# On the cohomology of $\operatorname{\mathsf{GL}}(N)$ and adjoint Selmer
groups

J. Tilouine and E. Urban

(Date: August 3, 2021)

###### Abstract.

We prove -under certain conditions (local-global compatibility and vanishing
of integral cohomology), a generalization of a theorem of Galatius and
Venkatesh. We consider the case of $\operatorname{\mathsf{GL}}(N)$ over a CM
field and we relate the localization of penultimate non vanishing cuspidal
cohomology group for a locally symmetric space to the Selmer group of the Tate
dual of the adjoint representation. More precisely we construct a Hecke-
equivariant injection from the divisible group associated to the first
fundamental group of a derived deformation ring to the Selmer group of the
twisted dual adjoint motive with divisible coefficients and we identify its
cokernel as its first Tate-Shafarevich group. Actually, we also construct
similar maps for higher homotopy groups with values in exterior powers of
Selmer groups, although with less precise control on their kernel and
cokernel. We generalize this to Hida families as well.

The first author is partially supported by the grants PerCoLaTor ANR-14-CE25,
Coloss AAPG2019, and NSF grant DMS 1464106. The second author is partially
funded by the grant DMS 1407239 from the National Science Foundation.

##  1\. Introduction

Let $G$ be a connected reductive group over ${\mathbb{Q}}$. Let $Z$ be its
center and $A\subset Z$ be the maximal split torus of $Z$. For any
${\mathbb{Q}}$-group $H$, we decompose its group of adelic points as
$H(\operatorname{A})=H_{f}\times H_{\infty}$ into its finite and archimedean
parts. Let $K_{\infty}$ be a maximal connected compact subgroup of
$G_{\infty}$. Let $X_{G}=G_{\infty}/C_{\infty}$ where
$C_{\infty}=A_{\infty}K_{\infty}$ be the riemannian space associated to $G$.

Let $d=\dim\,X_{G}=\dim\,G_{\infty}-\dim\,C_{\infty}$; let
$\ell_{0}=\dim\,T_{G_{\infty}}-\dim\,T_{C_{\infty}}$ where $T_{G_{\infty}}$,
resp. $T_{C_{\infty}}$ denote a maximal torus of $G_{\infty}$ resp.
$C_{\infty}$. If $G$ admits a Shimura variety, $\ell_{0}=0$ and $d$ is even.
Let $q_{0}=(d-\ell_{0})/2$. For instance, for $F$ a number field and
$G=\operatorname{Res}_{F/{\mathbb{Q}}}\operatorname{\mathsf{GL}}(N)$ with
$[F\colon{\mathbb{Q}}]=r_{1}+2r_{2}$, we have

$\ell_{0}=(N-[{N\over 2}])r_{1}+Nr_{2}-1,\quad q_{0}=[{N^{2}\over
4}]r_{1}+{N(N-1)\over 2}r_{2},\quad d=2q_{0}+\ell_{0}={N(N+1)\over
2}r_{1}+N^{2}r_{2}-1$

Let $U$ be a compact open subgroup of $G_{f}$ and

$Y_{G}(U)=G({\mathbb{Q}})\backslash(X_{G}\times G_{f}/U)$

be the locally symmetric space of level $U$ associated to $G$. Let $\lambda$
be a dominant weight for $G$ and $V_{\lambda}$ the irreducible algebraic
representation of $G$ of highest weight $\lambda$. Let
$\mathrm{H}^{\bullet}(Y_{G}(U),V_{\lambda}({\mathbb{C}}))$ be the graded space
of cohomology of the corresponding local system over $Y_{G}(U)$. Let
$H^{\bullet}_{temp}(Y_{G}(U),V_{\lambda}({\mathbb{C}}))$ be its tempered part
(corresponding to tempered automorphic algebraic representations). Let
$\mathrm{H}^{temp}_{\bullet}=\mathrm{H}^{d-\bullet}_{temp}(Y_{G}(U),V_{\lambda}({\mathbb{C}}))$.
Let $h_{\mathbb{Z}}$ be the spherical ${\mathbb{Z}}$-Hecke algebra outside
$Ram(U)$ acting faithfully on $\mathrm{H}^{temp}_{\bullet}$.

Recall that Prasanna-Venkatesh [PV16, Sect.3] defined a rank $\ell_{0}$
abelian Lie subalgebra ${\mathfrak{a}}_{G}\subset Lie(G_{\mathbb{C}})$ and an
action of the graded ring $\bigwedge^{\bullet}{\mathfrak{a}}_{G}$ on
$\mathrm{H}^{temp}_{\bullet}$; they conjectured that there is a Hecke
equivariant isomorphism

$\mathrm{H}^{temp}_{\bullet}\cong\mathrm{H}_{q_{0}}^{temp}\otimes_{{\mathbb{C}}}\bigwedge^{\bullet}{\mathfrak{a}}_{G}$

or

$\mathrm{H}^{temp}_{\bullet}\cong\mathrm{H}_{q_{0}}^{temp}\otimes_{h_{\mathbb{Z}}\otimes{\mathbb{C}}}\bigwedge_{h_{\mathbb{Z}}\otimes{\mathbb{C}}}^{\bullet}h_{\mathbb{Z}}\otimes{\mathfrak{a}}_{G}$

Let $p$ be a prime. We fix embeddings
$\overline{{\mathbb{Q}}}\hookrightarrow\overline{{\mathbb{Q}}}_{p}$ and
$\overline{{\mathbb{Q}}}\hookrightarrow{\mathbb{C}}$. Let $K$ be a
sufficiently big $p$-adic field, $\mathcal{O}$ its valuation ring, $\varpi$ a
uniformizing parameter, and $k=\mathcal{O}/(\varpi)$ its residue field.

Let $\mathrm{H}^{\bullet}(A)=\mathrm{H}^{\bullet}(Y_{G}(U),V_{\lambda}(A))$
for $A=\mathcal{O}$ or $k=\mathcal{O}/(\varpi)$. Note that
$\mathrm{H}^{\bullet}(\mathcal{O})$ may have torsion. Let $h$ be the
$\mathcal{O}$-Hecke algebra outside $Ram(U)$ acting faithfully on this module.
Let $S_{p}$ be the set of places of $F$ above $p$. Let $\pi$ be a cuspidal
representation occuring in $H^{q_{0}}(\mathcal{O})$, let $\lambda_{\pi}\colon
h\to\mathcal{O}$ be the corresponding Hecke eigensystem and
$\overline{\lambda}\colon h\to k$ its reduction modulo $\varpi$. We put
${\mathfrak{m}}=\operatorname{Ker}(\overline{\lambda})$ the corresponding
maximal ideal.

In what follos, we will assume that $\pi$ has an associated Galois
representation
$\rho_{\pi}\colon\operatorname{Gal}(\overline{F}/F)\to{}^{L}G(\mathcal{O})$
(unramified outside $Ram(U)\cup S_{p}$) satisfying:

$(RLI)$ The residual representation $\rho_{\pi}$ has residual large image.

Recall that this means that its image contains $\phi(H(k^{\prime}))$ where
$k^{\prime}\subset k$ is a subfield of $k$ and $\phi\colon
H\to\widehat{G}^{\prime}$ is the universal covering of the derived subgroup
$\widehat{G}^{\prime}$ of $\widehat{G}$). When this condition is satisfied, we
also say that ${\mathfrak{m}}$ is strongly non Eisenstein.

Let $H^{\bullet}_{\mathfrak{m}}$ (resp. $H^{\bullet}_{\mathfrak{m}}(k)$) be
the localization of $\mathrm{H}^{\bullet}(\mathcal{O})$ (resp.
$\mathrm{H}^{\bullet}(k)$) at ${\mathfrak{m}}$. Recall that Caraiani and
Scholze [CaSch15] have proven that if $Y_{G}$ is a unitary Shimura variety,
then $\mathrm{H}^{i}_{\mathfrak{m}}=0$ for $i\neq q_{0}={d\over 2}$ which
implies that $\mathrm{H}^{q_{0}}_{\mathfrak{m}}$ is torsion-free. In general,
we put $q_{m}=q_{0}$ and $q_{s}=q_{0}+\ell_{0}$ and consider the assumption

$(Van_{\mathfrak{m}})\quad\mathrm{H}^{i}_{\mathfrak{m}}(k)=0\ \mbox{\rm
for}\,i\notin[q_{m},q_{s}]$

When $(RLI)$ holds, it is generally believed that that $(Van_{\mathfrak{m}})$
is satisfied, but it seems very hard to prove in general. Nevertheless, it is
known if $q_{0}=1$. If $q_{0}\geq 2$, $\mathrm{H}^{1}_{\mathfrak{m}}=0$ can be
proven for many groups [Cai21, Section 1.2.4]. In particular, if $q_{0}\leq
2$, $(Van_{\mathfrak{m}})$ holds for many groups $G$. This is the case for:

  * 1)

$\operatorname{\mathsf{GL}}(2)_{/F}$ for $F$ imaginary quadratic (called in
the sequel the Bianchi case, for which $q_{0}=\ell_{0}=1,d=3$) or $F$ CM of
degree $4$ (called in the sequel the higher Bianchi case, where $q_{0}=2$,
$\ell_{0}=3$, $d=7$),

  * 2)

$\operatorname{\mathsf{GL}}(3)_{/{\mathbb{Q}}}$, ($q_{m}=2$,
$\ell_{0}=1,d=5$),

  * 3)

$O(p,q)$, $p+q\leq 5$ (see [Cai21, Section 7])

From now on, we furthermore assume that
$G=\operatorname{Res}_{F/{\mathbb{Q}}}\operatorname{\mathsf{GL}}_{N}$ and $F$
is CM with $[F\colon{\mathbb{Q}}]=2d_{0}$. Therefore, $\ell_{0}=Nd_{0}-1$ and
$q_{0}={N(N-1)\over 2}d_{0}$. Let $\operatorname{T}=h_{\mathfrak{m}}$ be the
localized Hecke algebra acting faithfully on
$\mathrm{H}^{\bullet}_{\mathfrak{m}}$. These modules are finite $\mathcal{O}$
but may contain torsion. We assume

$(\operatorname{Gal}_{\mathfrak{m}})$ There exists a continuous Galois
representation
$\rho_{\mathfrak{m}}\colon\operatorname{Gal}(\overline{F}/F)\to\operatorname{\mathsf{GL}}_{N}(\operatorname{T})$
unramified outside $Ram(U)\cup S_{p}$ and such that for any $v\notin
Ram(U)\cup S_{p}$,
$\operatorname{char}(\rho_{\mathfrak{m}}(\operatorname{Frob}_{v}))=Hecke_{v}(X)$
where the Hecke polynomial at $v$ is given by

$Hecke_{v}(X)=X^{N}-T_{v,1}X^{N-1}+\ldots+(-1)^{i}T_{v,i}\mathbb{N}v^{{i(i-1)\over
2}}X^{N-i}+\ldots+(-1)^{N}T_{v,N}\mathbb{N}v^{{N(N-1)\over 2}}.$

This conjecture is proven in [Sch15] if one replaces $\operatorname{T}$ by
$\operatorname{T}/I$ where $I$ is a nilpotent ideal of exponent bounded in
terms of $N$ and $[F\colon{\mathbb{Q}}]$. The exponent of $I$ is bounded by
$4$ in [NT16]. By [CGH+19], the ideal can be taken to be $0$ if $p$ splits
totally in $F$.

Let ${\mathfrak{n}}\subset\mathcal{O}_{F}$ be a squarefree ideal coprime to
$p$. We denote by $S$ the set of places of $F$ dividing ${\mathfrak{n}}$. Let
$U_{0}({\mathfrak{n}})\subset GL(N,\widehat{\mathcal{O}}_{F})$ be the Iwahori
subgroup of level ${\mathfrak{n}}$, and

$Y=Y_{0}({\mathfrak{n}})=\operatorname{\mathsf{GL}}_{N}(F)\backslash
GL_{N}(\operatorname{A}_{F})/U_{0}({\mathfrak{n}})C_{\infty}$

be the Shimura space of level $\Gamma_{0}({\mathfrak{n}})$ (here
$C_{\infty}={\mathbb{R}}^{\times}U_{N}(F^{+}_{\infty})$). We assume

$(MIN)$ $\rho_{\pi}$ is ${\mathfrak{n}}$-minimal.

Recall that this means that for any place $v|{\mathfrak{n}}$, the image
$\overline{\rho}_{\pi}(I_{v})$ of the inertia subgroup $I_{v}$ contains a
regular unipotent element. More precisely, let $J_{N}$ be the standard Jordan
block of size $N$ and $t_{v}\colon I_{v}\to{\mathbb{Z}}_{p}(1)$ be the
$p$-adic tame homomorphism of $F_{v}$ given by
$\tau(p_{v}^{1/p^{n}})=\zeta_{p^{n}}^{t_{v}(\tau)}\cdot p_{v}^{1/p^{n}}$ for
all $n$’s. Then the condition of ${\mathfrak{n}}$-minimality at $v$ is that
there exists $g_{v}\in\operatorname{\mathsf{GL}}_{N}(\mathcal{O})$ such that
for any $\tau\in I_{v}$,

$g_{v}\cdot\rho_{\pi}(\tau)\cdot g_{v}^{-1}=\exp(t_{v}(\tau)J_{N}).$

Let $\lambda=(\lambda_{\tau,i})$ where $\tau\colon
F\to\overline{{\mathbb{Q}}}$ and $i=1,\ldots,N$. Let
$\pi=\pi_{f}\otimes\pi_{\infty}$ be a cuspidal automorphic representation such
that $dim\,\pi_{f}^{U_{0}({\mathfrak{n}})}=1$ and of cohomological weight
$\lambda$. In other words, there are Hecke equivariant embeddings

$\pi_{f}^{U_{0}({\mathfrak{n}})}\subset\mathrm{H}^{i}(Y,V_{\lambda}({\mathbb{C}})),\quad
i=1,2$

Let $S_{p}=S_{F,p}$ be the set of places of $F$ above $p$. By [HLLT16], we
know that the Galois representation
$\rho_{\pi}\colon\operatorname{Gal}(\overline{F}/F)\to\operatorname{\mathsf{GL}}_{N}(\mathcal{O})$
associated to $\pi$ exists and by [Ca14], $\rho_{\pi}|_{G_{F_{w}}}$ is
Fontaine-Laffaille at $w\in S_{p}$. We will assume either of the two following
local conditions at $p$ :

$(FL)$ $p$ is unramified in $F$ and $\lambda_{\tau,1}-\lambda_{\tau,N}<p-N$
for any $\tau\in I_{F}$.

$(ORD_{\pi})$ $F$ contains an imaginary quadratic field in which $p$ splits,
$\pi$ is unramified and ordinary at all places $v\in S_{p}$.

To recall what the latter means, we follow the notations of [ACC+18, Section
5.1, before Th.5.5.1] (see also [Ge19, Lemma 2.7.6]). We assume $S\cap
S_{F,p}=\emptyset$. Then the ordinarity of $\pi$ at $v\in S_{p}$ means that
$u_{\lambda,\varpi_{v}}^{(i)}\in\mathcal{O}^{\times}$ for $i=1,\ldots,N$ where
the $u_{\lambda,\varpi_{v}}^{(i)}$’s are defined by

$Hecke_{v}(X)=\prod_{i=1}^{N}(X-\mathbb{N}v^{i-1}\frac{u_{\lambda,\varpi_{v}}^{(i)}}{u_{\lambda,\varpi_{v}}^{(i-1)}}\cdot\prod_{\tau\colon
F_{v}\to\overline{{\mathbb{Q}}}_{p}}\tau(\varpi_{v})^{\lambda_{\tau,N-i+1}})$

with $u_{\lambda,\varpi_{v}}^{(0)}=1$ by convention.

For $i=1,\ldots,N$, let $\chi_{\pi,v,i}\colon
G_{F_{v}}\to\mathcal{O}^{\times}$ be the character given by

$\chi_{\pi,v,i}\circ\operatorname{Art}_{F_{v}}(u)=\epsilon^{-(i-1)}(\operatorname{Art}_{F_{v}}(u))\prod_{\tau}\tau(u)^{\lambda_{\tau,N+1-i}}$

for all $u\in\mathcal{O}_{F_{v}}^{\times}$ , and by

$\chi_{\pi,v,i}\circ\operatorname{Art}_{F_{v}}(\varpi_{v})=\mathbb{N}v^{i-1}\frac{u_{\lambda,\varpi_{v}}^{(i)}}{u_{\lambda,\varpi_{v}}^{(i-1)}}\cdot\prod_{\tau\colon
F_{v}\to\overline{{\mathbb{Q}}}_{p}}\tau(\varpi_{v})^{\lambda_{\tau,N-i+1}}.$

We put

$\underline{\chi}_{\pi,v}=\operatorname{diag}(\chi_{\pi,v,i})_{i=1,\ldots,N}$

Then the translation of the condition $(ORD_{\pi})$ on the Galois side is as
follows. There exists $g_{v}\in\operatorname{\mathsf{GL}}_{N}(\mathcal{O})$
such that $g_{v}\cdot\rho_{\pi}|_{G_{F_{v}}}\cdot g_{v}^{-1}$ is upper
triangular. Let $B=TN$ be the Levi decomposition of the Borel $B$ of upper
triangular matrices with $T$ the subgroup of diagonal matrices. And if we
denote by $\underline{\chi}_{v}\colon G_{F_{v}}\to T(\mathcal{O})$ the
homomorphism $g_{v}\cdot\rho_{\pi}|_{G_{F_{v}}}\cdot g_{v}^{-1}$ modulo
$N(\mathcal{O})$. Then

$\underline{\chi}_{v}=\underline{\chi}_{\pi,v}$

In that situation, we consider the so-called $p$-distinguished condition

$(DIST)$ We assume that $\overline{\rho}_{\pi}$ is distinguished at each $v\in
S_{p}$ (or in brief, $p$-distinguished).

With the notations above, it means that for each $v\in S_{p}$, the residual
characters $\overline{\chi}_{v,i}$ modulo $\varpi$ for $i=1,\ldots,N$ are
mutually distinct. In other words, the assumption of $p$-distinguishability
expresses the fact that for all $v\in S_{p}$, the homomorphism
$\overline{\underline{\chi}}_{v}=(\overline{\chi}_{v,i})_{i=1,\ldots,N}$
separates the roots of $(\operatorname{\mathsf{GL}}_{N},B,T)$. We will also
need sometimes the assumption of strong $p$-distinguishability:

(STDIST) For all $i<j$’s, the quotients
$\overline{\chi}_{\widetilde{v},i}\cdot\overline{\chi}_{\widetilde{v},j}^{-1}$
are neither trivial nor equal to $\overline{\epsilon}^{-1}$.

We now recall briefly the deformation theory of Galois representations that
plays a central role in this paper. Let
$\mathcal{F}_{\overline{\rho}}\colon{}_{\mathcal{O}}\operatorname{Art}_{k}\to
Sets$ resp. $\mathcal{F}_{v}$, be the problem of deformations of
$\overline{\rho}=\overline{\rho}_{\pi}$, resp. of
$\overline{\rho}_{v}=\overline{\rho}_{\pi}|_{G_{F_{v}}}$. For $v\in S_{p}$, we
consider the subfunctors $\mathcal{F}^{?}_{v}\subset\mathcal{F}_{v}$ where
$?=\operatorname{ord},FL$; assuming $(ORD_{\pi})$,
$\mathcal{F}^{\operatorname{ord}}_{v}$ is the functor of
$\underline{\chi}_{\pi,v}$-ordinary deformations $\rho$’s, that is, those for
which $\underline{\chi}_{\rho}=\underline{\chi}_{\pi,v}$ (see Section 2.1.2);
similarly, assuming $(FL)$, $\mathcal{F}^{FL}_{v}$ is the functor of.
Fontaine-Laffaille local deformations (see Section 2.1.3). Similarly, for
$v\in S$, we consider the subfunctor
$\mathcal{F}_{v}^{min}\subset\mathcal{F}_{v}$ of minimal deformations (see
2.1.1). Let

$\mathcal{F}_{loc}=\prod_{v\in S\cup S_{p}}\mathcal{F}_{v}\quad\mbox{\rm and
}\mathcal{F}_{loc}^{min,?}=\prod_{v\in
S}\mathcal{F}_{v}^{min}\times\prod_{v\in S_{p}}\mathcal{F}_{v}^{?}$

and consider the global deformation problem defined as the fiber product

$\mathcal{D}^{?}=\mathcal{F}_{\overline{\rho}}^{min,?}=\mathcal{F}_{\overline{\rho}}\times_{\mathcal{F}_{loc}}\mathcal{F}_{loc}^{min,?}.$

It is pro-prepresentable by a pair $(R^{?},\rho^{?})$ where
$?=\operatorname{ord},FL$ (see for instance [CHT08, Prop.2.2.9 and Lemma
2.4.1] for $?=FL$ and [CHT08, Section 2.4.4] or [Ti96, Chapt.6] for
$?=\operatorname{ord}$). As in [CaGe18], we introduce allowable Taylor-Wiles
sets $Q$ disjoint of the set of places dividing ${\mathfrak{n}}p$ and consider
the functors $\mathcal{D}^{?}_{Q}$ similar to $\mathcal{D}^{?}$, where we
allow arbitrary ramification at places $v\in Q$. Similarly, let $R^{?}_{Q}$ be
the universal deformation ring of $\mathcal{D}^{?}_{Q}$ and
$(\operatorname{T}_{Q},{\mathfrak{m}}_{Q})$ be the analogue of
$(\operatorname{T},{\mathfrak{m}})$ with Taylor-Wiles auxiliary level at $Q$
(see [CaGe18, Section 9.1]). The rings $R_{Q}^{?}$ and $\operatorname{T}_{Q}$
both admit a natural structure of $\mathcal{O}[\Delta_{Q}]$-algebra where
$\Delta_{Q}$ is the $p$-Sylow of $\prod_{v\in Q}T(k_{v})$. By
$(\operatorname{Gal}_{\mathfrak{m}})$, there is a Galois representation
$\rho_{{\mathfrak{m}}_{Q}}\colon\operatorname{Gal}(\overline{F}/F)\to\operatorname{\mathsf{GL}}_{N}(\operatorname{T}_{Q})$
associated to $(\operatorname{T}_{Q},{\mathfrak{m}}_{Q})$. We will assume the
following

######  Conjecture 1.

(LLC) For any Taylor-Wiles set $Q$, the conjugacy class
$[\rho_{{\mathfrak{m}}_{Q}}]$ is in
$\mathcal{D}^{?}_{Q}(\operatorname{T}_{Q})$.

The local-global compatibility conjecture (LLC) for
$\rho_{{\mathfrak{m}}_{Q}}$ has been proven (at least if $F$ is not imaginary
quadratic but contains an imaginary quadratic field in which $p$ splits)
modulo a nilpotent ideal (see [ACC+18]) of exponent bounded by
$[F\colon{\mathbb{Q}}]$ and $N$. Assuming (LLC), for each Taylor-Wiles set
$Q$, there is a canonical surjective $\mathcal{O}[\Delta_{Q}]$-homomorphism
$\phi_{Q}\colon R^{?}_{Q}\to\operatorname{T}_{Q}$. Let
$\phi_{=}\phi_{\emptyset}$. By Taylor-Wiles-Kisin patching, the
$\mathcal{O}[\Delta_{Q}]$-homomorphism $\phi_{Q}$ give rise to morphism
$S_{\infty}\to
R_{\infty}^{?}\to\operatorname{T}_{\infty}\subset\operatorname{\mathsf{End}}_{S_{\infty}}(C^{\bullet}_{\infty})$
(see [CaGe18]). The graded module
$\operatorname{Tor}_{\bullet}^{S_{\infty}}(R_{\infty},\mathcal{O})$ has a
natural structure of graded algebra over
$\operatorname{Tor}_{0}^{S_{\infty}}(R_{\infty},\mathcal{O})=R_{\infty}\otimes_{S_{\infty}}\mathcal{O}=R^{?}$
. Let $\mathrm{H}_{\bullet}^{\mathfrak{m}}=H^{d-\bullet}_{\mathfrak{m}}$, then
we have:

######  Theorem 1.

([CaGe18]) Assume $\zeta_{p}\notin F$, $p>N$,
$(\operatorname{Gal}_{\mathfrak{m}})$, $(LLC)$ $(RLI)$, $(MIN)$, $(FL)$ or
$(ORD_{\pi})+(DIST)$. Then, the canonical map
$R_{\infty}^{?}\to\operatorname{T}_{\infty}$ is an isomorphism, and in
particular
$R^{?}_{\infty}\otimes_{S_{\infty}}\mathcal{O}=R^{?}\cong\operatorname{T}$.
Moreover, the module $\mathrm{H}^{\mathfrak{m}}_{q_{m}}$ is finite and free
over $\operatorname{T}$, and there is an isomorphism of graded modules:

$\mathrm{H}_{\bullet}^{\mathfrak{m}}\cong\mathrm{H}_{q_{0}}^{\mathfrak{m}}\otimes_{\operatorname{T}}\operatorname{Tor}_{\bullet}^{S_{\infty}}(\operatorname{T}_{\infty},\mathcal{O})$

in other words $\mathrm{H}_{\bullet}^{\mathfrak{m}}$ is a free graded
$\operatorname{Tor}_{\bullet}^{S_{\infty}}(\operatorname{T}_{\infty},\mathcal{O})$-module.

On the other hand, in [GV18], the authors introduced derived notions of
deformation problems for $\overline{\rho}_{{\mathfrak{m}}}$ denoted
$\mathcal{D}^{s}$, $\mathcal{D}^{s,?}$; they are pro-represented by simplicial
pro-artinian rings $\mathcal{R}$, resp. $\mathcal{R}^{s,?}$. Generalizing
their main result Th.12.1 and Th.14.1, Y. Cai [Cai21, Theorem 0.6] proved

######  Theorem 2.

([GV18, Cai21]) Assume that $p>N$, $\zeta_{p}\notin F$,
$(\operatorname{Gal}_{\mathfrak{m}})$, $(LLC)$ and $(Van_{\mathfrak{m}})$
hold. Under the assumptions $(RLI)$, $(MIN)$, $(FL)$ or $(ORD_{\pi})+(DIST)$,
we have an isomorphism of graded commutative rings
$\pi_{\bullet}\mathcal{R}\cong\operatorname{Tor}_{\bullet}^{S_{\infty}}(R_{\infty},\mathcal{O})$,
in particular we have an isomorphism of graded objects

$\mathrm{H}_{\bullet}^{\mathfrak{m}}\cong\mathrm{H}_{q_{0}}^{\mathfrak{m}}\otimes_{\operatorname{T}}\pi_{\bullet}(\mathcal{R})$

Our first main result concerns the relation between $\pi_{1}(\mathcal{R})$ and
the minimal and ordinary (resp. Fontaine-Laffaille) Selmer group of
$\operatorname{Ad}(\rho_{\pi})^{*}(1)$. Recall $\pi_{0}(\mathcal{R})=R$ is the
classical $(min,?)$-deformation ring. Let
$\mathcal{O}_{n}=\mathcal{O}/(\varpi^{n})$. We consider the simplicial ring
homomorphism

$\phi_{n}\colon\mathcal{R}\to R\to\mathcal{O}_{n}$

given by the universal property for the deformation
$\rho_{n}=\rho_{\pi}\pmod{(\varpi^{n})}$. Let
$M_{n}=\varpi^{-n}\mathcal{O}/\mathcal{O}$; it is a finite
$\mathcal{O}_{n}$-module. Consider the simplicial ring
$\Theta_{n}=\mathcal{O}_{n}\oplus M_{n}[1]$ concentrated in degrees $0$ and
$1$ up to homotopy. It is endowed with a simplicial ring homomorphism
$\operatorname{pr}_{n}\colon\Theta_{n}\to\mathcal{O}_{n}$ given by the first
projection. Let $L_{n}(\mathcal{R})$ be the set of homotopy equivalence
classes of simplicial ring homomorphisms $\Phi\colon\mathcal{R}\to\Theta_{n}$
such that $\operatorname{pr}_{n}\circ\Phi=\phi_{n}$. Following [GV18, Lemma
15.1], we define (see Section 4) compatible homomorphisms

$\pi(n,\mathcal{R})\colon
L_{n}(\mathcal{R})\to\operatorname{\mathsf{Hom}}_{R}(\pi_{1}(\mathcal{R}),M_{n})$

Let $\mathop{\rm
Sel}(F,\operatorname{Ad}\,\rho_{n}(1)\otimes_{\operatorname{T}}M_{n}^{\vee})$
be the minimal and $?=ord,FL$ Selmer group of
$\operatorname{Ad}\,\rho_{n}(1)\otimes_{\operatorname{T}}M_{n}^{\vee}$. Formal
arguments show that it is the Pontryagin dual of $L_{n}(\mathcal{R})$.
Therefore, by passing to Pontryagin dual, we obtain a homomorphism
$GV_{n}=\pi(n,\mathcal{R})^{\vee}$:

$GV_{n}\colon\operatorname{\mathsf{Hom}}_{\operatorname{T}}(\pi_{1}(\mathcal{R}),M_{n})^{\vee}\hookrightarrow
Sel(F,(\operatorname{Ad}\,\rho_{n})^{\ast}(1)\otimes M_{n}^{\vee})$

By taking the direct limit we obtain a homomorphism

$GV\colon\pi_{1}(\mathcal{R})\otimes_{R,\lambda}\mathcal{O}\otimes_{\mathcal{O}}K/\mathcal{O}\hookrightarrow\mathop{\rm
Sel}(F,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes K/\mathcal{O})$

Then our first result (Theorem 13 and Proposition 7 in the text) is

######  Theorem 3.

Assume $p>N$, $\zeta_{p}\notin F$, $(\operatorname{Gal}_{\mathfrak{m}})$,
$(LLC)$, $(RLI)$, $(MIN)$, $(FL)$ or $(ORD_{\pi})+(DIST)$. The natural
homomorphism

$GV\colon\pi_{1}(\mathcal{R})\otimes_{\operatorname{T}}K/\mathcal{O}\hookrightarrow\mathop{\rm
Sel}(\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes K/\mathcal{O})$

is injective. The left-hand side module is $\varpi$-divisible of
$\mathcal{O}$-corank $\ell_{0}$. If we are in the ordinary case, let us
furthermore assume $(STDIST)$, then the cokernel of $GV$ is the Tate-
Shafarevitch group ${}^{1}(F,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
K/\mathcal{O})$ and is finite. Moreover, its $\mathcal{O}$-Fitting ideal is
the same as the one of $\mathop{\rm
Sel}(F,\operatorname{Ad}(\rho_{\pi})\otimes K/\mathcal{O})$.

The key step to prove this theorem is Proposition 5. A similar result, under
stronger assumptions, is proven in the course of the proof of [GV18, Theorem
15.2]. However, under our assumptions, the proof given there doesn’t seem to
work. Our approach is different and involves only commutative algebra. The
last part of the statement is proved in Proposition 7.

Actually, under the same assumptions , we can also define a graded version
$GV^{\bullet}$ of the Galatius-Venkatesh homomorphism $GV$. Let us introduce
some notations before stating our result. For a graded algebra $G$, we
introduce the truncation

$\tau_{\leq\ell_{0}}G=\bigoplus_{j\leq\ell_{0}}G_{j}$

with a partial algebra structure defined only for $g_{j_{k}}\in G_{j_{k}}$
such that $j=j_{1}+j_{2}\leq\ell_{0}$. For a graded algebra $G$, we also
introduce its largest graded-commutative quotient $\widetilde{G}$, that is,
its largest (graded) quotient in which
$<a,b>_{j_{1},j_{2}}=(-1)^{j_{1}j_{2}}<b,a>_{j_{2},j_{1}}$ for
$a\in\widetilde{G}_{j_{1}}$, $b\in\widetilde{G}_{j_{2}}$. We endow
$\pi_{\bullet}(\mathcal{R}^{\otimes\bullet})$ with a structure of graded
algebra and we prove (see Theorem 14):

######  Theorem 4.

For every cuspidal representation $\pi^{\prime}$ occuring in
$H^{\bullet}_{\mathfrak{m}}$, for any $m\geq 1$, there is an homomorphism of
$A_{m}$-modules

$(HGV_{j})\quad GV_{m}^{j}\colon\widetilde{\pi}_{j}(\mathcal{R}^{\otimes
j})_{A_{m}}\to\bigwedge^{j}_{A_{m}}\mathop{\rm
Sel}(F,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes A_{m})^{\oplus j}$

It induces a morphism of truncated graded $A_{m}$-algebras

$(HGV_{\bullet})\quad
GV_{m}^{\bullet}\colon\tau_{\leq\ell_{0}}\widetilde{\pi}_{\bullet}(\mathcal{R}^{\otimes\bullet})\otimes
A_{m}\to\tau_{\leq\ell_{0}}\bigwedge^{\bullet}_{A_{m}}\mathop{\rm
Sel}(F,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes A_{m})^{\oplus\bullet}$

Moreover, for any $j$, the composition $a_{j}\circ GV_{m}^{j}\circ\mu_{j}$ of
$GV_{m}^{j}$ with the tensor shuffle multiplication map
$\mu_{j}\colon\bigwedge^{j}_{A}\pi_{1}(\mathcal{R})_{A}\to\widetilde{\pi}_{j}(\mathcal{R}^{\otimes
j})_{A}$ and the homomorphism $a_{j}$ induced by the addition

$\mathop{\rm Sel}(F,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes
A_{m})^{\oplus j}\to\mathop{\rm
Sel}(F,\operatorname{Ad}(\rho_{\pi})^{\ast}(1)\otimes A_{m})$

coincides with $\bigwedge^{j}GV_{m}^{1}$. For $j=\ell_{0}$, the cokernel of
$GV_{m}^{\ell_{0}}\circ\mu_{\ell_{0}}$ is annihilated by
$\operatorname{Fitt}_{\mathcal{O}}(\mathop{\rm
Sel}(F,Ad(\rho_{\pi^{\prime}}))$.

This theorem implies that for any $1\leq j\leq\ell_{0}$, the shuffle
multiplication $\mu_{j}$ is injective111However it would be more interesting
to study the injectivity of the homomorphisms $GV_{m}^{j}$..

We have similar results for Hida families in Section 7. Let $R_{h}$ be the
universal minimal $\Lambda$-ordinary deformation ring prorepresenting the
minimal $\Lambda$-ordinary deformation problem $\mathcal{D}_{h}$ (see Section
6.2). Let $\mathcal{R}_{h}$ the simplical proartinian ring prorepresenting the
simplicial minimal $\Lambda$-ordinary deformation problem
$\mathcal{D}_{h}^{s}$ (see Section 6.3). The rings $\mathcal{R}_{h}$ and
$R_{h}$ are naturally $\Lambda$-algebra, where $\Lambda$ is the Hida-Iwasawa
algebra (see Section 6.1). We have $\pi_{0}(\mathcal{R}_{h})=R_{h}$ ( Section
6.3), hence we have a surjective $\Lambda$-algebra homomorphism
$\mathcal{R}_{h}\to R_{h}$. Note that $\pi_{\bullet}(\mathcal{R}_{h})$ is a
graded algebra over $\pi_{0}(\mathcal{R}_{h})=R_{h}$. Let ${\mathfrak{m}}_{h}$
be the maximal ideal of $\Lambda$-ordinary Hida Hecke algebra acting
faithfully on
$\mathbb{H}_{h}^{\bullet}=e\mathrm{H}^{\bullet}(Y_{1}(p^{\infty}),\mathcal{O})$
associated to ${\mathfrak{m}}$ and let $\operatorname{T}_{h}$ be its
corresponding ${\mathfrak{m}}_{h}$-localization. Assuming
$(\operatorname{Gal}_{\mathfrak{m}})$ and (LLC), there is $\Lambda$-algebra
homomorphism $R_{h}\to\operatorname{T}_{h}$. We prove following the Calegari-
Geraghty method (Th. 15 and 16 in the text) :

######  Theorem 5.

Assuming $(\operatorname{Gal}_{\mathfrak{m}})$ and (LLC), $(MIN)$,
$(ORD_{\pi})+(DIST)$. Assume $(Van_{\mathfrak{m}})$ holds for a classical
specialization $\pi$ of $\operatorname{T}_{h}$ of level prime to $p$. Then

1) $R_{h}\to\operatorname{T}_{h}$ is an isomorphism,

2) there is a isomorphism of graded $\operatorname{T}_{h}$-modules

$\mathbb{H}_{h,{\mathfrak{m}}_{h}}^{\bullet}=\mathbb{H}^{q_{s}}_{h,{\mathfrak{m}}_{h}}\otimes_{\operatorname{T}_{h}}\pi_{d-\bullet}(\mathcal{R}_{h})$

Given a Hida family $\operatorname{T}_{h}\to\mathbf{I}$ of the
${\mathfrak{m}}_{h}$-localization of the ordinary Hida Hecke algebra, let
$\Lambda_{\mathbf{I}}$ be the image of $\Lambda$ in $\mathbf{I}$ and
$\mathcal{R}_{\mathbf{I}}=\mathcal{R}_{h}\underline{\otimes}_{\Lambda}\Lambda_{\mathbf{I}}$.
We prove (Theorem 17 in the text):

######  Theorem 6.

There are natural $\mathbf{I}$-linear injective homomorphisms

$GV_{h}\colon\pi_{1}(\mathcal{R}_{h})\otimes_{R_{h}}\widehat{\mathbf{I}}\hookrightarrow\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}^{\ast}\rho_{\theta}(1)\otimes_{\mathbf{I}}\widehat{\mathbf{I}}).$

and

$GV_{\mathbf{I}}\colon\pi_{1}(\mathcal{R}_{\mathbf{I}})\otimes_{R_{\mathbf{I}}}\widehat{\mathbf{I}}\hookrightarrow\mathrm{H}^{1}_{\mathcal{L}}(\Gamma,\operatorname{Ad}^{\ast}\rho_{\theta}(1)\otimes_{\mathbf{I}}\widehat{\mathbf{I}}).$

and we have $GV_{\mathbf{I}}\circ\pi_{\mathbf{I}}=GV_{h}$ for
$\pi_{\mathbf{I}}$ the homomorphism induced by the map
$\mathcal{R}_{h}\to\mathcal{R}_{\mathbf{I}}$.

The new feature in this Hida-theoretic context is that the Hida-theoretic
cohomology module
$\mathbb{H}_{h}^{\bullet}=e\mathrm{H}^{\bullet}(Y_{1}(p^{\infty}),\mathcal{O})$
is conjectured to be concentrated in the maximal degree $q_{s}$. This is one
form of the non-abelian Leopoldt conjecture due to Hida and one of the
authors. They are discussed in Section 6.4 in the text. Recall that a
simplicial ring is called discrete if it is weakly equivalent to a constant
ring. Then, this non-abelian Leopoldt conjecture is equivalent to the
following:

######  Conjecture 2.

The simplicial ring $\mathcal{R}_{h}$ is discrete and for any arithmetic
weight $\lambda^{\prime}$, there is an isomorphism of commutative graded rings

$\operatorname{Tor}_{\bullet}^{\Lambda}(R_{h},\Lambda/P_{\lambda^{\prime}})=\pi_{\bullet}(\mathcal{R}_{\lambda^{\prime}}).$

Because of this conjecture, the first statement of the theorem above is
probably empty. However, the second appears to be interesting and yields the
inconditional Theorem below (Theorem 18 in the text). Let $\mathcal{L}$ be the
minimal $\Lambda$-ordinary local deformation data. Let ${\mathop{\rm
Sel}}_{\mathbf{I}}=\mathrm{H}^{1}_{\mathcal{L}}(F,\operatorname{Ad}\rho_{\mathbf{I}}\otimes\widehat{\mathbf{I}})$
and $M_{\mathbf{I}}$ its Pontryagin dual. Let $\Phi_{\mathbf{I}}$ be the
Pontryagin dual of
$\pi_{1}(\mathcal{R}_{\mathbf{I}})\otimes_{R_{\mathbf{I}}}\widehat{\mathbf{I}}$.
Let finally
$N_{\mathbf{I}}=\operatorname{Ker}(M_{\mathbf{I}}\to\Phi_{\mathbf{I}})$ be the
Pontryagin dual of $\operatorname{Coker}\,GV_{\mathbf{I}}$.

######  Theorem 7.

Assume moreover (STDIST). Then

(1) The $\mathbf{I}$-modules $M_{\mathbf{I}}$ and $\Phi_{\mathbf{I}}$ have
rank $\ell_{0}$ and $\Phi_{\mathbf{I}}$ is free.

(2) The Poitou-Tate-Pontryagin duality induces an isomorphism

$N_{\mathbf{I}}\cong\varprojlim_{n}{\mathop{\rm
Sel}}_{\mathbf{I}}[{\mathfrak{m}}_{\mathbf{I}}^{n}]$

Let $\widetilde{\mathbf{I}}$ be the integral closure of $\mathbf{I}$ and let
$X=\operatorname{Supp}(\widetilde{\mathbf{I}}/\mathbf{I})$. For $P\notin X$,
we have $\mathbf{I}/P=\widetilde{\mathbf{I}}/P\widetilde{\mathbf{I}}$. For
such $P$, the reduction modulo $P$ of the exact sequence

$0\to N_{\mathbf{I}}\to M_{\mathbf{I}}\to\Phi_{\mathbf{I}}\to 0$

is exactly the Pontryagin dual of the exact sequence given by

$0\to\pi_{1}(\mathcal{R}_{P})\stackrel{{\scriptstyle
GV_{P}}}{{\rightarrow}}{\mathop{\rm Sel}}_{P}\to\operatorname{Coker}GV_{P}\to
0$

where the natural notations are explained before Corollary 4.

In learning the material used here, we benefited from a seminar on Galatius-
Venkatesh paper held in 2019 at Paris 13, and of discussions with Y. Harpaz
and A. Vezzani. Special thanks are due to Yichang Cai whose thesis provided
much of the framework for our work. He also taught us several technical
aspects of Model Categories and Derived Deformation theory, although
misunderstandings and mistakes left in the text are ours.

##  2\. Calegari-Geraghty Theory for $\operatorname{\mathsf{GL}}_{N}$

###  2.1. Local deformation conditions

For each finite place $v$ of $F$, let $\overline{F}_{v}$ be an algebraic
closure of $F_{v}$ and $G_{F_{v}}$ be the local Galois group of
$\overline{F}_{v}$ over $F_{v}$, identified to a decomposition group at $v$ in
$G_{F}$.

By definition, for an artinian local $\mathcal{O}$-algebra $A$ with residue
field $k$, let $\mathcal{F}_{\overline{\rho}}(A)$ (resp. $\mathcal{F}_{v}(A)$
) be the set of conjugacy classes of liftings $\rho\colon
G_{F}\to\operatorname{\mathsf{GL}}_{N}(A)$ of $\overline{\rho}_{\pi}$ (resp.
$\rho_{v}\colon G_{F_{v}}\to\operatorname{\mathsf{GL}}_{N}(A)$ of
$\overline{\rho}_{v}$). Let $\mathcal{F}_{v}^{\square}$ be the functor of
liftings of $\overline{\rho}_{v}$. Recall that it carries an action of the
formal group scheme $\widehat{\operatorname{PGL}_{N}}$ by conjugation and that
$\mathcal{F}_{v}^{\square}/\widehat{\operatorname{PGL}_{N}}\cong\mathcal{F}_{v}$.

We define below subfunctors $\mathcal{F}_{v}^{?,\square}$, resp.
$\mathcal{F}_{v}^{?}$ of local liftings, resp. deformations of
$\bar{\rho}_{v}$, for $?=min,\operatorname{ord},FL$.

######  Definition 1.

A functor
$\mathcal{F}\colon{}_{\mathcal{O}}\operatorname{Art}_{k}\to\operatorname{\mathsf{SETS}}$
is called liftable if for any surjection $\alpha\colon A_{1}\to A_{0}$, the
morphism $\mathcal{F}(A_{1})\to\mathcal{F}(A_{0})$ is surjective.

Note that if $\mathcal{F}_{v}^{?,\square}$ is formally smooth,
$\mathcal{F}_{v}^{?}$ is liftable. We prove this below for
$?=min,\operatorname{ord},FL$.

####  2.1.1. minimal case

For $v\in S$, let $t_{v}\colon I_{v}\to{\mathbb{Z}}_{p}^{\times}$ be the
$p$-adic tame character and let $q=\mathcal{O}_{F_{v}}/(\varpi_{v})$; for
$\tau\in I_{v}^{tame}$, and for $\phi_{v}$ an arithmetic Frobenius, we have
$\phi_{v}\circ\tau\circ\phi_{v}^{-1}=\tau^{q}$. We consider the problem
$\mathcal{F}_{v}^{min,\square}\subset\mathcal{F}_{v}^{\square}$, resp.
$\mathcal{F}_{v}^{min}\subset\mathcal{F}_{v}$ of $v$-minimal liftings, resp.
deformations:

$\mathcal{F}_{v}^{min,\square}(A)=\\{\rho_{v}\in\mathcal{F}_{v}^{\square}(A);\mbox{\rm
for}\,\tau\in I_{v},\mbox{\rm there
exists}\,g_{v}\in\operatorname{\mathsf{GL}}_{N}(A)\,\mbox{\rm such that for
any}\,\tau\in I_{v},$ $g_{v}\cdot\rho(\tau)\cdot
g_{v}^{-1}=\exp(t_{v}(\tau)J_{N})\\}$

and

$\mathcal{F}_{v}^{min}(A)=\mathcal{F}_{v}^{min,\square}(A)/\widehat{\operatorname{PGL}_{N}}(A).$

Let $\Phi_{v}=\operatorname{diag}(q^{N-1},\ldots,1)$.

######  Lemma 1.

The functor $\mathcal{F}_{v}^{min,\square}$ is prorepresented by a formally
smooth $\mathcal{O}$-algebra $R_{v}^{min,\square}$, isomorphic to a power
series ring in $N^{2}$ variables. The functor $\mathcal{F}_{v}^{min}$ is
liftable.

Comments:

1) Note that the proof uses the fact that the matrix $J_{N}$ is regular
nilpotent.

2) This result is also proven in [CHT08, Lemma 2.4.19 and Cor.2.4.20] and
[Bo19, Introduction].

3) The functor $\mathcal{F}_{v}^{min}$ is pro-representable if and only if $p$
does not divide $q_{v}^{i}-1$ ($i=1,\ldots,N-1$).

###### Proof.

Let $C_{0}$ be the centralizer of $J_{N}$ in the standard Borel $B\subset G$.
The formal scheme $\mathcal{C}$ of centralizers of liftings of
$\overline{J}_{N}$ is isomorphic to the principal homogeneous formal scheme
$\widehat{\operatorname{\mathsf{GL}}_{N}/C_{0}}$, hence is formally smooth of
dimension $N^{2}-N$. There is a fibration of functors
$\mathcal{F}_{v}^{min,\square}\to\mathcal{C}$ given by $\rho_{v}\to
g_{v}^{-1}C_{0}g_{v}$. Therefore, to study the formal smoothness of
$\mathcal{F}_{v}^{min,\square}$, one can fix $C\subset B$ and study its fiber
$\mathcal{F}_{v,[C]}^{min,\square}$ in $\mathcal{F}_{v}^{min,\square}$. In the
notations below, we take $C=C_{0}$. We therefore consider $\rho_{v}$ such that
for any $\tau\in I_{v}$, $\rho_{v}(\tau)=\exp(t_{v}(\tau)J_{N})$. Let
$\Phi_{v}=\operatorname{diag}(q_{v}^{N-1},\ldots,1)$. We have
$\rho_{v}(\phi_{v})=\Psi_{v}$ with $\Psi_{v}=\Phi_{v}g$ with $g\in C_{0}$. Let
$\widehat{C}_{0}$ be the formal group scheme associated to $C_{0}$. By the
considerations above, we see easily that the map $g\mapsto([\rho_{g}]$ where
$\rho_{g}(\phi_{v})=\Phi_{v}\cdot g$ and
$\rho_{g}(\tau)=\exp(t_{v}(\tau)J_{N})$ for $\tau\in I_{v}$, is an isomorphism
of functors

$\widehat{C}\cong\mathcal{F}_{v,[C]}^{min,\square}.$

This isomorphism is compatible to conjugation by $\widehat{G}$. It implies
that $\mathcal{F}_{v}^{min,\square}$ is a torsor under the smooth formal group
scheme $\widehat{G}$, hence is formally smooth of dimension $N^{2}$. This
proves the formal smoothness of
$R_{v}^{min,\square}\cong\mathcal{O}[[X_{1},\ldots,X_{N^{2}}]]$.

Any deformation $[\rho_{v}]\in\mathcal{F}_{v}^{min}(A)$ has a representative
$\rho_{v}$ such that for any $\tau\in I_{v}$,
$\rho_{v}(\tau)=\exp(t_{v}(\tau)J_{N})$. For such a representative,
$\rho_{v}(\phi_{v})=\Psi_{v}$ can be written $\Psi_{v}=\Phi_{v}g$ with $g\in
C_{0}$. If $\rho_{v}^{\prime}$ is another such representative, we see that
there exists $h\in\widehat{C}(A)$ such that

$\rho_{v}^{\prime}(\phi_{v})=h\Psi_{v}h^{-1}=\Phi_{v}g^{\prime}$

Therefore, $g^{\prime}=\Phi_{v}^{-1}h\Phi_{v}h^{-1}\cdot.g$ for an
$h\in\widehat{C}(A)$ and conversely. Conjugation by $\Phi_{v}$ is an
automorphism of $\widehat{C}(A)$. Thus we find that

$\mathcal{F}_{v}^{min}\cong\widehat{C}/(\Phi_{v}-1)\widehat{C}$

which is isomorphic to the functor

$A\mapsto A/(q-1)A\times A/(q^{2}-1)A\times\ldots A/(q^{N-1}-1)A$

This functor is not pro-representable unless $p$ does not divide the $q^{i}-1$
(for all $i=1,\ldots,N-1$). However, it is liftable. ∎

####  2.1.2. The ordinary case

For $v\in S_{p}$, we consider the problem
$\mathcal{F}_{v}^{\operatorname{ord}}\subset\mathcal{F}_{v}$ of ordinary
deformations of $\overline{\rho}_{v}$ without fixing the Hodge-Tate weights.
This means that $[\rho_{v}]\in\mathcal{F}_{v}(A)$ if and only if for a
representative $\rho_{v}$, there exists
$g_{v}\in\operatorname{\mathsf{GL}}_{N}(A)$ such that
$g_{v}\cdot\rho{|_{G_{F_{v}}}}\cdot g_{v}^{-1}$ takes values in $B(A)$ and
that its reduction $\underline{\chi}_{\rho,v}\colon G_{F_{v}}\to T(A)$ modulo
$N(A)$ is a lifting of $\overline{\underline{\chi}}_{v}\colon G_{F_{v}}\to
T(k)$.

Let $\mu=(\mu_{v})_{v\in S_{p}}$ where
$\mu_{v}\colon\mathcal{O}_{F_{v}}^{\times}\to T(\mathcal{O})$ is a continuous
character lifting the character
$\bar{\underline{\chi}}_{v}\colon\mathcal{O}_{F_{v}}^{\times}\to T(k)$ given
by $\bar{\rho}|_{I_{v}}$. We define the subfunctor
$\mathcal{F}_{v}^{\operatorname{ord},\mu}\subset\mathcal{F}_{v}^{\operatorname{ord}}$
of ordinary deformations of weight $\mu$ to be given by $[\rho_{v}]$’s for
which
$\underline{\chi}_{\rho,v}\circ\operatorname{Art}_{F_{v}}|_{\mathcal{O}_{F_{v}}^{\times}}=\mu_{v}$
where $\mu_{v}$ is considered as taking values in $T(A)$ via the canonical
morphism $T(\mathcal{O})\to T(A)$.

In this paper, besides the sections dealing with Hida theory, we only consider
the case

$\mu_{v}=\underline{\chi}_{\pi,v}\circ\operatorname{Art}_{F_{v}}|_{\mathcal{O}_{F_{v}}^{\times}}.$

The homomorphism $\mu$ is then algebraic, given by the Hodge-Tate weights of
$\rho_{\pi_{v}}$.

In the Lemma below, we also consider the subfunctor
$\mathcal{F}_{v}^{\det,\operatorname{ord}}\subset\mathcal{F}_{v}^{\operatorname{ord}}$
where the determinant of the deformations is fixed equal to $\det\,\rho_{v}$.

######  Lemma 2.

For $v\in S_{p}$, under the condition (STDIST) (and $p>N$) the functors
$\mathcal{F}_{v}^{\det,\operatorname{ord}}$,
$\mathcal{F}_{v}^{\operatorname{ord}}$,
$\mathcal{F}_{v}^{\operatorname{ord},\mu}$ are pro-representable by a complete
noetherian local ring $R_{v}^{\det,\operatorname{ord}}$,
resp.$R_{v}^{\operatorname{ord}}$, $R_{v}^{\operatorname{ord},\mu}$. These
problems are unobstructed, hence formally smooth. In particular, one has
$R_{v}^{\operatorname{ord}}=R_{v}^{\det,\operatorname{ord}}[[T_{1},\ldots,T_{f}]]$
(where $f=[F_{v}\colon{\mathbb{Q}}_{p}]$ is the degree of $F_{v}$).

###### Proof.

Let ${\mathfrak{b}}$ be the Lie algebra of the standard Borel $B$ and
${\mathfrak{b}}^{\prime}$ the subalgebra of trace zero elements. To prove that
$\mathcal{F}_{v}^{\det,\operatorname{ord}}$, resp.
$\mathcal{F}_{v}^{\operatorname{ord},\mu}$, is proprepresentable and
unobstructed, it is enough to show that
$H^{i}(\Gamma_{v},{\mathfrak{b}}^{\prime})=0$ for $i\neq 1$. By Tate duality,
this follows from the strong distinguishability condition. Since $p>N$, any
$[\rho]\in\mathcal{F}_{v}^{\operatorname{ord}}(A)$ has a twist $rho_{\chi}$ by
a $p$-power order character $\chi$ whose determinant is equal to
$\det\,\rho_{\pi,v}$. This character $\chi$ factors through the universal
character $G_{F_{v}}\to\mathcal{O}[[T_{1},\ldots,T_{f}]]^{\times}$. Hence
$\mathcal{F}_{v}^{\operatorname{ord}}$ is proprepresentable by
$R_{v}^{\operatorname{ord}}=R_{v}^{\det,\operatorname{ord}}[[T_{1},\ldots,T_{f}]]$.

∎

Comment: The dimension of $R_{v}^{\operatorname{ord}}$ (given in [CHT08, Lemma
2.4.8]) will be recalculated later when applying the Poitou-Tate formula.

######  Corollary 1.

The functor $\mathcal{F}_{v}^{\operatorname{ord},\square}$ is pro-
representable by a formally smooth ring $R_{v}^{\operatorname{ord},\square}$.

###### Proof.

The morphism
$\mathcal{F}_{v}^{\operatorname{ord},\square}\to\mathcal{F}_{v}^{\operatorname{ord}}$
is a $\widehat{G^{ad}}$-torsor, hence is smooth and the base is smooth. ∎

In all sections below (except those dealing with Hida theory), we fix
$\mu_{v}=\underline{\chi}_{\pi,v}\circ\operatorname{Art}_{F_{v}}|_{\mathcal{O}_{F_{v}}^{\times}}$
for each $v\in Sp$. We call the deformation problem
$\mathcal{F}_{v}^{\operatorname{ord},\mu}$ the
$\underline{\chi}_{\pi}$-ordinary deformation problem, or for brevity the
ordinary deformation problem (with Hodge-Tate weights fixed by $\pi$). Hence
we write only $\mathcal{F}_{v}^{\operatorname{ord}}$, although $\mu$ is fixed
by $\mu=\underline{\chi}_{\pi}$.

####  2.1.3. The Fontaine-Laffaille case

Finally, for $v\in S_{p}$, we consider the problem
$\mathcal{F}_{v}^{FL,\square}$, resp. $\mathcal{F}_{v}^{FL}$, of Fontaine-
Laffaille liftings $\rho_{v}$, resp. deformations $[\rho_{v}]$ of
$\overline{\rho}_{v}$. This means that there exists a $\phi$-filtered
$A$-module $M$ free of rank $N$ over $A$, such that $\rho_{v}$ is isomorphic
to $V_{crys}(M)$.

######  Lemma 3.

The functor $\mathcal{F}_{v}^{FL,\square}$ is pro-representable by a formally
smooth ring $R_{v}^{FL,\square}$ isomorphic to a power series
$\mathcal{O}$-algebra in $N^{2}+[F_{v}\colon F]{N(N-1)\over 2}$ variables.

###### Proof.

This is [CHT08, Coroll.2.4.3]. ∎

###  2.2. The Theorem of Calegari-Geraghty

For $?=\operatorname{ord},FL$, let

$\mathcal{F}_{loc}=\prod_{v\in S\cup S_{p}}\mathcal{F}_{v}\quad\mbox{\rm and
}\mathcal{F}_{loc}^{min,?}=\prod_{v\in
S}\mathcal{F}_{v}^{min}\times\prod_{v\in S_{p}}\mathcal{F}_{v}^{?}$

We define the global deformation problem as the fiber product

$\mathcal{D}^{?}=\mathcal{F}_{\overline{\rho}}^{min,?}=\mathcal{F}_{\overline{\rho}}\times_{\mathcal{F}_{loc}}\mathcal{F}_{loc}^{min,?}.$

By Schlessinger’s criterion, assuming $p$-distinguishedness, resp.
$p$-smallness, the functor $\mathcal{D}^{\operatorname{ord}}$, resp.
$\mathcal{D}^{FL}$, is proprepresentable by a pair $(R^{?},\rho^{?})$ where
$?=\operatorname{ord},FL$ (see for instance [CHT08, Prop.2.2.9 and Lemma
2.4.1] for $?=FL$ and [CHT08, Section 2.4.4] or [Ti96, Chapt.6] for
$?=\operatorname{ord}$). For any proartinian $\mathcal{O}$-algebra $A$ and any
object $[\rho]\in\mathcal{D}^{?}(A)$ there is a unique local
$\mathcal{O}$-algebra homomorphism $\phi_{\rho}\colon R\to A$ such that
$\phi_{\rho}\circ\rho^{?}\equiv\rho$. Assuming $(LLC)$, we have
$[\rho_{\pi}]\in\mathcal{D}^{?}(\mathcal{O})$ and
$[\rho_{\mathfrak{m}}]\in\mathcal{D}^{?}(T)$ and we therfore have the
following commutative diagram of local surjective $\mathcal{O}$-algebra
homomorphisms.

$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\rho_{\mathfrak{m}}}}$$\scriptstyle{\phi_{\rho_{\pi}}}$$\textstyle{\operatorname{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\pi}}$$\textstyle{\mathcal{O}}$

Recall that the local-global compatihility condition $(LLC)$ for
$\rho_{\mathfrak{m}}$ has been proven (see [ACC+18]), at least if $F$ is not
imaginary quadratic.

Let now $S_{\infty}$ be the Taylor-Wiles power series ring and let
$R_{\infty}\to\operatorname{T}_{\infty}$ be the surjective homomorphism of
$S_{\infty}$-algebras introduced by Calegari-Geraghty using Taylor-Wiles
systems (see [CaGe18]), and recalled in the introduction.

######  Theorem 8.

([CaGe18]) Assume $\zeta_{p}\notin F$, $p>N$,
$(\operatorname{Gal}_{\mathfrak{m}})$, $(LLC)$ $(RLI)$, $(MIN)$, $(FL)$ or
$(ORD_{\pi})+(DIST)$. Then, the canonical map
$R_{\infty}^{?}\to\operatorname{T}_{\infty}$ is an isomorphism, and in
particular
$R^{?}_{\infty}\otimes_{S_{\infty}}\mathcal{O}=R^{?}\cong\operatorname{T}$.
Moreover, the module $\mathrm{H}^{\mathfrak{m}}_{q_{m}}$ is finite and free
over $\operatorname{T}$, and there is an isomorphism of graded modules:

$\mathrm{H}_{\bullet}^{\mathfrak{m}}\cong\mathrm{H}_{q_{0}}^{\mathfrak{m}}\otimes_{\operatorname{T}}\operatorname{Tor}_{\bullet}^{S_{\infty}}(\operatorname{T}_{\infty},\mathcal{O})$

in other words $\mathrm{H}_{\bullet}^{\mathfrak{m}}$ is a free graded
$\operatorname{Tor}_{\bullet}^{S_{\infty}}(\operatorname{T}_{\infty},\mathcal{O})$-module.

Remarks:

1) Poincaré duality shows that $\mathrm{H}^{\mathfrak{m}}_{q_{s}}$ is
isomorphic to $\operatorname{\mathsf{Hom}}(T,\mathcal{O})^{m}$ as $T$-module
(this reflects also that it is torsion free as $\mathcal{O}$-module).

2) for $G=\operatorname{\mathsf{GL}}(2)$ over a number field $F$, the rank $m$
is $2^{r_{1}}$ (this follows from calculations of
$({\mathfrak{g}},C_{\infty})$-cohomology, see [BW00] or [H94a]); hence
$m=2^{[F\colon{\mathbb{Q}}]}$ if $F$ is totally real, or $1$ if $F$ is CM.

In the sequel we will focus on the case
$G=\operatorname{\mathsf{GL}}(N)_{/F}$, with $F$ CM, quadratic over the
totally real field $F^{+}$, assuming $(Van_{\mathfrak{m}})$ and $(LLC)$. More
general cases are treated in [Cai21].

##  3\. Simplicial Deformation problems and tangent complexes

We follow [GV18, Sections 5 and 7] and [Cai21]. Given a category
$\mathcal{C}$, we denote by $\mathcal{C}^{s}$ the category of simplicial
objects of $\mathcal{C}$. Note any object $C$ of $\mathcal{C}$ defines a
simplicial object $\mathcal{F}_{C}\in\mathcal{C}^{s}$ by putting
$\mathcal{F}_{C}([n])=C$ and $\mathcal{F}_{C}(\phi)=\operatorname{Id}_{C}$ for
any $\phi\in\operatorname{\mathsf{Hom}}_{\Delta}([n],[m])$. Such an object is
called discrete. We will be mostly concerned with
$\mathcal{C}=\operatorname{Mod}_{A}$, $\operatorname{\mathsf{SETS}}$ and the
category ${}_{A}\\!\operatorname{CR}_{B}$ of pairs $(X,\pi)$ where $X$ is an
$A$-algebra and $\pi\colon X\to B$ is an $A$-algebra homomorphism. The
resulting simplicial categories are model categories (see [Cai21, 3.1.2]).
Note it is not the case for the subcategory
${}_{A}\\!\operatorname{Art}_{B}\subset{}_{A}\\!\operatorname{CR}_{B}$ of
simplicial artinian rings. In the sequel, we mostly consider $A=\mathcal{O}$
and $B=\mathcal{O}/(\varpi)=k$.

A functor
$\mathcal{F}\colon{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}\to\operatorname{\mathsf{SETS}}^{s}$
is called homotopy invariant if it sends weakly equivalent objects to weakly
equivalent objects. Let $\ast\in\operatorname{\mathsf{SETS}}^{s}$. A
simplicial deformation problem of $\ast$ is a homotopy invariant functor

$\mathcal{F}\colon{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}\to\operatorname{\mathsf{SETS}}^{s}$

such that $\mathcal{F}(k)=\\{\ast\\}$. A diagram

$\begin{array}[]{ccc}A_{1}\times_{A_{0}}^{h}A_{2}&\stackrel{{\scriptstyle\pi_{2}}}{{\to}}&A_{2}\\\
\pi_{1}\downarrow&&f_{2}\downarrow\\\ A_{1}&\stackrel{{\scriptstyle
f_{1}}}{{\to}}&A_{0}\end{array}$

in ${}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}$ is a homotopy pullback if
$f_{1}\circ\pi_{1}$ is homotopic to $f_{2}\circ\pi_{2}$. It is unique up to
weak equivalence. A simplicial deformation problem $\mathcal{F}$ of an object
$\ast$ is formally cohesive [GV18, Definition 3.8] if it is homotopy invariant
and if it preserves homotopy pullback. Let us give an example of such a
functor.

######  Definition 2.

Given two simplicial rings $\mathcal{R},\mathcal{R}^{\prime}$, the simplicial
set
$\operatorname{\mathsf{sHom}}_{\operatorname{CR}^{s}}(\mathcal{R},\mathcal{R}^{\prime})$
is defined as
$[p]\mapsto\operatorname{\mathsf{Hom}}_{\operatorname{CR}^{s}}(\mathcal{R},(\mathcal{R}^{\prime})^{\Delta[p]})$.
Here, for each $p$, $(\mathcal{R}^{\prime})^{\Delta[p]}$ denotes the
simplicial ring given by
$[q]\mapsto\operatorname{\mathsf{Hom}}_{\operatorname{\mathsf{SETS}}^{s}}(\Delta[p]\times\Delta[q],\mathcal{R}^{\prime})$;
as a simplicial ring, it is weakly equivalent to $\mathcal{R}^{\prime}$ (see
cite[Lemma 2.13]GV18). The structure of simplicial set on
$[p]\mapsto\operatorname{\mathsf{Hom}}_{\operatorname{CR}^{s}}(\mathcal{R},(\mathcal{R}^{\prime})^{\Delta[p]})$
is given by the natural face and degeneracy maps coming from those between the
$\Delta[p]$’s.

Recall that $\mathcal{R}\in{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}$ is
cofibrant if for any $n\geq 0$, $\mathcal{R}_{n}$ is a free
$\mathcal{O}$-algebra. Note this is only a sufficient condition. For precise
definition and properties see [Cai21, Section 2.1]. One can show that for any
simplicial ring $\mathcal{R}$, there exists a fibration which is a weak
equivalence $c(\mathcal{R})\to\mathcal{R}$ where $c(\mathcal{R})$ is a
cofibrant simplicial ring. One can either assume that the
$c(\mathcal{R})_{n}$’s are noetherian for all $n$’s, or that
$\mathcal{R}\mapsto c(\mathcal{R})$ is a functorial construction, but in
general not both. Any such $c(\mathcal{R})$ is called a cofibrant replacement
of $\mathcal{R}$.

Let $\mathcal{R}\in{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}$ be cofibrant;
then the functor
${}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}\to\operatorname{\mathsf{SETS}}^{s}$
given by

$A\mapsto\operatorname{\mathsf{sHom}}_{{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}}(\mathcal{R},A):{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}\to\operatorname{\mathsf{SETS}}^{s}$

is formally cohesive. Any pro-representable (see [GV18, 2.14 and 2.19] and
Section 3.2 below) functor $\mathcal{F}$ should be formally cohesive. J. Lurie
gave a criterion for (pro)representability of a simplicial deformation functor
$\mathcal{F}$ of, say, a point $\ast\in\operatorname{\mathsf{SETS}}^{s}$ (see
[GV18, 4.6]). The first (necessary) condition is formal cohesiveness. The
second involves its tangent complex ${\mathfrak{t}}\mathcal{F}$ which we
recall below.

###  3.1. Tangent complex

Let $A$ be a commutative ring and $Z$ be a commutative $A$-algebra. Then for
any $Z$-module $M$ and $X\in{}_{A}\\!\operatorname{CR}_{Z}$, we have natural
isomorphisms

$\operatorname{\mathsf{Hom}}_{{}_{A}\\!\operatorname{CR}_{Z}}(X,Z\oplus
M)\cong\operatorname{Der}_{A}(X,M)\cong\operatorname{\mathsf{Hom}}_{Z}(\Omega_{X/A}\otimes_{X}Z,M).$

So the functor $X\mapsto\Omega_{X/A}\otimes_{X}Z$ is left ajoint to the
functor $M\mapsto Z\oplus M$. The functors $M\mapsto Z\oplus M$,
$X\to\Omega_{X/A}\otimes_{X}Z$ both have extensions to functors between
simplicial categories, and we have natural isomorphism

$\operatorname{\mathsf{Hom}}_{{}_{A}\\!\operatorname{CR}^{s}_{Z}}(X,Z\oplus
M)\cong\operatorname{\mathsf{Hom}}_{\operatorname{Mod}^{s}_{Z}}(\Omega_{X/A}\otimes_{X}Z,M).$

The functor $M\mapsto Z\oplus M$ preserves fibrations and weak equivalences.
So the total left derived functor of its left adjoint functor
$X\mapsto\Omega_{X/A}\otimes_{X}Z$ exists [GJ10, II.7.3]. It is denoted
$X\mapsto L_{X/Z}$. In order to give a concrete description of $L_{X/Z}$ and
its fundamental property, one often uses the Dold-Kan equivalence. Let
$\operatorname{Ch}(Z)$, resp. $\operatorname{Ch}_{\geq 0}(Z)$, be the category
of chain complexes of $Z$-modules, resp. the subcategory of effective
complexes $(\ldots\to V_{n}\to\ldots\to V_{1}\to V_{0})$. Recall that the
Dold-Kan equivalence is a functor [GV18, Section 4.3.1] and [Cai21, 3.1.4]

$\\\ DK\colon\operatorname{Ch}_{\geq 0}(Z)\to\operatorname{Mod}^{s}_{Z}$

inducing an equivalence of categories that sends weak equivalences to quasi-
isomorphisms. It therefore provides an identification222When no confusion is
possible, we will doing so without specifying it. of the categories of
simplicial $Z$-modules $\operatorname{Mod}^{s}_{Z}$ with the category
$\operatorname{Ch}_{\geq 0}(Z)$ of non-negative chain complexes $(\ldots
V_{n}\to\ldots\ldots V_{1}\to V_{0})$ of $Z$-modules. Now, let
$X\in{}_{A}\\!\operatorname{CR}_{Z}$; to construct $L_{X/Z}$, we choose a
cofibrant replacement $c(X)\to X$ and form the simplicial module
$\Omega_{c(X)/A}\otimes_{A}Z$. By the Dold-Kan equivalence, it comes from a
non-negative chain complex, well defined up to quasi-isomorphism, denoted
$L_{X/Z}$.

Moreover, given $M\in\operatorname{Ch}_{\geq 0}(Z)$, we can form the
simplicial ring $Z\oplus\operatorname{DK}(M)$ where the ideal
$\operatorname{DK}(M)$ is of square zero. We have a weak equivalence of
functors of $M\in\operatorname{Ch}_{\geq 0}(Z)$:

$\operatorname{\mathsf{sHom}}_{{}_{A}\\!\operatorname{CR}^{s}_{Z}}(X,Z\oplus\\\
DK(M))\cong\operatorname{\mathsf{sHom}}_{\operatorname{Mod}^{s}_{Z}}(L_{X/Z},M).$

For $A=\mathcal{O}$ and $Z=k$, we will use the following definition.

######  Definition 3.

The $k$-cotangent complex functor $\mathcal{R}\mapsto L_{\mathcal{R}/k}$ is
the total left derived functor of the functor
$\mathcal{R}\mapsto\Omega_{\mathcal{R}/\mathcal{O}}\otimes_{\mathcal{R}}k$.
More precisely, for
$\mathcal{R}\in{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}$, we have a
simplicial $k$-module
$L_{\mathcal{R}/k}=\Omega_{c(\mathcal{R})/\mathcal{O}}\otimes_{c(\mathcal{R})}k$,
where $c(\mathcal{R})$ is a cofibrant replacement of $\mathcal{R}$ in the
model category ${}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}$, and
$L_{\mathcal{R}/k}$ is well-defined in the homotopy category.

The simplicial $k$-module
$\operatorname{\mathsf{sHom}}_{\operatorname{Mod}^{s}_{k}}(L_{\mathcal{R}/k},k)$
may be viewed as an element333Given a non-positive finite dimensional chain
complex $V\in\operatorname{Ch}_{\leq 0}(k)$, its $k$-dual $V^{\vee}$ is a
finite dimensional non-negative chain complex
$V^{\vee}\in\operatorname{Ch}_{\geq 0}(Z)$. of $\operatorname{Ch}_{\geq
0}(k)$, we denote it by ${\mathfrak{t}}\mathcal{R}$ and call it the tangent
complex of $\mathcal{R}$ over $k$. It is well defined in the derived category
of $k$-vector spaces. We label its terms either as
${\mathfrak{t}}^{(i)}\mathcal{R}$ ($i\in{\mathbb{Z}}_{\geq 0}$), or as
${\mathfrak{t}}_{(-i)}\mathcal{R}={\mathfrak{t}}^{(i)}\mathcal{R}$. With these
notations, we have, via Dold-Kan identification of simplicial modules and
chain complexes:
$\pi_{-i}{\mathfrak{t}}\mathcal{R}=H_{-i}({\mathfrak{t}}\mathcal{R})=H^{i}({\mathfrak{t}}\mathcal{R})$.

More generally, if $\mathcal{R}_{1}\to\mathcal{R}_{2}$ is a morphism of
objects
$\mathcal{R}_{1},\mathcal{R}_{2}\in{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}$,
we fix compatible cofibrant replacements $c(\mathcal{R}_{1})\to
c(\mathcal{R}_{2})$ in the model category
${}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}$. The relative $k$-cotangent
complex
$L_{\mathcal{R}_{2}/\mathcal{R}_{1}}=\Omega_{c(\mathcal{R}_{2})/c(\mathcal{R}_{1})}\otimes_{c(\mathcal{R}_{2})}k$
is independent of the choice of the cofibrant replacements, hence is a well-
defined object in the homotopy category. The simplicial $k$-module
$\operatorname{\mathsf{sHom}}_{\operatorname{Mod}^{s}_{k}}(L_{\mathcal{R}_{2}/\mathcal{R}_{1}},k)$
may be viewed as an element of $\operatorname{Ch}_{\leq 0}(k)$, we denote it
by ${\mathfrak{t}}(\mathcal{R}_{2},\mathcal{R}_{1})$ and call it the relative
tangent complex of $\mathcal{R}_{2}$ with respect to $\mathcal{R}_{1}$ over
$k$ (we won’t specify this below). It is well defined in the derived category
of $k$-vector spaces.

######  Lemma 4.

If $\mathcal{R}_{1}\to\mathcal{R}_{2}$ is a morphism of objects
$\mathcal{R}_{1},\mathcal{R}_{2}\in{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}$,
we have a distinguished triangle

${\mathfrak{t}}(\mathcal{R}_{2},\mathcal{R}_{1})\to{\mathfrak{t}}\mathcal{R}_{2}\to{\mathfrak{t}}{\mathcal{R}_{1}}\stackrel{{\scriptstyle+1}}{{\rightarrow}}$

hence a long exact sequence

$\ldots H^{i}({\mathfrak{t}}(\mathcal{R}_{2},\mathcal{R}_{1}))\to
H^{i}({\mathfrak{t}}\mathcal{R}_{2})\to
H^{i}({\mathfrak{t}}\mathcal{R}_{1})\to
H^{i+1}({\mathfrak{t}}(\mathcal{R}_{2},\mathcal{R}_{1}))\to\ldots$

###### Proof.

Apply the exact functor
$\operatorname{\mathsf{sHom}}_{\operatorname{Mod}^{s}_{k}}(-,k)$ to the
distinguished triangle

$L_{\mathcal{R}_{1}/k}\underline{\otimes}_{\mathcal{R}_{1}}k\to
L_{\mathcal{R}_{2}/k}\to
L_{\mathcal{R}_{2}/\mathcal{R}_{1}}{\rightarrow}(L_{\mathcal{R}_{1}/k}\underline{\otimes}_{\mathcal{R}_{1}}k)[1]$

similar to [COT, Section 90.7]. To obtain this distinguished triangle, first,
take a cofibrant replacement $c(\mathcal{R}_{1})\to c(\mathcal{R}_{2})$ of
$\mathcal{R}_{1}\to\mathcal{R}_{2}$, then form the exact sequence of
simplicial $c(\mathcal{R}_{2})$-modules

$0\to\Omega_{c(\mathcal{R}_{1})/\mathcal{O}}\underline{\otimes}_{c(\mathcal{R}_{1})}c(\mathcal{R}_{2})\to\Omega_{c(\mathcal{R}_{2})/\mathcal{O}}\to\Omega_{c(\mathcal{R}_{2})/c(\mathcal{R}_{1})}{\rightarrow}0$

where the first tensor product is taken for simplicial
$c(\mathcal{R}_{1})$-modules; then, apply $-\otimes_{c(\mathcal{R}_{2})}k$. ∎

Let $k[n]$ be the object of $\operatorname{Ch}_{\geq 0}(k)$ defined by $k$ in
degree $n$ and $0$ elsewhere. Then, we have
$\operatorname{\mathsf{sHom}}_{{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}}(\mathcal{R},k\oplus
DK(k[n]))\cong\operatorname{\mathsf{sHom}}_{\operatorname{Mod}^{s}_{k}}(L_{\mathcal{R}/k},DK(k[n]))$
for $\mathcal{R}$ cofibrant, and hence for $i\geq 0$, we have

$\pi_{i}\operatorname{\mathsf{sHom}}_{{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{k}}(\mathcal{R},k\oplus
DK(k[n]))\cong H^{i+n}({\mathfrak{t}}\mathcal{R}).$

######  Proposition 1.

Let
$\mathcal{F}:{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}\to\operatorname{\mathsf{SETS}}^{s}$
be a formally cohesive functor. Then there exists an object
${\mathfrak{t}}\mathcal{F}\in\operatorname{Ch}(k)$, unique up to weak
equivalence, such that for any non-positive finite dimensional chain complex
$V\in\operatorname{Ch}_{\geq 0}(k)$, we have

$\operatorname{\mathsf{sHom}}_{\operatorname{Ch}(k)}(V,{\mathfrak{t}}\mathcal{F})\cong\mathcal{F}(k\oplus\\\
DK(V^{\vee})).$

###### Proof.

See [Cai21, Corollary 3.43]. ∎

The chain complex ${\mathfrak{t}}\mathcal{F}$ is called the tangent complex of
$\mathcal{F}$. We denote the terms of this complex as
${\mathfrak{t}}_{i}\mathcal{F}$ (with degree $-1$ differential) for
$i\in{\mathbb{Z}}$, or
${\mathfrak{t}}^{(i)}\mathcal{F}={\mathfrak{t}}_{(-i)}\mathcal{F}$ (with
degree $+1$ differential). We also write
${\mathfrak{t}}^{i}\mathcal{F}=\mathrm{H}^{i}({\mathfrak{t}}\mathcal{F})=\mathrm{H}_{-i}({\mathfrak{t}}\mathcal{F})$.
If we let $V=k[-n]$ and take the $i$-th homotopy group, then we get
$\mathrm{H}^{i+n}{\mathfrak{t}}\mathcal{F}=\pi_{-i}\mathcal{F}(k\oplus
DK(k[n]))$ for any $i,n\geq 0$. We write ${\mathfrak{t}}^{i}\mathcal{F}$ to
abbreviate
$\mathrm{H}^{i}({\mathfrak{t}}\mathcal{F})=\pi_{i}{\mathfrak{t}}\mathcal{F}$
(the last equality is via Dold-Kan, viewing ${\mathfrak{t}}\mathcal{F}$ as a
simplicial $k$-vector space). To end this section, we recall the important
conservativity property of the tangent complex functor.

######  Lemma 5.

Let $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ be formally cohesive functors from
${}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}$ to
$\operatorname{\mathsf{sSETS}}$. Then a natural transformation
$T:\mathcal{F}_{1}\to\mathcal{F}_{2}$ is a weak equivalence if and only if $T$
induces isomorphisms
${\mathfrak{t}}^{i}\mathcal{F}_{1}\cong{\mathfrak{t}}^{i}\mathcal{F}_{2}$ for
all $i$.

###### Proof.

See [Cai21, Corollary 3.46]. ∎

###  3.2. Representability

Given a projective system $\mathcal{R}=(\mathcal{R}_{\alpha})_{\alpha}$ with
$\mathcal{R}_{\alpha}\in{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}$, we put
for any $A\in{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}$

$\operatorname{\mathsf{sHom}}(\mathcal{R},A)=\operatorname{holim}_{\alpha}\operatorname{\mathsf{sHom}}(\mathcal{R}_{\alpha},A)$

######  Definition 4.

A functor
$\mathcal{F}\colon{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}\to\operatorname{\mathsf{sSETS}}$
is proprepresentable if there exists a projective system of cofibrant
simplical artinian rings $\mathcal{R}=(\mathcal{R}_{\alpha})_{\alpha}$ with
$\mathcal{R}_{\alpha}\in{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}$ and a
functorial weak equivalence:

$\operatorname{\mathsf{sHom}}(\mathcal{R},A)\to\mathcal{F}(A)$

for all $A\in{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}$.

Note that ${\mathfrak{t}}^{i}\mathcal{R}=H_{-i}({\mathfrak{t}}\mathcal{R})$
vanishes for $i<0$. So if a formally cohesive functor
$\mathcal{F}:{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}\to\operatorname{\mathsf{SETS}}^{s}$
is pro-representable, then ${\mathfrak{t}}^{i}\mathcal{F}$ should vanishes for
$i<0$. The converse statement is proved by Lurie.

######  Theorem 9 (Lurie’s derived Schlessinger criterion).

Let $\mathcal{F}$ be a formally cohesive functor. Then $\mathcal{F}$ is pro-
representable if and only if ${\mathfrak{t}}^{i}\mathcal{F}$ vanishes for each
$i<0$. In addition, if all the $k$-vector spaces
${\mathfrak{t}}^{i}\mathcal{F}$ are finite, the pro-artinian ring
$\mathcal{R}$ has a countable indexing category.

###  3.3. Simplicial Galois deformation functors

In this section we consider a smooth group scheme $G$ defined over
$\mathcal{O}$. For our application, $G=\operatorname{\mathsf{GL}}_{N}$. Given
an $\mathcal{O}$-algebra $A$, let ${\bf B}G(A)$ be the simplicial set
associated to the group $G(A)$ by the bar construction. By definition, it is
given by $[p]\mapsto N_{p}G(A)=G(A)^{p}$, with standard face and degeneracy
maps [Wei94, Example 8.1.7]. Note that for any fixed $p$,
$N_{p}G(A)=\operatorname{\mathsf{Hom}}_{rings}(\mathcal{O}_{G}^{\otimes
p},A)$. We can therefore say that the functor $N_{p}G\colon A\mapsto
N_{p}G(A)$ is represented by the ring
$\mathcal{O}_{N_{p}G}=\mathcal{O}_{G}^{\otimes p}$.

If now $A$ is a simplicial $\mathcal{O}$-algebra, we can define a bisimplicial
set $\operatorname{Bi}^{naive}_{G}(A)$, i.e. a contravariant functor

$\operatorname{Bi}^{naive}_{G}(A)\colon\Delta\times\Delta\to\operatorname{\mathsf{SETS}},\quad([p],[q])\mapsto\operatorname{\mathsf{Hom}}_{\operatorname{CR}}(\mathcal{O}_{N_{p}G},A_{q})=G(A_{q})^{p}$

where the $p$-degeneracy and $p$-face maps are given by the codegeneracy and
coface maps of $\mathcal{O}_{N_{\bullet}G}$ and the $q$-degeneracy and
$q$-face maps are given by the degeneracy and face maps of $A$. Recall that
given two simplicial rings $R,R^{\prime}$, we defined in Definition 2 the
simplicial set $\operatorname{\mathsf{sHom}}(R,R^{\prime})$ as
$[p]\mapsto\operatorname{\mathsf{Hom}}_{\operatorname{CR}^{s}}(R,(R^{\prime})^{\Delta[p]})$
with face and degeneracy maps coming from those between the $\Delta[p]$’s. We
can consider for each $p$ the bisimplicial set $\operatorname{Bi}_{G}(A)$
given by

$([p],[q])\mapsto\operatorname{\mathsf{Hom}}_{\operatorname{CR}^{s}}(c(\mathcal{O}_{N_{p}G}),A^{\Delta[q]})$

where $c(\mathcal{O}_{N_{p}G})$ denotes a cofibrant replacement of
$\mathcal{O}_{N_{p}G}$, with degeneracy and face maps as for the bisimplicial
set $\operatorname{Bi}^{naive}_{G}(A)$. Note that the natural morphism of
simplicial rings $A_{q}\to A^{\Delta[q]}$ induces an injective morphism of
bisimplicial sets

$\operatorname{Bi}^{naive}_{G}(A)\hookrightarrow\operatorname{Bi}_{G}(A).$

Consider the simplicial set ${\bf
B}^{\prime}G(A)=\operatorname{hocolim}_{q}\operatorname{Bi}_{G}(A)_{q}$. It
may not be fibrant. So, we fix a fibrant replacement

${\bf B}G(A)=F({\bf B}^{\prime}G(A)).$

For $A$ homotopically discrete, this obviously generalizes the usual
definition of ${\bf B}G(\pi_{0}(A))$.

Remark: Recall that in [GV18, Definition 5.1], the authors defined a
simplicial set ${\bf B}^{GV}G(A)$ as the fibrant replacement $Ex^{\infty}$ of
the simplicial set

$[p]\mapsto([p],[p])\mapsto\operatorname{Bi}_{G}(A)([p],[p])=\operatorname{\mathsf{Hom}}_{\operatorname{CR}^{s}}(c(\mathcal{O}_{N_{p}G}),A^{\Delta[p]}).$

The two definitions are weakly equivalent but there is a slight difference: in
our definition, we have a canonical fibration ${\bf B}G(A)\to{\bf B}G(k)$,
while for ${\bf B}G^{GV}(A)$, one has a canonical fibration ${\bf B}G(A)\to
Ex^{\infty}{\bf B}G(k)$.

The key properties of this construction are that:

  * •

The functor ${\bf B}G$ is invariant by homotopy : if $A\to B$ is a weak
equivalence, the resulting morphism of simplicial sets ${\bf B}G(A)\to{\bf
B}G(B)$ also is, by smoothness of $\mathcal{O}_{G}$ (see proof of [GV18,
Corollary 5.3]).

  * •

It preserves homotopy pullbacks.

  * •

The ${\bf B}G$ construction is functorial in $G$.

Let us write the $S\cup S_{p}$-ramified Galois group $\Gamma=G_{F,S\cup
S_{p}}$ as inverse limit of finite Galois groups
$\Gamma_{\alpha}=\operatorname{Gal}(F_{\alpha}/F)$, where $F_{\alpha}/F$ is a
finite $S\cup S_{p}$-ramified extension. Let ${\bf B}\Gamma$ be the
prosimplicial set given by the projective system $({\bf
B}\Gamma_{\alpha})_{\alpha}$ of simplicial sets ${\bf B}\Gamma_{\alpha}$. Note
that the residual Galois representation

$\overline{\rho}\colon\Gamma\to G(k)$

gives rise to an element $[\overline{\rho}]$ of
$\operatorname{\mathsf{sHom}}({\bf B}\Gamma_{\alpha},{\bf B}G(k))$ for some
$\alpha$. A natural idea to define the analogue of the classical deformation
functor

$\mathcal{F}_{\overline{\rho}}\colon{}_{\mathcal{O}}\\!\operatorname{Art}_{k}\to\operatorname{\mathsf{SETS}},A\mapsto\operatorname{\mathsf{Hom}}_{\overline{\rho}}(\Gamma,G(A))/adG(A)$

is to define for each $A\in
Ob({}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k})$, and any $\alpha$, the
simplicial set

$\mathcal{F}^{s}_{\alpha,\overline{\rho}}(A)=\operatorname{\mathsf{sHom}}_{[\overline{\rho}]}({\bf
B}\Gamma_{\alpha},{\bf B}G(A))$

which is the homotopy fiber in

$\mathcal{F}^{s}_{\alpha}(A)=\operatorname{\mathsf{sHom}}({\bf
B}\Gamma_{\alpha},{\bf B}G(A))$

of $[\overline{\rho}]\in\operatorname{\mathsf{sHom}}({\bf
B}\Gamma_{\alpha},{\bf B}G(k))$. Then, one would pass to the inductive limit
over $\alpha$ to obtain $\mathcal{F}^{s}_{\overline{\rho}}$ as the homotopy
fiber at $[\overline{\rho}]$ of

$\mathcal{F}^{s}=\varinjlim_{\alpha}\mathcal{F}^{s}_{\alpha}.$

It is homotopy invariant and preserves homotopy pullbacks. However, as
explained in [GV18, Section 5.4], when the center $Z$ of $G$ is non trivial,
the functor $\mathcal{F}^{s}_{\overline{\rho}}$ cannot be representable
because $Z$ gives rise to non trivial automorphisms. It can be modified in two
different ways to be made pro-representable: one is to fix the determinant

${\bf B}\det\colon{\bf B}G(A)\to{\bf B}{{\mathbb{G}}}_{m}(A)$

of the simplicial deformations, the other is to "quotient the action of $G$ by
the center $Z$" (with or without fixing the determinant) as in [GV18, Section
5.4]. The two modifications will be denoted
$\mathcal{F}^{s}_{\det,\overline{\rho}}$ and
$\mathcal{F}^{s}_{Z,\overline{\rho}}$. Let us recall their constructions.

###  3.4. Modification by the center

We first define $\mathcal{F}^{s}_{\det,\overline{\rho}}$.

Let $\rho_{\pi}\colon\Gamma\to G(\mathcal{O})$ be the Galois representation
that we fixed. For any $A\in
Ob({}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k})$, we have a morphism ${\bf
B}G(\mathcal{O})\to{\bf B}G(A)$. On the other hand, the determinant morphism
induces morphisms (both denoted by $\det$) of simplicial sets

${\bf B}G(A)\to{\bf B}{{\mathbb{G}}}_{m}(A)\quad\mbox{\rm
and}\,\mathcal{F}^{s}_{G}(A)=\operatorname{\mathsf{sHom}}({\bf B}\Gamma,{\bf
B}G(A))\to\mathcal{F}^{s}_{{{\mathbb{G}}}_{m}}(A)=\operatorname{\mathsf{sHom}}({\bf
B}\Gamma,{\bf B}{{\mathbb{G}}}_{m}(A))$

We recall now the definition of $\mathcal{F}^{s}_{Z,\overline{\rho}}$ ([GV18,
Section 5.4]). Consider the short exact sequence

$1\to Z(A)\to G(A)\to PG(A)\to 1$

It gives rise to a fibration sequence (see the section 3.1.3 of [Cai21]).

${\bf B}G(A)\to{\bf B}PG(A)\to{\bf B}^{2}Z(A).$

Let $\ast=\pi_{0}{\bf B}\Gamma$. Then the functor $\mathcal{F}^{s}_{Z}$ is
defined by the homotopy pullback diagram

$\begin{array}[]{ccc}\mathcal{F}^{s}_{Z}(A)&\to&\operatorname{\mathsf{Hom}}(\ast,{\bf
B}^{2}Z(A)\\\ \downarrow&&\downarrow\\\
\mathcal{F}^{s}(A)&\to&\operatorname{\mathsf{Hom}}({\bf B}\Gamma,{\bf
B}^{2}Z(A))\end{array}$

Its crucial property is that there is a functorial fibration sequence:

$\operatorname{\mathsf{sHom}}(\ast,{\bf
B}Z(A))\to\operatorname{\mathsf{sHom}}({\bf B}\Gamma,{\bf
B}G(A))\to\mathcal{F}^{s}_{Z}(A)$

We then define $\mathcal{F}^{s}_{Z,\overline{\rho}}$ as the homotopy fiber of
$[\overline{\rho}]$. It follows from the remarks above that

######  Lemma 6.

The functor $\mathcal{F}^{s}_{Z,\overline{\rho}}$ is homotopy invariant and
preserves homotopy pullbacks, hence is formally cohesive.

Let $\mathcal{F}^{s,S\cup
S_{p}}_{Z,\overline{\rho}}\colon{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}\to\operatorname{\mathsf{sSETS}}$
be the functor given by modifying as above the functor

$\mathcal{F}^{s}_{\overline{\rho}}(A)={\mathfrak{s}}_{[\overline{\rho}]}({\bf
B}\Gamma,{\bf
B}G(A))=\operatorname{holim}_{\alpha}\operatorname{\mathsf{sHom}}_{[\overline{\rho}]}({\bf
B}\Gamma_{\alpha},{\bf B}G(A))$

The functor $\mathcal{F}^{s,S\cup S_{p}}_{Z,\overline{\rho}}$ is called the
functor of simplicial deformations of $\overline{\rho}$ unramified outside
$S\cup S_{p}$. To study the representability of this deformation functor, we
now need to determine its tangent complex. The tangent complex of
$\mathcal{F}^{s}_{\alpha}$ can be computed as a Cech complex $C^{\bullet}({\bf
B}\Gamma_{\alpha},{\mathfrak{t}}{\bf B}G)$ ([GV18, Section 4.7] and [GV18,
Example 4.38]). Let ${\mathfrak{g}}_{k}=\operatorname{Lie}_{k}G$ be the Lie
algebra of $G$. The tangent complex ${\mathfrak{t}}{\bf B}G$ is concentrated
in degree $1$ and one has ${\mathfrak{t}}{\bf B}G={\mathfrak{g}}_{k}[1]$
([GV18, Lemma 5.5]). Therefore

${\mathfrak{t}}\mathcal{F}^{s}_{\alpha}=C^{\bullet+1}({\bf
B}\Gamma_{\alpha},{\mathfrak{g}}_{k})$

Let ${\mathfrak{z}}_{k}$ be the center of ${\mathfrak{g}}_{k}$. Then, the
functorial fibration 3.4 provides a distinguished triangle

$C^{\bullet+1}(\ast,{\mathfrak{z}}_{k})\to C^{\ast+1}({\bf
B}\Gamma,{\mathfrak{g}}_{k})\to{\mathfrak{t}}\mathcal{F}^{s}_{Z,\bar{\rho}}.$

From these facts it follows that the Cech complex $C^{\bullet}({\bf
B}\Gamma,{\mathfrak{t}}{\bf B}G)$ can be computed as the Galois cohomology
standard complex (see [Cai21]). Then by [GV18, Lemma 4.30, (iv)] (Mayer-
Vietoris long exact sequence) we have the following key proposition:

######  Proposition 2.

Suppose that $\mathrm{H}^{0}(\Gamma,{\mathfrak{g}}_{k})={\mathfrak{z}}_{k}$.
Then, the homology groups of the tangent complex
${\mathfrak{t}}\mathcal{F}^{s}_{Z,\bar{\rho}}$ satisfy
${\mathfrak{t}}^{i}\mathcal{F}^{s}_{Z,\bar{\rho}}=0$ when $i\leq 0$, and
${\mathfrak{t}}^{i}\mathcal{F}_{Z,\bar{\rho}}=\mathrm{H}^{i+1}(F_{S}/F,\operatorname{Ad}\bar{\rho})$
when $i\geq 0$.

and its corollary:

######  Corollary 2.

Suppose that the adjoint action of $\bar{\rho}$ fixes precisely the center of
the Lie algebra of $G(k)$. The functor $\mathcal{F}^{s}_{Z,\bar{\rho}}$ is
pro-representable by a simplicial pro-artinian ring $\mathcal{R}$. This means
that there exists a functorial weak equivalence

$\operatorname{\mathsf{sHom}}(\mathcal{R},A)\to\mathcal{F}^{s}_{Z,\bar{\rho}}(A)$

Note that, given a profinite group $\Gamma$, the question of
prorepresentability of the functor

$A\mapsto\mathcal{F}_{\bar{\rho}}^{s}(A)=\operatorname{\mathsf{sHom}}_{[\overline{\rho}]}({\bf
B}\Gamma,{\bf B}G(A))$

by a proartinian simplicial ring $\mathcal{R}$ can only be posed in terms of
the existence of a functorial weak equivalence

$\operatorname{\mathsf{sHom}}_{pro\operatorname{Art}^{s}_{k}}(R,A)\to\mathcal{F}_{\bar{\rho}}^{s}(A)$

and not of a functorial isomorphism, because even is $A$ is discrete, there is
a bijection between the set of conjugacy classes of homomorphisms $\Gamma\to
G(A)$ and the set of homotopy classes of the simplicial set of morphisms of
prosimplicial sets ${\bf B}\Gamma\to{\bf B}G(A)$, not with the simplicial set
of morphisms of prosimplicial sets itself.

######  Lemma 7.

The functor

$\pi_{0}\mathcal{F}^{s}_{Z,\overline{\rho}}\colon{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}\to\operatorname{\mathsf{SETS}}\quad
A\mapsto\pi_{0}\mathcal{F}^{s}_{Z,\overline{\rho}}(A)$

coincides with Mazur’s functor of $S\cup S_{p}$-ramified deformations of
$\overline{\rho}$.

###### Proof.

This is [GV18, Lemma 7.1]. It follows from the fact that for a discrete
artinian ring $A$, ${\bf B}G(A)$ is weakly equivalent to the classical
classifying space of $G(\pi_{0}(A))$ defined in [Wei94, Example 8.1.7].
Therefore $\pi_{0}(\operatorname{\mathsf{sHom}}({\bf B}\Gamma,{\bf B}G(A)))$
is $\operatorname{\mathsf{Hom}}({\bf B}\Gamma,{\bf B}G(\pi_{0}(A)))$, which is
naturally $\operatorname{\mathsf{Hom}}_{cont}(\Gamma,G(A))/G(A)$. By the
definition of $\mathcal{F}^{s}_{Z,\overline{\rho}}$, this implies that
$\pi_{0}\mathcal{F}^{s}_{Z,\overline{\rho}}(A)$ is naturally
$\operatorname{\mathsf{Hom}}_{cont}(\Gamma,G(A))/PG(A)=\operatorname{\mathsf{Hom}}_{cont}(\Gamma,G(A))/G(A)$,
as desired. ∎

###  3.5. Simplicial Galois deformation functors with local conditions

Let us define simplicial local deformation functors $\mathcal{F}_{v}^{s,?}$.
For $v\in S\cup S_{p}$, let us write $\Gamma_{v}=G_{F_{v}}$ as inverse limit
of finite Galois groups $\operatorname{Gal}(F^{\prime}_{v}/F_{v})$. Recall
that for $?=min,\operatorname{ord},\operatorname{ord}-\mu,FL$, under suitable
assumptions we have seen (Lemma 1, Lemma 2, Lemma 3, that the ring
$R_{v}^{?,\square}$ representing $\mathcal{F}_{v}^{?,\square}$ is formally
smooth. The functor $\mathcal{F}_{v}^{s}$ of arbitrary simplicial deformations
of $\overline{\rho}|_{\Gamma_{v}}$ is given by

$\mathcal{F}^{s}_{v}(A)=\operatorname{\mathsf{sHom}}_{\overline{\rho}_{v}}({\bf
B}\Gamma_{v},{\bf
B}G(A))=\operatorname{holim}_{F^{\prime}_{v}}\operatorname{\mathsf{sHom}}_{\overline{\rho}_{v}}({\bf
B}\operatorname{Gal}(F^{\prime}_{v}/F_{v}),{\bf B}G(A))$

We denote by $\mathcal{F}^{s}_{v,Z}$ its modification by the center as in
[GV18, Section 5.4] (or Section 2.4 above). Note that by [GV18, Lemma 5.2], we
have
${\mathfrak{t}}^{-1}\mathcal{F}_{v,Z}^{s}=\operatorname{Coker}(\mathrm{H}^{0}(\\{1\\},{\mathfrak{z}})\to\mathrm{H}^{0}(\Gamma_{v},{\mathfrak{g}}_{k}))$
which doesn’t vanish (except in certain FL cases). Therefore, by the converse
of Lurie’s criterion (Theorem 9 above), $\mathcal{F}_{v,Z}^{s}$ is not pro-
representable. However, by [GV18, Lemma 5.2] we have

######  Lemma 8.

For any $i\geq 0$,
${\mathfrak{t}}^{i}\mathcal{F}_{v,Z}^{s}={\mathfrak{t}}^{i}\mathcal{F}_{v}^{s}\cong\mathrm{H}^{i+1}(\Gamma_{v},{\mathfrak{g}}_{k}))$

For $?=min,\operatorname{ord},FL$, following [GV18, Section 9.2], we shall
define local deformation functors $\mathcal{F}_{v}^{s,?}$ with morphisms

$\mathcal{F}_{v}^{s,?}\to\mathcal{F}_{v}^{s}$

which induce embeddings of the tangent complexes. They are defined as "derived
quotients" of functors $\mathcal{F}_{v}^{s,?,\square}$ by the adjoint action
of $G$. Recall that for a simplicial set $X$ with action of a group $G$, the
simplicial quotient $G\backslash X$ is defined as the simplicial set
$N(\ast,G,X)$ given by the diagonal of the bisimplicial set
$([p],[q])\mapsto\ast\times G^{p}\times X_{q}$ where for $q$ fixed, the
$p$-faces and degeneracy maps are given by the bar construction $[p]\mapsto
N_{p}(\ast,G,X_{q})=\ast\times G^{p}\times X_{q}$ while the $q$-ones are given
by the simplicial set $X$. We now form, so to speak, a formally cohesive
replacement of the simplicial analogue of the quotient functor

$A\mapsto\widehat{G}(A)\backslash\mathcal{F}_{v,\ast}^{\square,?}(A).$

Namely, for each simplicial $\mathcal{O}$-algebra $A$ in
${}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}$, we define the simplicial set
$\mathcal{F}_{v}^{?}(A)$ as

$[p]\mapsto\operatorname{\mathsf{Hom}}_{{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}}(c(R_{v}^{?,\square}\otimes\mathcal{O}_{N_{p}G}),A^{\Delta[p]})$

By the use of the cofibrant replacement of
$R_{v}^{?,\square}\otimes\mathcal{O}_{N_{p}G}$, we know that
$\mathcal{F}_{v}^{?}$ is formally cohesive (by [Cai21, Example 2.2.9]). The
existence of the natural transformation of functors

$\mathcal{F}_{v}^{s,?}\to\mathcal{F}_{v}^{s}$

is established in great generality (which includes $?=min,ord$) in [Cai21,
Proposition 5.15].

######  Lemma 9.

We have a fibration

$\mathcal{F}_{v}^{s,\square,?}\to\mathcal{F}_{v}^{s,?}\to\operatorname{\mathsf{hofib}}_{\ast}({\bf
B}G(-)\to BG(k)).$

###### Proof.

By definition (see [GV18, Definition 5.4 (i)]),
$\mathcal{F}^{s,?,\square}_{v}(A)$ is the homotopy fiber of the composed
morphism

$\mathcal{F}^{s,?,\square}_{v}(A)\to\operatorname{\mathsf{sHom}}_{[\overline{\rho}_{v}]}(B\Gamma_{v},{\bf
B}G(A))\to\operatorname{\mathsf{hofib}}({\bf B}G(A)\to{\bf B}G(k))$

where the second map is the evaluation on the base point $e=(e_{n})_{n}\in
B\Gamma_{v}$ (with $e_{n}=(e,\ldots,e)\in\Gamma_{v}^{n}$). By definition,
$\mathcal{F}^{s,?}_{v}(A)$ is the diagonal of the bisimplicial set

$[p]\mapsto\operatorname{\mathsf{sHom}}_{{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}}(c(\mathcal{O}_{N_{p}G}\otimes
R^{?,\square}_{v}),A)\simeq\prod_{i=1}^{p}\operatorname{\mathsf{sHom}}_{{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}}(c(\mathcal{O}_{G}),A)\times\mathcal{F}_{v}^{s,\square,?}(A).$

Similarly, $\operatorname{\mathsf{hofib}}_{\ast}({\bf B}G(A)\to BG(k))$
(homotopy fiber of the standard marked point in $BG(k)$) is the diagonal of
the bisimplicial set

$[p]\mapsto\operatorname{\mathsf{sHom}}_{{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}}(c(\mathcal{O}_{N_{p}G}),A)\simeq\prod_{i=1}^{p}\operatorname{\mathsf{sHom}}_{{}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k}}(c(\mathcal{O}_{G}),A).$

Therefore, the morphism
$s\mathcal{F}^{s,\square,?}_{v}(A)\to\operatorname{\mathsf{sHom}}_{[\overline{\rho}_{v}]}(B\Gamma_{v},{\bf
B}G(A))$ factors through
$s\mathcal{F}^{s,\square,?}_{v}(A)\to\mathcal{F}^{s,?}_{v}(A)$. This gives the
desired fibration. ∎

######  Remark 1.

Note that in the simplicial context, the classical fibration
$\widehat{G}(A)\to\mathcal{F}_{v}^{\square,?}(A)\to\mathcal{F}_{v}^{?}(A)$ is
replaced by

$\mathcal{F}_{v}^{s,\square,?}\to\mathcal{F}_{v}^{s,?}\to\operatorname{\mathsf{hofib}}_{\ast}({\bf
B}G(-)\to BG(k))$

where, for a classical ring $A$, $\operatorname{\mathsf{hofib}}_{\ast}({\bf
B}G(A)\to BG(k))$ is to be thought of as $B\widehat{G}(A)[1]$.

Let us put

$\mathrm{H}^{1}_{min}(\Gamma_{v},{\mathfrak{g}}_{k}):=\mathrm{H}^{1}_{unr}(\Gamma_{v},{\mathfrak{g}}_{k}),\quad\mathrm{H}^{1}_{\operatorname{ord}}(\Gamma_{v},{\mathfrak{g}}_{k}):=\operatorname{Im}(L^{\prime}_{v}\to\mathrm{H}^{1}(\Gamma_{v},{\mathfrak{g}}_{k})),$

where
$L^{\prime}_{v}=\operatorname{Ker}(\mathrm{H}^{1}(\Gamma_{v},{\mathfrak{b}})\to\mathrm{H}^{1}(I_{v},{\mathfrak{b}}/{\mathfrak{n}}))$,
and

$\mathrm{H}^{1}_{FL}(\Gamma_{v},{\mathfrak{g}}_{k})=\mathrm{H}^{1}_{f}(\Gamma_{v},{\mathfrak{g}}_{k}).$

By the long exact sequence above, we conclude:

######  Lemma 10.

For $?=min,\operatorname{ord},\operatorname{ord}-\mu,FL$, we have

  1. (1)

${\mathfrak{t}}^{-1}\mathcal{F}_{v}^{s,?}\cong\mathrm{H}^{0}(\Gamma_{v},{\mathfrak{g}}_{k})$,

  2. (2)

${\mathfrak{t}}^{0}\mathcal{F}_{v}^{s,?}\cong\mathrm{H}^{1}_{?}(\Gamma_{v},{\mathfrak{g}}_{k})$,

  3. (3)

${\mathfrak{t}}^{i}\mathcal{F}_{v}^{s,?}=0$ for $i\geq 1$.

###### Proof.

Recall that for any deformation functor $\mathcal{F}$,
${\mathfrak{t}}^{n-i}\mathcal{F}\cong\pi_{i}\mathcal{F}(k\oplus k[n])$. By
Lemma 4.30 (4) of [GV18], or [GJ10, Lemma 7.3], the fibration of Lemma 9 gives
rise to a long exact sequence

$\displaystyle
0\to{\mathfrak{t}}^{-1}s\mathcal{F}_{v}^{s,?}\to{\mathfrak{t}}^{-1}\operatorname{\mathsf{hofib}}_{\ast}({\bf
B}G(-)\to BG(k))$ $\displaystyle\to$
$\displaystyle{\mathfrak{t}}^{0}\mathcal{F}_{v}^{s,\square,?}\to{\mathfrak{t}}^{0}s\mathcal{F}_{v}^{s,?}\to
0\to
t^{1}\mathcal{F}^{s,\square,?}\to{\mathfrak{t}}^{1}s\mathcal{F}_{v}^{s,?}\to
0$

(we have used that $\operatorname{\mathsf{hofib}}_{\ast}(BG(k[\epsilon])\to
BG(k))$ is a $K(\pi,1)$). Note that
${\mathfrak{t}}^{-1}\operatorname{\mathsf{hofib}}_{\ast}({\bf B}G(-)\to
BG(k))\cong\pi_{1}\operatorname{\mathsf{hofib}}_{\ast}(BG(k[\epsilon])\to
BG(k))$, and $\pi_{1}\operatorname{\mathsf{hofib}}_{\ast}(BG(k[\epsilon])\to
BG(k))$ is $\operatorname{Ker}(G(k[\epsilon])\to G(k))={\mathfrak{g}}_{k}$.
Therefore, the tangent spaces ${\mathfrak{t}}^{i}\mathcal{F}_{v}^{s,?}$ can be
calculated as follows. Since $\mathcal{F}_{v}^{s,\square,?}$ is prorepresented
by $c(R_{v}^{\square,?})$ which is weakly equivalent to the discrete ring
$R_{v}^{\square,?}$, its tangent complex is given by

${\mathfrak{t}}\mathcal{F}_{v}^{s,\square,?}={\mathfrak{t}}R_{v}^{\square,?}$

Moreover, since $R_{v}^{\square,?}$ is formally smooth, we know that
${\mathfrak{t}}R_{v}^{\square,?}$ is quasi-isomorphic to the complex
${\mathfrak{t}}^{0}R_{v}^{\square,?}[0]$ concentrated in degree $0$. It is
well-known that
${\mathfrak{t}}^{0}R_{v}^{\square,?}=Z^{1}_{?}(\Gamma_{v},{\mathfrak{g}}_{k})$.
Hence we can rewrite the long exact sequence above as

$\displaystyle
0\to{\mathfrak{t}}^{-1}\mathcal{F}_{v}^{s,?}\to{\mathfrak{g}}_{k}\to
Z^{1}_{?}(\Gamma_{v},{\mathfrak{g}}_{k})\to{\mathfrak{t}}^{0}\mathcal{F}_{v}^{s,?}\to
0\to 0\to{\mathfrak{t}}^{1}\mathcal{F}_{v}^{s,?}\to 0$

This yields the Lemma.

∎

As in the global case, we define also the deformation functors
$\mathcal{F}_{v,Z}^{s}$ and $\mathcal{F}_{v,Z}^{s,?}$ by modifying the center
$Z$. They are inserted in a fibration

$\operatorname{\mathsf{sHom}}(\ast,{\bf
B}Z)\to\mathcal{F}_{v}^{s,?}\to\mathcal{F}_{v,Z}^{s,?}$

similar to (3.4).

For $v\in S_{p}$ and $?=\operatorname{ord}$, we can give a more concrete
alternative definition of the local nearly ordinary deformation functor. We
define $\widetilde{\mathcal{F}}^{s,\operatorname{ord}}_{v}(A)$ as the set of
simplicial deformations $\phi\in\operatorname{\mathsf{sHom}}({\bf
B}G_{F_{v}},{\bf B}G(A))$ of $\overline{\rho}|_{G_{F_{v}}}$, which factor
through the morphism of simplicial sets ${\bf B}B(A)\to{\bf B}G(A)$ associated
to the inclusion of the standard Borel $B\subset G$. for
$?=\operatorname{ord}-\mu$, let
$\operatorname{Def}^{s}_{v,T}=\operatorname{\mathsf{sHom}}_{\operatorname{\mathsf{sSETS}}{}_{/BT(k)}}(BI_{v},{\bf
B}T(-))$ be the derived deformation functor of $\bar{\chi}_{v}|_{I_{v}}\colon
I_{v}\to T(k)$. Then there is a natural transformation
$\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord}}\to\operatorname{Def}^{s}_{v,T}$.
Note that $\mu_{v}$ defines a natural transformation
$\ast\to\operatorname{Def}^{s}_{v,T}$. We define

$\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord},\mu}=\operatorname{\mathsf{hofib}}_{\mu_{v}}(\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord}}\to\operatorname{Def}^{s}_{v,T}).$

In addition, the natural transformation
$\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord},\mu}\to\mathcal{F}^{s}_{v}$
induces
$\widetilde{\mathcal{F}}_{v,Z}^{s,\operatorname{ord},\mu}\to\mathcal{F}^{s}_{v,Z}$.

######  Lemma 11.

The functors $\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord}}$ and
$\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord},\mu}$ are formally
cohesive. Assuming $(STDIST)$, the tangent complex
${\mathfrak{t}}\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord}}\cong{\mathfrak{t}}\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord},\mu}$,
is quasi-isomorphic to
${\mathfrak{t}}\mathcal{F}^{s,\operatorname{ord}}_{v}\cong{\mathfrak{t}}\mathcal{F}_{v}^{s,\operatorname{ord},\mu}$
:

  * •

${\mathfrak{t}}^{-1}\widetilde{\mathcal{F}}_{v,\ast}^{s,\operatorname{ord}}\cong\mathrm{H}^{0}(\Gamma_{v},{\mathfrak{g}}_{k})$,

  * •

${\mathfrak{t}}^{0}\widetilde{\mathcal{F}}_{v,\ast}^{s,\operatorname{ord}}\cong\mathrm{H}^{1}_{?}(\Gamma_{v},{\mathfrak{g}}_{k}^{\prime})$,

  * •

${\mathfrak{t}}^{i}\widetilde{\mathcal{F}}_{v,\ast}^{s,\operatorname{ord}}=0$
for $i\geq 1$.

Moreover, we have

$\pi_{0}\widetilde{\mathcal{F}}^{s,\operatorname{ord}}_{v}=\mathcal{F}_{v}^{\operatorname{ord}}.$

###### Proof.

Note that $\operatorname{Def}^{s}_{v,T}$ is formally cohesive. We see that
$\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord}}$ is formally cohesive: the
functor

$A\mapsto\operatorname{\mathsf{sHom}}_{[\overline{\rho}_{v}]}({\bf
B}G_{F_{v}},{\bf B}B(A))$

is homotopy invariant and preserves homotopy pullbacks. From this, it follows
by definition that $\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord},\mu}$ is
also formally cohesive.

The tangent complex
${\mathfrak{t}}^{i}\widetilde{\mathcal{F}}_{v}^{\operatorname{ord}}$ is
calculated in [GV18, Lemma 5.6] (replacing $G$ by $B$). More precisely, let
${\mathfrak{b}}=Lie(B)$. Then we have a quasi-isomorphism of complexes

${\mathfrak{t}}\widetilde{\mathcal{F}}_{v}^{\operatorname{ord}}\sim
C^{\ast+1}({B}\Gamma_{v},{\mathfrak{b}})$

The assumption $(reg^{\ast})$ implies
$\mathrm{H}^{0}(\Gamma_{v},{\mathfrak{g}}_{k}/{\mathfrak{b}}_{k})=0$. Thus,
${\mathfrak{t}}\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord}}$ is quasi-
isomorphic to ${\mathfrak{t}}\mathcal{F}^{s,\operatorname{ord}}_{v}$. The same
holds for
${\mathfrak{t}}\widetilde{\mathcal{F}}_{v}^{s,\operatorname{ord},\mu}$. If
$A\in Ob({}_{\mathcal{O}}\\!\operatorname{Art}^{s}_{k})$ is a discrete
artinian ring, $\mathcal{F}^{s,\operatorname{ord}}_{v}(A)$ is the set of
$B$-conjugacy classes of liftings of $\overline{\rho}_{v}$; because of
$(reg)$, the set of $B$-conjugacy classes coincide with the set of
$G$-conjugacy classes of representations $\rho_{v}\colon G_{F_{v}}\to G(A)$,
such that after conjugation by some $g_{v}\in G(A)$, $\rho_{v}$ takes values
in $B(A)$ and such that the composition of ${}^{g_{v}}\\!\rho_{v}$ and the
reduction $B(A)\to T(A)=B(A)/U_{B}(A)$ is a lifting of
$\underline{\overline{\chi}_{v}}\colon G_{F_{v}}\to T(k)$. (see [Ti96, Claim
in Proof of Proposition 6.2]). Hence we conclude
$\widetilde{\mathcal{F}}^{s,\operatorname{ord}}_{v}(A)=\mathcal{F}^{\operatorname{ord}}_{v}(A)$.
Similarly for
$\widetilde{\mathcal{F}}^{s,\operatorname{ord},\mu}_{v}(A)=\mathcal{F}^{\operatorname{ord},\mu}_{v}(A)$.
∎

We define $\mathcal{F}_{loc}^{s}$, resp. $\mathcal{F}_{loc}^{s,min,?}$ (and
their center modified counterparts) as

$\mathcal{F}_{loc}^{s}=\prod_{v\in S\cup
S_{p}}\mathcal{F}^{s}_{v},\quad\mathcal{F}_{loc,Z}^{s}=\prod_{v\in S\cup
S_{p}}\mathcal{F}^{s}_{v,Z}$

resp.

$\mathcal{F}_{loc}^{s,min,?}=\prod_{v\in
S}\mathcal{F}^{s,min}_{v,Z}\times\prod_{v\in
S_{p}}\mathcal{F}^{s,?}_{v},\quad\mathcal{F}_{loc,Z}^{s,min,?}=\prod_{v\in
S}\mathcal{F}^{s,min}_{v,Z}\times\prod_{v\in S_{p}}\mathcal{F}^{s,?}_{v,Z}$

as in [GV18, Definition 9.1]. By definition, the morphisms

$\mathcal{F}_{\overline{\rho}}^{s}\to\mathcal{F}^{s}_{v}$

induce morphisms

$\mathcal{F}^{s}_{\overline{\rho},Z}\to\mathcal{F}^{s}_{v,Z}$

hence a morphism

$\mathcal{F}^{s}_{Z,\overline{\rho}}\to\mathcal{F}^{s}_{loc,Z}$

We finally put

$\mathcal{F}^{s,?}=\mathcal{F}^{s,gl,?}_{\overline{\rho}}=\mathcal{F}^{s}_{\overline{\rho}}\times^{h}_{\mathcal{F}^{s}_{loc}}\mathcal{F}_{loc}^{s,min,?},\quad\mathcal{F}_{Z}^{s,?}=\mathcal{F}^{s,gl,?}_{Z,\overline{\rho}}=\mathcal{F}^{s}_{Z,\overline{\rho}}\times^{h}_{\mathcal{F}^{s}_{loc,Z}}\mathcal{F}_{loc}^{s,min,?}$

By Lemma 3 and 4 we have $\pi_{0}\mathcal{F}_{Z}^{s,?}=\mathcal{F}^{?}_{Z}$.
For $?=\operatorname{ord},FL$, we put

$\mathrm{H}^{1}_{min,?}(\Gamma,\operatorname{Ad}\bar{\rho})=\operatorname{Ker}(\mathrm{H}^{1}(\Gamma,\operatorname{Ad}\bar{\rho})\to\bigoplus_{v\in
S}{\mathrm{H}^{1}(\Gamma_{v},\operatorname{Ad}\bar{\rho})\over\mathrm{H}^{1}_{unr}(\Gamma_{v},\operatorname{Ad}\bar{\rho})}\oplus\bigoplus_{v\in
S_{p}}{\mathrm{H}^{1}(\Gamma_{v},\operatorname{Ad}\bar{\rho})\over L^{?}_{v}}$

where $\operatorname{Ad}\bar{\rho}={\mathfrak{g}}_{k}$ with the adjoint Galois
action and if $?=\operatorname{ord}$,
$L^{?}_{v}=\operatorname{Im}(L^{\prime}_{v}\to\mathrm{H}^{1}(\Gamma_{v},{\mathfrak{g}}_{k}))$
for
$L_{v}^{\prime}=\operatorname{Ker}(\mathrm{H}^{1}(\Gamma_{v},{\mathfrak{b}}_{k}))\to\mathrm{H}^{1}(I_{v},{\mathfrak{b}}_{k}/{\mathfrak{n}}_{k}))$,
and if $?=FL$, $L_{v}^{?}=\mathrm{H}_{f}^{1}(\Gamma_{v},{\mathfrak{g}}_{k})$
(that is, the image of
$\mathrm{H}_{f}^{1}(\Gamma_{v},{\mathfrak{g}}_{\mathcal{O}})\to\mathrm{H}^{1}(\Gamma_{v},{\mathfrak{g}}_{k})$
where $\mathrm{H}_{f}^{1}(\Gamma_{v},{\mathfrak{g}}_{\mathcal{O}})$ is defined
by Bloch and Kato, [BK90, Definition (3.7.3)].

For $v\in S$, let $\widetilde{L}_{v}^{min}$ be the inverse image of
$\mathrm{H}^{1}_{unr}(\Gamma_{v},{\mathfrak{g}}_{k})$ by
$Z^{1}(\Gamma_{v},{\mathfrak{g}}_{k})\to\mathrm{H}^{1}(\Gamma_{v},{\mathfrak{g}}_{k})$,
and for $v\in S_{p}$, let $\widetilde{L}_{v}^{?}$ be the inverse image of
$L_{v}^{?}$ by
$Z^{1}(\Gamma_{v},{\mathfrak{g}}_{k})\to\mathrm{H}^{1}(\Gamma_{v},{\mathfrak{g}}_{k})$.

######  Definition 5.

Given a morphism of cochain complexes $f\colon A^{\bullet}\to B^{\bullet}$,
its mapping cone $MC(f)^{\bullet}$ is defined by $MC(f)^{n}=B^{n}\oplus
A^{n+1}$ with $d(b,a)=(db+fa,-da)$. This give rise to distinguished triangles

$A^{\bullet}\to B^{\bullet}\to MC(f)^{\bullet}\to A^{\bullet}[1]$

and

$MC(f)^{\bullet}[-1]\to A^{\bullet}\to B^{\bullet}\to MC(f)^{\bullet}$

We define the complex

$C^{\bullet}_{min,?}(\Gamma,{\mathfrak{g}}_{k})=MC(f)[-1]$

for the morphism

$f\colon C^{\bullet}(\Gamma,{\mathfrak{g}}_{k})\to\bigoplus_{v\in
S}C^{\bullet}(\Gamma_{v},{\mathfrak{g}}_{k}))/\widetilde{L}_{v}^{\bullet,min}\oplus\bigoplus_{v\in
S_{p}}C^{\bullet}_{?}(\Gamma_{v},{\mathfrak{g}}_{k}))/\widetilde{L}_{v}^{\bullet,?}$

where for $?=min,\operatorname{ord},FL$,
$\widetilde{L}_{v}^{0,?}=C^{0}(\Gamma_{v},{\mathfrak{g}}_{k})$,
$\widetilde{L}_{v}^{1,?}=\widetilde{L}_{v}^{?}$ and
$\widetilde{L}_{v}^{2,min}=0$.

We write its cohomology as
$\mathrm{H}^{\ast}_{min,?}(\Gamma,{\mathfrak{g}}_{k})=\mathrm{H}^{\ast}(C^{\bullet}_{min,?}(\Gamma,{\mathfrak{g}}_{k}))$.
In accordance with Wiles’ notations, we also write sometimes this cohomology
as $\mathrm{H}^{\ast}_{\mathcal{L}}(\Gamma,{\mathfrak{g}}_{k})$ where
$\mathcal{L}=(L_{v})_{v\in S\cup S_{p}}$. We have the following exact
sequence:

$\begin{array}[]{l}{0\to\mathrm{H}^{1}_{min,?}(\Gamma,\operatorname{Ad}\bar{\rho})\to\mathrm{H}^{1}(\Gamma,\operatorname{Ad}\bar{\rho})\to\bigoplus_{v\in
S}{\mathrm{H}^{1}(\Gamma_{v},\operatorname{Ad}\bar{\rho})\over\mathrm{H}^{1}_{unr}(\Gamma_{v},\operatorname{Ad}\bar{\rho})}\oplus\bigoplus_{v\in
S_{p}}{\mathrm{H}^{1}(\Gamma_{v},\operatorname{Ad}\bar{\rho})\over
L^{?}_{v}}}\\\
{\to\mathrm{H}^{2}_{min,?}(\Gamma,\operatorname{Ad}\bar{\rho})\to\mathrm{H}^{2}(\Gamma,\operatorname{Ad}\bar{\rho})\to\bigoplus_{v\in
S\cup S_{p}}\mathrm{H}^{2}(\Gamma_{v},\operatorname{Ad}\bar{\rho})\dots}\\\
\end{array}$

Recall that we are interested in the deformation functor
$\mathcal{F}_{Z}^{s,?}=\mathcal{F}^{s,gl,?}_{Z,\overline{\rho}}$.

######  Theorem 10.

We have ${\mathfrak{t}}^{-1}\mathcal{F}_{Z}^{s,?}=0$ and for any $i\geq 0$,
the $i$-th cohomology of the tangent complex
${\mathfrak{t}}\mathcal{F}_{Z}^{s,?}$ is naturally identified with the
cohomology with local conditions:

${\mathfrak{t}}^{i}\mathcal{F}^{s,?}\cong\mathrm{H}^{i+1}_{min,?}(\Gamma,\operatorname{Ad}\bar{\rho}).$

###### Proof.

Since
$\mathcal{F}_{Z}^{s,?}=\mathcal{F}^{s}_{Z,\overline{\rho}}\times^{h}_{\mathcal{F}^{s}_{loc,Z}}\mathcal{F}_{loc,Z}^{min,?,s}$,
we can apply [GV18, Lemma 4.30 (iv)] and we get a long exact sequence of
finite dimensional $k$-vector spaces

$0\to{\mathfrak{t}}^{-1}\mathcal{F}_{Z}^{s,?}\to{\mathfrak{t}}^{-1}\mathcal{F}^{s}_{Z,\overline{\rho}}\oplus{\mathfrak{t}}^{-1}\mathcal{F}_{loc,Z}^{min,?,s}\to{\mathfrak{t}}^{-1}\mathcal{F}^{s}_{loc,Z}\to$
$\to{\mathfrak{t}}^{0}\mathcal{F}_{Z}^{s,?}\to{\mathfrak{t}}^{0}\mathcal{F}^{s}_{Z,\overline{\rho}}\oplus{\mathfrak{t}}^{0}\mathcal{F}_{loc,Z}^{min,?,s}\to{\mathfrak{t}}^{0}\mathcal{F}^{s}_{loc,Z}\to$
$\to{\mathfrak{t}}^{1}\mathcal{F}_{Z}^{s,?}\to{\mathfrak{t}}^{1}\mathcal{F}^{s}_{Z,\overline{\rho}}\oplus{\mathfrak{t}}^{1}\mathcal{F}_{loc,Z}^{min,?,s}\to{\mathfrak{t}}^{1}\mathcal{F}^{s}_{loc,Z}\to{\mathfrak{t}}^{1}\mathcal{F}^{s,?}\ldots$

We know by Proposition 2 that
${\mathfrak{t}}^{-1}\mathcal{F}^{s}_{Z,\overline{\rho}}=0$ and
${\mathfrak{t}}^{i}\mathcal{F}^{s}_{Z,\overline{\rho}}\cong\mathrm{H}^{i+1}(\Gamma,{\mathfrak{g}}_{k})$
for $i\geq 0$. By Lemma 8, we have
${\mathfrak{t}}^{i}\mathcal{F}_{v}^{s}\cong\mathrm{H}^{i+1}(\Gamma_{v},{\mathfrak{g}}_{k})$
for all $i\geq-1$ and
${\mathfrak{t}}^{-1}\mathcal{F}_{v,Z}^{s}\cong\operatorname{Coker}({\mathfrak{z}}_{k}\to\mathrm{H}^{0}(\Gamma_{v},{\mathfrak{g}}_{k}))$.
By Lemma 10, we have for $v\in S\cup S_{p}$,
${\mathfrak{t}}^{-1}\mathcal{F}_{v}^{s,?}\cong\mathrm{H}^{0}(\Gamma_{v},{\mathfrak{g}}_{k})$,
${\mathfrak{t}}^{-1}\mathcal{F}_{v,Z}^{s,?}\cong\operatorname{Coker}({\mathfrak{z}}\to\mathrm{H}^{0}(\Gamma_{v},{\mathfrak{g}}_{k}))$,
${\mathfrak{t}}^{0}\mathcal{F}_{v}^{s,?}\cong\mathrm{H}^{1}_{?}(\Gamma_{v},{\mathfrak{g}}_{k})$,
${\mathfrak{t}}^{i}\mathcal{F}_{v}^{s,?}=0$ for $i\geq 1$. We have therefore
${\mathfrak{t}}^{-1}\mathcal{F}^{s,?}=0$ and

$0\to{\mathfrak{t}}^{0}\mathcal{F}_{Z}^{s,?}\to\mathrm{H}^{1}(\Gamma,{\mathfrak{g}}_{k})\oplus\bigoplus_{v\in
S\cup
S_{p}}\mathrm{H}^{1}_{?}(\Gamma_{v},{\mathfrak{g}}_{k})\to\bigoplus_{v\in
S\cup S_{p}}\mathrm{H}^{1}(\Gamma_{v},{\mathfrak{g}}_{k})\to$
$\to{\mathfrak{t}}^{1}\mathcal{F}_{Z}^{s,?}\to\mathrm{H}^{2}(\Gamma,{\mathfrak{g}}_{k})\to\bigoplus_{v\in
S\cup S_{p}}\mathrm{H}^{2}(\Gamma_{v},{\mathfrak{g}}_{k}).$

By comparing to the long exact sequence of cohomology, we conclude that for
any $i\geq 0$,

${\mathfrak{t}}^{i}\mathcal{F}_{Z}^{s,?}\cong\mathrm{H}^{i+1}_{min,?}(\Gamma,\operatorname{Ad}\bar{\rho}).$

∎

######  Proposition 3.

Under the assumption $(STDIST)$, the global deformation problem
$\mathcal{F}_{Z}^{s,\operatorname{ord}}$ is pro-representable. Under the
assumption $(FL)$, the deformation problem $\mathcal{F}_{Z}^{s,FL}$ is pro-
representable.

Note that $\mathcal{F}^{s}_{Z,\bar{\rho}}$ is pro-representable, but not
$\mathcal{F}_{loc,Z}^{min,?,s}$.

###### Proof.

We apply Lurie’s criterion Theorem 9. We know that
$\mathcal{F}^{s}_{Z,\bar{\rho}}$, $\mathcal{F}_{loc,Z}^{s}$ and
$\mathcal{F}_{loc,Z}^{min,?,s}$ are formally cohesive, hence so is their
homotopy fiber product $\mathcal{F}_{Z}^{s,?}$. By Theorem 10,
${\mathfrak{t}}^{-1}\mathcal{F}_{Z}^{s,?}=0$ and for any $i\geq 0$,
${\mathfrak{t}}^{i}\mathcal{F}_{Z}^{s,?}$ is finite dimensional over $k$ (and
vanishes for $i>1$). ∎

###  3.6. Theorems of Galatius-Venkatesh and Cai

In this section we state generalizations of the main theorems of Galatius and
Venkatesh [GV18] by Y. Cai [Cai21]. We recall the notations and assumptions.
Let $\pi$ be a cohomological cuspidal representation on
$\operatorname{\mathsf{GL}}_{N}(F)$ ($F$ a CM field), with squarefree
conductor ${\mathfrak{n}}$, and level group $U=U_{0}({\mathfrak{n}})$. Let $S$
be the set of places dividing ${\mathfrak{n}}$. It occurs in
$\mathrm{H}^{\bullet}(X_{U},V_{\lambda}({\mathbb{C}}))$ for a weight
$\lambda=(\lambda_{\tau,i})_{\tau\in I_{F},1\leq i\leq
N}\in(X^{\ast}(\operatorname{\mathbf{T}}_{K})^{+})^{\operatorname{\mathsf{Hom}}(F,{\mathbb{C}})}$,
i.e. $\lambda_{\tau,1}\geq\ldots\geq\lambda_{\tau,N}$ for any $\tau\in I_{F}$.
Let $p>2$ be a rational prime relatively prime to ${\mathfrak{n}}$, unramified
in $F$; let $S_{p}$ be the set of places of $F$ above $p$. We assume either

$(FL)\quad\lambda_{\tau,1}-\lambda_{\tau,N}<p-N,\forall\tau\in I_{F},$

or

$(\operatorname{ord})\quad\pi_{v}\,\mbox{\rm is ordinary for all}\,v\in
S_{p}.$

Let $\Gamma=\operatorname{Gal}(F_{S\cup S_{p}}/F)$. Let $K_{0}$ be a
sufficiently big number field containing the Hecke eigenvalues of $\pi$. Fix a
$p$-adic place $v_{0}$ and let $K$ be its $p$-adic completion, $\mathcal{O}$
its valuation ring, $\varpi$ a uniformizing parameter; $k$ its residue field.
Let $\rho_{\pi}\colon\Gamma\to\operatorname{\mathsf{GL}}_{N}(\mathcal{O})$ be
the Galois representation associated to $\pi$ and
$\bar{\rho}\colon\Gamma\to\operatorname{\mathsf{GL}}_{N}(k)$ its reduction
modulo $\varpi$. Assume

$(MIN)$ for any places $v\in S$, the image $\overline{\rho}_{\pi}(I_{v})$ of
the inertia subgroup $I_{v}$ contains a regular unipotent element.

Recall that for any place $v\in S_{p}$, it is known that $(FL)$, resp.
$(\operatorname{ord})$, implies that $\rho_{\pi}|_{\Gamma_{v}}$ and
$\bar{\rho}|_{\Gamma_{v}}$ are both Fontaine-Laffaille, resp. ordinary. Assume
that the image of $\bar{\rho}$ is enormous:

$(RLI)$ $\bar{\rho}(\operatorname{Gal}_{F(\zeta_{p})})\supset
k^{\times}\operatorname{\mathsf{SL}}_{N}(k^{\prime})$ for some subfield
$k^{\prime}$ of the residue field $k$.

Recall that a Taylor-Wiles prime $v$ is a place $v\notin S\cup S_{p}$ of
residual characteristic $q_{v}$ such that $q_{v}\equiv 1\pmod{p}$ and
$\bar{\rho}(\operatorname{Frob}_{v})$ is conjugated to a strongly regular
element of $T(k)$ (i.e. an element $t\in T(k)$ whose centralizer in $G(k)$
coincides with $T(k)$). In particular, for a Taylor-Wiles prime $v$, we may
choose a representation $\bar{\rho}_{v}:\Gamma_{v}\to T(k)$ such that the
composition with $T\hookrightarrow G$ is conjugate to
$\bar{\rho}|_{\Gamma_{v}}$. Note that by definition, $\bar{\rho}_{v}$ factors
through the Galois group $\Gamma_{v}^{\operatorname{unr}}$ of the maximal
unramified extension $F_{v}^{\operatorname{unr}}/F_{v}$. Let
$\mathcal{F}_{v}^{s}$, resp. $\mathcal{F}_{v}^{\operatorname{unr},s}$ be the
simplicial deformation functor of $\bar{\rho}|_{\Gamma_{v}}$. As in [GV18,
Section 8.1], in order to study them, one needs to compare them to the
(simpler) simplicial deformation functors of $\bar{\rho}_{v}$ with target in
$T$. Let $\mathcal{F}_{v}^{s,T}$, resp.
$\mathcal{F}_{v}^{s,T,\operatorname{unr}}$, be the $T$-valued derived
deformation functor, resp. its unramified analogue, of $\bar{\rho}_{v}$. Since
$T$ is commutative, its center is $T$ itself, so their center-modified
analogues make use of $T$; they are denoted by $\mathcal{F}_{v,T}^{s,T}$,
resp. $\mathcal{F}_{v,T}^{s,T,\operatorname{unr}}$. They are pro-
representable. Let $Q$ be a finite set of Taylor-Wiles primes; we consider the
functors

$\mathcal{F}_{Q,loc}^{s}=\prod_{v\in S\cup S_{p}\cup Q}\mathcal{F}^{s}_{v},$
$\mathcal{F}_{Q,loc}^{s,min,?}=\prod_{v\in
S}\mathcal{F}^{s,min}_{v}\times\prod_{v\in
S_{p}}\mathcal{F}^{s,?}_{v}\times\prod_{v\in Q}\mathcal{F}_{v}^{s},$

let $\Gamma_{Q}=\operatorname{Gal}(F_{S\cup S_{p}\cup Q}/F)$ and
$\mathcal{F}^{s}_{Q,\overline{\rho}}$ be the global derived deformation
problem $A\mapsto\operatorname{\mathsf{Hom}}_{[\bar{\rho}]}(B\Gamma_{Q},{\bf
B}G(A))$; we then define

$\mathcal{F}^{s,min,?}_{Q}=\mathcal{F}^{s,gl/loc,?}_{Q,\overline{\rho}}=\mathcal{F}^{s}_{Q,\overline{\rho}}\times^{h}_{\mathcal{F}^{s}_{Q,loc}}\mathcal{F}_{Q,loc}^{s,min,?}$

and their center-modified analogues $\mathcal{F}_{Z,Q}^{s,min,?}$ and
$\mathcal{F}_{Z,Q}^{s,min,?}$, involving products of center-modified global
and local factors. We also introduce $\mathcal{F}_{Q,T}^{s,T}=\prod_{v\in
Q}\mathcal{F}_{v,T}^{s,T}$ and
$\mathcal{F}_{Q,T}^{s,T,\operatorname{unr}}=\prod_{v\in
Q}\mathcal{F}_{v,T}^{s,T,\operatorname{unr}}$.

######  Definition 6.

An allowable Taylor-Wiles datum of level $n$ is a set of Taylor-Wiles primes
$Q=\\{v_{1},\dots,v_{r}\\}$, together with a strongly regular element
$t_{v_{i}}\in T(k)$ conjugate to $\bar{\rho}(\operatorname{Frob}_{v_{i}})$ for
each $i\in\\{1,\dots,r\\}$, such that:

  1. (1)

Each $q_{v_{i}}\equiv 1\pmod{p^{n}}$, $i\in\\{1,\dots,r\\}$.

  2. (2)

We have
$\mathrm{H}^{2}_{\mathcal{L}_{Q}}(\Gamma_{Q},\operatorname{Ad}\bar{\rho})=0$,
where $\mathcal{L}_{Q}=(L_{Q,v})_{v}$ with

$L_{Q,v}=\left\\{\begin{array}[]{l}{L_{v},\,v\in S\cup S_{p};}\\\
{\mathrm{H}^{1}(F_{v},\operatorname{Ad}\bar{\rho}),\,v\in
Q.}\end{array}\right.$

As noted in [GV18, Remark after Definition 6.2], we have

######  Remark 2.

Under the assumption that
$k^{\times}\cdot\operatorname{\mathsf{SL}}_{N}(k^{\prime})\subset\operatorname{Im}\bar{\rho}$,
for any level $n\geq 1$, there exist infinitely many allowable Taylor-Wiles
data $(Q,(t_{v})_{v\in Q})$ of degree $n$.

Let $\Gamma_{Q}=\operatorname{Gal}(F_{S\cup S_{p}\cup Q})/F)$ and for each
$v\in Q$, For an allowable Taylor-Wiles datum $Q$, we have the long exact
sequence:

$\begin{array}[]{l}{0\to\mathrm{H}^{1}_{\mathcal{L}_{Q}}(\Gamma_{Q},\operatorname{Ad}\bar{\rho})\to\mathrm{H}^{1}(\Gamma_{Q},\operatorname{Ad}\bar{\rho})\stackrel{{\scriptstyle
A}}{{\to}}\bigoplus\limits_{l\in
S}\mathrm{H}^{1}(\Gamma_{v},\operatorname{Ad}\bar{\rho})/L_{v})}\\\ {\to
0\to\mathrm{H}^{2}(\Gamma_{Q},\operatorname{Ad}\bar{\rho})\stackrel{{\scriptstyle
B}}{{\to}}\bigoplus\limits_{v\in S\cup
Q}\mathrm{H}^{2}(\Gamma_{v},\operatorname{Ad}\bar{\rho})}.\end{array}$

Moreover, the cokernel of $B$ is dual to
$\mathrm{H}^{0}(\mathbb{Z}[\frac{1}{S(Q)}],(\operatorname{Ad}\bar{\rho})^{\ast})$
by global duality. We want to approximate $\mathcal{R}$ by larger rings
allowing ramification at Taylor-Wiles primes. Let $(Q_{n},(t_{v})_{v\in
Q_{n}})$ be an allowable Taylor-Wiles datum of level $n$. Let
$\mathcal{F}^{s}_{n}=\mathcal{F}_{Q_{n},Z}^{s,?}$ and let $\mathcal{R}_{n}$ be
a representing ring for $\mathcal{F}_{n}^{s}$. Let
$\mathcal{F}_{n}^{s,\mathrm{loc}}:=\mathcal{F}_{Q_{n},T}^{s,T}$ with
representing ring $\mathcal{S}_{n}$. Let
$\mathcal{F}_{n}^{s,\mathrm{loc,ur}}:=\mathcal{F}_{Q_{n},T}^{s,T,\operatorname{unr}}$
with representing ring $\mathcal{S}_{n}^{\mathrm{ur}}$. As noticed in [Cai21],
$\mathcal{F}_{n}^{s,\mathrm{loc}}$ is objectwise weakly equivalent to the
derived framed deformation functor $\prod_{v\in
Q_{m}}\mathcal{F}_{v}^{s,T,\square}$, and similarly for
$\mathcal{F}_{Q_{n},T}^{s,T,\operatorname{unr}}$ and $\prod_{v\in
Q_{n}}\mathcal{F}_{v}^{s,T,\square}$. Therefore, the simplicial deformation
rings $\mathcal{S}_{n}$ and $\mathcal{S}_{n}^{\operatorname{unr}}$ can be
described explicitely as in [GV18, Remark 8.7].

######  Lemma 12.

We have a homotopy pullback square

$\textstyle{\mathcal{F}_{Z}^{s,?}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{n}^{s,\mathrm{loc,ur}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{n}^{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{n}^{s,\mathrm{loc}}.}$

We have an objectwise weak equivalence

$\mathcal{F}_{n}^{s,?}\times^{h}_{\mathcal{F}_{n}^{s,\mathrm{loc}}}\mathcal{F}_{n}^{s,\mathrm{loc,ur}}\sim\mathcal{F}_{Z}^{s,?}.$

###### Proof.

This is [Cai21, Lemma 6.6 and Corollary 6.7]. ∎

Now we pass to the level of rings. Recall the derived tensor product
$\underline{\otimes}$ of [GV18, Definition 3.3], we remark that
$\mathcal{R}_{1}\underline{\otimes}_{\mathcal{R}_{3}}\mathcal{R}_{2}$
represents the homotopy pullback of represented functors after cofibrant
replacements, and since all pro-rings for us are indexed by natural numbers,
the derived tensor product can be explicitly constructed levelwise (see [GV18,
Definition 3.3] for details). The above corollary gives a weak equivalence

$\mathcal{R}\sim\mathcal{R}_{n}\underline{\otimes}_{\mathcal{S}_{n}}\mathcal{S}_{n}^{\operatorname{unr}}.$

We say a pro-object $\mathcal{R}$ of $\operatorname{Art}^{s}$ is homotopy
discrete if the map $\mathcal{R}\to\pi_{0}\mathcal{R}$ induces an equivalence
on represented functors after applying level-wise cofibrant replacement (see
[GV18, Definition 7.4]). Let $F_{Q_{n}}^{\times}=\prod_{v\in
Q_{n}}F_{v}^{\times}$, $\mathcal{O}_{Q_{n}}^{\times}=\prod_{v\in
Q_{n}}\mathcal{O}_{v}^{\times}$. By choosing uniformizing parameters at $v$’s
in $Q_{n}$, we have an isomorphism

$F_{Q_{n}}^{(p)}/\mathcal{O}_{Q_{n}}^{(p)}\cong{\mathbb{Z}}^{Q_{n}}.$

hence a decomposition

$F_{Q_{n}}^{\times}=\mathcal{O}_{Q_{n}}^{\times}\times{\mathbb{Z}}^{Q_{n}}.$

Let $\Delta^{\prime}_{Q_{n}}=\mathcal{O}_{Q_{n}}^{(p)}$ be the pro-$p$
completion of $\mathcal{O}_{Q_{n}}^{\times}$. It is a finite group isomorphic
to the $p$-Sylow of $\prod_{v\in Q}k_{v}^{\times}$. Let
$\Delta_{Q_{n}}=\Delta^{\prime}_{Q_{n}}/(p^{n})$ and

$F_{Q_{n}}^{(p)}=\Delta_{Q_{n}}\times{\mathbb{Z}}^{Q_{n}}.$

By definition of the level, this group is free of rank $r$ over
${\mathbb{Z}}/p^{n}{\mathbb{Z}}$. Let $S_{n}=\mathcal{O}[\Delta_{Q_{n}}]$; it
is a complete intersection ring. Actually let
$S_{\infty}=\mathcal{O}[[Y_{1},\ldots,Y_{Nr}]]$. The ideal
$J_{n}=((1+Y_{1})^{n}-1,\ldots,(1+Y_{Nr})^{n}-1)$, and an isomorphism

$i_{n}\colon S_{\infty}/J_{n}\cong S_{n}$

Let

$\Sigma_{n}=\mathcal{O}[[X^{\ast}(T)\otimes
F_{Q_{n}}^{(p)}]]=S_{n}[[F_{Q_{n}}^{(p)}/\mathcal{O}_{Q_{n}}^{(p)}]]\stackrel{{\scriptstyle
i}}{{\cong}}S_{n}[[X_{1},\ldots,X_{Nr}]],$
$\Sigma_{\infty}=S_{\infty}[[X_{1},\ldots,X_{Nr}]]$

The isomorphism $i_{n}$ induces an isomorphism
$\Sigma_{\infty}/J_{n}\Sigma_{\infty}\cong\Sigma_{n}.$ Let also

$\Sigma_{n}^{\operatorname{unr}}=\mathcal{O}[[X^{\ast}(T)\otimes
F_{Q_{n}}^{(p)}/\mathcal{O}_{Q_{n}}^{(p)}]]\stackrel{{\scriptstyle
i_{\operatorname{unr}}}}{{\cong}}\mathcal{O}[[X_{1},\ldots,X_{Nr}]].$

Note that the isomorphisms $i$ and $i_{\operatorname{unr}}$ are compatible to
the augmentation homomorphism
$S_{n}=\mathcal{O}[\Delta_{Q_{n}}]\to\mathcal{O}$.

######  Lemma 13.

The simplicial rings $\mathcal{S}_{n}$ and
$\mathcal{S}_{n}^{\operatorname{unr}}$ are homotopy discrete. Actually
$\mathcal{S}_{n}$, resp. $\mathcal{S}_{n}^{\operatorname{unr}}$, is a
cofibrant replacement of the complete intersection classical ring
$\Sigma_{n}$, resp. the formally smooth classical ring
$\Sigma_{n}^{\operatorname{unr}}$.

###### Proof.

See [GV18] and [Cai21]. ∎

Since we can choose $\mathcal{S}_{n}$ and
$\mathcal{S}_{n}^{\operatorname{unr}}$ up to weak equivalence, we suppose in
the following that $\mathcal{S}_{n}=c(\Sigma_{n})$ and
$\mathcal{S}_{n}^{\operatorname{unr}}=c(\Sigma_{n}^{\operatorname{unr}})$. Let
$\bar{S}_{m}=S_{\infty}/(p^{m},J_{m})$ and
$\bar{\Sigma}_{m}=\Sigma_{\infty}/(p^{m},J_{m}\Sigma_{\infty})$. Note that
$\bar{S}_{m}$ is finite and if $M_{m}$ is a $\Sigma_{m}$-module,
$\bar{M}_{m}=M_{m}\otimes_{\Sigma_{m}}\bar{\Sigma}_{m}=M_{m}\otimes_{S_{m}}\bar{S}_{m}$
and
$\bar{M}_{m}\otimes_{\bar{\Sigma}_{m}}\Sigma_{m}^{\operatorname{unr}}=\bar{M}_{m}\otimes_{\bar{S}_{m}}\mathcal{O}/(p^{m})$.
This can be rewritten in terms of simplicial rings as follows. By quotienting
by the ideal $(p^{m},J_{m})$ the morphism $S_{\infty}\to\mathcal{S}_{m}$, we
get the commutative diagram

$\textstyle{\bar{S}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{S}_{m}/(p^{m},J_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}/p^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{S}_{m}^{\operatorname{unr}}/p^{m}}$

which induces a homotopy pullback square of represented functors. In
consequence, for any $\mathcal{S}_{m}/(p^{m},J_{m})$-module $\mathcal{M}_{m}$,
we have a weak equivalence
$\mathcal{M}_{m}\underline{\otimes}_{\bar{S}_{m}}\mathcal{O}/p^{m}\sim\mathcal{M}_{m}\underline{\otimes}_{\mathcal{S}_{m}/(p^{m},J_{m})}\mathcal{S}_{m}^{\operatorname{unr}}/p^{m}$.

Let $R_{m}$, resp. $\bar{R}_{m}$, be the quotient of the classical universal
deformation ring which represents the $(min,?)$-deformations with
$Q_{m}$-ramification, resp. the $(min,?)$-deformations modulo $p^{m}$ whose
$Q_{m}$-ramification has of inertial level $\leq m$ (i.e. such that the
restriction to inertia factors modulo $p^{m}$ for all $v\in Q_{m}$). By
definition, the ring $\bar{R}_{m}$ is a natural quotient of
$R_{m}/(p^{m},J_{m})$ and is an $\mathcal{S}_{m}/(p^{m},J_{m})$-algebra. The
map from
$\mathcal{R}_{m}\leftarrow\mathcal{S}_{m}\to\mathcal{S}_{m}^{\operatorname{unr}}$
to
$\bar{R}_{m}\leftarrow\mathcal{S}_{m}/(p^{m},J_{m})\to\mathcal{S}_{m}^{\operatorname{unr}}/p^{m}$
induces

(1)
$\mathcal{R}\sim\mathcal{R}_{m}\underline{\otimes}_{\mathcal{S}_{m}}\mathcal{S}_{m}^{\operatorname{unr}}\to\bar{R}_{m}\underline{\otimes}_{\mathcal{S}_{m}/(p^{m},J_{m})}\mathcal{S}_{m}^{\operatorname{unr}}/p^{m}\sim\bar{R}_{m}\underline{\otimes}_{\bar{S}_{m}}\mathcal{O}/p^{m}.$

A Taylor-Wiles patching argument provides projective systems out of the
$\bar{R}_{m}\underline{\otimes}_{\bar{S}_{m}}\mathcal{O}/p^{m}$’s. Let
$R_{\infty}$ be the projective limit for one of these projective systems (it
depends on the choice of such a projective system). The main result of [GV18]
(Theorem 14.1) compares $\pi_{\ast}\mathcal{R}$ to
$\operatorname{Tor}_{\ast}^{S_{\infty}}(R_{\infty},\mathcal{O})$ under certain
conditions.

For $n\geq m$, there is a natural
$\bar{R}_{n}\underline{\otimes}_{\bar{S}_{n}}\mathcal{O}/p^{n}\to\bar{R}_{m}\underline{\otimes}_{\bar{S}_{m}}\mathcal{O}/p^{m}$.
Let $f_{n,m}$ be the composition

$\mathcal{R}\to\bar{R}_{n}\underline{\otimes}_{\bar{S}_{n}}\mathcal{O}/p^{n}\to\bar{R}_{m}\underline{\otimes}_{\bar{S}_{m}}\mathcal{O}/p^{m}.$

In order to apply [GV18, Theorem 12.1], recall the key result [Cai21,
Proposition 6.11] generalizing [GV18, Theorem 11.1]:

######  Proposition 4.

Under the assumptions $(RLI)$, $(MIN)$, $(FL)$ or $(ORD_{\pi})$, let $n\geq
m$, the map $f_{n,m}$ induces an isomorphism on ${\mathfrak{t}}^{0}$ and a
surjection on ${\mathfrak{t}}^{1}$.

Based on the fact that the homology of
$\bar{R}_{m}\underline{\otimes}_{\bar{S}_{m}}\mathcal{O}/p^{m}$ is
$\operatorname{Tor}_{\bullet}^{\bar{S}_{m}}(\bar{R}_{m},\mathcal{O}/p^{m})$,
this Proposition yields the following

######  Theorem 11.

We have an isomorphism of graded commutative rings
$\pi_{\bullet}\mathcal{R}\cong\operatorname{Tor}_{\bullet}^{S_{\infty}}(R_{\infty},\mathcal{O})$.

###### Proof.

Recall that $\pi_{\bullet}(\mathcal{R})$ is a $\pi_{0}(\mathcal{R})$-algebra
(see [Cai21, Lemma 3.45]) and that $\pi_{0}(\mathcal{R})=R_{0}$. The theorem
is proven in [Cai21, Proposition 6.13] by applying Proposition 4 to [GV18,
Theorem 12.1]. Note that Theorem 12.1 requires a numerical coincidence (12.1),
which is satisfied when the (framed) local deformation rings are formally
smooth (here it is the case by Section 2.1). ∎

This Theorem implies the Main Theorem ([GV18, Th.14.1], generalized in [Cai21,
Theorem 7.3]) below. We need to add few more notations and assumptions. (this
implies Taylor-Wiles data exist, see [ACC+18, 6.2.28]). Let
$C^{\bullet}(X_{U},V_{\lambda}(\mathcal{O}))$ be the integral cochain complex;
its cohomology is $\mathrm{H}^{\bullet}(X_{U},V_{\lambda}(\mathcal{O}))$.
There is a natural homomorphism

$\mathcal{H}^{S,\operatorname{univ}}\to\operatorname{\mathsf{End}}_{D(\mathcal{O})}(C^{\bullet}(X_{U},V_{\lambda}(\mathcal{O})))$

from the abstract spherical Hecke algebra outside $S$ to the endomorphisms of
$C^{\bullet}$ in the derived category. Let
$\operatorname{T}(X_{U},V_{\lambda}(\mathcal{O}))$ be the image of this map.
Then $\operatorname{T}(X_{U},V_{\lambda}(\mathcal{O}))$ is finite over
$\mathcal{O}$. see [NT16]. Let ${\mathfrak{m}}$ be a non-Eisenstein maximal
ideal of $\operatorname{T}(X_{U},V_{\lambda}(\mathcal{O}))$ associated to
$\pi$ modulo $v_{0}$. For simplicity, we write
$C_{\lambda,{\mathfrak{m}}}^{\bullet}$ for
$C^{\bullet}(X_{U},V_{\lambda}(\mathcal{O}))_{{\mathfrak{m}}}$ and write
$\operatorname{T}_{\lambda,{\mathfrak{m}}}$ for
$\operatorname{T}(X_{U},V_{\lambda}(\mathcal{O}))_{{\mathfrak{m}}}$. We assume

$(\operatorname{Gal}_{\mathfrak{m}})$ For $?=FL,\operatorname{ord}$, there
exists a lifting $\rho_{\mathfrak{m}}\colon G_{F,S\cup
S_{p}}\to\operatorname{\mathsf{GL}}_{N}(\operatorname{T}_{\lambda,{\mathfrak{m}}})$
of $\bar{\rho}$ such that
$[\rho_{\mathfrak{m}}]\in\mathcal{F}^{?}(\operatorname{T}_{\mathfrak{m}})$. In
particular, there is a natural map

$\mathcal{R}\to\operatorname{T}_{\lambda,{\mathfrak{m}}}\hookrightarrow\operatorname{\mathsf{End}}_{D(\mathcal{O})}(C_{\lambda,{\mathfrak{m}}}^{\bullet}).$

Let
$\mathrm{H}^{\bullet}_{\mathfrak{m}}=\mathrm{H}^{\bullet}(X_{U},V_{\lambda}(\mathcal{O}))_{\mathfrak{m}}$.
Let $\mathrm{H}_{i}^{\mathfrak{m}}=\mathrm{H}^{d-i}_{\mathfrak{m}}$ for all
$i\geq 0$ and let $\mathrm{H}_{\bullet}^{\mathfrak{m}}=\bigoplus_{i\geq
0}\mathrm{H}_{i}^{\mathfrak{m}}$ the corresponding graded module. We assume

$(Van_{\mathfrak{m}})$ $\mathrm{H}^{i}_{\mathfrak{m}}(k)=0$ if
$i\notin[q_{0},q_{0}+\ell_{0}]$.

######  Theorem 12.

Assume $(RLI)$, $(MIN)$, $(FL)$ or $(ORD_{\pi})+(DIST)$,
$(\operatorname{Gal}_{\mathfrak{m}})$, $(Van_{\mathfrak{m}})$ as above. Then,

(i) we have an isomorphism of graded commutative rings
$\pi_{\ast}\mathcal{R}\cong\operatorname{Tor}_{\ast}^{S_{\infty}}(R_{\infty},\mathcal{O})$,

(ii)
$\pi_{0}(\mathcal{R})=R_{\infty}\otimes_{S_{\infty}}\mathcal{O}=R\cong\operatorname{T}_{\lambda,{\mathfrak{m}}}$,
$\mathrm{H}_{q_{0}}^{\mathfrak{m}}=\mathrm{H}^{q_{0}+\ell_{0}}_{\mathfrak{m}}$
is free of rank $1$ over $\operatorname{T}_{\lambda,{\mathfrak{m}}}$,

(iii) There is an isomorphism of graded $\pi_{\bullet}\mathcal{R}$-modules

$\mathrm{H}_{\bullet}^{\mathfrak{m}}\cong\mathrm{H}_{q_{0}}^{\mathfrak{m}}\otimes_{\operatorname{T}_{\lambda,{\mathfrak{m}}}}\pi_{\bullet}(\mathcal{R}).$

######  Remark 3.

As in [GV18, Theorem 14.1], this statement relies on [CaGe18, Theorem 12.1].
See examples where these assumptions are satisfied in [Cai21, Section 8].

###  3.7. Divisible part of the Dual Adjoint Selmer group

Let $P$ be a finite free $\mathcal{O}$-module with action of
$\Gamma=G_{F,S\cup S_{p}}$. For each $v\in S_{p}$, let $Fil_{v}^{+}P\subset
Fil_{v}P\subset P$ be direct factors submodules stable by $\Gamma_{v}$. We
define the minimal $(Fil_{v}^{+}P)_{v}$-ordinary Selmer group as

${\mathop{\rm
Sel}}_{\operatorname{ord}}(P)=\mathrm{H}^{1}_{min,\operatorname{ord}}(F,P\otimes{\mathbb{Q}}/{\mathbb{Z}})=\operatorname{Ker}(\mathrm{H}^{1}(F,P\otimes{\mathbb{Q}}/{\mathbb{Z}})\to\bigoplus_{v}\frac{\mathrm{H}^{1}(F_{v},P\otimes{\mathbb{Q}}/{\mathbb{Z}})}{\mathrm{H}^{1}_{\operatorname{ord}}(F_{v},P\otimes{\mathbb{Q}}/{\mathbb{Z}})}))$

where

$\mathrm{H}^{1}_{\operatorname{ord}}(F_{v},P\otimes{\mathbb{Q}}/{\mathbb{Z}})=\operatorname{Im}(L^{\prime}_{v}\to\mathrm{H}^{1}(F_{v},P\otimes{\mathbb{Q}}/{\mathbb{Z}}))$

$L^{\prime}_{v}=\operatorname{Ker}(\mathrm{H}^{1}(F_{v},Fil_{v}P\otimes{\mathbb{Q}}/{\mathbb{Z}})\to\mathrm{H}^{1}(F_{v},Fil_{v}P\otimes{\mathbb{Q}}/{\mathbb{Z}}))/\mathrm{H}^{1}(I_{v},Fil_{v}^{+}P\otimes{\mathbb{Q}}/{\mathbb{Z}}))$

Similarly,

${\mathop{\rm
Sel}}_{f}(P)=\mathrm{H}^{1}_{f}(F,P\otimes{\mathbb{Q}}/{\mathbb{Z}})=\operatorname{Ker}(\mathrm{H}^{1}(F,P\otimes{\mathbb{Q}}/{\mathbb{Z}})\to\bigoplus_{v}\frac{\mathrm{H}^{1}(F_{v},P\otimes{\mathbb{Q}}/{\mathbb{Z}})}{\mathrm{H}^{1}_{f}(F_{v},P\otimes{\mathbb{Q}}/{\mathbb{Z}})}))$

where

$\mathrm{H}^{1}_{f}(F_{v},P\otimes{\mathbb{Q}}/{\mathbb{Z}})=\operatorname{Im}(\mathrm{H}^{1}_{f}(F_{v},P\otimes{\mathbb{Q}})\to\mathrm{H}^{1}(F_{v},P\otimes{\mathbb{Q}}/{\mathbb{Z}})).$

In the sequel we shall take
$P=\operatorname{Ad}(\rho_{\pi})={\mathfrak{g}}_{\mathcal{O}}$,
$Fil_{v}P={\mathfrak{b}}_{\mathcal{O}}$,
$Fil_{v}^{+}P={\mathfrak{n}}_{\mathcal{O}}$, or
$P=\operatorname{Ad}(\rho_{\pi})^{\vee}(1)={\mathfrak{g}}_{\mathcal{O}}^{\vee}(1)$,
$Fil_{v}={\mathfrak{n}}_{\mathcal{O}}^{\perp}(1)$,
$Fil_{v}^{+}P={\mathfrak{b}}_{\mathcal{O}}^{\perp}(1)$. For
$?=\operatorname{ord},f$, Theorem (12) implies that $\mathop{\rm
Sel}_{?}(\operatorname{Ad}(\rho_{\pi}))$ is finite. Let

${}_{?}(P)=\mathrm{H}^{1}_{?}(F,P\otimes{\mathbb{Q}}/{\mathbb{Z}})/\mathrm{H}^{1}_{?}(F,P\otimes{\mathbb{Q}}/{\mathbb{Z}})_{\varpi-
div}$

It is the torsion quotient of
$\mathrm{H}^{1}_{?}(F,P\otimes{\mathbb{Q}}/{\mathbb{Z}})$. For
$P={\mathfrak{g}}_{\mathcal{O}}^{\vee}(1)$, We will recall below, using
Poitou-Tate duality, that it is Pontryagin dual to $\mathop{\rm
Sel}_{?}(\operatorname{Ad}(\rho_{\pi}))$. Recall that Beilinson conjecture
predicts that

$(B)\quad cork_{\mathcal{O}}\mathop{\rm
Sel}_{?}({\mathfrak{g}}_{\mathcal{O}}^{\vee}(1))=\ell_{0}.$

In the next section, we’ll see that this conjecture follows from Theorem 12.
In the next section, assuming the assumption of Theorem 12, we will provide an
automorphic description of a $\varpi$-divisible subgroup of a corank
$\ell_{0}$ of $\mathop{\rm Sel}_{?}({\mathfrak{g}}_{\mathcal{O}}^{\vee}(1))$.
By Proposition 7 below, $\mathop{\rm
Sel}_{?}({\mathfrak{g}}_{\mathcal{O}}^{\vee}(1))$, assuming Theorem 12,
Conjecture $(B)$ holds. It implies that the cokernel of this embedding is
therefore ${}_{?}({\mathfrak{g}}_{\mathcal{O}}^{\vee}(1))$, which is
Pontryagin dual to $\mathop{\rm Sel}_{?}({\mathfrak{g}}_{\mathcal{O}})$.

##  4\. The Galatius-Venkatesh homomorphism

We redefine the map of Lemma 15.1 of [GV18] as follows. Let
$\mathcal{O}_{n}=\mathcal{O}/(\varpi^{n})$. We consider the simplicial ring
homomorphism

$\phi_{n}\colon\mathcal{R}\to R\to\mathcal{O}_{n}$

given by the universal property for the deformation
$\rho_{n}=\rho_{\pi}\pmod{(\varpi^{n})}$. Let $M_{n}$ be a finite
$\mathcal{O}_{n}$-module. Consider the simplicial ring
$\Theta_{n}=\mathcal{O}_{n}\oplus{\operatorname{DK}}(M_{n}[1])$ where
$M_{n}[1]$ is the chain complex concentrated in degree $1$ up to homotopy. For
an explicit description of the simplicial module ${\\\ DK}(M_{n}[1])$, see
[Cai21, Section 3.4]. The simplicial ring $\Theta_{n}$ is endowed with a
simplicial ring homomorphism
$\operatorname{pr}_{n}\colon\Theta_{n}\to\mathcal{O}_{n}$ given by the first
projection. Let $L_{n}(\mathcal{R})$ be the set of homotopy equivalence
classes of simplicial ring homomorphisms $\Phi\colon\mathcal{R}\to\Theta_{n}$
such that $\operatorname{pr}_{n}\circ\Phi=\phi_{n}$. There is a canonical
bijection

$L_{n}(\mathcal{R})\cong\mathrm{H}^{2}_{f}(F,\operatorname{Ad}(\rho_{n})\otimes
M_{n})$

Moreover, as in [GV18, Lemma 15.1], there is a map

$\pi(n,\mathcal{R})\colon
L_{n}(\mathcal{R})\to\operatorname{\mathsf{Hom}}_{R}(\pi_{1}(\mathcal{R}),M_{n})$

sending (the homotopy class of) $\Phi$ to the homomorphism
$\pi(n,\mathcal{R})(\Phi)$ which sends the homotopy class $[\gamma]$ of a loop
$\gamma$ to
$\Phi\circ\gamma\in\operatorname{\mathsf{Hom}}_{\operatorname{\mathsf{sSETS}}}(\Delta[1],M_{n}[1])=M_{n}$.
Recall a loop $\gamma$ is a morphism of $\operatorname{\mathsf{sSETS}}$

$\gamma\colon\Delta[1]\to\Theta_{n}$

from the simplicial interval $\Delta[1]$ to the simplicial set $\Theta_{n}$
which sends the boundary $\partial\Delta[1]$ to $0$. Note that
$\pi(n,\mathcal{R})(\Phi)$ is $R$-linear by definition of the structure of
$\pi_{0}(\mathcal{R})$-module on $\pi_{1}(\mathcal{R})$.

######  Proposition 5.

For any $n\geq 1$, the map $\pi(n,\mathcal{R})$ is surjective.

We’ll give two proofs of this Proposition. Note first that by the isomorphisms
(14.4), we have a diagram

$\begin{array}[]{ccccc}\pi(n,\mathcal{R})&\colon&L_{n}(\mathcal{R})&\rightarrow&\operatorname{\mathsf{Hom}}_{\pi_{0}(\mathcal{R})}(\pi_{1}(\mathcal{R}),M_{n})\\\
&&\uparrow&&\uparrow\\\ \pi(R_{n}\stackrel{{\scriptstyle
L}}{{\otimes}}_{S_{n}}\mathcal{O}_{n})&\colon&L_{n}(R_{n}\stackrel{{\scriptstyle
L}}{{\otimes}}_{S_{n}}\mathcal{O}_{n})&\rightarrow&\operatorname{\mathsf{Hom}}_{\pi_{0}(\mathcal{R})}(\pi_{1}(R_{n}\stackrel{{\scriptstyle
L}}{{\otimes}}_{S_{n}}\mathcal{O}_{n}),M_{n})\\\ &&\downarrow&&\downarrow\\\
\pi(R_{\infty}\stackrel{{\scriptstyle
L}}{{\otimes}}_{S_{\infty}}\mathcal{O})&\colon&L_{n}(R_{\infty}.\stackrel{{\scriptstyle
L}}{{\otimes}}_{S_{\infty}}\mathcal{O})&\rightarrow&\operatorname{\mathsf{Hom}}_{\pi_{0}(\mathcal{R})}(\pi_{1}(R_{\infty}\stackrel{{\scriptstyle
L}}{{\otimes}}_{S_{\infty}}\mathcal{O}),M_{n})\end{array}$

Note that $\pi_{1}\mathcal{R}=\pi_{1}(R_{\infty}\stackrel{{\scriptstyle
L}}{{\otimes}}_{S_{\infty}}\mathcal{O})$ by [GV18, Theorem 14.1]. So it is
enough to prove that the map on the last line is surjective. In the sequel,
since $n$ is fixed, we drop the indices $n$ and write simply $A=A_{n}$,
$\Theta=\Theta_{n}$, $M=M_{n}$.

###### Proof.

First Proof: Recall that by finiteness of the Calegari-Geraghty complex of
finite free $S_{\infty}$-modules constructed in [CaGe18], $R_{\infty}$ is a
finite $S_{\infty}$ algebra. Let be a cofibrant replacement $P_{\bullet}\to
R_{\infty}$ by noetherian polynomial rings $\mathbf{P}_{n}$ over $S_{\infty}$.
Then, one can take as $R_{\infty}\stackrel{{\scriptstyle
L}}{{\otimes}}_{S_{\infty}}\mathcal{O}$ the complex associated to the
simplicial ring
${\bf{R}}_{\bullet}=\mathbf{P}_{\bullet}\otimes_{S_{\infty}}\mathcal{O}$,
which consists in (noetherian) polynomial $\mathcal{O}$-algebras and is a
cofibrant fibration of $R_{\infty}\otimes_{S_{\infty}}\mathcal{O}$ (but is not
a weak equivalence). By the adjunction formula of Section 3.1, we have

$\operatorname{\mathsf{sHom}}_{{}_{\mathcal{O}}\\!\operatorname{CR}^{s}_{A}}({\bf{R}}_{\bullet},A\oplus{\operatorname{DK}}(M[1]))\cong\operatorname{\mathsf{sHom}}_{\operatorname{Mod}^{s}_{A}}(L_{{\bf{R}}_{\bullet}/\mathcal{O}}\otimes_{{\bf{R}}_{\bullet}}A,M[1]).$

Note that by construction, since ${\bf{R}}_{\bullet}$ is cofibrant, we have

$L_{{\bf{R}}_{\bullet}/\mathcal{O}}=\Omega_{{\bf{R}}_{\bullet}/\mathcal{O}}.$

Therefore,the datum of a homomorphism of chain complexes $\Phi\colon
L_{{\bf{R}}_{\bullet}/\mathcal{O}}\otimes_{{\bf{R}}_{\bullet}}A\to M[1]$ is
equivalent to that of a homomorphism of modules
$\psi_{1}\colon\Omega^{1}_{{\bf{R}}_{1}/\mathcal{O}}\otimes_{{\bf{R}}_{1}}A\to
M$ placed in degree $1$ which is itself equivalent to the datum of a classical
$\mathcal{O}$-algebra homomorphism $\Phi_{1}\colon{\bf{R}}_{1}\to A\oplus M$.
Thus we need to lift a given $\varphi_{1}\colon\pi_{1}({\bf{R}}_{\bullet})\to
M$ to a homomorphism $\Phi_{1}$.

Following [COT, Section 09D4 Example 5.9], we give an explicit description of
$\mathbf{P}_{i}$ ($i=0,1,2$) and their face and degeneracy maps. We denote
$\mathbf{P}_{0}=S_{\infty}[u_{1},\ldots,u_{k}]$ which we abbreviate as
$\mathbf{P}_{0}=S_{\infty}[u_{j}]$, and similarly for the other rings:
$\mathbf{P}_{1}=S_{\infty}[u_{j},x_{t}]$,
$\mathbf{P}_{2}=S_{\infty}[u_{j},x_{t},v_{r},y_{t},z_{t},w_{t,t^{\prime}}]$,
where $u_{i},x_{t},v_{r},y_{t},z_{t},w_{t,t^{\prime}}$ are independent
variables. One fixes $\mathbf{P}_{0}\to R_{\infty}$ a surjective
$S_{\infty}$-algebra homomorphism, and a system $(f_{t})$ of generators of
$\operatorname{Ker}(\mathbf{P}_{0}\to R_{\infty})$. Here $r=(r_{t})$ runs over
the (finitely generated) module of systems of relations between the $f_{t}$’s
in $\mathbf{P}_{0}$ : $\sum_{t}r_{t}\cdot f_{t}=0$.

Let $d_{0},d_{1}\colon\mathbf{P}_{1}\to\mathbf{P}_{0}$ be the two face maps
given by $u_{i}\mapsto u_{i}$ and $x_{t}\mapsto 0$, resp. $x_{t}\mapsto
f_{t}$. Let
$\delta_{0},\delta_{1},\delta_{2}\colon\mathbf{P}_{2}\to\mathbf{P}_{1}$ the
three face maps. See their definitions in [COT, Section 09D4 Example 5.9]. We
simply recall that $\delta_{0}(v_{r})=\delta_{1}(v_{r})=0$ and
$\delta_{2}(v_{r})=\sum_{t}r_{t}x_{t}$.

By tensoring by $\mathcal{O}$ over $\mathbf{S}_{\bullet}$, these maps yield
the face and degeneracy maps of $\bf{R}_{i}$, $i=0,1,2$. We still denote by
$d_{0},d_{1}\colon\bf{R}_{1}\to\bf{R}_{0}$ and
$\delta_{0},\delta_{1},\delta_{2}\colon\bf{R}_{2}\to\bf{R}_{1}$ the resulting
face maps. We have
$\pi_{1}({\bf{R}}_{\bullet})=\operatorname{Ker}(d_{0}-d_{1})/\operatorname{Im}(\delta_{0}-\delta_{1}+\delta_{2})$.
For any homomorphism $\varphi_{1}\colon\operatorname{Ker}(d_{0}-d_{1})\to M$,
such that $\varphi_{1}\circ(\delta_{0}-\delta_{1}+\delta_{2})=0$, we consider
$\Phi_{0}\colon\mathbf{P}_{0}\otimes_{\mathbf{S}_{0}}\mathcal{O}\to A$ as the
composition of $\mathbf{P}_{0}\otimes_{S_{\infty}}\mathcal{O}\to
R_{\infty}\otimes_{S_{\infty}}\mathcal{O}=R$ and $R\to A$. We need to define
$\Phi_{1}\colon\mathbf{P}_{1}\otimes_{\mathbf{S}_{1}}\mathcal{O}\to A\oplus
M$, compatible to $\Phi_{0}$ for the $d_{i}$ ($i=0,1$) and $s_{0}$, in terms
of $\varphi_{1}$.

One can rewrite this more explicitely,
${\bf{R}}_{0}=\mathbf{P}_{0}\otimes_{S_{\infty}}\mathcal{O}=\mathcal{O}[[u_{j}]]$
and
${\bf{R}}_{1}=\mathbf{P}_{1}\otimes_{S_{\infty}}\mathcal{O}=\mathcal{O}[[u_{j},x_{t}]]$.
Then $\operatorname{Ker}(d_{0}-d_{1})$ is the $\mathcal{O}[[u_{j}]]$-module of
$P(u_{j},x_{t})$ such that

$P(u_{j},0)=P(u_{j},f_{t}).$

Moreover, $R_{\infty}\otimes_{S\infty}\mathcal{O}$ is a quotient of
$\bf{R}_{0}$. We can extend $\varphi_{1}$ to $\mathcal{O}[[u_{j},x_{t}]]$ as
follows. We first write $P=P_{1}+K_{1}$ where
$P_{1}=P(u_{j},0)+\sum_{t}Q_{t}(u_{j})\cdot x_{t}$ and
$K_{1}\in\operatorname{Ker}(d_{0}-d_{1})$; indeed, if we write
$P(u_{j},f_{t})-P(u_{j},0)=\sum_{t}Q_{t}(u_{j})\cdot f_{t}$, then if we put
$P_{1}(u_{j},x_{t})=P(u_{j},0)+\sum_{t}Q_{t}(u_{j})\cdot x_{t}$, we see that
$K_{1}=P-P_{1}\in\operatorname{Ker}(d_{0}-d_{1})$. Then we put

$\Phi_{1}(P)=\Phi_{0}(d_{0}(P_{1}))+\varphi_{1}(K_{1})$

Note that $\Phi_{0}(d_{0}(P_{1}))=\Phi_{0}(d_{0}(P))$. Let us check that
$\Phi_{1}$ is well defined. If we write

$P(u_{j},f_{t})-P(u_{j},0)=\sum_{k}Q_{t}(u_{j})\cdot
f_{t}=\sum_{k}Q^{\prime}_{t}(u_{j})\cdot f_{t}$

for another system $(Q^{\prime}_{t})$ of elements of $\mathcal{O}[[u_{j}]]$;
let $K_{1}^{\prime}=P-P^{\prime}_{1}$ where
$P_{1}^{\prime}=P(u_{j},0)+\sum_{t}Q_{t}^{\prime}x_{t}$. We need to show that
$\varphi_{1}(K_{1})=\varphi_{1}(K^{\prime}_{1})$. The system $r=(r_{t})$
defined by $r_{t}=Q_{t}-Q_{t}^{\prime}$ is a relation between the $f_{t}$’s:
$\sum_{t}r_{t}f_{t}=0$. By [COT, 09D4], we see that

$\varphi_{1}(K_{1})-\varphi_{1}(K^{\prime}_{1})=\varphi_{1}(\sum_{t}r_{t}x_{t})=\varphi_{1}(\delta_{0}-\delta_{1}+\delta_{2})(v_{r}))=0$

hence $\Phi_{1}$ is well defined. This yields the desired lifting

$\Phi\colon R_{\bullet}\to A\oplus{\operatorname{DK}}(M[1]))$

of $\varphi_{1}\colon\pi_{1}(R_{\bullet})\to M$.

To conclude the proof, we have compatible morphisms $\mathcal{R}\to
R_{\bullet}/{\mathfrak{a}}_{n}$ by [GV18, (14.4)] which induce an isomorphism
$\pi_{1}(\mathcal{R})\cong\pi_{1}(R_{\bullet})$. Therefore, the surjectivity
of $\pi(n,\mathcal{R})$ follows from that of $\pi(n,R_{\bullet})$. ∎

###### Proof.

Second Proof: For a $B$-algebra $C$, we denote by $B[C]^{(j)}$ the ring
defined by induction as follows : $B[C]^{(0)}$ is the polynomial $B$-algebra
with variables the elements of $C$, and for $j\geq 1$, $B[C]^{(j)}$ is the
polynomial $B$-algebra with variables the elements of $B[C]^{(j-1)}$. As
explained in [SIMP, Section 14.34.5], these sets form a simplicial set. In
particular the $d_{k}$’s, $k=0,\ldots,j$ from $B[C]^{(j)}$ to $B[C]^{(j-1)}$
are given as follows. Let

$P_{j}=P_{j}([P_{j-1}(\ldots([P_{0}([\underline{c})])]\ldots])\in B[C]^{(j)}.$

Then, $d_{k}P_{j}$ is obtained by removing the $k$-th bracket (the zeroth
being the innermost one).

Let $B=S_{\infty}$ and $C=R_{\infty}$ We choose the functorial cofibrant
replacement $P_{\bullet}$ of $R_{\infty}$ by rings
$\mathbf{P}_{j}=S_{\infty}[R_{\infty}]^{(j)}$ Then, one can take as
$R_{\infty}\stackrel{{\scriptstyle L}}{{\otimes}}_{S_{\infty}}\mathcal{O}$ the
complex associated to the simplicial ring
${\bf{R}}_{\bullet}=\mathbf{P}_{\bullet}\otimes_{S_{\infty}}\mathcal{O}$; tt
follows from [Gil13, Corollary 7.10.5] that ${\bf{R}}_{\bullet}$ is a
cofibrant fibration of $R_{\infty}\otimes_{S_{\infty}}\mathcal{O}$ (but is not
a weak equivalence). It can also be proven by verifying first that the
morphism of simplicial rings $\mathcal{O}\to{\bf{R}}_{\bullet}$ is free in the
sense of [Gil13, Definition 7.6.2] (that is, by verifying Conditions (1)-(3)
which follow this definition), then, by quoting [Gil13, Theorem 7.6.13]
stating that any free simplicial morphism is a cofibration.

Let $P_{1}=P_{1}([P_{0}([\underline{c}])])\in{\bf{R}}_{1}$ where $P_{1}$,
resp. $P_{0}$, has coefficients in $\mathcal{O}$, resp. in $B$. Let
$P^{(1,0)}_{0},\ldots,P^{(m,0)}_{0}$ be the finite list of polynomials $P_{0}$
which actually occur as variables in $P_{1}$. We write
$P_{1}=P_{1}([P_{0}(\underline{c})])+\sum_{s=1}^{m}([P^{(s,0)}_{0}]-[P^{(s,0)}(\underline{c})])E_{s}$,
where $E_{s}\in{\bf{R}}_{1}$. Let $P^{(1,1)}_{0},\ldots,P^{(n,1)}_{0}$ be the
list of polynomials $P_{0}$ which occur as variables in $E_{s}$. Then we
redecompose $E_{s}$ as
$E_{s}=E_{s}(P_{0}([\underline{c}]))+\sum_{t=1}^{n}([P^{(t,1)}_{0}]-P^{(t,1)}([\underline{c}]))F_{s,t}$.
If we put

$Q_{1}=P_{1}([P_{0}(\underline{c})])+\sum_{s=1}^{m}([P^{(s,0)}_{0}]-[P^{(s,0)}(\underline{c})])\cdot
E_{s}(P_{0}([\underline{c}]))$

and

$K_{1}=\sum_{s,t}([P^{(s,0)}_{0}]-[P^{(s,0)}(\underline{c})])\cdot([P^{(t,1)}_{0}]-[P^{(t,1)}(\underline{c})])\cdot
F_{s,t}$

we see that $P_{1}=Q_{1}+K_{1}$, with $Q_{1}\in{\bf{R}}_{0}$ and $K_{1}\in