varpi]$ be the closed point corresponding to a tuple of absolutely irreducible representations, so that $(E_{\mathfrak{B}})_{\mathfrak{p}}^{\wedge}\cong M_{|\mathfrak{B}|}(R_{p,\rho})$ . Set $R_{p,\rho}(\sigma)=R_{p}(\sigma)_{\mathfrak{p}}^{\wedge}$ . Then $P_{\mathfrak{B}}(\sigma)_{\mathfrak{p}}^{\wedge}$ is isomorphic to $R_{p,\rho}(\sigma)^{\oplus|\mathfrak{B}|}$ as an $(E_{\mathfrak{B}})_{\mathfrak{p}}^{\wedge}$ -module, with $(E_{\mathfrak{B}})_{\mathfrak{p}}^{\wedge}$ -action given by matrix multiplication.

Proof.

This follows from Theorem 6.11 and the discussion at the beginning of this section. Indeed, we have

(P_{\mathfrak{B}}(\sigma)_{\mathfrak{p}}^{\wedge}\cong\big{(}P({\sigma^{\circ}})_{\mathfrak{p}}^{\wedge}\big{)}^{\oplus|\mathfrak{B}|}\cong R_{p,\rho}(\sigma)^{\oplus|\mathfrak{B}|},

where the right-most module has the usual right action of $M_{|\mathfrak{B}|}(R_{p,\rho})$ . ∎

6.6 Completed homology

In this section, we introduce the $p$ -adically completed homology of the group $\mathbf{G}$ . For a survey, see [9].

Fix a tame level $U^{p}\subset\mathbf{G}(\mathbb{A}^{\infty,p}_{F})$ , and let $K_{1}\subset K$ be chosen so that $K_{1}U^{p}$ is good. Consider the tower of Galois covers

(X_{U_{p}U^{p}})_{U_{p}},

where $U_{p}$ runs over a countable basis of neighbourhoods of the identity, normal in $K_{1}$ . This defines a projective system of continuous $K_{1}$ -spaces, and $X_{U_{p}U^{p}}$ is a $K_{1}/U_{p}$ -torsor over $X_{K_{1}U^{p}}$ . Since singular homology is a covariant functor, for any choice of coefficients one has a corresponding projective system of homology groups.

Definition 6.13.

We define the completed homology with tame level $U^{p}$ as the projective limit

\widetilde{H}_{\ast}(X_{U^{p}},\mathcal{O})=\varprojlim_{U_{p}}H_{\ast}(X_{U_{p}U^{p}},\mathcal{O})

where $U_{p}$ runs over a countable basis of neighbourhoods of the identity, normal in $K_{1}$ for any choice of $K_{1}\subset K$ such that $K_{1}U^{p}$ is a good subgroup.

Since $H_{\ast}(X_{U_{p}U^{p}},\mathcal{O})$ is an $\mathcal{O}[K/U_{p}]$ -module, $\widetilde{H}_{\ast}(X_{U^{p}},\mathcal{O})$ is an $\mathcal{O}[[K]]$ -module.

Proposition 6.14.

Suppose $K_{0}\subset K$ is a subgroup such that $K_{0}U^{p}$ is good. Then there is a canonical isomorphism

\widetilde{H}_{\ast}(X_{U^{p}},\mathcal{O})\cong H_{\ast}(X_{K_{0}U^{p}},\mathcal{O}[[K_{0}]]).

Moreover, for any compact open $K_{1}\subset K$ , $\widetilde{H}_{\ast}(X_{U^{p}},\mathcal{O})$ is a finitely generated $\mathcal{O}[[K_{1}]]$ -module.

Proof.

Let $K_{0}$ be as in the statement and consider the natural Hecke-equivariant map

C^{\operatorname{ad}}_{\bullet}(K_{0}U^{p},\mathcal{O}[[K]])\to\varprojlim_{U_{p}}C^{\operatorname{ad}}_{\bullet}(KU^{p},\mathcal{O}[K/U_{p}]).

Using our fixed choice of chain homotopy equivalence between the adèlic complex and the Borel-Serre complex, the map above corresponds to a map

C^{\operatorname{BS}}_{\bullet}(KU^{p},\mathcal{O}[[K]])\to\varprojlim_{U_{p}}C^{\operatorname{BS}}_{\bullet}(KU^{p},\mathcal{O}[K/U_{p}])

which is an isomorphism since $C^{\operatorname{BS}}_{\bullet}(K)$ consists of free and finitely generated $\mathbb{Z}[K]$ -modules. Now,

C^{\operatorname{BS}}_{\bullet}(KU^{p},\mathcal{O}[K/U_{p}])\cong\varprojlim_{U_{p}}C^{\operatorname{BS}}_{\bullet}(U_{p}U^{p},\mathcal{O})

and since $C^{\operatorname{BS}}_{\bullet}(U_{p}U^{p},\mathcal{O})$ consists of compact $\mathcal{O}$ -modules, Lemma 3.2 implies

H_{\ast}\big{(}\varprojlim_{U_{p}}C^{\operatorname{BS}}_{\bullet}(U_{p}U^{p},\mathcal{O})\big{)}\cong\varprojlim_{U_{p}}H_{\ast}\big{(}C^{\operatorname{BS}}_{\bullet}(U_{p}U^{p},\mathcal{O}))\cong H_{\ast}(X_{U^{p}},\mathcal{O}\big{)},

as required. The finite generation of $\widetilde{H}_{\ast}(X_{U^{p}},\mathcal{O})$ follows from the previous statement, together with the fact that if $K_{1}\subset K_{0}$ is normal, then $\mathcal{O}[[K_{0}]]$ is finitely generated (in fact, free) over $\mathcal{O}[[K_{1}]]$ . ∎

Let $G=\prod_{v\mid p}\mathbf{G}(F_{v})$ . The action of $K$ on $\widetilde{H}_{\ast}(X_{U^{p}},\mathcal{O})$ coming from the Iwasawa algebra $\mathcal{O}[[K]]$ can be extended to an action of $G$ . For any $g\in G$ and $U_{p}\subseteq K$ open compact, right multiplication by $g$ gives homomorphism

H_{\ast}(X_{U_{p}U^{p}},\mathcal{O})\to H_{\ast}(X_{(gU_{p}g^{-1})U^{p}},\mathcal{O}),

which induces an endomorphism of the projective limit $\widetilde{H}_{\ast}(X_{U^{p}},\mathcal{O})$ , extending the $\mathcal{O}[K]$ -action (see e.g. [16, Remark 3.4.13]). Since we have already established that $\widetilde{H}_{\ast}(X_{U^{p}},\mathcal{O})$ is finitely generated over $\mathcal{O}[[K]]$ , we obtain the following.

Proposition 6.15.

[9, Theorem 1.1(1)] With $G$ -action defined as above, $\widetilde{H}_{\ast}(X_{U^{p}},\mathcal{O})$ is a object of $\operatorname{Mod}_{G}^{\operatorname{fga}}(\mathcal{O})$ . In particular, $\widetilde{H}_{\ast}(X_{U^{p}},\mathcal{O})\in\mathfrak{C}_{G}(\mathcal{O})$ .

Almost by definition, the big Hecke algebra acts on completed homology. The action is equivariant for the $G$ -action; there exists a map

\mathbb{T}^{S}(U^{p})_{\mathfrak{m}}\to\operatorname{End}_{\mathfrak{C}_{G}(\mathcal{O})}(\widetilde{H}_{\ast}(X_{U^{p}},\mathcal{O})_{\mathfrak{m}}).

Proposition 6.16.

Let $\sigma=\sigma(\mathbf{w},\tau)=\sigma(\mathbf{w})\otimes\sigma(\tau)$ and $K_{\tau}=\ker(\sigma(\tau))\subset K$ . Suppose $KU^{p}$ is good. There is a Hecke-equivariant homological first-quadrant spectral sequence

E^{2}_{i,j}=\operatorname{Tor}_{j}^{\mathcal{O}[[K]]}\big{(}\widetilde{H}_{i}(X_{U^{p}},\mathcal{O}),\sigma)\implies H_{i+j}(X_{K_{\tau}U^{p}},\sigma(\mathbf{w})\big{)}[\tau^{\ast}]

where $[\tau^{\ast}]$ denotes the $\sigma(\tau)^{\ast}$ -isotypic component.

Proof.

The quotient $K/K_{\tau}$ is a finite group. By Proposition 6.14 and our choice of chain homotopy equivalence between the adèlic complex and the Borel-Serre complex, we have isomorphisms

\begin{split}\widetilde{H}_{\ast}(X_{U^{p}},\mathcal{O})&\cong H_{\ast}\big{(}C^{\operatorname{ad}}_{\bullet}(KU^{p})\otimes_{\mathcal{O}[K]}\mathcal{O}[[K]]\big{)}\\ &\cong H_{\ast}\big{(}C^{\operatorname{BS}}_{\bullet}(KU^{p})\otimes_{\mathcal{O}[K]}\mathcal{O}[[K]]\big{)}.\end{split}

Let $\widetilde{C}_{\bullet}=C^{\operatorname{BS}}_{\bullet}(KU^{p})\otimes_{\mathcal{O}[K]}\mathcal{O}[[K]]$ . This is a complex of finitely generated projective $\mathcal{O}[[K]]$ -modules, so there is a hyperhomology spectral sequence (see [39, Theorem 5.7.6])

E^{2}_{i,j}=\operatorname{Tor}_{j}^{\mathcal{O}[[K]]}\big{(}\widetilde{H}_{i}(X_{U^{p}},\mathcal{O}),\sigma\big{)}\implies H_{i+j}\big{(}\widetilde{C}_{\bullet}\otimes_{\mathcal{O}[[K]]}\sigma(\mathbf{w})\otimes_{L}\sigma(\mathbf{\tau})\big{)}.

Note that

\widetilde{C}_{\bullet}\otimes_{\mathcal{O}[[K]]}\sigma(\mathbf{w})\otimes_{L}\sigma(\mathbf{\tau})\cong\big{(}\widetilde{C}_{\bullet}\otimes_{[}\mathcal{O}[[K_{\tau}]]]\sigma(\mathbf{w})\otimes_{L}\sigma(\mathbf{\tau})\big{)}_{K/K_{\tau}}

where $(-)_{K/K_{\tau}}$ denotes taking coinvariants with respect to the action defined by

k(u\otimes u^{\prime})=uk\otimes k^{-1}u^{\prime},

where $u\in\widetilde{C}_{n}$ and $u^{\prime}\in\sigma(\mathbf{w})\otimes_{L}\sigma(\mathbf{\tau})$ . In characteristic 0, taking coinvariants is an exact functor, and since $K_{\tau}$ acts trivially on $\sigma(\tau)$ we have

H_{i+j}\big{(}\widetilde{C}_{\bullet}\otimes_{\mathcal{O}[[K]]}\sigma(\mathbf{w})\otimes_{L}\sigma(\mathbf{\tau})\big{)}\cong\big{(}H_{i+j}(\widetilde{C}_{\bullet}\otimes_{\mathcal{O}[[K_{\tau}]]}\sigma(\mathbf{w}))\otimes_{L}\sigma(\tau)\big{)}_{K/K\tau},

which by Schur’s lemma is precisely $H_{i+j}(X_{K_{\tau}U^{p}},\sigma(\mathbf{w}))[\tau^{\ast}]$ . This completes the proof. ∎

In our setting, one expects the spectral sequence above to degenerate at the $E^{2}$ -page after localisation at a ‘non-Eisenstein ideal’ $\mathfrak{m}\subset\mathbb{T}^{S}(U^{p})$ . The condition that $\mathfrak{m}$ is non-Eisenstein means that the representation $\bar{\rho}_{\mathfrak{m}}$ introduced in the next section is absolutely irreducible.

Conjecture 6.17.

Let $\mathfrak{m}\subset\mathbb{T}^{S}(U^{p})$ be a non-Eisenstein maximal ideal. Then

\widetilde{H}_{\ast}(X_{U^{p}},\mathcal{O})_{\mathfrak{m}}=\widetilde{H}_{q_{0}}(X_{U^{p}},\mathcal{O})_{\mathfrak{m}}.

Remark 6.18.

As is shown in [16, Proposition 4.2.1(1)], this vanishing follows from conjectures of Calegari-Emerton [9, Conjecture 1.5] and Calegari-Geraghty [10, Conjecture B(4)(a)]. These conjectures are open in general, but known to hold when $l_{0}=1$ , i.e. when $F$ is an imaginary quadratic field.

An immediate consequence of the conjecture and Proposition 6.16 is the isomorphism

\operatorname{Tor}_{i}^{\mathcal{O}[[K]]}\big{(}\widetilde{H}_{q_{0}}(X_{U^{p}},\mathcal{O}),\sigma\big{)}\cong H_{q_{0}+i}(X_{K_{\tau}U^{p}},\sigma(\mathbf{w}))[\tau^{\ast}].

6.7 Galois representations

Suppose that $\Pi$ is a regular algebraic cuspidal automorphic representation which is cohomological with respect to an algebraic weight, and let $S$ be the set of ramified places of $\Pi$ and the places above $p$ . The Hecke-equivariance of the spectral sequence of Proposition 6.16 implies that the action of the abstract Hecke algebra $\mathbb{T}^{S}$ on the homology $H_{\ast}(X_{U_{p}U^{p}},\sigma(\mathbf{w}))$ factors through the big Hecke algebra $\mathbb{T}^{S}(U^{p})$ . The assumption that $\Pi$ is cohomological means there is a $\sigma(\mathbf{w})$ such that the $\Pi$ -part

H_{\ast}(X_{U_{p}U^{p}},\sigma(\mathbf{w}))_{\mathfrak{N}_{\Pi,\iota}}

is non-zero in degrees $[q_{0},q_{0}+l_{0}]$ . By the observation above, if we let

\mathfrak{p}=\mathfrak{p}_{\Pi,\iota}=\ker\big{(}\mathbb{T}^{S}(U^{p})\to\operatorname{End}(H_{\ast}(X_{U_{p}U^{p}},\sigma(\mathbf{w}))_{\mathfrak{N}_{\Pi,\iota}})\big{)}

then we have

H_{\ast}(X_{U_{p}U^{p}},\sigma(\mathbf{w}))_{\mathfrak{N}_{\Pi,\iota}}=H_{\ast}(X_{U_{p}U^{p}},\sigma(\mathbf{w}))_{\mathfrak{p}}.

Moreover, we have a maximal ideal

\mathfrak{m}=\mathfrak{m}_{\Pi,\iota}=(\mathfrak{p},\varpi)\subset\mathbb{T}^{S}(U^{p}),

to which we associate a residual representation

\bar{\rho}_{\mathfrak{m}}:G_{F,S}\to\operatorname{GL}_{2}(\mathbb{T}^{S}(U^{p})/\mathfrak{m})

using the main result of [34]. By [12, Theorem 6.1.4], we may lift $\bar{\rho}_{\mathfrak{m}}$ to a representation

\rho_{\mathfrak{m}}:G_{F,S}\to\operatorname{GL}_{2}(\mathbb{T}^{S}(U^{p})_{\mathfrak{m}})

with determinant $\varepsilon^{-1}$ . This defines a surjective map

R_{\mathcal{S}}\twoheadrightarrow\mathbb{T}^{S}(U^{p})_{\mathfrak{m}},

where $\mathcal{S}$ is the global deformation problem $\mathcal{S}=(\bar{\rho}_{\mathfrak{m}},S,\varepsilon,D_{v}^{\varepsilon,\square})$ corresponding to deformations of $\bar{\rho}_{\mathfrak{m}}$ that are unramified outside of $S$ . We denote by $\rho=\rho_{\Pi,\iota}$ the composition

G_{F,S}\stackrel{{\scriptstyle\rho_{\mathfrak{m}}}}{{\to}}\operatorname{GL}_{2}(\mathbb{T}^{S}(U^{p})_{\mathfrak{m}})\to\operatorname{GL}_{2}(L),

where $L$ is the residue field of $\mathbb{T}^{S}(U^{p})_{\mathfrak{m}}$ at $\mathfrak{p}$ (if necessary, we replace $L$ by a finite extension). Thus, $\rho$ defines a maximal ideal $\mathfrak{p}\subset R_{\mathcal{S}}[1/\varpi]$ .

For every $v\in S_{p}$ , after twisting with $\bar{\varepsilon}$ we obtain a pseudorepresentation $\operatorname{tr}(\bar{\rho}_{\mathfrak{m},v}\otimes\bar{\varepsilon})$ of trivial determinant which defines a block $\mathfrak{B}_{\mathfrak{m},v}$ of $\operatorname{Mod}^{\operatorname{pfa}}_{\operatorname{GL}_{2}(F_{v}),1}(\mathcal{O})$ . We obtain a block of $\mathfrak{C}_{G}(\mathcal{O})$ by forming the tensor product as in Proposition 6.6, i.e.

\mathfrak{B}_{\mathfrak{m}}=\otimes_{v\mid p}\mathfrak{B}_{\mathfrak{m},v}.

7 Main results

We keep the notation from the preceding section, and from now on make the following assumptions.

(i)

$\bar{\rho}_{\mathfrak{m}}:G_{F,S}\to\operatorname{GL}_{2}(k)$ is absolutely irreducible and the restriction $\bar{\rho}_{\mathfrak{m}}|_{G_{F(\zeta_{p})}}$ has adequate image (see [37, Definition 2.3]).
(ii)

The characteristic 0 representations $\rho_{v}$ for all $v\in S_{p}$ are irreducible of $v$ -adic Hodge type ${(\mathbf{w}_{v},\tau_{v},\chi_{v})}$ .
(iii)

The local characteristic 0 representations $\rho_{v}$ have generic associated Weil-Deligne representations for all $v\in S$ .

7.1 The patching argument

The goal of this subsection is to prove the following theorem. The notation $R_{\mathcal{S},\rho}$ was introduced in Theorem 5.18.

Theorem 7.1.

With the same notation as in Section 6.7 and under Conjectures 6.17 and Conjecture 7.5 (see below), consider the $E_{\mathfrak{B}}$ -module

m_{0}=\operatorname{Hom}_{\mathfrak{C}_{\mathfrak{B}}(\mathcal{O})}\big{(}P_{\mathfrak{B}},\widetilde{H}_{q_{0}}(X_{U^{p}},\mathcal{O})_{\mathfrak{m}}\big{)}.

Then $(m_{0})_{\mathfrak{p}}^{\wedge}$ is free as an $R_{\mathcal{S},\rho}$ -module, and hence isomorphic as an $(E_{\mathfrak{B}})_{\mathfrak{p}}^{\wedge}$ -module to a direct sum of modules of the form $R_{\mathcal{S},\rho}^{\oplus|\mathfrak{B}|}$ with $(E_{\mathfrak{B}})_{\mathfrak{p}}^{\wedge}$ -action defined by matrix multiplication.

This result is a characteristic 0 analogue of [16, Conjecture 5.1.2]. Note that our setting differs in that we have no ‘minimal level’ assumption ensuring the smoothness of the local framed deformation rings $R_{v}^{\square}$ for $v\in S\setminus S_{p}$ . For our purposes, it is sufficient to assume that the associated Weil-Deligne representations of $\rho_{v}$ are generic at all places $v\in S\setminus S_{p}$ , as it ensures the restrictions $\rho_{v}$ define smooth points in the generic fibers of $R_{v}^{\square}$ (Theorem 5.12).

Our proof of Theorem 7.1 is based on the construction of ‘patched completed homology’ in [16]. The strategy is to first prove an analogous result (Theorem 7.9) at ‘patched’ level and then ‘unpatch’ to deduce Theorem 7.1. Before we can state the patched analogue, we need to recall the construction of patched completed homology. We follow [16] with only slight adjustments.

To begin, we note that the assumption that $\bar{\rho}(G_{F(\zeta_{p})})$ is adequate is equivalent to it being enormous ([16, Lemma 3.2.3]), and we let $q$ be an integer large enough to guarantee the existence of Taylor-Wiles primes as in [16, Lemma 3.3.1]. We let $\mathcal{T}$ be the power series ring over $\mathcal{O}$ in the $S$ -frame variables, i.e.

\mathcal{T}=\mathcal{O}[[\{X^{v}_{i,j}\mid v\in S,\ i,j=1,2\}]]/(X^{v_{0}}_{1,1}),

where $v_{0}$ is an arbitrary element of $S$ . Then $\mathcal{T}$ is of relative dimension $4|S|-1$ over $\mathcal{O}$ , and we define

\mathcal{O}_{\infty}=\mathcal{T}\widehat{\otimes}\mathcal{O}[[(\mathbb{Z}_{p}^{\times})^{q}]],

a power series ring over $\mathcal{O}$ in $4|S|-1+q$ variables. Let $\mathbf{a}=\ker(\mathcal{O}_{\infty}\to\mathcal{O})$ be the augmentation ideal of $\mathcal{O}_{\infty}$ , and set

S_{\infty}=R^{\operatorname{ps}}_{p}\widehat{\otimes}\mathcal{O}_{\infty}.

The construction features a second ring denoted $R_{\infty}$ which is as a power series ring over $R^{S,\operatorname{loc}}_{\mathcal{S}}=\widehat{\bigotimes}_{v\in S}R_{v}^{\square}$ in

|S|-1+q-[F:\mathbb{Q}]-l_{0}

variables, and it is constructed in such a way that there are maps

\mathcal{O}_{\infty}\to R_{\infty}\twoheadrightarrow R_{\mathcal{S}},

where $R_{\mathcal{S}}$ is the ring representing the functor of type $\mathcal{S}$ deformations. Since $R_{\infty}$ is a $R_{p}^{\square}$ -algebra, it is also an $R^{\operatorname{ps}}_{p}$ -algebra, and using this map we replace the diagram above with

S_{\infty}=R^{\operatorname{ps}}_{p}\widehat{\otimes}_{\mathcal{O}}\mathcal{O}_{\infty}\stackrel{{\scriptstyle\varphi}}{{\to}}R_{\infty}\twoheadrightarrow R_{\mathcal{S}}\twoheadrightarrow\mathbb{T}^{S}(U^{p}).

The proof of Theorem 7.1 is based on analysing this diagram. One expects the final arrow to be an isomorphism, and we will confirm this expectation after localisation and completion at the point $\mathfrak{p}\subset R_{\mathcal{S}}[1/\varpi]$ corresponding to $\rho$ .

Consider the diagram

S_{\infty}\stackrel{{\scriptstyle\varphi}}{{\to}}R_{\infty}\twoheadrightarrow R_{\mathcal{S}}\to L

where the final map corresponds to $\rho$ . We define

	$\displaystyle\mathfrak{p}_{\infty}$	$\displaystyle=\ker(R_{\infty}\to L),$
	$\displaystyle\mathfrak{q}_{\infty}$	$\displaystyle=\ker(S_{\infty}\to R_{\infty}\to L),$

so that $\mathfrak{q}_{\infty}=\varphi^{-1}(\mathfrak{p}_{\infty})$ and we have a homomorphism $(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}\to(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ .

Lemma 7.2.

The rings $(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}$ and $(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ are regular and

\dim(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}=\dim(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}-[F:\mathbb{Q}]-l_{0}.

Proof.

For the purpose of this proof only, define the auxiliary ring

T=\mathcal{O}[[y_{1},\dots,y_{|S|-1+q-[F:\mathbb{Q}]-l_{0}}]],

so that $R_{\infty}=R^{S,\operatorname{loc}}_{\mathcal{S}}\widehat{\otimes}T$ and $S_{\infty}=R^{\operatorname{ps}}_{p}\widehat{\otimes}\mathcal{O}_{\infty}$ are rings of formal power series with coefficients in $R^{S,\operatorname{loc}}_{\mathcal{S}}$ and $R^{\operatorname{ps}}_{p}$ , respectively. The regularity of the points $\mathfrak{p}_{\infty},\mathfrak{q}_{\infty}$ follows from that of the corresponding points of $R^{S,\operatorname{loc}}_{\mathcal{S}}$ and $R^{\operatorname{ps}}_{p}$ . Using [3, Lemma 3.3] and Theorem 5.13, we find that the dimensions are

\begin{split}\dim(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}&=\sum_{v\in S}\dim(R_{v}^{\square})_{\mathfrak{p}_{v}}+|S|-1+q-[F:\mathbb{Q}]-l_{0}]\\ &=3|S\setminus S_{p}|+3|S_{p}|+\sum_{v\mid p}\dim R_{\rho_{v}}+|S|-1+q-[F:\mathbb{Q}]-l_{0}\\ &=\sum_{v\mid p}\dim R_{\rho_{v}}+4|S|-1+q-[F:\mathbb{Q}]-l_{0},\end{split}

and

\begin{split}\dim(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}&=\sum_{v\mid p}\dim R_{\rho_{v}}+\dim\mathcal{O}_{\infty}-1\\ &=\sum_{v\mid p}\dim R_{\rho_{v}}+4|S|-1+q.\end{split}

The result follows. ∎

So far, we have only mentioned the rings involved in the patching argument. Let us now recall the key features for us of the complex on which these rings act. Note that the ‘minimal level’ assumption present in [16] is not required to prove the results cited below.

Theorem 7.3.

There exists a perfect complex $\widetilde{\mathcal{C}}(\infty)$ of $\mathcal{O}_{\infty}[[K]]$ -modules with a compatible $G$ -action such that the following hold.

(i)

The action of $\mathcal{O}_{\infty}$ on $\widetilde{\mathcal{C}}(\infty)$ factors through the map $\mathcal{O}_{\infty}\to R_{\infty}$ .
(ii)

$H_{\ast}(\widetilde{\mathcal{C}}(\infty))$ lies in $\mathfrak{C}_{G}(\mathcal{O})$ .

(iii)

Let $\mathbf{a}=\ker(\mathcal{O}_{\infty}\to\mathcal{O})$ . There is a $G$ -equivariant isomorphism of $\mathcal{O}_{\infty}[[K]]$ -modules.

H_{i}(\widetilde{\mathcal{C}}(\infty)\otimes_{\mathcal{O}_{\infty}}\mathcal{O}_{\infty}/\mathbf{a})\cong\widetilde{H}_{i}(X_{U^{p}},\mathcal{O})_{\mathfrak{m}}.

(iv)

Assume Conjecture 6.17. Then $\widetilde{\mathcal{C}}(\infty)$ has homology concentrated in degree $q_{0}$ .

Proof.

The complex is constructed in [16, Section 3.4]. For (i), (iii) and (iv), see [16, Proposition 3.4.16(2), Remark 3.4.17, Proposition 3.4.19, Proposition 4.2.1]. To prove (ii), first note that since $\widetilde{\mathcal{C}}(\infty)$ is a perfect complex of $\mathcal{O}_{\infty}[[K]]$ -modules,

H_{\ast}(\widetilde{\mathcal{C}}(\infty))\cong\varprojlim_{n}H_{\ast}(\widetilde{\mathcal{C}}(\infty)\otimes_{\mathcal{O}_{\infty}}\mathcal{O}_{\infty}/\mathbf{a}^{n}).

The category $\mathfrak{C}_{G}(\mathcal{O})$ has projective limits ([32, p.14]) and hence it is enough to prove that each term in the above limit is an element of $\mathfrak{C}_{G}(\mathcal{O})$ . We proceed by induction on $n$ , noting that the case $n=1$ follows from (iii) and Proposition 6.15. For every $n\geq 1$ , we have a short eact sequence of chain complexes

0\to\widetilde{\mathcal{C}}(\infty)\otimes_{\mathcal{O}_{\infty}}\mathbf{a}^{n-1}/\mathbf{a}^{n}\to\widetilde{\mathcal{C}}(\infty)\otimes_{\mathcal{O}_{\infty}}\mathcal{O}_{\infty}/\mathbf{a}^{n}\to\widetilde{\mathcal{C}}(\infty)\otimes_{\mathcal{O}_{\infty}}\mathcal{O}_{\infty}/\mathbf{a}^{n-1}\to 0

inducing a long exact sequence of homology groups

\dots\to H_{i}(\widetilde{\mathcal{C}}(\infty)\otimes_{\mathcal{O}_{\infty}}\mathbf{a}^{n-1}/\mathbf{a}^{n})\to H_{i}(\widetilde{\mathcal{C}}(\infty)\otimes_{\mathcal{O}_{\infty}}\mathcal{O}_{\infty}/\mathbf{a}^{n})\to H_{\ast}(\widetilde{\mathcal{C}}(\infty)\otimes_{\mathcal{O}_{\infty}}\mathcal{O}_{\infty}/\mathbf{a}^{n-1})\to\dots,

which in turn decomposes into short exact sequences in the usual way. The category $\mathfrak{C}_{G}(\mathcal{O})$ is closed under kernels, cokernels and extensions (in $\operatorname{Mod}^{\operatorname{pfa}}_{G,\zeta}(\mathcal{O})$ ). Indeed, $\mathfrak{C}_{G}(\mathcal{O})$ is abelian and moreover the inclusion $\operatorname{Mod}^{\operatorname{ladm}}_{G}(\mathcal{O})\hookrightarrow\operatorname{Mod}^{\operatorname{sm}}_{G,\zeta}(\mathcal{O})$ preserves injectives (see [32, Corollary 5.18]), so that for any locally admissible $V,W$ we have a canonical isomorphism $\operatorname{Ext}^{1}_{\operatorname{ladm}}(V,W)\cong\operatorname{Ext}^{1}_{\operatorname{sm}}(V,W)$ . Viewing these groups as parametrising extensions, we obtain the claimed closedness of $\mathfrak{C}_{G}(\mathcal{O})$ under extensions. Finally, by direct observation we see that

\widetilde{\mathcal{C}}(\infty)\otimes_{\mathcal{O}_{\infty}}\mathbf{a}^{n-1}/\mathbf{a}\cong(\widetilde{\mathcal{C}}(\infty)\otimes_{\mathcal{O}_{\infty}}\mathcal{O}_{\infty}/\mathbf{a})^{\oplus r}

for some $r$ depending on $n$ . The right-hand side has homology equal to a direct sum of copies of $\widetilde{H}_{q_{0}}(X_{U^{p}},\mathcal{O})_{\mathfrak{m}}$ , and the theorem now follows by induction on $n$ . ∎

Remark 7.4.

In light of item (ii), we think of $\widetilde{\mathcal{C}}(\infty)$ as a complex computing ‘patched completed homology’.

Neither $\widetilde{H}_{q_{0}}(X_{U^{p}},\mathcal{O})$ nor $H_{q_{0}}(\widetilde{\mathcal{C}}(\infty))$ are finitely generated over the big Hecke algebra, so to carry out the depth estimate part of the Taylor-Wiles method, we work instead with the respective images in the category of $E_{\mathfrak{B}}$ -modules. This idea can only work under the following additional assumption,

Conjecture 7.5.

The two actions of $R^{\operatorname{ps}}_{p}$ on $H_{q_{0}}(\widetilde{\mathcal{C}}(\infty))$ – one coming from the map $R^{\operatorname{ps}}_{p}\to E_{\mathfrak{B}}$ and the other from the map $R^{\operatorname{ps}}_{p}\to\mathbb{T}^{S}(U^{p})_{\mathfrak{m}}$ – coincide.

Remark 7.6.

In [31, §3.5], a similar statement in a different setting is interpreted as a form of local-global compatibility at $v\mid p$ . It seems a reasonable guess that under suitable assumptions the conjecture can be verified in our setting using recent results of Hevesi on local-global compatibility for completed homology [20].

From now on, we assume Conjecture 7.5. We define

	$\displaystyle m_{0}$	$\displaystyle=\operatorname{Hom}_{\mathfrak{C}_{G}(\mathcal{O})}\big{(}P_{\mathfrak{B}},\widetilde{H}_{q_{0}}(X_{U^{p}},\mathcal{O})_{\mathfrak{m}}\big{)},$
	$\displaystyle m_{\infty}$	$\displaystyle=\operatorname{Hom}_{\mathfrak{C}_{G}(\mathcal{O})}\big{(}P_{\mathfrak{B}},H_{q_{0}}(\widetilde{\mathcal{C}}(\infty))\big{)}.$

Note that, as an $R^{\operatorname{ps}}_{p}$ -module,

m_{0}\cong\bigoplus_{\pi\in\mathfrak{B}}\operatorname{Hom}_{\mathfrak{C}_{G}(\mathcal{O})}\big{(}P_{\pi},\widetilde{H}_{q_{0}}(X_{U^{p}},\mathcal{O})_{\mathfrak{m}}\big{)}

and similarly for $m_{\infty}$ . By Theorem 7.3(i) and Conjecture 7.5, $m_{\infty}$ is an $S_{\infty}$ -module such that the action of $S_{\infty}$ factors through the map $S_{\infty}\stackrel{{\scriptstyle\varphi}}{{\to}}R_{\infty}$ .

Proposition 7.7.

With $m_{0},m_{\infty}$ as above, we have:

(i)

$m_{\infty}\otimes^{\mathbf{L}}_{\mathcal{O}_{\infty}}\mathcal{O}_{\infty}/\mathbf{a}\cong m_{0}$ .
(ii)

$m_{0}$ is a finitely generated $R^{\operatorname{ps}}_{p}$ -module.
(iii)

$m_{\infty}$ is finitely generated as an $S_{\infty}$ -module and as an $R_{\infty}$ -module.

Proof.

(i) follows from Theorem 7.3. Note that since $\widetilde{H}_{q_{0}}(X_{U^{p}},\mathcal{O})_{\mathfrak{m}}$ is finitely generated over $\mathcal{O}[[K]]$ , $m_{0}$ is finitely generated over $E_{\mathfrak{B}}$ by Lemma 6.5. Now, $E_{\mathfrak{B}}$ is finitely generated over $R^{\operatorname{ps}}_{p}$ (Proposition 6.6) and (ii) follows. Since the $S_{\infty}$ -action factors through $S_{\infty}\to R_{\infty}$ , the first statement implies the second. From (i) we know that $m_{\infty}\otimes_{\mathcal{O}_{\infty}}\mathcal{O}\cong m_{0}$ , and therefore by Nakayama’s lemma for compact modules (see [8, Corollary 1.5]) it suffices to prove that $m_{\infty}$ is a compact module, which follows from that $m_{\infty}$ is an inverse limit of finitely generated modules. Thus, we have proved (iii). ∎

We obtain a corresponding diagram to what we had before:

S_{\infty}\to R_{\infty}\to\operatorname{End}_{E_{\mathfrak{B}}}(m_{\infty}).

Localising and completing the diagram we obtain

(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}\to(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}\to\operatorname{End}_{(E_{\mathfrak{B}})_{\mathfrak{p}}^{\wedge}}((m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}).

Lemma 7.8.

The module $(m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ is finitely generated over $(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ and $(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}$ .

Proof.

The first statement follows from the fact that $m_{\infty}$ is finitely generated over $R_{\infty}$ , which is Proposition 7.7(ii). For the second, note that since $\mathfrak{p}_{\infty}$ is maximal in $R_{\infty}[1/\varpi]$ , we have

(m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}\cong\varprojlim_{r}m_{\infty}[1/\varpi]/\mathfrak{p}_{\infty}^{r}.

Now, by definition $\mathfrak{q}_{\infty}=\varphi^{-1}(\mathfrak{p}_{\infty})$ and hence $\varphi(\mathfrak{q}_{\infty}^{r})\subseteq\mathfrak{p}_{\infty}^{r}$ , so that the finitely generated $(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}$ -module $(m_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}$ surjects onto $(m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ . The statement follows. ∎

The following theorem is the patched counterpart of Theorem 7.1.

Theorem 7.9.

$(m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ is a finitely generated and free $(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ -module.

Proof.

Let $(\mathbf{w},\tau,\varepsilon)_{v\in S_{p}}$ be the $v$ -adic Hodge types such that

\sigma=\otimes_{v\mid p}\ \sigma(\mathbf{w}_{v},\tau_{v},\varepsilon)

is the $K$ -module for which $\Pi$ contributes (in degrees $[q_{0},q_{0}+l_{0}]$ ) to $H_{\ast}(X_{U_{p}U^{p}},\sigma)$ . We define, as before,

R_{p,\rho}(\sigma)=\big{(}\widehat{\bigotimes}_{v\mid p}R_{v}(\sigma_{v},\tau_{v})\big{)}_{\mathfrak{p}}^{\wedge}

and set

\mathbf{a}_{\sigma,\mathfrak{p}}=\ker\big{(}(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}\twoheadrightarrow R_{p,\rho}(\sigma)\big{)}.

By Theorem 5.16 and Lemma 7.2, the rings $(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}$ and $R_{p,\rho}(\sigma)$ are regular. Thus $\mathbf{a}_{\sigma,\mathfrak{p}}$ is generated by a regular sequence of length

\dim(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}-\dim R_{p,\rho}(\sigma)=\dim(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}-[F:\mathbb{Q}].

We are going to combine Theorem 2.10 and Lemma 3.10 to prove that $(m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ is maximal Cohen-Macaulay over the regular local ring $(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ .

Lemma 7.10.

We have a canonical isomorphism

\big{(}(m_{0})_{\mathfrak{p}}^{\wedge}\otimes^{\mathbf{L}}_{(E_{\mathfrak{B}})_{\mathfrak{p}}^{\wedge}}P_{\mathfrak{B}}(\sigma)_{\mathfrak{p}}^{\wedge})\big{)}^{\oplus|\mathfrak{B}|^{2}}\cong(m_{0})_{\mathfrak{p}}^{\wedge}\otimes^{\mathbf{L}}_{R_{p,\rho}}P_{\mathfrak{B}}(\sigma)_{\mathfrak{p}}^{\wedge}.

Proof.

Since $P_{\mathfrak{B}}=\oplus_{\pi\in\mathfrak{B}}P_{\pi}$ , we have

m_{0}=\oplus_{\pi\in\mathfrak{B}}\operatorname{Hom}_{G}(P_{\pi},\widetilde{H}_{q_{0}}(X_{U^{p}},\mathcal{O})_{\mathfrak{m}}).

By Proposition 6.6, $(E_{\mathfrak{B}})_{\mathfrak{p}}^{\wedge}$ is a rank $|\mathfrak{B}|$ matrix algebra over $R_{p,\rho}$ and consequently the module $(m_{0})_{\mathfrak{p}}^{\wedge}$ is a sum of $|\mathfrak{B}|$ pairwise isomorphic $R_{p,\rho}$ -modules. The same is true for $P_{\mathfrak{B}}(\sigma)_{\mathfrak{p}}^{\wedge}$ . For any commutative ring $R$ , matrix algebra $E=M_{n}(R)$ , and $R$ -modules $M,N$ , one has

M\otimes^{\mathbf{L}}_{R}N\cong M^{\oplus n}\otimes^{\mathbf{L}}_{E}N^{\oplus n}

where $M^{\oplus n},N^{\oplus n}$ are right and left $E$ -modules, respectively. Thus, we have

\begin{split}(m_{0})_{\mathfrak{p}}^{\wedge}\otimes^{\mathbf{L}}_{R_{p,\rho}}P_{\mathfrak{B}}(\sigma)_{\mathfrak{p}}^{\wedge}&\cong\bigoplus_{\pi_{1}\in\mathfrak{B}}\bigoplus_{\pi_{2}\in\mathfrak{B}}\operatorname{Hom}_{\mathfrak{C}_{G}(\mathcal{O})}\big{(}P_{\pi_{1}},\widetilde{H}_{q_{0}}(X_{U^{p}},\mathcal{O})_{\mathfrak{m}}\big{)}_{\mathfrak{p}}^{\wedge}\otimes^{\mathbf{L}}_{R_{p,\rho}}P_{\pi_{2}}(\sigma)_{\mathfrak{p}}^{\wedge}\\ &\cong\bigoplus_{\pi_{1}\in\mathfrak{B}}\bigoplus_{\pi_{2}\in\mathfrak{B}}(m_{0})_{\mathfrak{p}}^{\wedge}\otimes^{\mathbf{L}}_{(E_{\mathfrak{B}})_{\mathfrak{p}}^{\wedge}}P_{\mathfrak{B}}(\sigma)_{\mathfrak{p}}^{\wedge}\\ &\cong((m_{0})_{\mathfrak{p}}^{\wedge}\otimes^{\mathbf{L}}_{(E_{\mathfrak{B}})_{\mathfrak{p}}^{\wedge}}P_{\mathfrak{B}}(\sigma)_{\mathfrak{p}}^{\wedge}))^{\oplus|\mathfrak{B}|^{2}},\end{split}

as claimed. ∎

Taking the derived quotient of $(m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ over $(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}$ with respect to $\mathbf{a}_{\sigma,\mathfrak{p}}$ gives the following chain of isomorphisms:

\begin{split}&(m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}\otimes^{\mathbf{L}}_{(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}}(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}/\mathbf{a}_{\sigma,\mathfrak{p}}\\ \cong\ &(m_{0})_{\mathfrak{p}}^{\wedge}\otimes^{\mathbf{L}}_{R_{p,\rho}}R_{p,\rho}(\sigma)\\ \cong\ &\big{(}(m_{0}\otimes^{\mathbf{L}}_{E_{\mathfrak{B}}}P_{\mathfrak{B}}\otimes^{\mathbf{L}}_{\mathcal{O}[[K]]}{\sigma^{\circ}})_{\mathfrak{p}}^{\wedge}\big{)}^{\oplus|\mathfrak{B}|^{2}}\\ \cong\ &\big{(}(\widetilde{H}_{q_{0}}(X_{U^{p}},\mathcal{O})_{\mathfrak{m}}\otimes^{\mathbf{L}}_{\mathcal{O}[[K]]}{\sigma^{\circ}})_{\mathfrak{p}}^{\wedge}\big{)}^{\oplus|\mathfrak{B}|^{2}}\end{split}

By Proposition 6.16 and Theorem 2.10, this complex has homology concentrated in degrees $[q_{0},q_{0}+l_{0}]$ , and thus we have proved

\operatorname{Tor}_{i}^{(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}}\big{(}(m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge},R_{p,\rho}(\sigma)\big{)}=0\text{ for }i\notin[0,l_{0}].

Applying Lemma 3.10, we obtain

\begin{split}\operatorname{dp}_{(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}}(m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}&\geq\operatorname{dp}_{\mathbf{a}_{\sigma,\mathfrak{p}}}(m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}\\ &=\dim(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}-[F:\mathbb{Q}]-l_{0}\\ &=\dim(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}.\end{split}

Since the action of $(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}$ on $(m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ factors through $(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ , any regular sequence in $(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}$ gives rise to a regular sequence in $(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ , and hence

\begin{split}\operatorname{dp}_{(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}}(m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}&\geq\operatorname{dp}_{(S_{\infty})_{\mathfrak{q}_{\infty}}^{\wedge}}(m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}\\ &\geq\dim(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}.\end{split}

Thus, $(m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ is a maximal Cohen-Macaulay module over the regular local ring $(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ , and Theorem 7.9 now follows from 3.13. ∎

Having proved Theorem 7.9, we now deduce Theorem 7.1.

Corollary 7.11.

The natural surjections

(R_{\infty}/\mathbf{a})_{\mathfrak{p}_{\infty}}^{\wedge}\twoheadrightarrow R_{\mathcal{S},\rho}\twoheadrightarrow\mathbb{T}^{S}(U^{p})_{\mathfrak{p}}^{\wedge}

are isomorphisms. In particular, Theorem 7.1 holds.

Proof.

By Theorem 7.9, $(m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ is free over $(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}$ , and hence the unpatched module

(m_{0})_{\mathfrak{p}}^{\wedge}=(m_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}\otimes_{\mathcal{O}_{\infty}}\mathcal{O}_{\infty}/\mathbf{a}

is free over $(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}/\mathbf{a}$ . But the action of $(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}/\mathbf{a}$ on $(m_{0})_{\mathfrak{p}}^{\wedge}$ factors through the maps displayed in the statement of the theorem, which forces them to be injective. Thus both maps are bijections and $(m_{0})_{\mathfrak{p}}^{\wedge}$ is free over $(R_{\infty})_{\mathfrak{p}_{\infty}}^{\wedge}/\mathbf{a}\cong R_{\mathcal{S},\rho}$ . ∎

7.2 The main theorem

In this section, we deduce our main theorem from Theorem 7.9.

Theorem 7.12.

Let $F$ be a CM field, suppose $p\geq 5$ is totally split in $F$ and let $\Pi$ be a regular algebraic cuspidal automorphic representation of $\mathbf{G}=\operatorname{PGL}_{2}/F$ which contributes to homology with weight $\sigma(\mathbf{w})$ , and fix an isomorphism $\iota:\overline{\mathbb{Q}}_{p}\to\mathbb{C}$ . Let $\tau$ be the inertial type of $\Pi$ determined by the local Langlands correspondence, set $K_{\tau}=\ker(\sigma(\tau))$ and suppose $U^{p}\subset\mathbf{G}(\mathbb{A}^{\infty}_{F})$ is a good tame level subgroup. Let $\mathfrak{m}=\mathfrak{m}_{\Pi,\iota}\subset\mathbb{T}^{S}(U^{p})$ be the maximal ideal of the big Hecke algebra associated to $(\Pi,\iota)$ and its associated Galois representation by $\rho_{\mathfrak{m}}:G_{F,S}\to\operatorname{GL}_{2}(\mathbb{T}^{S}(U^{p})_{\mathfrak{m}})$ . Let $\rho=\rho_{\Pi,\iota}:G_{F,S}\to\operatorname{GL}_{2}(L)$ be the characteristic 0 representation associated to $(\Pi,\iota)$ .

Suppose the following statements hold.

(i)

The residual representation $\bar{\rho}_{\mathfrak{m}}:G_{F,S}\to\operatorname{GL}_{2}(k)$ is absolutely irreducible and the restriction $\bar{\rho}_{\mathfrak{m}}|_{G_{F(\zeta_{p})}}$ has adequate image.
(ii)

The characteristic 0 representations $\rho_{v}$ are irreducible of $v$ -adic Hodge type ${(\mathbf{w}_{v},\tau_{v},\chi_{v})}$ , for all $v\in S_{p}$ .
(iii)

The local characteristic 0 representations $\rho_{v}$ have generic associated Weil-Deligne representations for all $v\in S$ .
(iv)

$\widetilde{H}_{\ast}(X_{U^{p}},\mathcal{O})_{\mathfrak{m}}$ vanishes outside degree $q_{0}$ (Conjecture 6.17).
(v)

Local-global compatibility in the sense of Conjecture 7.5 holds.
(vi)

The adjoint Bloch-Kato Selmer group vanishes, i.e. $H^{1}_{f}(G_{F},\operatorname{ad}^{0}\rho)=0$ .

Then the graded $L$ -vector space

H_{\ast}(X_{K_{\tau}U^{p}},\sigma)_{\mathfrak{p}}[\tau^{\ast}]

has a canonical structure of finitely generated and free graded module over the $\operatorname{Tor}$ -algebra

\operatorname{Tor}_{\ast}^{R_{p,\rho}}\big{(}R_{\mathcal{S},\rho},R_{p,\rho}(\sigma)\big{)}.

Suppose in addition the following.

(vii)

$H^{2}(G_{F,S},\operatorname{ad}^{0}\rho)=0$ , i.e. the ring $R_{\mathcal{S},\rho}$ is smooth.

Then there is a canonical isomorphism of graded-commutative rings

\operatorname{Tor}_{\ast}^{R_{p,\rho}}\big{(}R_{\mathcal{S},\rho},R_{p,\rho}(\sigma)\big{)}\cong\wedge^{\ast}H^{1}_{f}\big{(}G_{F,S},\operatorname{ad}^{0}\rho(1)\big{)}.

Remark 7.13.

We have discussed assumptions (iv) and (v) where they appeared above. The assumption (vi) is known to hold in many cases, see [1]. Assumption (vii) is a special case of a conjecture of Jannsen [21].

Proof.

By Theorem 7.1, Proposition 6.10 and Theorem 6.12,

\begin{split}(\widetilde{H}_{q_{0}}(X_{U^{p}},\mathcal{O})_{\mathfrak{m}}\otimes^{\mathbf{L}}_{\mathcal{O}[[K]]}\sigma)\otimes^{\mathbf{L}}_{R^{\operatorname{ps}}_{p}}(R^{\operatorname{ps}}_{p})_{\mathfrak{p}}^{\wedge}&\cong\big{(}(m_{0}\otimes^{\mathbf{L}}_{E_{\mathfrak{B}}}P_{\mathfrak{B}})\otimes^{\mathbf{L}}_{\mathcal{O}[[K]]}\sigma\big{)}\otimes^{\mathbf{L}}_{R^{\operatorname{ps}}_{p}}(R^{\operatorname{ps}}_{p})_{\mathfrak{p}}^{\wedge}\\ &\cong(m_{0})_{\mathfrak{p}}^{\wedge}\otimes^{\mathbf{L}}_{(E_{\mathfrak{B}})_{\mathfrak{p}}^{\wedge}}P_{\mathfrak{B}}(\sigma)_{\mathfrak{p}}^{\wedge}\\ &\cong R_{\mathcal{S},\rho}^{\oplus m}\otimes^{\mathbf{L}}_{R_{p,\rho}}R_{p,\rho}(\sigma)\end{split}

for some $m\geq 1$ . By assumption (iv), the spectral sequence of Proposition 6.16 degenerates at the $E^{2}$ -page, hence we obtain

\begin{split}H_{q_{0}+i}(X_{K_{\tau}U^{p}},\sigma(\mathbf{w}))_{\mathfrak{p}}[\tau^{\ast}]&\cong\operatorname{Tor}_{i}^{R_{p,\rho}}\big{(}R_{\mathcal{S},\rho},R_{p,\rho}(\sigma)\big{)}^{\oplus m},\end{split}

where $K_{\tau}=\ker(\sigma(\tau))$ . This proves the first part of the theorem.

Under the additional assumption that $H^{2}(G_{F},\operatorname{ad}^{0}\rho)=0$ , all three of $R_{p,\rho},R_{\mathcal{S},\rho}$ and $R_{p,\rho}(\sigma)$ are formally smooth $L$ -algebras. Moreover, the exact sequence of Theorem 4.2 simplifies to a short exact sequence

0\to H^{1}\big{(}G_{F,S},\operatorname{ad}^{0}\rho\big{)}\to\prod_{v\mid p}\frac{H^{1}\big{(}G_{F_{v}},\operatorname{ad}^{0}\rho\big{)}}{H^{1}_{f}\big{(}G_{F_{v}},\operatorname{ad}^{0}\rho\big{)}}\to H^{1}_{f}\big{(}G_{F,S},\operatorname{ad}^{0}\rho(1)\big{)}^{\vee}\to 0,

and the final term has dimension $h^{1}_{f}(G_{F},\operatorname{ad}^{0}\rho(1))=l_{0}$ by Proposition 4.1.

Dualising the sequence and interpreting the cohomology groups as tangent spaces, we see that the closed subschemes of $\operatorname{Spec}R_{p,\rho}$ corresponding to $R_{\mathcal{S},\rho}$ and $R_{p,\rho}(\sigma)$ share no tangent directions at $\mathfrak{p}$ , and thus we may express the maximal ideal of $R_{p,\rho}$ as the sum

\mathfrak{m}_{R_{p,\rho}}=\ker(R_{p,\rho}\to R_{\mathcal{S},\rho})+\ker(R_{p,\rho}\to R_{p,\rho}(\sigma)).

Furthermore, the intersection of the summands is generated by $l_{0}$ elements, corresponding to a basis of the dual Selmer group $H^{1}_{f}\big{(}G_{F,S},\operatorname{ad}^{0}\rho(1)\big{)}$ . A choice of generators $\underline{y}=y_{1},\dots,y_{l_{0}}$ thus determines an isomorphism (using Proposition 3.8)

\begin{split}\operatorname{Tor}_{\ast}^{R_{p,\rho}}\big{(}R_{\mathcal{S},\rho},R_{p,\rho}(\sigma)\big{)}&\cong\operatorname{Tor}_{\ast}^{L[[\underline{y}]]}(L,L)\\ &\cong\wedge^{\ast}H^{1}_{f}\big{(}G_{F,S},\operatorname{ad}^{0}\rho(1)\big{)}.\end{split}

By general properties of the $\operatorname{Tor}$ -product, these are isomorphisms of graded algebras (see [13, XI.§2]). Furthermore, the isomorphism does not depend on $\underline{y}$ , since any viable choice yields the same power series ring. ∎

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