# Congruence modules in higher codimension and zeta lines in Galois cohomology

Srikanth B. Iyengar  Department of Mathematics, University of Utah, Salt Lake
City, UT 84112, U.S.A<EMAIL_ADDRESS>,  Chandrashekhar B. Khare
Department of Mathematics, University of California, Los Angeles, CA 90095,
U.S.A<EMAIL_ADDRESS>,  Jeffrey Manning  Mathematics Department,
Imperial College, London, SW7 2RH, UK<EMAIL_ADDRESS>and  Eric Urban
Mathematics Department, Columbia University, New York, U. S. A.
<EMAIL_ADDRESS>
###### Abstract.

This work builds on that in [Iyengar/Khare/Manning:2022a] where a notion of
congruence modules in higher codimension is introduced. The main new results
are a criterion for detecting regularity of local rings in terms of congruence
modules, and a more refined version of a result tracking the change of
congruence modules under deformation is proved. Number theoretic applications
include the construction of canonical lines in certain Galois cohomology
groups arising from adjoint motives of Hilbert modular forms.

###### Key words and phrases:

congruence module, Hecke module, modularity lifting, Wiles defect

###### 2020 Mathematics Subject Classification:

11F80 (primary); 13C10, 13D02 (secondary)

##  1\. Introduction

Let $p$ be a prime number, $\mathcal{O}$ the ring of integers of a finite
extension of $\mathbb{Q}_{p}$, and $R$ a complete noetherian local
$\mathcal{O}$-algebras with an $\mathcal{O}$-algebra morphism $\lambda\colon
R\to\mathcal{O}$ such that the local ring $R_{\mathfrak{p}}$, where
$\mathfrak{p}\colonequals\operatorname{Ker}(\lambda)$, is regular.

In [Iyengar/Khare/Manning:2022a] we develop an analog of the Wiles-Lenstra-
Diamond numerical criterion in arbitrary codimension, with the original
criterion (see [Diamond:1997, Wiles:1995]) corresponding to the codimension 0
case. This gives a criterion for finitely generated $R$-module $M$ to have a
free direct summand and for $R$ to be a complete intersection ring. This
involves two invariants associated to $\lambda$: the torsion
$\Phi_{\lambda}(R)$ of the cotangent space of $\lambda$, and the congruence
module $\Psi_{\lambda}(M)$. The codimension $c$ of the augmentation $\lambda$
is the height of $\mathfrak{p}$, or equivalently under our regularity
assumption, the dimension of $R_{\mathfrak{p}}$. We denote the category of
such pairs $(R,\lambda)$ by $\operatorname{C}_{\mathcal{O}}(c)$. The earlier
work of [Diamond:1997] corresponds to the case $c=0$ in which case
$\mathfrak{p}/\mathfrak{p}^{2}=\Phi_{\lambda}(R)$.

Wiles uses the criterion in his work on the modularity of elliptic curves over
$\mathbb{Q}$ to go from modularity lifting theorems in the minimal case to
those in the non-minimal case. In [Iyengar/Khare/Manning:2022a] the numerical
criterion in higher codimension is used along the same lines to prove integral
modularity lifting results for non-minimal lifts in situations of positive
defect which arise in considering Galois representations over imaginary
quadratic fields.

In this work we explore the meaning of the invariants $\Phi_{\lambda}(R)$ and
$\Psi_{\lambda}(M)$ in certain number theoretic situations, relating them to
the index of zeta elements in global Galois cohomology groups.

To begin with we focus on one of the key ideas of
[Iyengar/Khare/Manning:2022a], namely the definition and properties of
congruence modules, and congruence ideals, associated to an augmentation
$\lambda$ in arbitrary codimension. This is the content of Section 2. The
highlights are a characterisation, Theorem 2.6, of regularity of rings
$(A,\lambda)$ in our category $\operatorname{C}_{\mathcal{O}}(c)$ in terms of
vanishing of the invariants $\Phi_{\lambda}(A)$ and $\Psi_{\lambda}(A)$); a
more transparent description, in §2.10, of the connection between duality and
our congruence modules than in the earlier paper, and a refinement, Theorem
2.32, of a result about deformation invariance of Wiles defect.

Section 3 focuses on number theoretic applications, and concerns the
congruence ideal attached to an augmentation of Hida’s ordinary Hecke algebra
$\mathbb{T}^{\rm ord}$, or ordinary deformation ring $R^{\rm ord}$, arising
from cohomological Hilbert modular forms $f$ over totally real fields $F$. The
functorial properties of the congruence ideal are used to relate it to lines
(that is to say, free $\mathcal{O}$-modules of rank one) in the Galois
cohomology with coefficients in $\operatorname{ad}\rho_{f}$; see Theorem 3.7.
The index of their image under global-to-local restriction maps to singular
local Galois cohomology $H^{1}_{{\rm ord}/f}(G_{p},\operatorname{ad}\rho_{f})$
is related to classical congruence module of $f$, and to special values of
adjoint $L$-function

$L^{\rm alg}(1,\operatorname{ad}\rho_{f})\,,$

by work of Hida [Hida:1981] and its generalization by Dimitrov
[Dimitrov:2009]. This is connected to the “zeta elements” of [Urban:2021,
Theorem 1.1]. The terminology is due to Kato [Kato:2004], who used it for
elements of Galois cohomology he constructs in a related context, arising from
Beilinson-Flach classes. In [Urban:2021] exact sequences of Selmer groups are
used while here we use in addition our congruence modules in higher
codimension, allowing one to eliminate some hypotheses which arise from
relying on $R=\mathbb{T}$ theorems.

This connection arises from the following circumstance. Let
$\boldsymbol{t}\colonequals t_{1},\dots,t_{c}$ be indeterminates,
$\Lambda_{c}\colonequals\mathcal{O}[\\![{\boldsymbol{t}}]\\!]$ the power-
series ring, and $\Lambda_{c}\to\mathcal{O}$ the natural augmentation. Fix
$(A,\lambda)$ in $\operatorname{C}_{\mathcal{O}}(c)$ equipped with a finite
flat map $\iota\colon\Lambda_{c}\to A$ of $\mathcal{O}$-algebras over
$\mathcal{O}$, so that composite map

$\Lambda_{c}\xrightarrow{\ \iota\ }A\xrightarrow{\ \lambda\ }\mathcal{O}\,,$

is the augmentation. For $A_{0}=A/(\boldsymbol{t})$ one has a map

$\operatorname{Ext}^{c}_{A}(\mathcal{O},A)(=A_{0}[\ker\lambda])\to\bigwedge^{c}_{\mathcal{O}}\operatorname{Hom}(\mathfrak{p}/\mathfrak{p}^{2},\mathcal{O})\,,$

whose cokernel is $\Psi_{\lambda}(A)$. Abstractly both domain and range are
simply $\mathcal{O}$’s. In applications when $A$ is a nearly ordinary
deformation ring and $\lambda$ arises from the classical form $f$, the range
is a Selmer group as
$\operatorname{Hom}(\mathfrak{p}/\mathfrak{p}^{2},\mathcal{O})=H^{1}_{\mathcal{L}}(G_{F,S},\operatorname{ad}\rho_{f})$,
with local conditions $\mathcal{L}=(\mathcal{L}_{v})$ such that
$\mathcal{L}_{v}\subset H^{1}(G_{v},\operatorname{ad}\rho_{f})$ for $v\in S$
and $\mathcal{L}_{v}$ is the unramified subspace $H^{1}_{\rm
unr}(G_{v},\operatorname{ad}\rho_{f})$ for $v$ not in $S$. When $A$ is a
nearly ordinary Hecke algebra
$\operatorname{Hom}(\mathfrak{p}/\mathfrak{p}^{2},\mathcal{O})$ is a subspace
of $H^{1}(G_{F,S},\operatorname{ad}\rho_{f})$. This gives a “pure thought”
construction of canonical lines in Galois cohomology; see Theorem 3.7.

##  2\. Higher congruence modules and Wiles defects

We being by recalling the setup of [Iyengar/Khare/Manning:2022a]. This section
complements the material presented in [Iyengar/Khare/Manning:2022a, Part 1],
where the commutative algebraic aspects of the theory of congruence modules is
developed. There are some new results, the main ones being Theorems 2.6 and
2.32, and Proposition 2.20. Along the way we provide also a different
perspective and new proofs of some key results from
[Iyengar/Khare/Manning:2022a].

######  2.1.

Let $\mathcal{O}$ be a complete discrete valuation ring, with valuation
$\operatorname{ord}(-)$ and uniformizer $\varpi$. Throughout we fix a complete
local $\mathcal{O}$-algebra $A$ and a finitely generated $A$-module $M$. Given
a map $\lambda\colon A\to\mathcal{O}$ of $\mathcal{O}$-algebras, set

$\mathfrak{p}_{\lambda}\colonequals\operatorname{Ker}\lambda\quad\text{and}\quad
c\colonequals 0pt{\mathfrak{p}_{\lambda}}\,.$

For any finitely generated $A$-module $M$, set

$\operatorname{F}^{i}_{\lambda}(M)\colonequals{\operatorname{Ext}^{i}_{A}(\mathcal{O},M)}^{\operatorname{tf}}$

the torsion-free quotient of the $\mathcal{O}$-module
$\operatorname{Ext}^{i}_{A}(\mathcal{O},M)$. Here $\mathcal{O}$ is viewed as
an $A$-module via $\lambda$. The _congruence module_ of $M$ at $\lambda$ is
the $\mathcal{O}$-module

$\Psi_{\lambda}(M)\colonequals\operatorname{coker}\left(\operatorname{F}^{c}_{\lambda}(M)\xrightarrow{\
\operatorname{F}^{c}_{\lambda}(\lambda\otimes M)\
}\operatorname{F}^{c}_{\lambda}(M/\mathfrak{p}_{\lambda}M)\right)\,.$

We have also to consider $\mathcal{O}$-module

$\Phi_{\lambda}(A)\colonequals\operatorname{tors}(\mathfrak{p}_{\lambda}/\mathfrak{p}_{\lambda}^{2})\,,$

namely, the torsion part of the cotangent module
$\mathfrak{p}_{\lambda}/\mathfrak{p}_{\lambda}^{2}$ of $\lambda$.

We say an $A$-module $M$ has a certain property _at $\lambda$_ if the
$A_{\mathfrak{p}_{\lambda}}$-module $M_{\mathfrak{p}_{\lambda}}$ has the
stated property. For instance we say $A$ is regular at $\lambda$ to mean that
the local ring $A_{\mathfrak{p}_{\lambda}}$ is regular. The starting point of
our work is the following result; see [Iyengar/Khare/Manning:2022a, Theorem
2.5 and Lemma 2.6].

######  Theorem 2.2.

With $\lambda\colon A\to\mathcal{O}$ as above, the following conditions are
equivalent:

  1. (1)

The local ring $A$ is regular at $\lambda$.

  2. (2)

The rank of the $\mathcal{O}$-module
$\mathfrak{p}_{\lambda}/\mathfrak{p}^{2}_{\lambda}$ is
$0pt\mathfrak{p}_{\lambda}$.

  3. (3)

The $\mathcal{O}$-module $\Psi_{\lambda}(A)$ is torsion.

  4. (4)

The $\mathcal{O}$-module $\Psi_{\lambda}(M)$ is torsion for each finitely
generated $A$-module $M$.

Moreover, when these conditions hold the $\mathcal{O}$-module
$\Psi_{\lambda}(A)$ is cyclic. ∎

Condition (2) is that the embedding dimension of the ring
$A_{\mathfrak{p}_{\lambda}}$ equals its Krull dimension, so
(1)$\Leftrightarrow$(2) is one definition of regularity; see
[Bruns/Herzog:1998, Definition 2.2.1]. The key input in proving
(1)$\Leftrightarrow$(3) is the following result due to Lescot [Lescot:1983];
see also [Avramov/Iyengar:2013].

######  2.3.

A noetherian local ring $R$ is regular if and only if the map

$\operatorname{Ext}_{R}(k,R)\longrightarrow\operatorname{Ext}_{R}(k,k)$

induced by the canonical surjection $R\to k$ to the residue field of $R$, is
nonzero. When this is the case, the map above is nonzero in (upper) degree
$\dim R$.

The result below is implicit in the proof of (1)$\Rightarrow$(4) in Theorem
2.2, in [Iyengar/Khare/Manning:2022a]. We make it explicit, for it is used
also in proving Lemma 2.5 and Theorem 2.6 below.

######  Lemma 2.4.

Let $\varepsilon\colon R\to S$ be a surjective map of noetherian rings such
that the ideal $\operatorname{Ker}(\varepsilon)$ is generated by a regular
sequence, and set $c\colonequals\dim R-\dim S$. For any $R$-module $M$ the map
below is bijective:

$\operatorname{Ext}^{c}_{R}(S,\varepsilon\otimes_{R}M)\colon\operatorname{Ext}^{c}_{R}(S,M)\longrightarrow\operatorname{Ext}^{c}_{R}(S,S\otimes_{R}M)\,.$

###### Proof.

The Koszul complex, say $K$, on any regular sequence generating
$\operatorname{Ker}(\varepsilon)$ is a minimal resolution of $S$ as an
$R$-module. The map $\operatorname{Ext}^{c}_{R}(S,\varepsilon\otimes_{R}M)$ is
the one obtained in cohomology in (upper) degree $c$ from the morphism in the
upper row of the following commutative diagram of complexes:

${\operatorname{Hom}_{R}(K,M)}$${\operatorname{Hom}_{R}(K,S\otimes_{R}M)}$${\operatorname{Hom}_{R}(K,R)\otimes_{R}M}$${\operatorname{Hom}_{R}(K,R)\otimes_{R}(S\otimes_{R}M)}$$\scriptstyle{\operatorname{Hom}_{R}(K,\varepsilon\otimes
M)}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{Hom}_{R}(K,R)\otimes(\varepsilon\otimes
M)}$$\scriptstyle{\cong}$

The vertical maps are isomorphisms because $K$ is a finite free $R$-complex.
It is clear that the map in the lower row induces a bijection in cohomology in
the top degree, $c$. Thus the same holds for the one in the upper row, as
claimed. ∎

We denote $\operatorname{C}_{\mathcal{O}}$ the category whose objects are
pairs $(A,\lambda)$ satisfying the equivalent conditions in Theorem 2.2. A
morphism $\varphi\colon(A,\lambda)\to(A^{\prime},\lambda^{\prime})$ in this
category is a map of $\mathcal{O}$-algebras $\varphi\colon A\to A^{\prime}$
over $\mathcal{O}$; that is to say, with
$\lambda^{\prime}\circ\varphi=\lambda$. We write
$\operatorname{C}_{\mathcal{O}}(c)$ for the subcategory of
$\operatorname{C}_{\mathcal{O}}$ consists of pairs $(A,\lambda)$ such that
$0pt\mathfrak{p}_{\lambda}=c$.

######  Lemma 2.5.

For any $(A,\lambda)$ in $\operatorname{C}_{\mathcal{O}}(c)$ and finitely
generated $A$-module $M$ the map

$\operatorname{F}^{c}_{\lambda}(\lambda\otimes_{A}M)\colon\operatorname{F}^{c}_{\lambda}(M)\longrightarrow\operatorname{F}^{c}_{\lambda}(M/\mathfrak{p}_{\lambda}M)$

is one-to-one.

###### Proof.

Set $R\colonequals A_{\mathfrak{p}_{\lambda}}$ and let $\varepsilon\colon R\to
E$ be the map obtained by localizing $\lambda$ at $\mathfrak{p}_{\lambda}$;
here $E$ is the residue field of $R$, which is also the field of fractions of
$\mathcal{O}$. Since injectivity of a map of torsion-free
$\mathcal{O}$-modules can be detected after passing to the field of fractions,
it suffices to check that the map

$\operatorname{F}^{c}_{\lambda}(\lambda\otimes_{A}M)_{\mathfrak{p}_{\lambda}}\cong\operatorname{Ext}_{R}^{c}(E,\varepsilon\otimes_{R}M)$

is one-to-one. Since $R$ is regular the ideal
$\operatorname{Ker}(\varepsilon)$ is generated by a regular sequence of length
$c$; see [Bruns/Herzog:1998, Proposition 2.2.4]. It remains to apply Lemma
2.4. ∎

In the work of Hida [Hida:1981] and Ribet [Ribet:1983] congruence modules (for
codimension $c=0$) are attached to augmentations $\mathbb{T}\to\mathcal{O}$ of
Hecke algebras $\mathbb{T}$ that act faithfully on certain localized Betti
cohomology groups $H^{1}(X_{1}(N),\mathcal{O})_{\mathfrak{m}}$. They measure
the complexity of $\mathbb{T}$ and their vanishing is equivalent to
$\mathbb{T}$ being smooth, namely just $\mathcal{O}$. Analogously we show in
the result below that for rings $A\in\operatorname{C}_{\mathcal{O}}$ the
vanishing of either the congruence module $\Psi_{\lambda}(A)$ or
$\Phi_{\lambda}(A)$, the torsion part of the cotangent module, at any
augmentation $\lambda\colon A\to\mathcal{O}$ implies $A$ is smooth.

Unlike most results in [Iyengar/Khare/Manning:2022a, Part 1], the following
theorem does not make assumptions on the depth of the ring $A$.

######  Theorem 2.6.

For $(A,\lambda)$ in $\operatorname{C}_{\mathcal{O}}$, the local ring $A$ is
regular if and only if $\Phi_{\lambda}(A)=0$, if and only if
$\Psi_{\lambda}(A)=0$.

###### Proof.

We first verify that $A$ is regular if and only if $\Phi_{\lambda}(A)=0$. As
$A$ is a complete $\mathcal{O}$-algebra, one has $A\cong P/I$ where
$P\colonequals\mathcal{O}[\\![{t_{1},\dots,t_{n}}]\\!]$, a ring of formal
power series over $\mathcal{O}$, the ideal
$I\subseteq(\varpi)(\boldsymbol{t})+(\boldsymbol{t})^{2}$, and $\lambda\colon
A\to\mathcal{O}$ is quotient by $(\boldsymbol{t})$.

Let $\boldsymbol{f}\colonequals f_{1},\dots,f_{m}$ be a minimal generating set
for $I$. The cotangent module
$\mathfrak{p}_{\lambda}/\mathfrak{p}_{\lambda}^{2}$ depends only on $n$ and
the linear part of the $f_{i}$, in the following sense: By our assumption on
$I$, each $f_{i}$ has an unique expression of the form

(2.7) $f_{i}\colonequals\sum_{j=1}^{n}{u_{ij}}t_{j}+g_{i}\qquad\text{with
$u_{ij}\in(\varpi)\mathcal{O}$ and $g_{i}\in(\boldsymbol{t})^{2}$.}$

Then one has a presentation

$\mathcal{O}^{m}\xrightarrow{\ (u_{ij})\
}\mathcal{O}^{n}\longrightarrow\mathfrak{p}_{\lambda}/\mathfrak{p}_{\lambda}^{2}\longrightarrow
0\,.$

The torsion part of $\mathfrak{p}_{\lambda}/\mathfrak{p}_{\lambda}^{2}$ is
zero if and only if $(u_{ij})=0$, that is to say,
$(\boldsymbol{f})\subseteq(\boldsymbol{t})^{2}$. Since $A$ is regular at
$\lambda$ this condition is equivalent to $\boldsymbol{f}=0$, as desired.

Next we verify the claim that $A$ is regular if and only if
$\Psi_{\lambda}(A)=0$.

When $A$ is regular, $\operatorname{Ker}(\lambda\colon A\to\mathcal{O})$ is
generated by a regular sequence of length $c\colonequals
0pt(\operatorname{Ker}\lambda)$; see [Bruns/Herzog:1998, Proposition 2.2.4].
Thus Lemma 2.4 yields that the map
$\operatorname{Ext}^{c}_{A}(\mathcal{O},\lambda)$ is one-to-one so
$\Psi_{\lambda}(A)=0$.

Assume $\Psi_{\lambda}(A)=0$. To verify that $A$ is regular it suffices to
verify that the map

$\operatorname{Ext}_{A}(k,\varepsilon)\colon\operatorname{Ext}_{A}(k,A)\to\operatorname{Ext}_{A}(k,k)\,,$

induced by the quotient map $\varepsilon\colon A\to k$, is non-zero, for then
Lescot’s result 2.3.

Let $M$ be a finitely generated $A$-module. The exact sequence

(2.8) $0\longrightarrow\mathcal{O}\xrightarrow{\ \varpi\
}\mathcal{O}\longrightarrow k\longrightarrow 0$

of $A$-modules induces exact sequences of $k$-modules

$0\longrightarrow
k\otimes_{\mathcal{O}}\operatorname{Ext}_{A}^{i}(\mathcal{O},M)\xrightarrow{\
\eth^{i+1}(M)\
}\operatorname{Ext}_{A}^{i}(k,M)\longrightarrow\operatorname{Ext}_{A}^{i+1}(\mathcal{O},M)[\varpi]\longrightarrow
0$

For what follows the relevant point is that the maps $\eth^{i}(M)$ are
inclusions. Set $c\colonequals 0pt(\mathfrak{p}_{\lambda})$ and consider the
following commutative diagram of $k$-vector spaces:

${k\otimes_{\mathcal{O}}\operatorname{Ext}^{c}_{A}(\mathcal{O},A)}$${k\otimes_{\mathcal{O}}\operatorname{Ext}^{c}_{A}(\mathcal{O},\mathcal{O})}$${\operatorname{Ext}^{c+1}_{A}(k,A)}$${\operatorname{Ext}^{c+1}_{A}(k,\mathcal{O})}$${\operatorname{Ext}^{c+1}_{A}(k,k)}$$\scriptstyle{\eth(A)}$$\scriptstyle{\eth(\mathcal{O})}$$\scriptstyle{\operatorname{Ext}^{c+1}_{A}(k,\varepsilon)}$

The map in the top row is induced by $\lambda\colon A\to\mathcal{O}$ and the
ones in the lower row are induced by $A\to\mathcal{O}\to k$. That the map in
the lower right is one-to-one follows by considering the long exact sequence
in cohomology that arises by applying $\operatorname{Hom}_{A}(k,-)$ to the
exact sequence (2.8). It is easy to verify that the hypothesis
$\Psi_{\lambda}(A)=0$ implies the map in the top row is nonzero, and hence so
is the diagonal map. It then follows from the commutative diagram that the map
$\operatorname{Ext}^{c+1}_{A}(k,\varepsilon)$ is nonzero. This is as desired.
∎

######  Remark 2.9.

Consider the ordinary Hida Hecke algebra $\mathbb{T}^{\operatorname{ord}}$ of
tame level $N$. It is finite flat over
$\Lambda\colonequals\mathcal{O}[\\![{t}]\\!]$, with $t$ the weight variable,
and $\mathbb{T}^{\operatorname{ord}}/(t)=\mathbb{T}$ is a classical Hecke
algebra, acting faithfully on $H^{1}(X_{1}(Np),\mathcal{O})_{\mathfrak{m}}$.
Consider an augmentation
$\lambda\colon\mathbb{T}^{\operatorname{ord}}\to\mathbb{T}\to\mathcal{O}$
arising from a ($p$-stabilized) newform $f\in S_{1}(\Gamma_{1}(Np^{r}))$. The
vanishing of the congruence module of $\mathbb{T}$ for the augmentation
$\mathbb{T}\to\mathcal{O}$ implies $\mathbb{T}=\mathcal{O}$ and
$\mathbb{T}^{\operatorname{ord}}=\Lambda$ while the vanishing of the
congruence module for $\mathbb{T}^{\operatorname{ord}}\to\mathcal{O}$ implies
that $\mathbb{T}^{\operatorname{ord}}$ is smooth, while $\mathbb{T}$ may not
be smooth. In other words $\Psi_{\lambda}(\mathbb{T}^{\operatorname{ord}})=0$
implies $\mathbb{T}^{\operatorname{ord}}=\mathcal{O}[\\![{x}]\\!]$,
furthermore $x$ can be taken to be the weight variable $t$ if and only if the
classical congruence module $\Psi_{\lambda}(\mathbb{T})=0$ also vanishes.

Next we describe a pairing associated with the definition of congruence
modules. This too appears in [Iyengar/Khare/Manning:2022a], but does not play
a major role in the development there. The presentation below is more
transparent, and is used to give another perspective on some of the subsequent
results.

######  2.10.

For any finitely generated $A$-module $M$, one has natural isomorphisms

$\displaystyle{\operatorname{Ext}^{c}_{A}(\mathcal{O},M/\mathfrak{p}_{\lambda}M)}^{\operatorname{tf}}$
$\displaystyle\cong{\operatorname{Ext}^{c}_{A}(\mathcal{O},\mathcal{O})}^{\operatorname{tf}}\otimes_{\mathcal{O}}{(M/\mathfrak{p}_{\lambda}M)}^{\operatorname{tf}}$
$\displaystyle\cong\operatorname{Hom}_{\mathcal{O}}(\operatorname{Hom}_{\mathcal{O}}(M,\mathcal{O}),{\operatorname{Ext}^{c}_{A}(\mathcal{O},\mathcal{O})}^{\operatorname{tf}})\,.$

Thus, the map

$\operatorname{F}^{c}_{\lambda}(\lambda\otimes
M)\colon\operatorname{Ext}^{c}_{A}(\mathcal{O},M)\longrightarrow{\operatorname{Ext}^{c}_{A}(\mathcal{O},M/\mathfrak{p}_{\lambda}M)}^{\operatorname{tf}}$

whose cokernel is the congruence module of $M$, is adjoint to the map

$\langle-,-\rangle_{M}\colon{\operatorname{Ext}^{c}_{A}(\mathcal{O},M)}^{\operatorname{tf}}\otimes_{\mathcal{O}}\operatorname{Hom}_{A}(M,\mathcal{O})\longrightarrow{\operatorname{Ext}^{c}_{A}(\mathcal{O},\mathcal{O})}^{\operatorname{tf}}\,.$

The _congruence ideal_ of $M$, with respect to the augmentation $\lambda$, is
the image of this pairing:

$\eta_{\lambda}(M)\colonequals\mathrm{Image}\langle-,-\rangle_{M}\,.$

Localizing at $\mathfrak{p}_{\lambda}$, it is easy to verify that the free
$\mathcal{O}$-modules $\operatorname{F}^{c}_{\lambda}(M)$ and
$\operatorname{Hom}_{A}(M,\mathcal{O})$ have the same rank and that
$\operatorname{F}^{c}_{\lambda}(\mathcal{O})$ has rank one, so

$\operatorname{length}_{\mathcal{O}}(\mathcal{O}/\eta_{\lambda}(M))\leq\operatorname{length}_{\mathcal{O}}\Psi_{\lambda}(M)\leq\operatorname{rank}_{\lambda}(M)\cdot\operatorname{length}_{\mathcal{O}}(\mathcal{O}/\eta_{\lambda}(M))\,.$

Here $\operatorname{rank}_{\lambda}(M)$ denotes the rank of $M$ at $\lambda$,
that is to say, the rank of the $A_{\lambda}$-module
$M_{\mathfrak{p}_{\lambda}}$. In particular, when this rank equals $1$, the
length of the congruence module can be computed from the pairing.

The pairing above is induced–by passage to torsion-free quotients–by the
natural pairing given by composition of morphisms:

(2.11)
$\langle-,-\rangle_{M}\colon\operatorname{Ext}^{c}_{A}(\mathcal{O},M)\otimes_{\mathcal{O}}\operatorname{Hom}_{A}(M,\mathcal{O})\longrightarrow\operatorname{Ext}^{c}_{A}(\mathcal{O},\mathcal{O})\,.$

Namely, $\operatorname{Ext}^{c}_{A}(\mathcal{O},M)$ can be realized as
$\operatorname{Hom}_{\mathsf{D}}(\mathcal{O},M[c])$, the morphisms in the
derived category of $A$ from $\mathcal{O}$ to $M[c]$, and given such a
morphism $f$ and a map $g\colon M\to\mathcal{O}$, the pairing above is

$\langle f,g\rangle\colonequals g\circ f\colon\mathcal{O}\to\mathcal{O}[c]\,.$

In terms of the Yoneda interpretation of
$\operatorname{Ext}^{c}_{A}(\mathcal{O},M)$ as equivalence classes

$0\longrightarrow M\longrightarrow X_{c-1}\longrightarrow
X_{c-1}\longrightarrow\cdots\longrightarrow
X_{0}\longrightarrow\mathcal{O}\longrightarrow 0$

of exact sequences, the pairing is given by taking push-out along $g\colon
M\to\mathcal{O}$.

#### Cohen-Macaulay modules

When $M$ is Cohen-Macaulay of dimension $c+1$, local duality yields an
identification

$\operatorname{Hom}_{A}(\mathcal{O},M)\cong\operatorname{Ext}^{c}_{A}(\mathcal{O},M^{\vee})\quad\text{where
$M^{\vee}\cong H_{c+1}(\mathrm{RHom}_{A}(M,\omega_{A}))$}$

Here $\omega_{A}$ is the dualizing complex of $A$, normalized as in [stacks-
project]; see [Iyengar/Khare/Manning:2022a, §4] for details. With this
identification, the pairing (2.11) takes the form

(2.12)
$\langle-,-\rangle\colon\operatorname{Ext}^{c}_{A}(\mathcal{O},M)\otimes_{\mathcal{O}}\operatorname{Ext}^{c}_{A}(\mathcal{O},M^{\vee})\longrightarrow\operatorname{Ext}^{c}_{A}(\mathcal{O},\mathcal{O})\,.$

See [Iyengar/Khare/Manning:2022a, Proposition 4.7].

The pairing (2.11) can be described concretely when $c\leq 1$. One
simplification that occurs then is that
$\operatorname{Ext}^{1}_{A}(\mathcal{O},\mathcal{O})$ is already torsion-free,
as is explained below.

Consider the exact sequence

(2.13) $0\longrightarrow\mathfrak{p}_{\lambda}\longrightarrow
A\longrightarrow\mathcal{O}\longrightarrow 0\,.$

Applying $\operatorname{Hom}_{A}(-,M)$ yields the exact sequence

$M\cong\operatorname{Hom}_{A}(A,M)\longrightarrow\operatorname{Hom}_{A}(\mathfrak{p}_{\lambda},M)\longrightarrow\operatorname{Ext}^{1}_{A}(\mathcal{O},M)\longrightarrow\operatorname{Ext}_{A}^{1}(A,M)=0\,.$

This justifies the following result.

######  Lemma 2.14.

For any $c\geq 0$, and any $A$-module $M$, there is a natural isomorphism of
$\mathcal{O}$-modules

$\operatorname{Ext}^{1}_{A}(\mathcal{O},M)\cong\operatorname{coker}(M\to\operatorname{Hom}(\mathfrak{p}_{\lambda},M))\,.\qed$

The isomorphism above assigns to any $A$-linear map
$f\colon\mathfrak{p}_{\lambda}\to M$ the exact sequence obtained by push-out
of the exact sequence in (2.13) along $f$:

${0}$${\mathfrak{p}_{\lambda}}$${A}$${\mathcal{O}}$${0}$${0}$${M}$${X}$${\mathcal{O}}$${0}$$\scriptstyle{f}$

The natural map
$\mathcal{O}\to\operatorname{Hom}_{A}(\mathfrak{p}_{\lambda},\mathcal{O})$ is
zero, so for $M\colonequals\mathcal{O}$ the isomorphism in Lemma 2.14 becomes

(2.15)
$\operatorname{Ext}^{1}_{A}(\mathcal{O},\mathcal{O})\cong\operatorname{Hom}_{A}(\mathfrak{p}_{\lambda},\mathcal{O})\cong\operatorname{Hom}_{\mathcal{O}}(\mathfrak{p}_{\lambda}/\mathfrak{p}^{2}_{\lambda},\mathcal{O})$

which is already torsion-free.

Now we return to the pairing (2.11).

### The case $c=0$

Since $\operatorname{Hom}_{A}(\mathcal{O},M)=M[\mathfrak{p}_{\lambda}]$, the
$\mathfrak{p}_{\lambda}$-torsion submodule of $M$, the pairing (2.11) becomes

$\displaystyle
M[\mathfrak{p}_{\lambda}]\otimes_{\mathcal{O}}\operatorname{Hom}_{A}(M,\mathcal{O})\longrightarrow\mathcal{O}$
$\displaystyle m\otimes f\mapsto f(m)$

When $0pt_{A}M\geq 1$, one has
$M^{\vee}\cong\operatorname{Hom}_{\mathcal{O}}(M,\mathcal{O})$ this pairing is
equivalent to the one given by the composition

$M[\mathfrak{p}_{\lambda}]\otimes_{\mathcal{O}}M^{\vee}[\mathfrak{p}_{\lambda}]\longrightarrow
M\otimes_{\mathcal{O}}M^{\vee}\longrightarrow\mathcal{O}$

where the map on the right is the obvious one.

### The case $c=1$

With this description, for $c=1$ the pairing (2.11) is induced by the obvious
pairing

$\operatorname{Hom}_{A}(\mathfrak{p},M)\otimes_{A}\operatorname{Hom}_{A}(M,\mathcal{O})\longrightarrow\operatorname{Hom}_{A}(\mathfrak{p},\mathcal{O})\cong\operatorname{Hom}_{\mathcal{O}}(\mathfrak{p}/\mathfrak{p}^{2},\mathcal{O})$

given by composition of maps. Since
$\operatorname{Ext}^{1}_{A}(\mathcal{O},\mathcal{O})$ is torsion-free, as in
the case $c=0$ the ideal $\eta_{\lambda}(M)$ is just the image of the pairing
above.

### Structure of $\operatorname{F}^{*}_{A}(\mathcal{O})$

A key input in the development of the commutative algebraic properties of the
congruence module is a structure theorem for
$\operatorname{F}^{*}_{A}(\mathcal{O})$. The Yoneda product gives
$\operatorname{Ext}^{*}_{A}(\mathcal{O},\mathcal{O})$ the structure of a
graded $\mathcal{O}$-algebra, and this is inherited by its torsion-free
quotient, $\operatorname{F}^{*}_{\lambda}(\mathcal{O})$. The remarkable fact
[Iyengar/Khare/Manning:2022a, Theorem 6.8] is that although the Ext-algebra
itself can be highly non-commutative, and infinite,
$\operatorname{F}^{*}_{\lambda}(\mathcal{O})$ is just an exterior algebra
generated by its degree one component

$\operatorname{F}^{1}_{\lambda}(\mathcal{O})\cong\operatorname{Hom}_{\mathcal{O}}(\mathfrak{p}_{\lambda}/\mathfrak{p}^{2}_{\lambda},\mathcal{O})\,.$

See 2.15 for the isomorphism above. As explained in
[Iyengar/Khare/Manning:2022a, Introduction], this may be seen an an integral
version of a result, due to Serre, on the structure of the Ext algebra of a
regular local ring. The proof of this structure theorem for
$\operatorname{F}^{*}_{\lambda}(\mathcal{O})$ uses ideas from the theory of
differential graded algebras. For the present purpose the important takeaway
is that there is an natural isomorphism of $\mathcal{O}$-modules

(2.16)
$\bigwedge^{c}\operatorname{Hom}_{\mathcal{O}}(\mathfrak{p}_{\lambda}/\mathfrak{p}_{\lambda}^{2},\mathcal{O})\xrightarrow{\
\cong\ }\operatorname{F}^{c}_{\lambda}(\mathcal{O})\,.$

The naturality assertion is that given any morphism
$\varphi\colon(A,\lambda)\to(A^{\prime},\lambda^{\prime})$ in
$\operatorname{C}_{\mathcal{O}}(c)$, the induced map
$\mathfrak{p}_{\lambda}/\mathfrak{p}_{\lambda}^{2}\to\mathfrak{p}_{\lambda^{\prime}}/\mathfrak{p}_{\lambda^{\prime}}$
gives rise to commutative square

${\bigwedge^{c}\operatorname{Hom}_{\mathcal{O}}(\mathfrak{p}_{\lambda}/\mathfrak{p}_{\lambda}^{2},\mathcal{O})}$${\operatorname{F}^{c}_{\lambda}(\mathcal{O})}$${\bigwedge^{c}\operatorname{Hom}_{\mathcal{O}}(\mathfrak{p}_{\lambda^{\prime}}/\mathfrak{p}_{\lambda^{\prime}}^{2},\mathcal{O})}$${\operatorname{F}^{c}_{\lambda^{\prime}}(\mathcal{O})}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

of maps of $\mathcal{O}$-modules. This leads to the following _invariance of
domain_ property for congruence modules; see [Iyengar/Khare/Manning:2022a,
Theorem 7.4].

######  Theorem 2.17.

Given a surjective map
$\varphi\colon(A,\lambda)\to(A^{\prime},\lambda^{\prime})$ in
$\operatorname{C}_{\mathcal{O}}(c)$, and an $A^{\prime}$-module $M^{\prime}$
with $0pt_{A^{\prime}}M^{\prime}\geq c$, there is a natural isomorphism of
$\mathcal{O}$-modules
$\Psi_{\lambda^{\prime}}(M^{\prime})\cong\Psi_{\lambda}(M^{\prime})$. ∎

Since $\varphi$ is surjective, $\mathfrak{p}_{\lambda}\cdot
A^{\prime}=\operatorname{Ker}\lambda^{\prime}$. The hypotheses in the
statement above imply that $\varphi_{\mathfrak{p}_{\lambda}}\colon
A_{\mathfrak{p}_{\lambda}}\to A^{\prime}_{\mathfrak{p}_{\lambda}}$ is
surjective map of regular local rings of dimension $c$ and hence an
isomorphism.

### Freeness criterion

Fix $(A,\lambda)$ in $\operatorname{C}_{\mathcal{O}}$ and a finitely generated
$A$-module $M$. For any $A$-module $X$ one has a map

$\operatorname{Ext}^{c}_{A}(\mathcal{O},X)\otimes_{\mathcal{O}}(M/\mathfrak{p}_{\lambda}M)\cong\operatorname{Ext}^{c}_{A}(\mathcal{O},X)\otimes_{A}M\longrightarrow\operatorname{Ext}^{c}_{A}(\mathcal{O},X\otimes_{A}M)$

where the one on the right is a Künneth map. This is functorial in $X$, and
one gets the commutative diagram below:

${{\operatorname{Ext}^{c}_{A}(\mathcal{O},A)}^{\operatorname{tf}}\otimes_{\mathcal{O}}{(M/\mathfrak{p}_{\lambda}M)}^{\operatorname{tf}}}$${{\operatorname{Ext}^{c}_{A}(\mathcal{O},\mathcal{O})}^{\operatorname{tf}}\otimes_{\mathcal{O}}{(M/\mathfrak{p}_{\lambda}M)}^{\operatorname{tf}}}$${{\operatorname{Ext}^{c}_{A}(\mathcal{O},M)}^{\operatorname{tf}}}$${{\operatorname{Ext}^{c}_{A}(\mathcal{O},M/\mathfrak{p}_{\lambda}M)}^{\operatorname{tf}}}$

The horizontal maps are one-to-one, by Lemma 2.5. Moreover, the one on right
is an isomorphism, as can be verified easily. It follows that the map on the
left is one-to-one. This justifies the following result.

######  Lemma 2.18.

The diagram above induces a natural surjective map of $\mathcal{O}$-modules

$a_{\lambda}(M)\colon\Psi_{\lambda}(A)^{\mu}\twoheadrightarrow\Psi_{\lambda}(M)\,,\quad\text{where
$\mu\colonequals\operatorname{rank}_{\lambda}(M)$.}$

In particular there is an equality

$\operatorname{length}_{\mathcal{O}}\Psi_{\lambda}(M)=\mu\cdot\operatorname{length}_{\mathcal{O}}\Psi_{\lambda}(A)-\operatorname{length}_{\mathcal{O}}\operatorname{Ker}(a_{\lambda}(M))\,.\qed$

When $A$ is Gorenstein and $M$ is maximal Cohen-Macaulay,
$\operatorname{Ker}(a_{\lambda}(M))$ can be interpretted as a“stable”
cohomology module of the pair $(A,M)$. This identification leads to the
criterion below for detecting free summands of $M$; see
[Iyengar/Khare/Manning:2022a, Theorem 9.2].

######  Theorem 2.19.

With notation as above, when $A$ Gorenstein and $M$ is maximal Cohen–Macaulay,
$\operatorname{length}_{\mathcal{O}}\Psi_{\lambda}(M)=\mu\cdot\operatorname{length}_{\mathcal{O}}\Psi_{\lambda}(A)$
if and only if

$M\cong A^{\mu}\oplus W\quad\text{and $W_{\mathfrak{p}_{\lambda}}=0$,}$

as $A$-modules. In this case, when $\mu\neq 0$ the $A$-module $M$ is faithful.
∎

### Isomorphism criteria

The preceding results leads to a criterion for detecting isomorphisms between
rings, in terms of congruence modules.

######  Proposition 2.20.

Let $\varphi\colon A\to B$ be a surjective map of complete local
$\mathcal{O}$-algebra. Assume there exists an augmentation $\lambda\colon
B\to\mathcal{O}$ such that $(A,\lambda\varphi)$ and $(B,\lambda)$ are in
$\operatorname{C}_{\mathcal{O}}(c)$ for some $c\geq 0$, and either of the
following conditions hold:

  1. (1)

The ring $A$ is Gorenstein, $B$ is Cohen-Macaulay, and

$\operatorname{length}_{\mathcal{O}}\Psi_{\lambda\varphi}(A)=\operatorname{length}_{\mathcal{O}}\Psi_{\lambda}(B)\,;$

  2. (2)

The ring $B$ is complete intersection and

$\operatorname{length}_{\mathcal{O}}\Phi_{\lambda\varphi}(A)=\operatorname{length}_{\mathcal{O}}\Phi_{\lambda}(B)\,.$

Then the map $\varphi$ is an isomorphism.

###### Proof.

(1) The hypotheses imply that $\varphi$ is an isomorphism at $\lambda\varphi$
so $\operatorname{rank}_{\lambda\varphi}B=1$. Thus Theorem 2.19 implies that
$B$ is a faithful $A$-module, so $\operatorname{Ker}\varphi=(0)$.

(2) is an simple argument using the Jacobi-Zariski sequence arising from maps
$A\to B\to\mathcal{O}$ and Nakayama’s Lemma; see [Iyengar/Khare/Manning:2022a,
Lemma 5.10] for details. ∎

The isomorphism (2.16) is also a critical input in tracking the behavior of
congruence modules under deformations.

### Deformations

Fix $(A,\lambda)$ in $\operatorname{C}_{\mathcal{O}}(c)$ and elements
$\boldsymbol{f}\colonequals f_{1},\dots,f_{n}$ in $\mathfrak{p}_{\lambda}$
such that their residue classes in the $\mathcal{O}$-module
$\mathfrak{p}_{\lambda}/\mathfrak{p}_{\lambda}^{2}$ form a linearly
independent set. Set $\overline{A}\colonequals A/\boldsymbol{f}A$. The
augmentation $\lambda\colon A\to\mathcal{O}$ factors through $\overline{A}$ so
we an augmentation $\overline{\lambda}\colon\overline{A}\to\mathcal{O}$. The
hypotheses on $\boldsymbol{f}$ is equivalent to saying that the pair
$(\overline{A},\overline{\lambda})$ is in
$\operatorname{C}_{\mathcal{O}}(c-n)$; see [Iyengar/Khare/Manning:2022a, §8].
A straightforward computation yields an equality

(2.21)
$\operatorname{length}_{\mathcal{O}}\Phi_{\overline{\lambda}}(\overline{A})=\operatorname{length}_{\mathcal{O}}\Phi_{\lambda}(A)+\sum_{i}\operatorname{ord}(f_{i})\,,$

where $\operatorname{ord}(f_{i})$ is the order of $f_{i}$ in
$\mathfrak{p}_{\lambda}/\mathfrak{p}_{\lambda}^{2}$, defined by

$(\varpi^{\operatorname{ord}(f_{i})})\mathcal{O}=\\{\alpha(f_{i})|\alpha\in\operatorname{Hom}_{\mathcal{O}}(\mathfrak{p}_{\lambda}/\mathfrak{p}_{\lambda}^{2},\mathcal{O})\\};$

see [Iyengar/Khare/Manning:2022a, §8.5].

######  Theorem 2.22.

In the context above, let $M$ be a finitely generated $A$-module with
$0pt_{A}M\geq c+1$ and set $\overline{M}\colonequals M/\boldsymbol{f}M$. If
$\boldsymbol{f}$ is $M$-regular, then

$\operatorname{length}_{\mathcal{O}}\Psi_{\overline{\lambda}}(\overline{M})=\operatorname{length}_{\mathcal{O}}\Psi_{\lambda}(M)+(\operatorname{rank}_{\lambda}M)\sum_{i}\operatorname{ord}(f_{i})\,.$

###### Sketch of proof.

It is enough to consider the case when $n=1$. One first reduces to the case
when $f$ is not a zerodivisor on $A$ as well; this uses the invariance of
domain property for congruence modules, Theorem 2.17. See
[Iyengar/Khare/Manning:2022a, Proof of Theorem 8.2] for details. The essence
of the argument is captured $c=1$, so we start with a sketch of the proof in
that context. Since $f$ is in $\mathfrak{p}_{\lambda}$, and it is not a
zerodivisor on $A$ nor on $M$ one gets the isomorphism on the right:

$\operatorname{coker}(M\to\operatorname{Hom}(\mathfrak{p}_{\lambda},M))\xrightarrow{\cong\
}\operatorname{Ext}^{1}_{A}(\mathcal{O},M)\xrightarrow{\ \cong\
}\operatorname{Hom}_{A_{0}}(\mathcal{O},M_{0})$

The one on the left is from Lemma 2.14. It is straightforward to check that
the composite isomorphism is induced by the assignment

$\alpha\mapsto-\alpha(f)\mod fM\quad\text{for
$\alpha\in\operatorname{Hom}_{A}(\mathfrak{p}_{\lambda},M)$.}$

Consider the commutative diagram

${\operatorname{Hom}_{A}(\mathfrak{p}_{\lambda},M)\otimes_{A}\operatorname{Hom}_{A}(M,\mathcal{O})}$${\operatorname{Hom}_{\mathcal{O}}(\mathfrak{p}_{\lambda}/\mathfrak{p}_{\lambda}^{2},\mathcal{O})}$${\phantom{needspace}\overline{M}[\mathfrak{p}_{\lambda}]\otimes_{A}\operatorname{Hom}_{\overline{A}}(\overline{M},\mathcal{O})}$${\mathcal{O}}$$\scriptstyle{\alpha\mapsto\alpha(f)}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\beta\mapsto\beta(f)}$

The image of the vertical map on the right is precisely the order ideal of
$f$, that is to say, $(\varpi^{\operatorname{ord}(f)})$. Since
$\Psi_{\lambda}(M)$ and $\Psi_{\overline{\lambda}}(\overline{M})$ are cokernel
of the maps adjoint to the top and bottom row, respectively, the desired
equality follows.

To tackle the general case where $c\geq 2$, consider a commutative diagram
analogous to the one above:

${\operatorname{Ext}^{c}_{A}(\mathcal{O},M)\otimes_{A}\operatorname{Hom}_{A}(M,\mathcal{O})}$${\operatorname{Ext}^{c}_{A}(\mathcal{O},\mathcal{O})}$${\operatorname{F}^{c}_{\lambda}(\mathcal{O})}$${\operatorname{Ext}^{c-1}_{\overline{A}}(\mathcal{O},\overline{M})\otimes_{A}\operatorname{Hom}_{\overline{A}}(\overline{M},\mathcal{O})}$${\operatorname{Ext}^{c-1}_{\overline{A}}(\mathcal{O},\mathcal{O})}$${\operatorname{F}^{c-1}_{\overline{\lambda}}(\mathcal{O})}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

The isomorphism on the left is by [Bruns/Herzog:1998, Lemma 1.2.4]. What is
left is to identify the vertical map on the right, and this exploits the
isomorphism (2.16). ∎

### Wiles defect

Fix a pair $(A,\lambda)$ in $\operatorname{C}_{\mathcal{O}}$ and a finitely
generated $A$-module $M$. Since $A$ is regular at $\mathfrak{p}_{\lambda}$,
and in particular a domain, the $A_{\mathfrak{p}_{\lambda}}$-module
$M_{\mathfrak{p}_{\lambda}}$ has a rank. The _Wiles defect_ of $M$ at
$\lambda$ is the integer

$\delta_{\lambda}(M)\colonequals\operatorname{rank}_{\lambda}(M)\cdot\operatorname{length}_{\mathcal{O}}\Phi_{\lambda}(A)-\operatorname{length}_{\mathcal{O}}\Psi_{\lambda}(M)\,.$

In particular the Wiles defect of $A$ at $\lambda$ is

$\operatorname{length}_{\mathcal{O}}\Phi_{\lambda}(A)-\operatorname{length}_{\mathcal{O}}\Psi_{\lambda}(A)\,.$

We refer to [Iyengar/Khare/Manning:2022a, Introduction] for a discussion on
precedents to this definition.

Theorem 2.22 and (2.21) give the following result, which is
[Iyengar/Khare/Manning:2022a, Theorem 8.2]:

######  Theorem 2.23.

One has $\delta_{\lambda}(M/\boldsymbol{f}M)=\delta_{\lambda}(M)$ for
$M,\boldsymbol{f}$ as in Theorem 2.22. ∎

Also, Theorem 2.17 implies the following (which is
[Iyengar/Khare/Manning:2022a, Theorem 7.4]):

######  Lemma 2.24.

If $\varphi\colon(A^{\prime},\lambda^{\prime})\to(A,\lambda)$ is a surjective
map in $\operatorname{C}_{\mathcal{O}}(c)$, then

$\delta_{\lambda^{\prime}}(M)\geq\delta_{\lambda}(M)$

with equality if and only if
$\Phi_{\lambda^{\prime}}(A^{\prime})\cong\Phi_{\lambda}(A)$ holds. ∎

With $a_{\lambda}(M)$ as in (2.25), one gets a “defect formula”:

(2.25)
$\delta_{\lambda}(M)=\operatorname{rank}_{\lambda}(M)\cdot\delta_{\lambda}(A)+\operatorname{length}_{\mathcal{O}}\operatorname{Ker}(a_{\lambda}(M))\,.$

In particular $\delta_{\lambda}(M)\geq 0$ for all $M$ if and only if
$\delta_{\lambda}(A)\geq 0$.

######  Theorem 2.26.

When $(A,\lambda)\in\operatorname{C}_{\mathcal{O}}(c)$ with $0ptA\geq c+1$ one
has $\delta_{\lambda}(A)\geq 0$, and equality holds if and only if $A$ is
complete intersection.

In [Iyengar/Khare/Manning:2022a] this result was proved by reduction to the
case $c=0$, using Theorem 2.22. Here is an alternative argument, under the
slightly more restrictive case where $A$ is Cohen-Macaulay (so $\dim A=c+1$),
that argues by “going up” to a regular ring.

###### Proof.

First we verify that $\delta_{\lambda}(A)=0$ when
$A\in\operatorname{C}_{\mathcal{O}}(c)$ is complete intersection, that is to
say, isomorphic to

$\mathcal{O}[\\![{t_{1},\dots,t_{n}}]\\!]/(f_{1},\dots,f_{m})$

for some regular sequence $\boldsymbol{f}\colonequals f_{1},\dots,f_{m}$ in
$(\boldsymbol{t})$. Since $A$ is in $\operatorname{C}_{\mathcal{O}}(c)$ it
follows that $n-m=c$ and that $\boldsymbol{f}$ satisfies the hypothesis of
Theorem 2.22, so we get the first equality below

$\delta_{\lambda}(A)=\delta_{\lambda\varepsilon}(\mathcal{O}[\\![{\boldsymbol{t}}]\\!])=0\,.$

The second equality is by Theorem 2.6. This is as desired.

Next we verify that when $A$ is Cohen-Macaulay $\delta_{\lambda}(A)\geq 0$,
and that if equality holds $A$ is complete intersection. Since $\dim A=c+1$
one can find a surjection $\varepsilon\colon C\to A$ where $C$ is a complete
intersection in $\operatorname{C}_{\mathcal{O}}(c)$ and $\varepsilon$ induces
an isomorphism $\Phi_{\lambda\varepsilon}(C)\cong\Phi_{\lambda}(A)$; see
[Iyengar/Khare/Manning:2022a, Theorem 5.6]. Thus

$\displaystyle\operatorname{length}_{\mathcal{O}}\Psi_{\lambda\varepsilon}(C)$
$\displaystyle=\operatorname{length}_{\mathcal{O}}\Phi_{\lambda\varepsilon}(C)$
$\displaystyle=\operatorname{length}_{\mathcal{O}}\Phi_{\lambda}(A)$
$\displaystyle=\operatorname{length}_{\mathcal{O}}\Psi_{\lambda}(A)\,,$

where the first equality holds because $C$ is complete intersection; the
second is by the invariance of domain property 2.17, and the last one is the
hypothesis $\delta_{\lambda}(A)=0$. Thus Proposition 2.20 yields that
$\varepsilon$ is an isomorphism. ∎

Theorem 2.26 extends to modules, in the following sense; this is
[Iyengar/Khare/Manning:2022a, Theorem 9.6].

######  Theorem 2.27.

When $0pt_{A}M\geq c+1$ and $M_{\mathfrak{p}_{\lambda}}\neq 0$ one has
$\delta_{\lambda}(M)\geq 0$, and equality holds if and only if $A$ is complete
intersection and

$M\cong A^{\mu}\oplus W\quad\text{and $W_{\mathfrak{p}_{\lambda}}=0$.}$

###### Sketch of proof.

When $0ptA\geq c+1$ also holds, the inequality $\delta_{\lambda}(M)\geq 0$
follows from (2.25) and Theorem 2.26. Given this, the other part of the
statement follows from Theorems 2.26 and 2.19.

The argument in the general case is a reduction to the case where $A$ has
positive depth, and an induction on $c$. This uses the invariance of domain
property and the behavior of defects under deformations, stated below. ∎

###  2.1. $\Lambda$-structures

Motivated by number theory, we consider a setting where the algebra $A$ in
$\operatorname{C}_{\mathcal{O}}$ has additional structure, and give a variant
of the computation of change of congruence modules in §2 on going modulo
regular sequences.

Let $\boldsymbol{t}\colonequals t_{1},\dots,t_{c}$ be indeterminates,
$\Lambda_{c}\colonequals\mathcal{O}[\\![{\boldsymbol{t}}]\\!]$ the power-
series ring, and $\Lambda_{c}\to\mathcal{O}$ the natural augmentation. Fix
$(A,\lambda)$ in $\operatorname{C}_{\mathcal{O}}(c)$ equipped with a finite
flat map $\iota\colon\Lambda_{c}\to A$ of $\mathcal{O}$-algebras over
$\mathcal{O}$, so that composite map

$\Lambda_{c}\xrightarrow{\ \iota\ }A\xrightarrow{\ \lambda\ }\mathcal{O}\,,$

is the augmentation (that is, so that $\iota$ is a morphism in
$\operatorname{C}_{\mathcal{O}}(c)$). Since $\iota$ is flat the sequence
$\iota(t_{1}),\dots,\iota(t_{c})$ is $A$-regular. We assume also that the
residue classes of $\boldsymbol{t}$ in
$\mathfrak{p}_{\lambda}/\mathfrak{p}_{\lambda}^{2}$ form a linearly
independent set. Thus, setting $A_{0}\colonequals A/\boldsymbol{t}A$, the map
$\lambda$ factors through $A_{0}$, yielding an augmentation $\lambda_{0}\colon
A_{0}\to\mathcal{O}$, and $(A_{0},\lambda_{0})$ is in
$\operatorname{C}_{\mathcal{O}}(0)$. One gets a commutative diagram of
$\mathcal{O}$-algebras

(2.28)
${\Lambda_{c}}$${A}$${\mathcal{O}}$${A_{0}}$${\mathcal{O}}$$\scriptstyle{\varepsilon}$$\scriptstyle{\iota}$$\scriptstyle{\alpha}$$\scriptstyle{\lambda_{0}}$

all augmented to $\mathcal{O}$, via $\lambda_{0}$. We wish to track the change
in contangent modules and congruence modules along $\alpha$, and we do that by
using the diagram above, to reducing the problem to one about the map
$\varepsilon$, where it is trivial, and the map $\iota$, where it is easier to
handle.

In the rest of this discussion we write $\mathfrak{p}$ and $\mathfrak{p}_{0}$
instead of $\mathfrak{p}_{\lambda}$ and $\mathfrak{p}_{\lambda_{0}}$,
respectively.

We first discuss the change in cotangent modules in passing from $\lambda$ to
$\lambda_{0}$. Since $A_{0}$ is regular at $\lambda_{0}$, the
$\mathcal{O}$-module $\mathrm{D}_{2}(\mathcal{O}/A_{0},\mathcal{O})$, the
second André-Quillen homology of the map $A_{0}\to\mathcal{O}$, is torsion.
Moreover one has

$\mathrm{D}_{1}(A_{0}/A,\mathcal{O})\cong\mathrm{D}_{1}(\mathcal{O}/\Lambda_{c},\mathcal{O})\cong\mathfrak{m}/\mathfrak{m}^{2}\quad\text{where
$\mathfrak{m}\colonequals(\boldsymbol{t})\Lambda_{c}$.}$

In particular, this is a free $\mathcal{O}$-module, of rank $c$. Thus the
Jacobi-Zariski sequence arising from the maps $A\to A_{0}\to\mathcal{O}$
yields an exact sequence of $\mathcal{O}$-modules

$0\longrightarrow\mathfrak{m}/\mathfrak{m}^{2}\xrightarrow{\ \iota\
}\mathfrak{p}/\mathfrak{p}^{2}\longrightarrow\mathfrak{p}_{0}/\mathfrak{p}_{0}^{2}\longrightarrow
0\,.$

where we use $\iota$ also to denote the map induced on cotangent modules by
the ring map $\iota$. One gets an exact sequence

$0\longrightarrow\mathfrak{m}/\mathfrak{m}^{2}\xrightarrow{\ \iota\
}{(\mathfrak{p}/\mathfrak{p}^{2})}^{\operatorname{tf}}\longrightarrow\Phi_{\lambda_{0}}(A_{0})/\Phi_{\lambda}(A)\longrightarrow
0\,.$

From a number theory perspective, it is more natural to consider the exact
sequence obtained by applying
$(-)^{*}\colonequals\operatorname{Hom}_{\mathcal{O}}(-,\mathcal{O})$, namely
the sequence

(2.29) $0\longrightarrow(\mathfrak{p}/\mathfrak{p}^{2})^{*}\xrightarrow{\
\iota^{*}\
}(\mathfrak{m}/\mathfrak{m}^{2})^{*}\longrightarrow\operatorname{Ext}^{1}_{\mathcal{O}}(\Phi_{\lambda_{0}}(A_{0})/\Phi_{\lambda}(A),\mathcal{O})\longrightarrow
0\,.$

On the subcategory of torsion $\mathcal{O}$-modules one has an isomorphism of
functors

$\operatorname{Ext}^{1}_{\mathcal{O}}(-,\mathcal{O})\cong\operatorname{Hom}_{\mathcal{O}}(-,E/\mathcal{O})\,.$

where $E$ is the field of fractions of $\mathcal{O}$. Since the functor on the
right preserves lengths, the computations above yield

(2.30)
$\displaystyle\operatorname{length}_{\mathcal{O}}\Phi_{\lambda_{0}}(A_{0})-\operatorname{length}_{\mathcal{O}}\Phi_{\lambda}(A)$
$\displaystyle=\operatorname{length}_{\mathcal{O}}(\Phi_{\lambda_{0}}(A_{0})/\Phi_{\lambda}(A))$
$\displaystyle=\operatorname{length}_{\mathcal{O}}\operatorname{coker}(\iota^{*})$
$\displaystyle=\operatorname{length}_{\mathcal{O}}\operatorname{coker}(\wedge^{c}\iota^{*})\,.$

The second equality holds because $\iota^{*}$ is a map between free
$\mathcal{O}$-modules of rank $c$.

Now we move on to the congruence modules. Given commutative diagram 2.28 of
algebras over $\mathcal{O}$ and the functorial properties of
$\operatorname{F}^{-}_{-}(\mathcal{O})$ one gets a commutative diagram of
$\mathcal{O}$-modules

${\operatorname{F}^{c}_{\lambda\iota}(\mathcal{O})}$${\operatorname{F}^{c}_{\lambda}(\mathcal{O})}$${\mathcal{O}=\operatorname{F}^{0}_{\operatorname{id}}(\mathcal{O})}$${\operatorname{F}^{0}_{\lambda_{0}}(\mathcal{O})=\mathcal{O}}$$\scriptstyle{\cong}$

The isomorphism in the lower row is clear from the definitions; the vertical
isomorphism is by a direct computation. The identity map on $\mathcal{O}$ is a
canonical generator for
$\operatorname{F}^{0}_{B}(\mathcal{O})=\operatorname{Hom}_{B}(\mathcal{O},\mathcal{O})$,
for any $B$ in $\operatorname{C}_{\mathcal{O}}$; this is why we write
equalities in the last row. Using the commutative diagram above and the
functoriality of the map (2.16), one gets a commutative diagram

${\bigwedge^{c}(\mathfrak{p}/\mathfrak{p}^{2})^{*}}$${\operatorname{F}^{c}_{\lambda}(\mathcal{O})}$${\bigwedge^{c}(\mathfrak{m}/\mathfrak{m}^{2})^{*}}$${\operatorname{F}^{0}_{\lambda_{0}}(\mathcal{O})}$$\scriptstyle{\cong}$$\scriptstyle{\wedge^{c}\iota^{*}}$$\scriptstyle{\cong}$

Consider a finitely generated $A$-module $M$ such that $\boldsymbol{t}$ is
also regular on $M$ and $0pt_{A}M\geq c+1$. Setting $M_{0}\colonequals
M/(\boldsymbol{t})M$ and using the identifications above, one gets a
commutative diagram like so:

(2.31)
${\operatorname{Ext}^{c}_{A}(\mathcal{O},M)\otimes_{\mathcal{O}}\operatorname{Hom}_{A}(M,\mathcal{O})}$${\bigwedge^{c}(\mathfrak{p}/\mathfrak{p}^{2})^{*}}$${\operatorname{Ext}^{0}_{A_{0}}(\mathcal{O},M_{0})\otimes_{\mathcal{O}}\operatorname{Hom}_{A_{0}}(M_{0},\mathcal{O})}$${\bigwedge^{c}(\mathfrak{m}/\mathfrak{m}^{2})^{*}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\wedge^{c}\iota^{*}}$

All these lead to the following structural refinement of Theorem 2.22.

######  Theorem 2.32.

Viewing $\eta_{\lambda}(M)$ and $\eta_{\lambda_{0}}(M_{0})$ as submodules of
$\wedge^{c}(\mathfrak{p}/\mathfrak{p}^{2})^{*}$ and
$\wedge^{c}(\mathfrak{m}/\mathfrak{m}^{2})^{*}$, respectively, there is an
equality

$\eta_{\lambda_{0}}(M_{0})=(\wedge^{c}\iota^{*})(\eta_{\lambda}(M))\,.$

Moreover, with $\mu\colonequals\operatorname{rank}_{\lambda}M$ there are
equalities

$\displaystyle\operatorname{length}_{\mathcal{O}}\Psi_{\lambda_{0}}(M_{0})-\operatorname{length}_{\mathcal{O}}\Psi_{\lambda}(M)$
$\displaystyle=\mu\cdot\operatorname{length}_{\mathcal{O}}\operatorname{coker}(\wedge^{c}\iota^{*})$
$\displaystyle=\mu\cdot(\operatorname{length}_{\mathcal{O}}\Phi_{\lambda_{0}}(A_{0})-\operatorname{length}_{\mathcal{O}}\Phi_{\lambda}(A))\,.$

###### Proof.

The first part of the proposition is immediate from the commutative diagram
(2.31). The second part then follows, given also (2.30). ∎

##  3\. Zeta lines and congruence modules

We focus on number theory applications of the results in Section 2, notably
the exact sequence (2.29) and Theorem 2.32. The main result is Theorem 3.7. We
begin with Proposition 3.3, which is a simple consequence of the Poitou-Tate
exact sequence and is used to prove Proposition 3.10.

Let $F$ be a number field, $S$ a finite set of places of $F$, and $G_{F,S}$
the Galois group of $F_{S}/F$, the maximal extension of $F$ unramified outside
the places above $S$ in an algebraic closure of $F$. Fix a prime number $p$, a
finite extension $E/\mathbb{Q}_{p}$, and let $\mathcal{O}$ denote the ring of
integers of $E$. Let $A$ be a $\mathcal{O}$-module, which is finitely or
cofinitely generated, with an action of $G_{F,S}$. The Pontryagin dual and the
twisted Pontryagin dual of $A$, respectively, are the $G_{F,S}$-modules

$A^{\vee}\colonequals\operatorname{Hom}_{\mathcal{O}}(A,E/\mathcal{O})\quad\text{and}\quad
A^{\prime}=A^{\vee}(1)=\operatorname{Hom}_{\mathcal{O}}(A,E/\mathcal{O}(1))\,.$

A _Selmer datum_ for $S$ and $A$ is a collection
$\mathcal{L}=\\{\mathcal{L}_{v}\\}_{v}$, where $\mathcal{L}_{v}$ is an
$\mathcal{O}$-submodule of $H^{1}(G_{v},A)$ for each $v\in S$. The
corresponding _Selmer group_ is

$H^{1}_{\mathcal{L}}(F,A)\colonequals\operatorname{Ker}\big{(}H^{1}(G_{F,S},A)\longrightarrow\prod_{v\in
S}H^{1}(G_{v},A)/\mathcal{L}_{v}\big{)}\,.$

Local Tate-duality induces the perfect pairing

(3.1) $H^{1}(G_{v},A)\times H^{1}(G_{v},A^{\prime})\to
H^{2}(G_{v},E/\mathcal{O}(1))\cong E/\mathcal{O}.$

The _dual Selmer datum_ $\mathcal{L}^{\perp}$ (for $S$ and $A^{\prime}$) is
defined with $\mathcal{L}^{\perp}_{v}\subset H^{1}(G_{v},A^{\prime})$ the
annihilator of $\mathcal{L}_{v}$ under this pairing. The _dual_ Selmer group
of $A$ is $H^{1}_{\mathcal{L}^{\perp}}(F,A^{\prime})$.

For $i=1,2$, set
${}_{S}^{i}(F,A)\colonequals\operatorname{Ker}\big{(}H^{i}(G_{F,S},A)\to\prod_{v\in
S}H^{i}(G_{v},A)\big{)}$. The result below is standard; see
[Neukirch/Schmidt/Wingberg:2008, 8.7.9]. The argument is based on notes of
Boeckle.

######  Lemma 3.2.

One has an exact sequence

$0\to H^{1}_{\mathcal{L}}(F,A)\to H^{1}(G_{F,S},A)\to\prod_{v\in
S}\frac{H^{1}(G_{v},A)}{\mathcal{L}_{v}}\to
H^{1}_{\mathcal{L}^{\perp}}(F,A’)^{\vee}\to{}_{S}^{2}(F,A)\to 0\,.$

If $A$ is finite, then

$\frac{\\#H^{1}_{\mathcal{L}}(F,A)}{\\#H^{1}_{\mathcal{L}^{\perp}}(F,A^{\prime})}=\frac{\\#H^{0}(F,A)}{\\#H^{0}(F,A^{\prime})}\cdot\prod_{v\in
S}\frac{\\#\mathcal{L}_{v}}{\\#H^{0}(F_{v},A)}\,.\qed$

Given Selmer datum $\mathcal{L}$ and $\tilde{\mathcal{L}}$ for $S$ and $A$, we
write $\mathcal{L}\subseteq\tilde{\mathcal{L}}$ if
$\mathcal{L}_{v}\subseteq\tilde{\mathcal{L}}_{v}$ for all $v$.

######  Proposition 3.3.

Suppose that $A$ is compact or cocompact and that
$\mathcal{L}\subseteq\tilde{\mathcal{L}}$ are Selmer data. Then one has a
natural exact sequence

$0\longrightarrow\frac{H^{1}_{\tilde{\mathcal{L}}}(S,A)}{H^{1}_{\mathcal{L}}(S,A)}\longrightarrow\prod_{v\in
S}\frac{\tilde{\mathcal{L}}_{v}}{\mathcal{L}_{v}}\longrightarrow\bigg{(}\frac{H^{1}_{\mathcal{L}^{\perp}}(S,A^{\prime})}{H^{1}_{\tilde{\mathcal{L}}^{\perp}}(S,A^{\prime})}\bigg{)}^{\vee}\longrightarrow
0$

###### Proof.

From the definitions one gets that the following natural maps are injective:

$H^{1}_{\mathcal{L}}(S,A)\longrightarrow
H^{1}_{\tilde{\mathcal{L}}}(S,A)\quad\text{and}\quad
H^{1}_{\tilde{\mathcal{L}}^{\perp}}(S,A^{\prime})\longrightarrow
H^{1}_{\mathcal{L}^{\perp}}(S,A^{\prime})\,.$

Setting
$K\colonequals\operatorname{Ker}(H^{1}_{\mathcal{L}^{\perp}}(F,A^{\prime})^{\vee}\to{}_{S}^{2}(F,A))$
and similarly $\widetilde{K}$ with $\tilde{\mathcal{L}}$ in place of
$\mathcal{L}$, the naturality of the exact sequence in Lemma 3.2 yields the
commutative diagram

${0}$${\displaystyle{\frac{H^{1}(G_{F,S},A)}{H^{1}_{\mathcal{L}}(F,A)}}}$${\prod_{v\in
S}\displaystyle\frac{H^{1}(G_{v},A)}{\mathcal{L}_{v}}}$${K}$${0}$${0}$${\displaystyle{\frac{H^{1}(G_{F,S},A)}{H^{1}_{\tilde{\mathcal{L}}}(F,A)}}}$${\prod_{v\in
S}\displaystyle\frac{H^{1}(G_{v},A)}{\tilde{\mathcal{L}}_{v}}}$${\widetilde{K}}$${0}$

The Snake Lemma yields the exact sequence

$0\longrightarrow\displaystyle\frac{H^{1}_{\tilde{\mathcal{L}}}(F,A)}{H^{1}_{\mathcal{L}}(F,A)}\longrightarrow\prod_{v\in
S}\displaystyle\frac{\tilde{\mathcal{L}}_{v}}{\mathcal{L}_{v}}\longrightarrow
K\longrightarrow\widetilde{K}\longrightarrow 0\,.$

By applying the Snake Lemma to the commutative diagram

${0}$${K}$${H^{1}_{\tilde{\mathcal{L}}^{\perp}}(F,A^{\prime})^{\vee}}$${{}_{S}^{2}(F,A)}$${0}$${0}$${\widetilde{K}}$${H^{1}_{\mathcal{L}^{\perp}}(F,A^{\prime})^{\vee}}$${{}_{S}^{2}(F,A)}$${0}$

and using Pontryagin duality, we obtain isomorphisms

$\displaystyle\operatorname{Ker}(K\to\widetilde{K})$
$\displaystyle\cong\operatorname{Ker}(H^{1}_{\mathcal{L}^{\perp}}(F,A^{\prime})^{\vee}\to
H^{1}_{\tilde{\mathcal{L}}^{\perp}}(F,A^{\prime})^{\vee})$
$\displaystyle\cong\bigg{(}\frac{H^{1}_{\mathcal{L}^{\perp}}(F,A^{\prime})}{H^{1}_{\tilde{\mathcal{L}}^{\perp}}(F,A^{\prime})}\bigg{)}^{\vee},$

concluding the proof of the proposition. ∎

### Nearly ordinary Hilbert modular forms

We specialize to a situation which corresponds to that of [Urban:2021]. Thus
$F$ is a totally real number field, $p$ an odd prime and $f$ a nearly ordinary
at $p$, holomorphic and cohomological cuspidal Hilbert modular newform for
$\operatorname{GL}_{2}(\mathbb{A}_{F})$. Our coefficients $A$ as in the
section above arise from the adjoint $\operatorname{ad}\rho_{f}$ (and
$\operatorname{ad}(\rho_{f}\otimes_{\mathcal{O}}E/\mathcal{O})$) of (an)
integral Galois representation

$\rho_{f}\colon G_{F,S}\to\operatorname{GL}_{2}(\mathcal{O})$

associated to $f$ and an embedding of
$\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}_{p}}$. We assume the
residual representation $\overline{\rho}_{f}\colon
G_{F,S}\to\operatorname{GL}_{2}(k)$ is irreducible (which implies that it is
absolutely irreducible as $\rho_{f}$ is totally odd and $p>2$), and thus there
is a unique integral representation $\rho_{f}$ associated to $f$ (by a well-
known result of Carayol).

We apply the results of the previous section to study the Galois cohomology of
the adjoint representation of $\rho_{f}$ with several local conditions at
places dividing $p$. For each $v|p$, we fix a decomposition subgroup $D_{v}$
at $v$. We call $I_{v}\subset D_{v}$ the inertia subgroup, $F_{v}$ the
completion of $F$ at $v$, and $d_{v}$ its degree over $\mathbb{Q}_{p}$. By
nearly ordinarity of $f$, for each $v|p$ there exists
$g_{v}\in\operatorname{GL}_{2}(\mathcal{O})$ such that the restriction to the
decomposition subgroup $D_{v}$ at $v$ of $g_{v}\rho_{f}g_{v}^{-1}$ is upper
triangular. We also assume that it is $v$-distinguished (that is, the
characters appearing on the diagonal are distinct modulo the uniformizer
$\varpi$ of $\mathcal{O}$) and indecomposable. We then consider the following
summands:

$\mathcal{F}^{+}_{v}\colonequals\\{g_{v}\begin{pmatrix}0&*\\\
0&0\end{pmatrix}g_{v}^{-1}\\}\subset\mathcal{F}^{0}_{v}\colonequals\\{g_{v}\begin{pmatrix}*&*\\\
0&*\end{pmatrix}g_{v}^{-1}\\}\subset\operatorname{ad}\rho_{f}$

We denote $Gr^{0}_{v}\colonequals\mathcal{F}^{0}_{v}/\mathcal{F}^{+}_{v}$ and
fix an isomorphism of $D_{v}$-modules $Gr^{0}_{v}\cong\mathcal{O}$. Let $B$ be
a $\mathcal{O}$-module. The _ordinary_ condition
$H^{1}_{\operatorname{ord}}(F_{v},\operatorname{ad}\rho_{f}\otimes B)$ at $v$
is given by the image of $H^{1}(F_{v},\mathcal{F}_{v}^{0}\otimes B)$ in
$H^{1}(F_{v},\operatorname{ad}\rho_{f}\otimes B)$, and (in the terminology of
Wiles [Wiles:1995]) the _Selmer_ condition
$H^{1}_{\mathrm{Sel}}(F_{v},\operatorname{ad}\rho_{f}\otimes B)$ at $v$ is
given by the image of

$\operatorname{Ker}(H^{1}(F_{v},\mathcal{F}^{+}_{v}\otimes B)\longrightarrow
H^{1}(I_{v},Gr^{0}_{v}\otimes B))$

in $H^{1}_{\operatorname{ord}}(F_{v},\operatorname{ad}\rho_{f}\otimes B)$.
Since the representation $\rho_{f}$ is $v$-distinguished, we get an exact
sequence:

$0\to H^{1}_{\mathrm{Sel}}(F_{v},\operatorname{ad}\rho_{f}\otimes B)\to
H^{1}_{\operatorname{ord}}(F_{v},\operatorname{ad}\rho_{f}\otimes B)\to
H^{1}(I_{v},B)^{\frac{D_{v}}{I_{v}}}\to 0$

The orthogonal of the finite _Selmer_ condition
$H^{1}_{\mathrm{Sel}^{\perp}}(F_{v},\operatorname{ad}\rho_{f}\otimes B(1))$
and of the ordinary condition
$H^{1}_{\operatorname{ord}^{\perp}}(F_{v},\operatorname{ad}\rho_{f}\otimes
B(1))$ are respectively given by the images of
$H^{1}(F_{v},\mathcal{F}_{v}^{0}\otimes B(1))$ and of
$H^{1}(F_{v},\mathcal{F}^{+}_{v}\otimes B(1))$ in
$H^{1}(F_{v},\operatorname{ad}\rho_{f}\otimes B(1))$.

######  Remark 3.4.

If the action of $D_{v}$ on $\mathcal{F}^{+}_{v}$ is distinct from the
cyclotomic character, then the finite _Selmer_ condition is nothing else but
the finite Bloch-Kato condition. In that case one has

$\displaystyle H^{1}_{\mathrm{Sel}}(F_{v},\operatorname{ad}\rho_{f}\otimes B)$
$\displaystyle=H^{1}_{f}(F_{v},\operatorname{ad}\rho_{f}\otimes B)$
$\displaystyle
H^{1}_{\mathrm{Sel}^{\perp}}(F_{v},\operatorname{ad}\rho_{f}\otimes B(1))$
$\displaystyle=H^{1}_{f}(F_{v},\operatorname{ad}\rho_{f}\otimes B(1))\,.$

Next we interpret some higher congruence modules in terms of Galois cohomology
(local and global) and by applying Theorem 2.32 in the situation described
below.

### Congruence modules and Galois cohomology

We recall some ingredients of the set up of [Urban:2021]; most of the notation
is borrowed from it.

Let
$\kappa\colonequals(\sum_{\sigma}k_{\sigma}.\sigma,\sum_{\sigma}l_{\sigma}.\sigma,)\in\mathbb{Z}[\Sigma_{F}]^{2}$
be the weight of the cohomological Hilbert modular cusp form $f$. We have
$k_{\sigma}\geq 2$ for all $\sigma\in\Sigma_{F}$ and
$w=k_{\sigma}+2l_{\sigma}$ is independent of $\sigma$. For such a weight and a
$O_{F^{\prime}}$-algebra $S$ with $F^{\prime}$ the normal closure of $F$, we
consider the algebraic representation of $\operatorname{GL}_{2}(O_{F’})$:

$L(\kappa,S)\colonequals\bigotimes_{\sigma}\mathrm{Sym}^{k_{\sigma}-2}(S^{2})\otimes{\det}^{l_{\sigma}}\,.$

For each neat open compact subgroup
$K\subset\operatorname{GL}_{2}(\mathbb{A}_{f}\otimes F)$, this defines a local
system $L(\kappa,\mathbb{C})$ on the Hilbert modular variety

$X(K)\colonequals\operatorname{GL}_{2}(F)\backslash\operatorname{GL}_{2}(\mathbb{A}\otimes
F)/KK_{\infty}Z(F)$

where $Z$ stands for the center of $\operatorname{GL}_{2}$ and $K_{\infty}$ is
the maximal compact (modulo the center) subgroup of
$\operatorname{GL}_{2}(\mathbb{R}\otimes F)$.

Let $\mathfrak{n}\subset O_{F}$ be the tame conductor of $f$. It is a nonzero
integral ideal of $O_{F}$ prime to $p$. Let
$K^{p}_{11}(\mathfrak{n})\subset\operatorname{GL}_{2}(\widehat{\mathbb{Z}}^{p}\otimes
O_{F})$ be the subgroup of matrices which are upper unipotent modulo
$\mathfrak{n}$ and where we have written $\widehat{\mathbb{Z}}^{p}$ for the
prime-to-$p$ part of the profinite completion of $\mathbb{Z}$. We will assume
that $K_{11}^{p}(\mathfrak{n})$ is neat. Let $\omega$ be the central character
of the cuspidal representation attached to $f$. It is an idèle class character
of conductor dividing $\mathfrak{n}p^{\infty}$ and infinity type
$|\cdot|^{w}$.

For each positive integer $n$, we denote $K_{0}(p^{n})$ the subgroup of
$\operatorname{GL}_{2}(O_{F}\otimes\mathbb{Z}_{p})$ of matrices which are
upper triangular modulo $p^{n}$ and by $K_{1}(p^{n})$ its subgroup of those
such that the diagonal entries are congruent modulo $p^{n}$. We identify
$K_{0}(p^{n})/K_{1}(p^{n})$ with $(O_{F}/p^{n}O_{F})^{\times}$ via the map
$\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\mapsto a^{-1}d$.

Let $h_{\kappa}^{\operatorname{ord}}(\mathfrak{n}p^{n},\omega)$ be the nearly
ordinary Hecke algebra of level $K_{11}^{p}(\mathfrak{n})K_{1}(p^{n})$, weight
$\kappa$ and central character $\omega$. We then consider the universal nearly
ordinary Hecke algebra of weight $\kappa$ and tame level $K^{p}$ and action of
the center given by $\omega$.

$\mathbf{h}^{\operatorname{ord}}=\mathbf{h}^{\operatorname{ord}}_{\kappa}(\mathfrak{n})\colonequals\lim_{\underset{n}{\leftarrow}}h_{\kappa}^{\operatorname{ord}}(\mathfrak{n}p^{n})$

The Hecke ring $h_{\kappa}^{\operatorname{ord}}(\mathfrak{n}p^{n})$ has a
natural structure of $\mathcal{O}[(O_{F}/p^{n}O_{F})^{\times}]$-algebra which
induces a structure of $\Lambda_{F}$-algebra on
$\mathbf{h}^{\operatorname{ord}}$ with

$\Lambda_{F}\colonequals\mathcal{O}[\\![{O_{F,p}^{1}}]\\!]\cong\mathcal{O}[\\![{t_{1},\dots,t_{d}}]\\!]$

and $O_{F,p}^{1}\cong\mathbb{Z}_{p}^{d}$ the subgroup of $O_{F,p}^{\times}$ of
local units congruent to 1 modulo $p$. Let $\mathfrak{m}$ be the kernel of the
map $\Lambda_{F}\to\mathcal{O}$ corresponding to the trivial character of
$O_{F,p}^{1}$.

The Hecke eigensystem attached to our nearly ordinary Hilbert modular form $f$
gives us an homomorphism:‘1

$\lambda_{f}\colon\mathbf{h}^{\operatorname{ord}}\rightarrow
h^{\operatorname{ord}}_{\kappa}(\mathfrak{n}p^{r})\rightarrow\mathcal{O}.$

with $r$ the smallest integer so that $f$ is $K_{1}(p^{r})$-invariant.

We now denote by $\mathbb{T}^{\operatorname{ord}}$ (resp. $\mathbb{T}_{0}$)
the localisation of $\mathbf{h}^{\operatorname{ord}}$ (resp.
$h^{\operatorname{ord}}_{\kappa}(\mathfrak{n}p^{r})$ ) at its maximal ideal
$\mathfrak{m}_{f}$ containing $\ker\lambda_{f}$. It is known thanks to the
work of Hida that $\mathbb{T}^{\operatorname{ord}}$ is free of finite rank
over $\Lambda_{F}$. Moreover, we have a canonical isomorphism

$\mathbb{T}^{\operatorname{ord}}\otimes_{\Lambda_{F}}\mathcal{O}\cong\mathbb{T}_{0}\,.$

We now construct a $\mathbb{T}^{\operatorname{ord}}$-module which is free over
$\Lambda_{F}$ and interpolates the nearly ordinary cohomology of the Hilbert
modular variety localized at the maximal ideal associated to $f$. For any
$p$-adically complete $\mathcal{O}$-algebra $A$ and $n\geq r$, let

$\mathcal{C}_{n}(\kappa,A)\colonequals
Ind_{K_{1}(p^{n})}^{K_{1}(p^{r})}L(\kappa,A)$

and write $\mathcal{C}(\kappa,A)$ for the direct limit of the
$\mathcal{C}_{n}(\kappa,A)$ for the obvious transition maps, and
$\mathcal{C}(\kappa,\mathcal{O})$ for the inverse limit of the
$\mathcal{C}(\kappa,\mathcal{O}/p^{m}\mathcal{O})$ as $m$ varies. It is
clearly a $\Lambda_{F}[K_{1}(p^{r})]$-module.

Let $\eta_{\lambda_{f}}(M)$ be the congruence ideal of $M$ with respect to
$\lambda_{f}$ introduced in 2.10.

######  Proposition 3.5.

Assume the image of $\bar{\rho}_{f}$ is not solvable. Then for any $\epsilon$
in $\\{\pm 1\\}^{\Sigma_{F}}$, the $\mathbb{T}^{\operatorname{ord}}$-module

$\mathbf{M}^{\epsilon}\colonequals
H^{d}(X(K_{11}(\mathfrak{n}p^{r}),\mathcal{C}(\kappa,\mathcal{O}))_{\mathfrak{m}_{f}}^{\epsilon}$

is free of finite rank over $\Lambda_{F}$, and

$\mathbf{M}^{\epsilon}/\mathfrak{m}\mathbf{M}^{\epsilon}=\mathbf{M}^{\epsilon}_{0}\colonequals
H^{d}(X(K_{1}(\mathfrak{n}p^{r}),\mathcal{L}(\kappa,\mathcal{O}))_{\mathfrak{m}_{f}}^{\epsilon}\,.$

Moreover $\eta_{\lambda_{f}}(\mathbf{M}^{\epsilon}_{0})=(\xi^{\epsilon}_{f})$,
where

$\xi_{f}^{\epsilon}\colonequals\frac{\Gamma(\operatorname{ad}\rho_{f},1)L^{S_{f}}(\operatorname{ad}\rho_{f},1)}{\Omega_{f}^{\epsilon}\Omega_{f}^{-\epsilon}}$

where $S_{f}$ is the set of finite places where $\rho_{f}$ is ramified and
$(\Omega_{f}^{\epsilon})_{\epsilon\in\\{\pm 1\\}^{\Sigma_{F}}}$ are the
canonical complex periods attached to the Hilbert modular form $f$ in111In
[Dimitrov:2009], the $\epsilon$-parts of the cohomology and the periods are
indexed by the subsets $J\subset\Sigma_{F}$ corresponding to the character
$\epsilon_{J}$ [Dimitrov:2009, §7.1].

###### Proof.

This is a classical exercise in Hida theory since the localization at
$\mathfrak{m}_{f}$ capture a direct factor of the nearly ordinary part of the
cohomology. The fact that the module is free over $\Lambda_{F}$ follows from a
control theorem and the vanishing Theorem 7.1.1 of Caraiani and Tamiozzo
[Caraiani/Tamiozzo:2023]. The last part of the proposition follows from a
computation of Dimitrov in the sections 7.2 and 7.3 of [Dimitrov:2009],and in
particular its equations (50) and (51). ∎

We use the inclusion $\eta_{\lambda_{f}}(\mathbf{M}^{\epsilon})\subset
F^{d}_{\lambda_{f}}(\mathcal{O})=\bigwedge_{\mathcal{O}}^{d}(\mathfrak{p}/\mathfrak{p}^{2})^{*}$
to define the zeta $\mathcal{O}$-module associated to $f$.

### Construction of zeta lines

Given an $\mathcal{O}$-module $A$ we set

$A^{*}\colonequals\operatorname{Hom}_{\mathcal{O}}(A,\mathcal{O})\,.$

Let $R^{\operatorname{ord}}$ (resp. $R_{0}$) be the universal deformation ring
of $\bar{\rho}_{f}$ with fixed determinant equal to $\det\rho_{f}$ and with
nearly ordinary conditions (resp. with ordinary condition of weight $\kappa$)
at places dividing $p$ and the unramified condition at finite places away from
those dividing $\mathfrak{n}p\infty$.

We have a canonical surjective map
$R^{\operatorname{ord}}\to\mathbb{T}^{\operatorname{ord}}$. Set

$\mathfrak{p}\colonequals\operatorname{Ker}(\mathbb{T}^{\operatorname{ord}}\to\mathcal{O})\quad\text{and}\quad\mathfrak{p}_{R}\colonequals\operatorname{Ker}(R^{\operatorname{ord}}\to\mathcal{O})\,,$

and consider the natural surjection

$H^{1}_{\mathrm{full},\operatorname{ord}}(F,\operatorname{ad}\rho_{f}\otimes
E/\mathcal{O})^{\vee}\cong\mathfrak{p}_{R}/\mathfrak{p}_{R}^{2}\twoheadrightarrow\mathfrak{p}/\mathfrak{p}^{2}\,.$

The subscript $\mathrm{full}$ means that no local conditions are required at
places dividing $\mathfrak{n}$. The isomorphism is standard; see, for example,
[Urban:2021, Lemma 3.3]. This induces maps

$F^{d}_{\lambda_{f}}(\mathcal{O})=\bigwedge_{\mathcal{O}}^{d}(\mathfrak{p}/\mathfrak{p}^{2})^{*}\longrightarrow\bigwedge_{\mathcal{O}}^{d}(\mathfrak{p}_{R}/\mathfrak{p}_{R}^{2})^{*}\cong\bigwedge^{d}_{\mathcal{O}}H^{1}_{\mathrm{full},\operatorname{ord}}(F,\operatorname{ad}(\rho_{f}))\,.$

######  Definition 3.6.

The image of the submodule
$\eta_{\lambda_{f}}(\mathbf{M}^{\epsilon})\subset\bigwedge^{d}(\mathfrak{p}/\mathfrak{p}^{2})^{*}$
under the composition of maps above is a cyclic $\mathcal{O}$-submodule; we
write it as $(z_{f}^{\epsilon})$ and call it the zeta line. Thus
$z_{f}^{\epsilon}$ is well-defined only up to multiplication by a unit in
$\mathcal{O}$.

For each $v$, the quotient map $\mathcal{F}^{0}_{v}\to Gr^{0}_{v}$ induces the
map

$H^{1}(F_{v},\mathcal{F}^{0}_{v})\longrightarrow H^{1}(F_{v},Gr^{0}_{v})\cong
H^{1}(I_{v},Gr^{0}_{v})^{D_{v}/I_{v}}\cong\mathcal{O}^{d_{v}}\,.$

Since $\sum_{v|p}d_{v}=[F:\mathbb{Q}]=d$ these induce the isomorphism

$\bigwedge^{d}(\prod_{v|p}H^{1}(F_{v},\mathcal{F}^{0}_{v}))\xrightarrow{\
\cong\
}\bigotimes_{v|p}\bigwedge_{\mathcal{O}}^{d_{v}}H^{1}(I_{v},Gr^{0}_{v})^{D_{v}/I_{v}}\cong\mathcal{O}\,.$

Precomposing this with the $d$th exterior power of the restriction map

$\mathrm{res}_{p}\colon
H^{1}_{\mathrm{full},\operatorname{ord}}(F,\operatorname{ad}\rho_{f})\to\prod_{v|p}H^{1}(F_{v},\mathcal{F}^{0}_{v})\,.$

yields the map

$\bigwedge^{d}\mathrm{res}_{p}\colon\bigwedge^{d}H^{1}_{\mathrm{full},\operatorname{ord}}(F,\operatorname{ad}\rho_{f})\longrightarrow\bigotimes_{v|p}\bigwedge_{\mathcal{O}}^{d_{v}}H^{1}(I_{v},Gr^{0}_{v})^{D_{v}/I_{v}}\,.$

The following theorem is the main result of this section.

######  Theorem 3.7.

Assume that the residual representation $\bar{\rho}_{f}$ has non solvable
image and choose $\epsilon\in\\{\pm 1\\}^{\Sigma_{F}}$. Then

$(\bigwedge^{d}\mathrm{res}_{p})(z^{\epsilon}_{f})=(\xi_{f}^{\epsilon})$

where as before

$\xi_{f}^{\epsilon}\colonequals\frac{\Gamma(\operatorname{ad}\rho_{f},1)L^{S_{f}}(\operatorname{ad}\rho_{f},1)}{\Omega_{f}^{\epsilon}\Omega_{f}^{-\epsilon}}.$

###### Proof.

Let $R_{0}$ be the universal deformation ring of $\bar{\rho}_{f}$ with fixed
determinant equal to $\det\rho_{f}$ and with ordinary condition of weight
$\kappa$ at places dividing $p$ and the unramified condition at finite places
away from those dividing $\mathfrak{n}p\infty$, and $R_{0}\to\mathbb{T}_{0}$
the canonical surjection. The restriction of the universal deformations to the
decomposition subgroups at places dividing $p$ gives an homomorphism
$\Lambda_{F}\to R^{\operatorname{ord}}$ making the map
$R^{\operatorname{ord}}\to\mathbb{T}^{\operatorname{ord}}$ an
$\Lambda_{F}$-algebra homomorphim and a canonical isomorphism
$R^{\operatorname{ord}}\otimes_{\Lambda_{F}}\mathcal{O}\cong
R^{\operatorname{ord}}_{0}$. Setting

$\mathfrak{p}_{0}\colonequals\operatorname{Ker}(\mathbb{T}^{\operatorname{ord}}_{0}\to\mathcal{O})=\mathfrak{p}/\mathfrak{m}\quad\text{and}\quad\mathfrak{p}_{R,0}\colonequals\operatorname{Ker}(R_{0}^{\operatorname{ord}}\to\mathcal{O})=\mathfrak{p}_{R}/\mathfrak{m}$

the isomorphism and surjective maps induces the following commutative diagram
of Kähler differentials (see §2.1).

${0}$${\mathfrak{m}/\mathfrak{m}^{2}}$${\mathfrak{p}/\mathfrak{p}^{2}}$${\mathfrak{p}_{0}/\mathfrak{p}_{0}^{2}}$${0}$${0}$${\mathfrak{m}/\mathfrak{m}^{2}}$${\mathfrak{p}_{R}/\mathfrak{p}_{R}^{2}}$${\mathfrak{p}_{R,0}/\mathfrak{p}_{R,0}^{2}}$${0}$${0}$${\bigoplus_{v|p}(H^{1}(I_{v},\frac{E}{\mathcal{O}})^{\frac{D_{v}}{I_{v}}})^{\vee}}$${H^{1}_{\mathrm{full},\operatorname{ord}}(F,\operatorname{ad}\rho_{f}\otimes\frac{E}{\mathcal{O}})^{\vee}}$${H^{1}_{\mathrm{full},\mathrm{Sel}}(F,\operatorname{ad}\rho_{f}\otimes\frac{E}{\mathcal{O}})^{\vee}}$${0}$

Here
$H^{1}_{\mathrm{full},\mathrm{Sel}}(F,\operatorname{ad}\rho_{f}\otimes\frac{E}{\mathcal{O}})$
means no condition at primes dividing $\mathfrak{n}$, and the Selmer condition
at places of $F$ above $p$. The exactness on the left in the top row follows
from Hida’s theorem that $\mathbb{T}^{\operatorname{ord}}$ is unramified over
the weight space $\Lambda_{F}$ at the augmentation $\lambda_{f}$ arising from
the holomorphic cohomological newform $f$. The vertical arrows are surjective
and that the $\mathcal{O}$-module
$\mathfrak{p}_{0}/\mathfrak{p}_{0}^{2}\cong\Omega_{\mathbb{T}_{0}/\mathcal{O}}\otimes_{\lambda_{f}}\mathcal{O}$
is torsion, and therefore both $\mathfrak{m}/\mathfrak{m}^{2}$ and
$(\mathfrak{p}/\mathfrak{p}^{2})^{t\mathrm{f}}$ are free of rank $d$ over
$\mathcal{O}$. The diagram above yields the commutative diagram

${\bigwedge^{d}(\mathfrak{m}/\mathfrak{m}^{2})^{*}}$${\bigwedge^{d}(\mathfrak{p}/\mathfrak{p}^{2})^{*}}$${\bigwedge^{d}(\mathfrak{m}/\mathfrak{m}^{2})^{*}}$${\bigwedge^{d}(\mathfrak{p}_{R}/\mathfrak{p}_{R}^{2})^{*}}$${\bigotimes_{v|p}\bigwedge^{d_{v}}H^{1}(I_{v},{\mathcal{O}})^{\frac{D_{v}}{I_{v}}}}$${\bigwedge^{d}H^{1}_{\mathrm{full},\operatorname{ord}}(F,\operatorname{ad}\rho_{f})}$

Given this diagram, Proposition 3.5, and Theorem 2.32, it follows that the
image of $z_{f}^{\epsilon}$ under the local restriction map at $p$ is
$\xi_{f}^{\epsilon}$, fixing an isomorphism with $\mathcal{O}$. ∎

######  Remark 3.8.

Our hypothesis here are less restrictive than in [Urban:2021]. Moreover, the
method used here allows us to bypass the use of local complete intersection
results on the corresponding Hecke ring used in [Urban:2021], and therefore to
remove some hypotheses; in particular, it does not require us to have a
$R=\mathbb{T}$ theorem.

As explained in [Urban:2021], $(z_{f}^{\epsilon})$ is the bottom class of an
Euler system of rank $d$. It would be interesting to extend our new method to
construct the other classes using higher congruence modules for the base
change of $f$ to abelian extensions of $F$. We have shown that
$(z_{f}^{\epsilon})$ is related to the $L$-value $\xi_{f}$. If we could extend
our method we would be able to construct the $p$-adic $L$-function
$L^{S_{f}}_{p}(\operatorname{ad}\rho_{f},s)$.

######  Remark 3.9.

By (2.29), cokernel of
$(\mathfrak{p}/\mathfrak{p}^{2})^{*}\to(\mathfrak{m}/\mathfrak{m}^{2})^{*}$
has length equal to

$\operatorname{length}_{\mathcal{O}}(\Phi_{\lambda_{f}}(\mathbb{T}))-\operatorname{length}_{\mathcal{O}}(\Phi_{\lambda_{f}}(\mathbb{T}^{\operatorname{ord}}))\,.$

This results in “factorizing” classical Selmer groups $\Phi_{\lambda}(R)$ into
a part coming from $\Phi_{\lambda}(R^{\operatorname{ord}})$ and a part coming
from the cokernel of
$(\mathfrak{p}/\mathfrak{p}^{2})^{*}\to(\mathfrak{m}/\mathfrak{m}^{2})^{*}$.

### The co-torsion in Galois cohomology

In this section, we assume that $\rho_{f}$ is a minimal deformation of
$\bar{\rho}_{f}$ as in [Dimitrov:2009, §4.2]. We replace the rings
$R^{\operatorname{ord}}$ and $R_{0}$ by their minimal deformation analogues.
Then it is known that the maps
$R_{\mathrm{min}}^{\operatorname{ord}}\to\mathbb{T}^{\operatorname{ord}}$ and
$R_{0,\mathrm{min}}\to\mathbb{T}_{0}$ are isomorphisms of complete
intersection rings thanks to the works of Fujiwara [Fujiwara] and Dimitrov
[Dimitrov:2009] (note that we can and do replace the big image assumption of
Dimitrov for $\bar{\rho}_{f}$ by the much weaker one of
[Caraiani/Tamiozzo:2023] of being non solvable). In addition, we make the
following hypothesis justified by the Remark 3.4. For all $v|p$, we assume
that the action of $D_{v}$ on $\mathcal{F}^{+}_{v}$ is distinct from the
cyclotomic character. By our minimality assumption, the local conditions at
places away from $p$ are the finite Bloch-Kato conditions for all the Galois
cohomology groups considered in this section, so that we can now make the
following identifications:

  * •

$\operatorname{Hom}(\mathfrak{p}/\mathfrak{p}^{2},\mathcal{O})=H^{1}_{\operatorname{ord}}(F,\operatorname{ad}\rho_{f})$

  * •

$\operatorname{Hom}(\mathfrak{m}/\mathfrak{m}^{2},\mathcal{O})=H^{1}_{\operatorname{ord}/f}(F_{p},\operatorname{ad}\rho_{f})=\oplus_{v|p}H^{1}(I_{v},\mathcal{O})^{\frac{D_{v}}{I_{v}}}$

  * •

$H^{1}_{f}(F,\operatorname{ad}\rho_{f}\otimes_{\mathcal{O}}E/\mathcal{O})=\mathfrak{p}_{0}/\mathfrak{p}_{0}^{2}\colonequals\Phi_{\lambda_{f}}$
since $\mathbb{T}_{0}$ is finite over $\mathcal{O}$,

  * •

${\mathrm{cotors}}(H^{1}_{\mathcal{L}^{\operatorname{ord}}}(\mathbb{Q},\operatorname{ad}\rho_{f}\otimes_{\mathcal{O}}E/\mathcal{O}))=\operatorname{tors}(\mathfrak{p}/\mathfrak{p}^{2})=\Phi_{\lambda_{f}}^{\operatorname{ord}}\colonequals\Phi_{\lambda_{f}}(\mathbb{T}^{\operatorname{ord}})$

We abbreviate $\eta_{\lambda_{f}}(\mathbb{T}_{0})$ and
$\eta_{\lambda_{f}}(\mathbb{T}^{\operatorname{ord}})$ to $\eta_{\lambda_{f}}$
and $\eta_{\lambda_{f}}^{\operatorname{ord}}$, respectively, and view them as
ideals of $\mathcal{O}$. Since we have assumed that the Hecke rings are
complete intersection we have

$\eta_{\lambda_{f}}=\mathrm{Fitt}_{\mathcal{O}}(\Phi_{\lambda_{f}})\;\subset\;\eta_{\lambda_{f}}^{\operatorname{ord}}=\mathrm{Fitt}_{\mathcal{O}}(\Phi_{\lambda_{f}}^{\operatorname{ord}})$

Here is an interpretation of the invariants
$\Phi^{\operatorname{ord}}_{\lambda_{f}}$ and
$\Psi^{\operatorname{ord}}_{\lambda_{f}}$.

######  Proposition 3.10.

With the minimality assumptions as above we have

  1. (1)

the isomorphisms:

$\Phi_{\lambda_{f}}^{\operatorname{ord}}\cong{\mathrm{cotors}}(H^{1}_{{\operatorname{ord}}}(F,\operatorname{ad}\rho_{f}\otimes_{\mathcal{O}}E/\mathcal{O}))$

and an equality
$\operatorname{length}_{\mathcal{O}}(\Phi_{\lambda_{f}}^{\operatorname{ord}})=\operatorname{length}_{\mathcal{O}}(H^{1}_{{\operatorname{ord}}^{\perp}}(F,{\operatorname{ad}\rho_{f}}\otimes_{\mathcal{O}}E/\mathcal{O}(1)))$.

  2. (2)

an isomorphism:

$\Psi_{\lambda_{f}}^{\operatorname{ord}}\cong\frac{\bigwedge^{d}H^{1}_{\operatorname{ord}}(F,\operatorname{ad}\rho_{f})}{(z_{f}^{\epsilon})}\,.$

###### Proof.

From (2.29) we get the exact sequence

$0\longrightarrow(\mathfrak{p}/\mathfrak{p}^{2})^{*}\longrightarrow(\mathfrak{m}/\mathfrak{m}^{2})^{*}\longrightarrow\left(\frac{\Phi_{\lambda_{f}}}{\Phi^{\operatorname{ord}}_{\lambda_{f}}}\right)^{\vee}\longrightarrow
0\,.$

Comparing with the following Poitou-Tate duality exact sequence

$0\to H^{1}_{{\operatorname{ord}}}(F,\operatorname{ad}\rho_{f})\to
H^{1}_{\operatorname{ord}/f}(F_{p},\operatorname{ad}\rho_{f})\to\left(\frac{H^{1}_{f}(F,{\operatorname{ad}\rho_{f}}\otimes_{\mathcal{O}}E/\mathcal{O}(1))}{H^{1}_{{\operatorname{ord}}^{\perp}}(F,{\operatorname{ad}\rho_{f}}\otimes_{\mathcal{O}}E/\mathcal{O}(1))}\right)^{\vee}\to
0$

from Proposition 3.3 and the identifications recalled above, we get

$\frac{\Phi_{\lambda_{f}}}{\Phi^{\operatorname{ord}}_{\lambda_{f}}}\cong\frac{H^{1}_{f}(F,{\operatorname{ad}\rho_{f}}\otimes_{\mathcal{O}}E/\mathcal{O}(1))}{H^{1}_{{\operatorname{ord}}^{\perp}}(F,{\operatorname{ad}\rho_{f}}\otimes_{\mathcal{O}}E/\mathcal{O}(1))}$

On the other hand, we have the canonical isomorphisms

$\Phi_{\lambda_{f}}\cong H^{1}_{f}(F,\operatorname{ad}\rho_{f}\otimes
E/\mathcal{O})^{\vee}\,.$

It remains to note that by the balanced properties of Bloch-Kato Selmer groups
and Lemma 3.2, the $\mathcal{O}$-module above has finite length, equal to

$\operatorname{length}_{\mathcal{O}}(H^{1}_{f}(F,\operatorname{ad}\rho_{f}\otimes
E/\mathcal{O}(1)))\,.$

(2): From Theorem 2.32 one gets an exact sequence

$0\longrightarrow\bigwedge^{d}(\mathfrak{p}/\mathfrak{p}^{2})^{*}\longrightarrow\bigwedge^{d}(\mathfrak{m}/\mathfrak{m}^{2})^{*}\longrightarrow\frac{\Psi_{\lambda_{f}}}{\Psi^{\operatorname{ord}}_{\lambda_{f}}}\longrightarrow
0.$

and therefore

$0\longrightarrow\frac{\bigwedge^{d}(\mathfrak{p}/\mathfrak{p}^{2})^{*}}{(z_{f}^{\epsilon})}\longrightarrow\frac{\bigwedge^{d}(\mathfrak{m}/\mathfrak{m}^{2})^{*}}{(\bigwedge^{d}\mathrm{res}_{p})(z_{f}^{\epsilon})}\longrightarrow\frac{\Psi_{\lambda_{f}}}{\Psi^{\operatorname{ord}}_{\lambda_{f}}}\longrightarrow
0\,.$

The desired isomorphism follows since after identification of
$\wedge^{d}(\mathfrak{m}/\mathfrak{m}^{2})^{*}$ with $\mathcal{O}$, we have
$\bigwedge^{d}\mathrm{res}_{p}(z_{f}^{\epsilon})=(\xi_{f}^{\epsilon})$ which
is the same as the ideal $\eta_{\lambda_{f}}$ by construction. ∎

The next result is immediate from Proposition 3.10 and Theorem 2.6.

######  Corollary 3.11.

With our previous assumptions, the following are equivalent

  1. (1)

$\mathbb{T}^{\operatorname{ord}}$ is regular

  2. (2)

$H^{1}_{\operatorname{ord}^{\perp}}(F,\operatorname{ad}\bar{\rho}_{f}(1))=0$

  3. (3)

$(z_{f}^{\epsilon})=\bigwedge^{d}H^{1}_{\operatorname{ord}}(F,\operatorname{ad}\rho_{f})$.
∎

######  Remark 3.12.

This corollary is an analog of the well-known fact that the $p$-part of the
class group of $\mathbb{Q}(\zeta_{p})^{+}$ being trivial (Vandiver’s
conjecture is that this should always be the case) is equivalent to the group
of cyclotomic units having index prime to $p$ inside the global units of
$\mathbb{Q}(\zeta_{p})$.

We end this section with the the following proposition whose proof we owe to
Gebhard Boeckle. It should be compared to the statement in Proposition 3.10
for Selmer groups arising from motives associated to adjoints of modular
forms. Recall that the Leopoldt conjecture state that if $F$ is a number field
and $F^{{\rm ab},p}/F$ is the maximal abelian $p$ extension of $F$ unramified
outside the primes above $p$ and $\infty$, then the $\mathbb{Z}_{p}$-rank of
$\operatorname{Gal}(F^{{\rm ab},p}/F)$ is $r_{2}+1$.

######  Proposition 3.13.

Let $F$ be a number field and assume the Leopoldt Conjecture. One has an exact
sequence

$0\to\frac{\prod_{v\in S_{p}}\mu_{p^{\infty}}(F_{v})}{\mu_{p^{\infty}}(F)}\to
A\to B\to 0\,,$

where $A\colonequals\operatorname{tors}(\operatorname{Gal}(F^{{\rm ab},p}/F))$
and $B\colonequals H^{1}_{S_{p}-\rm
split}(F_{S_{p}}/F,\mathbb{Q}_{p}/\mathbb{Z}_{p}(1))$.

###### Proof.

Note that $A^{*}\colonequals{\rm
cotor}(H^{1}(F_{S_{p}}/F,\mathbb{Q}_{p}/\mathbb{Z}_{p}))$. Consider

$H^{1}(G_{F,S_{p}},\mathbb{Q}_{p}/\mathbb{Z}_{p})=\operatorname{Hom}(G_{F,S_{p}}^{ab},\mathbb{Q}_{p}/\mathbb{Z}_{p})=(G_{F,S_{p}}^{ab})^{*}=(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{r_{2}+1}\times
A^{*}\,.$

By [Neukirch/Schmidt/Wingberg:2008, 2.7.11] the kernel of the map

$H^{1}(G_{F,S_{p}},\mathbb{Q}_{p}/\mathbb{Z}_{p})\to
H^{2}(G_{F,S_{p}},\mathbb{Z}_{p})$

is the divisible part of $H^{1}(G_{F,S_{p}},\mathbb{Q}_{p}/\mathbb{Z}_{p})$,
that is to say, $(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{r_{2}+1}$, and its image is
the torsion subgroup of $H^{2}(G_{F,S_{p}},\mathbb{Z}_{p})$, that is to say,
all of $H^{2}(G_{F,S_{p}},\mathbb{Z}_{p})$, under the Leopoldt conjecture.

Further note that by Poitou-Tate one gets the isomorphism

$B={}^{1}(G_{F,S_{p}},\mathbb{Q}_{p}/\mathbb{Z}_{p}(1))\cong{}^{2}(G_{F,S_{p}},\mathbb{Z}_{p})^{*}\,.$

So there is a map $B^{*}\to A^{*}$ that is the inclusion
${}^{2}(G_{F,S_{p}},\mathbb{Z}_{p})\to H^{2}(G_{F,S_{p}},Z_{p})$. The cokernel
is isomorphic to the dual of the cokernel of
$\mu_{p^{\infty}}(F)\to\prod_{v\in S_{p}}\mu_{p^{\infty}}(F_{v})$, by Poitout-
Tate and local Tate duality. ∎

## Acknowledgements

This work is partly supported by National Science Foundation grants DMS-200985
(SBI) and DMS-2200390 (CBK), and by a Simons Fellowship (CBK). The second and
fourth authors thank TIFR, and the third author received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No. 884596). The paper has
its origins in conversations between the second and fourth author in New York
and Mumbai, during visits to Columbia University and Tata Institute, in which
we tried to understand more concretely the results of
[Iyengar/Khare/Manning:2022a] about congruence modules in the codimension 1
case. The second author would like to thank Gebhard Boeckle and Chris Skinner
for helpful discussions.

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