# Block theory and Brauer’s first main theorem for profinite
groups11footnotemark: 1

Ricardo J. Franquiz Flores222The author was financially supported by a CAPES
doctoral scholarship<EMAIL_ADDRESS>John W. MacQuarrie<EMAIL_ADDRESS>Universidade Federal de Minas Gerais, Belo Horizonte, Brazil

###### Abstract

We develop the local-global theory of blocks for profinite groups. Given a
field $k$ of characteristic $p$ and a profinite group $G$, one may express the
completed group algebra $k[[{G}]]$ as a product $\prod_{i\in I}B_{i}$ of
closed indecomposable algebras, called the blocks of $G$. To each block $B$ of
$G$ we associate a pro-$p$ subgroup of $G$, called the defect group of $B$,
unique up to conjugacy in $G$. We give several characterizations of the defect
group in analogy with defect groups of blocks of finite groups. Our main
theorem is a version of Brauer’s first main theorem: a correspondence between
the blocks of $G$ with defect group $D$ and the blocks of the normalizer
$\textnormal{N}_{G}(D)$ with defect group $D$.

###### keywords:

block theory , modular representation theory , profinite group, defect group ,
Brauer’s first main theorem.

###### MSC:

[2020]20C20, 20E18

††journal: Advances in Mathematics

##  1 Introduction

Let $G$ be a finite group and $k$ a field of characteristic $p$. The algebra
$kG$, as with any finite dimensional associative algebra, has a unique
decomposition as a direct product of indecomposable algebras
$kG=B_{1}\times\cdots\times B_{n}$, the $B_{i}$ being known in group
representation theory as the _blocks_ of $G$. Rather than try to deal with
$kG$ all at once, the philosophy of block theory is to study the blocks of $G$
individually, as the representation theory of certain $B_{i}$ may be
considerably more tractable than the representation theory of $kG$.

The difficulty of a block $B$ is measured by a $p$-subgroup of $G$ (unique up
to $G$-conjugacy) called the _defect group_ of $B$. As examples of this
“measure”: blocks with cyclic defect group are completely understood, while
there are structural conjectures about blocks whose defect group is abelian
that are false for arbitrary blocks, perhaps the most famous being Broué’s
Abelian Defect Group Conjecture (for a light-hearted survey of the state-of-
the-art in modular representation theory and block theory see [5]).

The most fundamental theorem of the block theory of finite groups is Brauer’s
first main theorem (see [3, Theorem 5] for the original statement or for
instance [2, Theorem 6.2.6] for a more modern one), which gives a natural
bijection between the blocks of $G$ with defect group $D$ and the blocks of
the normalizer $\textnormal{N}_{G}(D)$ of $D$ in $G$ with defect group $D$.
Our main theorem is Brauer’s first main theorem for profinite groups:

######  Theorem 1.1.

Let $k$ be a field of characteristic $p$, $G$ a profinite group and $D$ a
closed subgroup of $G$ that is open in a $p$-Sylow subgroup that contains it.
There is a natural bijection between the blocks of $G$ over $k$ with defect
group $D$, and the blocks of $\textnormal{N}_{G}(D)$ over $k$ with defect
group $D$.

The hypothesis that $D$ be open in a $p$-Sylow subgroup is necessary, but is
not a restriction because we will show that the defect group of every block of
$G$ has this property.

More broadly, we begin a systematic study of the local-global theory of blocks
of a profinite group $G$ over a field $k$ of characteristic $p$. We define the
defect group of a block of $G$, prove its basic properties and give several
alternative characterizations (Theorem 5.18), before proving Brauer’s first
main theorem.

The most striking feature of the discussion is the generality that block
theory permits: all results apply to arbitrary profinite groups. This suggests
that the block theoretic approach to the representation theory of profinite
groups will be as central to the theory as it is with finite groups. For
comparison, we note that with modules, conditions on the group are frequently
required to prove the most powerful results – see for instance [14, 15].

Because we do not demand that the field $k$ be finite, our algebras and
modules are not compact, but they are _pseudocompact_ , thus maintaining a
weaker version of compactness (linear compactness) that suffices for our
purposes. While pseudocompact objects are perhaps less familiar than compact
ones, the advantage of allowing this greater generality is undeniable, as it
gives a representation theorist access to algebraically closed fields.

In a future paper, the tools developed here will be used to provide a complete
classification of the blocks of a profinite group having (finite or infinite)
cyclic defect group, in close analogy with the classification of blocks of
finite groups with cyclic defect group.

The background to the material discussed in this article is well served by the
literature. For an introduction to profinite groups see [18] or [23]. For
basic properties of pseudocompact algebras see [4] or [17]. For introductory
expositions on the modular representation theory of finite groups see [2] or
[1]. For an excellent and encyclopedic introduction to the block theory of
finite groups see [12, 13].

## Acknowledgements:

We thank Jeremy Rickard, Peter Symonds and Pavel Zalesskii for helpful
conversations, and the anonymous referee for pertinent comments.

##  2 Preliminaries

###  2.1 Pseudocompact algebras and modules

Let $k$ be a discrete topological field of characteristic $p$. In what
follows, when the coefficient ring is unspecified it is assumed to be $k$ (so
for example “algebra” means “$k$-algebra”). Throughout this section, unless
otherwise specified, all modules are topological left modules. For a general
introduction to pseudocompact objects, see [4].

######  Definition 2.1.

A _pseudocompact algebra_ is an associative, unital, Hausdorff topological
$k$-algebra $A$ possessing a basis of $0$ consisting of open ideals $I$ having
cofinite dimension in $A$ that intersect in 0 and such that
$A=\varprojlim_{I}A/I$.

Equivalently, a pseudocompact algebra is an inverse limit of discrete finite
dimensional algebras in the category of topological algebras. If $G$ is a
profinite group, an inverse system of finite continuous quotients $G/N$ of $G$
induces an inverse system of finite dimensional algebras $k[G/N]$, whose
inverse limit $k[[{G}]]$, the _completed group algebra_ of $G$, is a
pseudocompact algebra.

We will occasionally make use of the Jacobson radical $J(A)$ of a
pseudocompact algebra $A$: that is, the intersection of the maximal open left
ideals of $A$. Essentially the only property we require of the Jacobson
radical is the easily checked fact that if $\rho:A\to B$ is a surjective
homomorphism of pseudocompact algebras, then $\rho(J(A))\subseteq J(B)$ (cf.
[4, §1] for more on $J(A)$).

If $A$ is a pseudocompact algebra, a _pseudocompact $A$-module_ is a
topological $A$-module $U$ possessing a basis of 0 consisting of open
submodules $V$ of finite codimension that intersect in 0 and such that
$U=\varprojlim_{V}U/V$. The category of pseudocompact modules for a
pseudocompact algebra has exact inverse limits.

If $V$ is a topological vector space, cosets of closed subspaces of $V$ are
called _closed linear varieties_. We say that $V$ is _linearly compact_ if for
every family $\mathcal{F}=\\{W_{i}\mbox{ : }i\in I\\}$ of closed linear
varieties of $V$ with the finite intersection property, we have that
$\bigcap_{i\in I}W_{i}\neq\emptyset$.

######  Lemma 2.2.

Let $A$ be a pseudocompact algebra and $U,V$ pseudocompact $A$-modules.

  1. 1.

The module $U$ is linearly compact.

  2. 2.

If ever $\rho:U\to V$ is a continuous homomorphism, then $\rho(U)$ is linearly
compact and hence closed in $V$.

  3. 3.

The submodule abstractly generated by a finite subset of $U$ is closed.

  4. 4.

If $U$ is finitely generated as an $A$-module, then every $A$-module
homomorphism $U\to V$ is continuous.

###### Proof.

Items 1 to 3 are [9, Lemma 2.2] and Item 4 is [20, Proposition 3.5]. ∎

We collect here several properties of linearly compact vector spaces, given in
[11, II, §6.27], which will be used frequently:

######  Lemma 2.3.

  1. 1.

Discrete finite dimensional $k$-vector spaces are linearly compact.

  2. 2.

A product of linearly compact $k$-vector spaces is linearly compact.

  3. 3.

Let $\\{V_{i},\varphi_{ij},I\\}$ be an inverse system of topological vector
spaces and for each $i$, let $W_{i}$ be a non-empty closed linear variety in
$V_{i}$. If $\varphi_{ij}(W_{i})\subseteq W_{j}$, so that
$\\{W_{i},\varphi_{ij},I\\}$ forms an inverse system, then its inverse limit
is non-empty.

  4. 4.

A continuous linear bijective map between two linearly compact spaces is an
isomorphism.

Several times in this section we will give results for pseudocompact algebras
and cite for the proof a reference whose statement is for profinite algebras
(that is, compact rather than just linearly compact algebras). When we do
this, what we mean is that the proof in the given reference goes through
without change for pseudocompact algebras.

###  2.2 Induction and restriction

The following definition makes use of the _completed tensor product_ (cf. [17,
§2.2]). Let $G$ be a profinite group and $H$ a closed subgroup of $G$. If $V$
is a pseudocompact $k[[{H}]]$-module, then the _induced $k[[{G}]]$-module_ is
defined by $V\uparrow^{G}=k[[{G}]]\widehat{\otimes}_{k[[{H}]]}V$, with action
from $G$ on the left factor. If $U$ is a $k[[{G}]]$-module, then the
_restricted $k[[{H}]]$-module_ $U\downarrow_{H}$ is the original
$k[[{G}]]$-module $U$ with coefficients restricted to the subalgebra
$k[[{H}]]$. Induction is left (but not right) adjoint to restriction [17,
§2.2]. If $H$ is a closed subgroup of $G$, denote by $k[[{G/H}]]$ the left
pseudocompact permutation $k[[{G}]]$-module given as the inverse limit of the
modules $k[G/HN]$, as $N$ runs through the open normal subgroups of $G$ (cf.
[17, §2.3]).

######  Lemma 2.4.

([18, Proposition 5.8.1]) Let $G$ be a profinite group, $H$ a closed subgroup
of $G$ (denoted $H\leq_{C}G$) and $U$ a pseudocompact $k[[{G}]]$-module. The
map $g\widehat{\otimes}_{k[[{H}]]}u\mapsto gH\widehat{\otimes}_{k}gu$ induces
a natural continuous isomorphism of left $k[[{G}]]$-modules

$k[[{G}]]\widehat{\otimes}_{k[[{H}]]}U\cong k[[{G/H}]]\widehat{\otimes}_{k}U,$

where the action of $k[[{G}]]$ on $k[[{G/H}]]\widehat{\otimes}_{k}U$ is
diagonal.

######  Lemma 2.5.

Let $G$ be a profinite group, $H\leq_{C}G$ and $U$ a pseudocompact
$k[[{G}]]$-module. Then
$U\downarrow_{{H}}\uparrow^{{G}}\cong\varprojlim_{N}U\downarrow_{{HN}}\uparrow^{{G}}$.

###### Proof.

This follows from [18, Proposition 5.2.2, Lemma 5.5.2, Proposition 5.8.1]. ∎

###  2.3 Homomorphisms and coinvariants

Let $U$ be a pseudocompact $k[[{G}]]$-module and $N$ a closed normal subgroup
of $G$. The module of _$N$ -coinvariants_ $U_{N}$ is defined as
$k\widehat{\otimes}_{k[[{N}]]}U\cong U/I_{N}U$, where $I_{N}$ denotes the
augmentation ideal of $k[[{N}]]$ – that is, the kernel of the continuous map
$k[[{N}]]\to k$ sending $n\mapsto 1$ for each $n\in N$. The action of $G$ on
$U_{N}$ is given by $g(\lambda\widehat{\otimes}u)=\lambda\widehat{\otimes}gu$.
The subgroup $N$ acts trivially on $U_{N}$, so we can treat $U_{N}$ as a
$k[G/N]$-module if we wish. Properties of the completed tensor product imply
that $(-)_{N}$ is a right exact functor from the category of finitely
generated $k[[{G}]]$-modules to the category of finitely generated
$k[[{G/N}]]$-modules. If $U$ is a finitely generated pseudocompact
$k[[{G}]]$-module and $N$ is an open normal subgroup of $G$, denoted
$N\trianglelefteq_{O}G$, then $U_{N}$ is finite dimensional.

The module $U_{N}$, together with the canonical quotient map $\varphi_{N}:U\to
U_{N}$, satisfies the following universal property:

_Every continuous $k[[{G}]]$-module homomorphism $\rho$ from $U$ to a
pseudocompact $k[[{G}]]$-module $X$ on which $N$ acts trivially, factors
uniquely through $\varphi_{N}$. That is, there is a unique continuous
homomorphism $\rho^{\prime}:U_{N}\to X$ such that
$\rho=\rho^{\prime}\varphi_{N}$._

We collect here several technical properties of coinvariant modules:

######  Lemma 2.6.

([15, Lemma 2.6]) Let $G$ be a profinite group, $N,M$ closed normal subgroups
of $G$ with $N\leq M$ and $H$ a closed subgroup of $G$. Let $U,W$ be
pseudocompact $k[[{G}]]$-modules and let $V$ be a $k[[{H}]]$-module. Then

  1. 1.

$(U_{N})_{M}$ is naturally isomorphic to $U_{M}$.

  2. 2.

$(U\oplus W)_{N}\cong U_{N}\oplus W_{N}$.

  3. 3.

$V_{H\cap N}$ is naturally a $k[[{HN/N}]]$-module.

  4. 4.

$(V\uparrow^{G})_{N}\cong V_{H\cap N}\uparrow^{G/N}$.

  5. 5.

$U_{N}\downarrow_{HN/N}\cong(U\downarrow_{HN})_{N}$.

######  Proposition 2.7.

([15, Proposition 2.7]) If $U$ is a pseudocompact $k[[{G}]]$-module, then
$\\{U_{N}\,:\,N\trianglelefteq_{O}G\\}$ together with the set of canonical
quotient maps forms a surjective inverse system with inverse limit $U$.

######  Remark 2.8.

Throughout this text, whenever $N\leqslant M$ are closed normal subgroups of
the profinite group $G$, the notations $\varphi_{N},\varphi_{MN}$ will be
reserved exclusively for the canonical maps between coinvariant modules. In
the special case of $k[[{G}]]$ we have $k[[{G}]]_{N}=k[[{G/N}]]$ and the
corresponding maps $\varphi_{N},\varphi_{MN}$ are in fact algebra
homomorphisms.

Let $A$ be a pseudocompact algebra. If $U$ and $W$ are topological
$A$-modules, then $\textnormal{Hom}_{A}(U,W)$ denotes the topological
$k$-vector space of continuous $A$-module homomorphisms from $U$ to $W$ with
the compact-open topology. If $U,W$ are pseudocompact, and
$W=\varprojlim_{i}W_{i}$, then
$\textnormal{Hom}_{A}(U,W)=\varprojlim_{i}\textnormal{Hom}_{A}(U,W_{i})$. In
particular, when $U$ is finitely generated as an $A$-module, then
$\textnormal{Hom}_{A}(U,W)$ is a pseudocompact vector space. For more details
see [17, §2.2]. In the special case where $A=k[[{G}]]$, the natural
isomorphism
$\textnormal{Hom}_{k[[{G}]]}(U,W_{N})\cong\textnormal{Hom}_{k[[{G}]]}(U_{N},W_{N})$
implies that
$\textnormal{Hom}_{k[[{G}]]}(U,W)\cong\varprojlim_{N}\textnormal{Hom}_{k[[{G}]]}(U_{N},W_{N})$.

######  Proposition 2.9.

Let $G$ be a profinite group and let $U$ be an indecomposable finitely
generated $k[[{G}]]$-module. Then $\textnormal{End}_{k[[{G}]]}(U)$ is a local
pseudocompact algebra.

###### Proof.

The algebra $\textnormal{End}_{k[[{G}]]}(U)$ is pseudocompact by [17, Lemma
2.3]. By [6], linearly compact modules are algebraically compact (see for
instance [6, Proposition 4.11]) and so the ring of abstract endomorphisms of
$U$ is local [6, Theorem 4.15 and Proposition 4.10]. But since $U$ is finitely
generated, the rings of abstract and continuous endomorphisms coincide by
Lemma 2.2. ∎

######  Lemma 2.10.

Let $U,W$ be pseudocompact $k[[{G}]]$-modules with $U$ finitely generated. If
$\pi:W\to U$ is a surjective $k[[{G}]]$-module homomorphism such that the
induced map $\pi_{N}:W_{N}\to U_{N}$ splits as a $k[G/N]$-module homomorphism
for each $N\trianglelefteq_{O}G$, then $\pi$ splits as a $k[[{G}]]$-module
homomorphism.

###### Proof.

For each $N\trianglelefteq_{O}G$ we have by hypothesis a non-empty set
$X_{N}\subseteq\textnormal{Hom}_{k[[{G}]]}(U_{N},W_{N})$ of splittings of
$\pi_{N}$. For $N\leqslant M$ the map $X_{N}\to X_{M}$ given by
$\alpha\mapsto\alpha_{M}$ makes the $X_{N}$ into an inverse system. Each
$X_{N}$ is a non-empty closed linear variety, being equal to
$\alpha+\textnormal{Ker}(\Gamma)$, where $\alpha$ is an arbitrary element of
$X_{N}$ and
$\Gamma:\textnormal{Hom}_{k[[{G}]]}(U_{N},W_{N})\to\textnormal{End}_{k[[{G}]]}(U_{N})$
is the continuous linear map sending $\gamma\mapsto\pi_{N}\gamma$. It follows
from Lemma 2.3 that $\varprojlim X_{N}\neq\varnothing$. An element
$\iota\in\varprojlim X_{N}$ is a splitting of $\pi$. ∎

######  Remark 2.11.

Lemma 2.10 may be compared with [15, Lemma 3.4], which says that when $k$ is
finite and $U,W$ are finitely generated, then if $U_{N}$ is isomorphic to a
direct summand of $W_{N}$ for each $N$, then $U$ is isomorphic to a direct
summand of $W$. On one hand Lemma 2.10 is stronger, as it does not require
that $W$ be finitely generated. On the other hand, even when $W$ is assumed to
be finitely generated, the proof of [15, Lemma 3.4] does not obviously carry
through when $k$ is infinite. This weaker version (wherein we must specify our
candidate $\pi$ for the split surjection) will be sufficient for our purposes.
If $W$ is assumed to be finitely generated, one may alternatively fix the
candidate $\iota:U\to W$ for a split injection and find a corresponding split
surjection $\pi$.

###  2.4 Relative Projectivity

Most of the results of this section were proved for $k$ finite in [15]. We
restate them here in the greater generality we require, observing any
modifications required in the proof. A pseudocompact $k[[{G}]]$-module $U$ is
_relatively $H$-projective_ if given any diagram of pseudocompact
$k[[{G}]]$-modules and continuous $k[[{G}]]$-module homomorphisms of the form

such that there exists a continuous $k[[{H}]]$-module homomorphism $U\to V$
making the triangle commute, then there exists a continuous $k[[{G}]]$-module
homomorphism $U\to V$ making the triangle commute. Observe that a projective
module is the same thing as a 1-projective module. Several familiar
characterizations of relatively projective modules follow exactly as with
finite groups:

######  Lemma 2.12.

Let $G$ be a profinite group, $H$ a closed subgroup of $G$ and let $U$ be a
pseudocompact $k[[{G}]]$-module. The following are equivalent.

  1. 1.

$U$ is relatively $H$-projective,

  2. 2.

If a continuous $k[[{G}]]$-module epimorphism $V\to U$ splits as a
$k[[{H}]]$-module homomorphism, then it splits,

  3. 3.

$U$ is isomorphic to a direct summand of $U\downarrow_{{H}}\uparrow^{{G}}$,

  4. 4.

The natural projection $\pi:U\downarrow_{{H}}\uparrow^{{G}}\to U$ sending
$g\widehat{\otimes}u\mapsto gu$ splits.

  5. 5.

$U$ is isomorphic to a direct summand of $X\uparrow^{{G}}$ for some
$k[[{H}]]$-module $X$.

###### Proof.

This goes through as for finite groups (cf. [2, Proposition 3.6.4]). We show
only that 3 implies 4, since it is not completely standard (though still well-
known). Note that $\pi$ is the component at $U$ of the counit of the
induction-restriction adjunction. If $\alpha:U\to
U\downarrow_{{H}}\uparrow^{{G}}$ and $\beta:U\downarrow_{{H}}\uparrow^{{G}}\to
U$ are the given splitting maps then, by formal properties of adjoint
functors, $\beta=\pi\circ\gamma\uparrow^{{G}}$ for some endomorphism $\gamma$
of $U\downarrow_{{H}}$. But then
$\textnormal{id}_{U}=\beta\alpha=\pi(\gamma\uparrow^{{G}}\alpha)$, showing
that $\pi$ is split. ∎

######  Proposition 2.13.

Let $U$ be a finitely generated pseudocompact $k[[{G}]]$-module, and
$H\leq_{C}G$. Then $U$ is relatively $H$-projective if, and only if, $U$ is
relatively $HN$-projective for every $N\trianglelefteq_{O}G$.

###### Proof.

One may follow the proof of [15, Proposition 3.3], noting as in the proof of
Lemma 2.10 that the subsets $I_{N}$ of
$\textnormal{Hom}_{k[[{G}]]}(U,U\downarrow_{{HN}}\uparrow^{{G}})$ are closed
linear varieties, so that the inverse limit of the $I_{N}$ is non-empty by
Lemma 2.3. ∎

Recall that a subset $J$ of the directed set $I$ is _cofinal_ if, for every
$i\in I$ there is $j\in J$ with $j\geqslant i$. In this case whenever
$\\{X_{i}\,|\,i\in I\\}$ is an inverse system indexed by $I$, then
$\\{X_{i}\,|\,i\in J\\}$ is also an inverse system and
$\underleftarrow{\textnormal{lim}}\,_{i\in
I}X_{i}=\underleftarrow{\textnormal{lim}}\,_{i\in J}X_{i}$. Our directed set
will usually be the set of open normal subgroups of a profinite group $G$,
ordered by reverse inclusion, so that cofinal means that for each open normal
subgroup $M$ of $G$, there is $N$ in the cofinal subset with $N\leqslant M$.

######  Proposition 2.14.

Let $U$ be a finitely generated pseudocompact $k[[{G}]]$-module, and
$H\leq_{C}G$. Then $U$ is relatively $H$-projective if, and only if, $U_{N}$
is relatively $HN$-projective for every $N\trianglelefteq_{O}G$.

###### Proof.

We essentially follow the proof of [15, Proposition 3.5]. Fix
$M\trianglelefteq_{O}G$ and consider the cofinal system of open normal
subgroups $N$ of $G$ contained in $M$. The module $U_{N}$ is relatively
$HM$-projective and hence by Lemma 2.12, the canonical projection
$U_{N}\downarrow_{{HM}}\uparrow^{{G}}\to U_{N}$ splits. But
$U_{N}\downarrow_{{HM}}\uparrow^{{G}}\cong(U\downarrow_{{HM}}\uparrow^{{G}})_{N}$
and hence by Lemma 2.10, $U$ is relatively $HM$-projective. The result now
follows from Proposition 2.13. ∎

######  Definition 2.15.

If $H\leq_{O}G$ and $U$, $W$ are $k[[{G}]]$-modules, then _the trace map_

$\textnormal{Tr}_{H}^{G}:\textnormal{Hom}_{k[[{H}]]}(U\downarrow_{H},W\downarrow_{H})\to\textnormal{Hom}_{k[[{G}]]}(U,W),$

is defined by $\alpha\mapsto\sum_{s\in G/H}s\alpha s^{-1}$.

Collecting the above results and mimicking the finite case for the final item,
we have:

######  Theorem 2.16.

Let $G$ be a profinite group, $H\leq_{C}G$, and $U$ a finitely generated
pseudocompact $k[[{G}]]$-module. The following are equivalent:

  1. 1.

$U$ is relatively $H$-projective,

  2. 2.

If a continuous $k[[{G}]]$-module epimorphism $V\to U$ splits as a
$k[[{H}]]$-module homomorphism, then it splits,

  3. 3.

$U$ is isomorphic to a continuous direct summand of
$U\downarrow_{{H}}\uparrow^{{G}}$,

  4. 4.

The natural projection $\pi:U\downarrow_{{H}}\uparrow^{{G}}\to U$ sending
$g\widehat{\otimes}u\mapsto gu$ splits.

  5. 5.

$U$ is isomorphic to a continuous direct summand of $X\uparrow^{{G}}$ for some
$k[[{H}]]$-module $X$.

  6. 6.

$U$ is relatively $HN$-projective for every $N\trianglelefteq_{O}G$.

  7. 7.

$U_{N}$ is relatively $HN$-projective for every $N\trianglelefteq_{O}G$.

  8. 8.

For every $N\trianglelefteq_{O}G$ there exists a continuous
$k[[{HN}]]$-endomorphism $\alpha_{N}$ of $U$ such that
$\textnormal{id}_{U}=\textnormal{Tr}_{HN}^{G}(\alpha_{N})$.

###  2.5 Vertices

Let $G$ be a profinite group and $U$ a finitely generated indecomposable
pseudocompact $k[[{G}]]$-module. A _vertex_ of $U$ is a closed subgroup $Q$ of
$G$ such that $U$ is projective relative to $Q$, but not projective relative
to any proper closed subgroup of $Q$.

The following results were proved for profinite $k[[{G}]]$-modules in [15,
Section 4], but both statements and proofs carry through for pseudocompact
$k[[{G}]]$-modules, given Proposition 2.9.

######  Proposition 2.17.

([15, Section 4]) If $U$ is an indecomposable finitely generated pseudocompact
$k[[{G}]]$-module, then a vertex of $U$ exists. The vertices of $U$ are a
$G$-conjugacy class of pro-$p$ subgroups of $G$.

Here and elsewhere, when $U,V$ are $k[[{G}]]$-modules, we write
$U\raisebox{0.5pt}{\,\big{|}\,}V$ to mean that $U$ is isomorphic to a
continuous direct summand of $V$.

######  Lemma 2.18.

Let $G$ be a profinite group, $N\trianglelefteq_{C}G$ and $U$ a finitely
generated indecomposable $k[[{G}]]$-module on which $N$ acts trivially. If $U$
has vertex $Q$ as a $k[[{G}]]$-module, then $U$ has vertex $QN/N$ as a
$k[G/N]$-module.

###### Proof.

The module $U$ has vertex $R/N$ as a $k[G/N]$-module, for some $N\leq R\leq
QN$. Note that $N$ acts trivially on $U\downarrow_{{R}}\uparrow^{{G}}$, so
that by Lemma 2.6

$U\downarrow_{{R}}\uparrow^{{G}}\cong(U\downarrow_{{R}}\uparrow^{{G}})_{N}\cong
U_{N}\downarrow_{{R/N}}\uparrow^{{G/N}}\cong
U\downarrow_{{R/N}}\uparrow^{{G/N}}.$

Hence $U\raisebox{0.5pt}{\,\big{|}\,}U\downarrow_{R}\uparrow^{G}$ as a
$k[[{G}]]$-module. Thus $U$ is relatively $R$-projective, and hence
${}^{g}\\!QN\leqslant R\leqslant QN$ for some $g\in G$ by Proposition 2.17, so
that $R=QN$. ∎

######  Proposition 2.19.

Let $U$ be an indecomposable finite dimensional $k[[{G}]]$-module with vertex
$Q$. If $P$ is a $p$-Sylow subgroup of $G$ containing $Q$, then $Q$ is open in
$P$.

###### Proof.

We must check that the index of $Q$ in $P$ is finite. Since $U$ is finite
dimensional, there is a cofinal system of open normal subgroups $N$ of $G$
acting trivially on $U$. By Lemma 2.18, $U$ has vertex $QN/N$ as a
$k[G/N]$-module. Furthermore, by [23, Proposition 2.2.3] $PN/N$ is a Sylow
$p$-subgroup of $G/N$ and so by [8, Theorem 9 and Corollary 2], $|PN/N:QN/N|$
divides the dimension of $U$ for each $N$, and hence so does $|P:Q|$, showing
that $Q$ is open in $P$. ∎

##  3 Trace maps

Let $G$ be a profinite group and $A$ a pseudocompact $k$-algebra. We say that
$A$ is a pseudocompact _$G$ -algebra_ if $A$ is endowed with a continuous
action $G\times A\to A$ of $G$ on $A$, written $(g,a)\mapsto{}^{g}\\!a$ for
all $a\in A$ and $g\in G$, such that the continuous map sending $a\in A$ to
${}^{g}\\!a$ is a $k$-algebra automorphism. If $A,B$ are pseudocompact
$G$-algebras, a continuous $k$-algebra homomorphism $f:A\to B$ is a
_homomorphism of $G$-algebras_ if $f({}^{g}\\!a)={}^{g}\\!f(a)$ for all $a\in
A$ and $g\in G$.

If $H$ is a closed subgroup of $G$, the _subalgebra of $H$-fixed points_ of
$A$ is defined by

$A^{H}:=\\{a\in A\,|\,{}^{h}a=a\,\forall h\in H\\}.$

######  Definition 3.1.

Let $A$ be a pseudocompact $G$-algebra and $H$ an open subgroup of $G$. The
corresponding _trace map_ is the continuous linear map

$\displaystyle\textnormal{Tr}_{H}^{G}:A^{H}$ $\displaystyle\to$ $\displaystyle
A^{G}$ $\displaystyle a$ $\displaystyle\mapsto$ $\displaystyle\sum_{g\in
G/H}{}^{g}\\!a,$

where $G/H$ denotes a set of left coset representatives of $H$ in $G$.

######  Remark 3.2.

When $U$ is a finite dimensional $k[[{G}]]$-module, $\textnormal{End}_{k}(U)$
is a finite dimensional $G$-algebra with action
$({}^{g}\\!\rho)(u):=g\rho(g^{-1}u)$, and furthermore when $H$ is a closed
subgroup of $G$ then
$\textnormal{End}_{k}(U)^{H}=\textnormal{End}_{k[[{H}]]}(U)$. It follows in
particular that the notations $\textnormal{Tr}_{H}^{G}$ used in Definitions
2.15 and 3.1 are consistent for finite groups. Formalizing the consistency for
profinite groups is made awkward by the fact that $\textnormal{End}_{k}(U)$ is
not pseudocompact when $U$ has infinite dimension, so to avoid unnecessary
technicalities we will not attempt to do so. The overlap of notation will not
cause any difficulties.

We list some basic properties of the trace map, whose proofs carry through
just as for finite groups, see for instance [2, Lemma 3.6.3].

######  Lemma 3.3.

Let $G$ be a profinite group and $A$ a pseudocompact $G$-algebra. Then

  1. 1.

If $H$ is an open subgroup of $G$, for any $a\in A^{H}$ and $b\in A^{G}$ we
have $b\,\textnormal{Tr}_{H}^{G}(a)=\textnormal{Tr}_{H}^{G}(ba)$ and
$\textnormal{Tr}_{H}^{G}(a)b=\textnormal{Tr}_{H}^{G}(ab)$.

  2. 2.

If $H,L$ are open subgroups of $G$ with $L\leq H$, then
$\textnormal{Tr}_{H}^{G}\circ\textnormal{Tr}_{L}^{H}=\textnormal{Tr}_{L}^{G}$.

  3. 3.

(Mackey’s Formula) If $H,L$ are closed subgroups of $G$ with $L$ open, then
for any $a\in A^{L}$ we have

$\textnormal{Tr}_{L}^{G}(a)=\sum\limits_{g\in H\backslash
G/L}\textnormal{Tr}_{H\cap{}^{g}\\!L}^{H}({}^{g}\\!a),$

where $H\backslash G/L$ denotes a set of double coset representatives.

Definition 3.1 does not make sense for subgroups of $G$ that are not open. We
present two alternatives. Firstly:

######  Definition 3.4.

Let $A$ be a pseudocompact $G$-algebra. If $H$ is a closed subgroup of $G$,
define

$\textnormal{oTr}_{H}^{G}(A^{H}):=\bigcap_{N\trianglelefteq_{O}G}\textnormal{Tr}_{HN}^{G}(A^{HN}).$

The letter “o” indicates “open subgroups” and is only included to minimize
notational confusion. Observe that, while $\textnormal{oTr}_{H}^{G}(A^{H})$ is
a well-defined subset of $A^{G}$, one cannot really regard
$\textnormal{oTr}_{H}^{G}$ as a map. In order to obtain a map, for simplicity
we restrict to the case where $A=k[[{G}]]$, a $G$-algebra with action from $G$
given by conjugation, so for $g\in G$ and $x\in k[[{G}]]$ we have
${}^{g}\\!x:=gxg^{-1}$. The reason to restrict is that $k[[{G}]]$ gives us
access to the algebra homomorphisms $\varphi_{MN},\varphi_{N}$ of Remark 2.8,
and because we will require only this special case. The definition relies on
Sylow’s Theorem for profinite groups, for which see for instance [18,
Corollary 2.3.6]. We use the standard trick of realizing an inverse limit
$A=\underleftarrow{\textnormal{lim}}\,_{I}\\{A_{i},\varphi_{ij}\\}$ as

$A=\bigg{\\{}(a_{i})_{i\in
I}\in\prod_{I}A_{i}\,|\,\varphi_{ij}(a_{j})=a_{i}\,\forall i\leqslant
j\bigg{\\}}.$

If ever $n$ is a natural number, write $n=n_{p}\cdot n_{p^{\prime}}$ where
$n_{p}$ is a power of $p$ and $p$ does not divide $n_{p^{\prime}}$. Recall
[18, §2.3] that if ever $H\leqslant L$ are closed subgroups of the profinite
group $G$, the index of $H$ in $L$ is a well-defined supernatural number, with
a well-defined decomposition

$|L:H|=|L:H|_{p}\cdot|L:H|_{p^{\prime}}.$

######  Definition 3.5.

  1. 1.

Let $G$ be a finite group and $H\leqslant L$ subgroups of $G$. The modified
trace function $\widetilde{\textnormal{Tr}}_{H}^{L}:kG^{H}\to kG^{L}$ is
defined as follows:

$\widetilde{\textnormal{Tr}}_{H}^{L}(x):=\frac{1}{|L:H|_{p^{\prime}}}\textnormal{Tr}_{H}^{L}(x).$

  2. 2.

Let $G$ be a profinite group and $H\leqslant L$ closed subgroups of $G$.
Suppose that $|L:H|_{p}$ is a finite number. The modified trace function
$\widetilde{\textnormal{Tr}}_{H}^{L}:k[[{G}]]^{H}\to k[[{G}]]^{L}$ is defined
as follows:

$\widetilde{\textnormal{Tr}}_{H}^{L}(x):=\left(\widetilde{\textnormal{Tr}}_{HN/N}^{LN/N}(\varphi_{N}(x))\right)_{N\in\mathcal{N}}=\left(\frac{1}{|LN:HN|_{p^{\prime}}}\textnormal{Tr}_{HN/N}^{LN/N}(\varphi_{N}(x))\right)_{N\in\mathcal{N}},$

where $\mathcal{N}$ is the set of open normal subgroups of $G$ such that
$|LN:HN|_{p}=|L:H|_{p}$.

Some observations are called for:

  * 1.

The set $\mathcal{N}$ is non-empty (and hence cofinal in the set of all open
normal subgroups of $G$) because $|L:H|_{p}$ is finite. We do not define a
trace function between subgroups without this property. In practice the
subgroups $H$ to which we will apply $\widetilde{\textnormal{Tr}}_{H}^{L}$ are
defect groups of a block of $G$, which we will see in Proposition 5.8 are open
subgroups of a $p$-Sylow subgroup of $G$.

  * 2.

We prove only those properties of $\widetilde{\textnormal{Tr}}_{H}^{L}$ that
we require to show Brauer’s first main theorem in Section 6. There may be some
value to a more systematic study of the linear map
$\widetilde{\textnormal{Tr}}_{H}^{L}$.

We must check that the element $\widetilde{\textnormal{Tr}}_{H}^{L}(x)$
defined in the second part of the above definition is a well-defined element
of $k[[{G}]]^{L}$. This is a formal consequence of the first part of the
following lemma (it is to make the vector compatible that we restrict our $N$
to the cofinal set $\mathcal{N}$).

######  Lemma 3.6.

Let $H\leqslant L$ be closed subgroups of $G$ with $|L:H|_{p}$ finite.

  1. 1.

Given $N\leqslant M$ in $\mathcal{N}$, we have

$\varphi_{MN}\widetilde{\textnormal{Tr}}_{HN/N}^{LN/N}=\widetilde{\textnormal{Tr}}_{HM/M}^{LM/M}\varphi_{MN}.$

  2. 2.

If $|G:H|_{p}$ is finite, then

$\widetilde{\textnormal{Tr}}_{L}^{G}\widetilde{\textnormal{Tr}}_{H}^{L}=\widetilde{\textnormal{Tr}}_{H}^{G}.$

  3. 3.

If $a\in k[[{G}]]^{H}$ and $b\in k[[{G}]]^{L}$ then
$b\,\widetilde{\textnormal{Tr}}_{H}^{L}(a)=\widetilde{\textnormal{Tr}}_{H}^{L}(ba)$
and
$\widetilde{\textnormal{Tr}}_{H}^{L}(a)b=\widetilde{\textnormal{Tr}}_{H}^{L}(ab)$.

###### Proof.

  1. 1.

Note first that $(HM)(LN)=LM$ and so, by the second isomorphism theorem,

$|LN:HM\cap LN|=|LM:HM|.$

We have

$|LN:HN|=|LN:HM\cap LN|\cdot|HM\cap LN:HN|=|LM:HM|\cdot|HM\cap LN:HN|$

and hence, since $|LN:HN|_{p}=|LM:HM|_{p}$, we have

$|LN:HN|_{p^{\prime}}=|LM:HM|_{p^{\prime}}\cdot|HM\cap LN:HN|.$

Now

$\displaystyle\varphi_{MN}\widetilde{\textnormal{Tr}}_{HN/N}^{LN/N}(x)$
$\displaystyle=\frac{1}{|LN:HN|_{p^{\prime}}}\sum_{g\in
LN/HN}{}^{gM}\varphi_{MN}(x)$ $\displaystyle=\frac{|HM\cap
LN:HN|}{|LN:HN|_{p^{\prime}}}\cdot\textnormal{Tr}_{HM/M}^{LM/M}\varphi_{MN}(x)$
$\displaystyle=\frac{1}{|LM:HM|_{p^{\prime}}}\cdot\textnormal{Tr}_{HM/M}^{LM/M}\varphi_{MN}(x)$
$\displaystyle=\widetilde{\textnormal{Tr}}_{HM/M}^{LM/M}\varphi_{MN}(x)$

as required.

  2. 2.

For finite groups this is an easy check using the transitivity of the normal
trace function (cf. Lemma 3.3). The general case follows from this and Part 1.

  3. 3.

This also follows from the corresponding property for finite groups.

∎

######  Lemma 3.7.

Let $G$ be a profinite group and $D$ a closed subgroup of $G$ such that
$|G:D|_{p}$ is finite. Then
$\textnormal{oTr}_{D}^{G}(k[[{G}]]^{D})\subseteq\textnormal{Im}(\widetilde{\textnormal{Tr}}_{D}^{G})$.

###### Proof.

It is sufficient to check that $\varphi_{N}(y)$ is in
$\textnormal{Tr}_{DN/N}^{G/N}(k[G/N]^{DN/N})$ for any $y$ in
$\textnormal{oTr}_{D}^{G}(k[[{G}]]^{D})$ and open normal subgroup $N$ of $G$.
But by definition $y=\sum_{g\in G/DN}{}^{g}\\!a$ for some $a\in k[[{G}]]^{DN}$
and so

$\varphi_{N}(y)=\varphi_{N}\left(\sum_{g\in
G/DN}{}^{g}\\!a\right)=\textnormal{Tr}_{DN/N}^{G/N}(\varphi_{N}(a))\in\textnormal{Tr}_{DN/N}^{G/N}(k[G/N]^{DN/N}),$

as required. ∎

##  4 Blocks of profinite groups

Throughout this section, $k$ is a discrete field of characteristic $p$ and $A$
is a pseudocompact $k$-algebra. An element $e\in A$ is _idempotent_ if
$e^{2}=e$. Two idempotents $e,f$ of $A$ are _orthogonal_ if $ef=fe=0$. A non-
zero idempotent is _primitive_ if it cannot be written as the sum of two non-
zero orthogonal idempotents. We denote by $\textnormal{Z}(A)$ the center of
$A$. An idempotent $e\in A$ is _centrally primitive_ if $e$ is a primitive
idempotent of $\textnormal{Z}(A)$.

By [7, IV. §3, Corollary 1,2] applied to $\textnormal{Z}(A)$, there is a
unique set of pairwise ortogonal centrally primitive idempotents
$E=\\{e_{i}\mbox{ : }i\in I\\}$ in $A$ such that

$\displaystyle A=\prod\limits_{i\in I}Ae_{i}.$ (2)

We call $E$ the complete set of centrally primitive orthogonal idempotents of
$A$. Any subset $F$ of $E$ is _summable_ , in the sense that each open ideal
$I$ of $A$ contains all but finitely many of the elements of $F$, so that
$\sum_{f\in F}f$ is a well-defined element of $A$. We have $\sum_{e\in
E}e=1_{A}$.

Each $Ae_{i}$ is a pseudocompact algebra (with identity $e_{i}$), called a
_block_ of $A$. The elements of $E$ are called _block idempotents_. The
decomposition (2) is called the _block decomposition_ of $A$. Note that the
block decomposition of $A$ is unique.

In what follows, $\overline{Y}$ denotes the topological closure of the subset
$Y$ of $A$.

######  Lemma 4.1.

Let $A$ be a pseudocompact algebra and let $e\in A$ be an idempotent. If $X$
is a not necessarily closed ideal of $A$, then
$e\overline{X}e=\overline{eXe}$.

###### Proof.

The map $\alpha:A\to eAe$ sending $a$ to $eae$ is continuous so that by Lemma
2.2, $e\overline{X}e$ is a closed subset of $eAe$ containing $eXe$, and hence
$\overline{eXe}\subseteq e\overline{X}e$. The other inclusion is direct from
the continuity of $\alpha$. ∎

######  Lemma 4.2.

Let $A$ be a pseudocompact $k$-algebra and $e$ a primitive idempotent of $A$.
Then

  1. 1.

$eAe$ is a local algebra.

  2. 2.

(Rosenberg’s Lemma) If $\mathcal{J}$ is a set of closed ideals of $A$ and
$e\in\overline{\sum_{I\in\mathcal{J}}I}$, then there is some $I\in\mathcal{J}$
such that $e\in I$.

###### Proof.

  1. 1.

This follows as in [2, Lemma 1.3.3], using Lemma 2.9.

  2. 2.

By Lemma 4.1,
$e(\overline{\sum_{I\in\mathcal{J}}I})e=\overline{e(\sum_{I\in\mathcal{J}}I)e}=\overline{\sum_{I\in\mathcal{J}}eIe}$,
so that if $e\in\overline{\sum_{I\in\mathcal{J}}I}$ then
$e\in\overline{\sum_{I\in\mathcal{J}}eIe}$. Since the algebra $eAe$ is local,
each ideal $eIe$, with $I\in\mathcal{J}$, is either contained in the Jacobson
radical of $eAe$ or is equal to $eAe$. Thus there is at least one
$I\in\mathcal{J}$ such that $eIe=eAe$, and $e$ is contained in this $I$.

∎

We say that the pseudocompact $A$-module $U$ _lies_ in the block $B$ if $BU=U$
and $B^{\prime}U=0$ for any block $B^{\prime}$ different from $B$. If $B$ has
unity $e$, then $U$ lies in $B$ if, and only if, $eU=U$ and $fU=0$ for each
centrally primitive idempotent $f$ distinct from $e$. If $U$ lies in $B$ and
$V$ is a closed submodule of $U$, then $V$ and $U/V$ lie in $B$. If
$U_{1},U_{2}$ are $A$-modules lying in $B$ then $U_{1}\oplus U_{2}$ lies in
$B$.

######  Proposition 4.3.

Let $A=\prod B_{i}$ be the block decomposition of $A$ and let $U$ be a
pseudocompact $A$-module. Then $U$ has a unique decomposition of the form
$\prod\limits_{i\in I}U_{i},$ where $U_{i}$ is a submodule of $U$ lying in the
block $B_{i}$.

###### Proof.

Let $e_{i}$ be the block idempotent of $B_{i}$. Define $U_{i}=e_{i}U$, an
$A$-module lying in $B_{i}$. By Lemmas 2.2 and 2.3, $\prod\limits_{i\in
I}U_{i}$ is a pseudocompact $A$-module.

Define for each $i\in I$ the continuous homomorphism $\rho_{i}:U\to U_{i}$ by
$u\mapsto e_{i}u$, and for each finite subset $K$ of $I$, the continuous
homomorphism $\rho_{K}:U\to\prod\limits_{i\in K}U_{i}$ by
$\rho_{K}(u)=(\rho_{i}(u))_{i\in K}$. The homomorphisms $\rho_{K}$ induce a
continuous surjective homomorphism $\rho:U\to\prod\limits_{i\in I}U_{i}$,
which is injective since $\sum_{e\in E}e=1_{A}$. As continuous linear
bijections between pseudocompact algebras are isomorphisms by Lemma 2.3,
$U\cong\prod\limits_{i\in I}U_{i}$. The uniqueness of the decomposition
follows from the uniqueness of the complete set of centrally primitive
orthogonal idempotents. ∎

In particular, indecomposable pseudocompact modules lie in a unique block.

###  4.1 Finite dimensional modules and blocks

Recall our convention that the symbols $\varphi_{N}$ and $\varphi_{MN}$ are
reserved for the canonical projections $\varphi_{N}:k[[{G}]]\to
k[G/N],\varphi_{MN}:k[G/N]\to k[G/M]$.

######  Remark 4.4.

By [16, Proposition 6.4], the complete set of centrally primitive orthogonal
idempotents $E$ of $k[[{G}]]$ is a discrete set and can be obtained as the
direct limit of the corresponding complete sets $E_{N}$ for the $k[G/N]$. We
make the direct system explicit, as we will use it in future: if $N\leq M$ are
open normal subgroups of $G$, define $\psi_{MN}:E_{M}\to E_{N}$ by sending
$c\in E_{M}$ to the unique centrally primitive idempotent $d$ of
$k\left[G/N\right]$ such that $\varphi_{MN}(d)c\neq 0$. Note that as $c$ is
primitive as a central idempotent, this is equivalent to saying that
$\varphi_{MN}(d)c=c$.

######  Lemma 4.5.

Let $U,V$ be finite dimensional $k[[{G}]]$-modules lying in the same block of
$G$. Then there is $N_{0}\trianglelefteq_{O}G$ acting trivially on $U$, $V$
and such that $U,V$ lie in the same block of $G/N_{0}$.

###### Proof.

Since $U,V$ are finite dimensional, it is sufficient to prove the lemma
supposing that both are indecomposable. Let $e$ be the block idempotent such
that $eU=U,eV=V$ and let $M$ be an open normal subgroup of $G$ acting
trivially on both $U$ and $V$, so that both can be treated as
$k[G/M]$-modules. Note that they are indecomposable as such.

We use the notation from Remark 4.4. If $f,g$ are the block idempotents of
$U,V$ respectively in $k[G/M]$ then, since $\varphi_{M}(e)f=f$ and
$\varphi_{M}(e)g=g$, we have that $\psi_{M}(f)=\psi_{M}(g)=e$. It follows from
the construction of the direct limit that there is an open normal subgroup
$N_{0}$ of $G$ contained in $M$ such that $\psi_{MN_{0}}(f)=\psi_{MN_{0}}(g)$.
Both $U$ and $V$ are in the block of $k[G/N_{0}]$ with block idempotent
$\psi_{MN_{0}}(f)$. ∎

###  4.2 Inverse systems of blocks

Consider $k[[{G}]]$ as a module for the completed group algebra $k[[{G\times
G}]]$ with the following continuous multiplication:

$\displaystyle k[[{G\times G}]]\times k[[{G}]]$ $\displaystyle\to$
$\displaystyle k[[{G}]]$ $\displaystyle((g_{1},g_{2}),x)$
$\displaystyle\mapsto$ $\displaystyle g_{1}xg_{2}^{-1},$

for $g_{1},g_{2}\in G$ and $x\in k[[{G}]]$. The canonical projections
$\varphi_{MN}:k[G/N]\to k[G/M],\varphi_{N}:k[[{G}]]\to k[G/N]$ are
$k[[{G\times G}]]$-module homomorphisms.

The blocks of $G$ are precisely the indecomposable summands of $k[[{G}]]$ as a
$k[[{G\times G}]]$-module and they are pairwise non-isomorphic as $k[[{G\times
G}]]$-modules, since their annihilators as $k[[{G\times 1}]]$-modules are
pairwise non-equal.

Let $B$ be a block of $G$ treated as a $k[[{G\times G}]]$-module, and
$N\trianglelefteq_{O}G$. Define $B_{N}$ to be the coinvariant module
$B_{N\times N}$, but note that $B_{N\times N}=B_{N\times 1}$, since given
$m,n\in N$ and $x\in B_{N\times 1}$ there is $n^{\prime}\in N$ such that

$(m,n)x=mxn^{-1}=mn^{\prime}x=(mn^{\prime},1)x=x,$

so that $N\times N$ acts trivially on $B_{N\times 1}$. By Proposition 2.7,
$B=\varprojlim_{N}\\{B_{N},\varphi_{MN}\\}$ where, as ever, $\varphi_{MN}$ is
the canonical quotient map.

Denote by $\delta:G\to G\times G$ the diagonal map $g\mapsto(g,g)$.

######  Lemma 4.6.

The $k[[{G\times G}]]$-module $k[[{G}]]$ is isomorphic to the induced module
$k\uparrow_{\delta(G)}^{G\times G}$.

###### Proof.

The $k[[{G\times G}]]$-module homomorphisms $\rho:k[[{G}]]\to
k\uparrow_{\delta(G)}^{G\times G}$ defined by
$g\mapsto(g,1)\widehat{\otimes}1$, and $\gamma:k\uparrow_{\delta(G)}^{G\times
G}\to k[[{G}]]$ defined by $(g,h)\widehat{\otimes}\lambda\mapsto g\lambda
h^{-1}$, are continuous. The checks that they are well-defined and mutually
inverse are as in [12, Proposition 5.11.6]. ∎

######  Proposition 4.7.

Let $B$ be a block of $G$. Then $B$ is the inverse limit of a surjective
inverse system of blocks of $G/N$ and algebra homomorphisms, for some cofinal
set of open normal subgroups $N$ of $G$.

###### Proof.

Let $E$ be the set of block idempotents of $k[[{G}]]$. We use the direct
system $\\{E_{N},\psi_{MN}\\}$ described in Remark 4.4. If $e\in E$ is the
block idempotent of $B$, there is $N_{0}\trianglelefteq_{O}G$ and $e_{0}\in
E_{N_{0}}$ such that $e=\psi_{N_{0}}(e_{0})$. Consider the cofinal system
$\mathcal{N}$ of open normal subgroups $N$ of $G$ with $N\leq N_{0}$.

For each $N\leq N_{0}$, let $e_{N}=\psi_{N_{0}N}(e_{0})$, the unique centrally
primitive idempotent of $k[G/N]$ such that
$\varphi_{N_{0}N}(e_{N})e_{0}=e_{0}$. Observe that if ever $N\leqslant
M\leqslant N_{0}$, then $\varphi_{MN}(e_{N})e_{M}=e_{M}$. The block $X_{N}$ of
$G/N$ with block idempotent $e_{N}$ is a direct summand of $B_{N}$ as a
$k[[{G\times G}]]$-module, so there are $k[[{G\times G}]]$-module
homomorphisms $\iota_{N}:X_{N}\to B_{N}$ inclusion, and $\pi_{N}:B_{N}\to
X_{N}$ multiplication by $e_{N}$, such that
$\pi_{N}\iota_{N}=\textnormal{id}_{X_{N}}$. Now we can form a new inverse
system of blocks $X_{N}$ with maps
$\gamma_{MN}:=\pi_{M}\varphi_{MN}\iota_{N}$. Note that
$\gamma_{MN}(e_{N})=\varphi_{MN}(e_{N})e_{M}=e_{M}$, which implies in
particular that the $\gamma_{MN}$ are surjective algebra homomorphisms. Then
$\\{X_{N},\gamma_{MN}\\}$ forms an inverse system, because for all $N\leqslant
M\leqslant L$ and all $x=xe_{N}\in X_{N}$ we have

$\gamma_{LM}\gamma_{MN}(x)=\pi_{L}\varphi_{LM}(\varphi_{MN}(x)e_{M})=\varphi_{LN}(x)\varphi_{LM}(e_{M})e_{L}=\gamma_{LN}(x).$

Denote by $X$ the inverse limit of this inverse system.

The split homomorphisms $\pi_{N}:B_{N}\to X_{N}$ are the components of a map
of inverse systems, so by Lemma 2.10, the induced continuous map
$\pi=\varprojlim_{N}\pi_{N}$ is a split $k[[{G\times G}]]$-module
homomorphism. Hence $X$ is a direct summand of $B$. But $B$ is indecomposable
as a $k[[{G\times G}]]$-module and $X$ is not $0$, so

$B=X=\varprojlim_{N}X_{N}.$

∎

######  Remark 4.8.

Throughout this article, given a block $B$ of $G$ with block idempotent $e$,
the notation $\\{X_{N},\gamma_{MN},N\in\mathcal{N}\\}$ will refer to a fixed
inverse system of blocks as constructed in Proposition 4.7. This inverse
system is not unique, due to the choice of the open normal subgroup $N_{0}$
and of the block idempotent $e_{0}$ of $G/N_{0}$, but it will be important
that once these choices are made, the maps $\gamma_{MN}$ are canonical.

######  Corollary 4.9.

Let $B$ be a block of $G$ with block idempotent $e$. Let $S$ be a finite
dimensional $k[[{G}]]$-module lying in $B$. Then there is an open normal
subgroup $N^{\prime}$ of $G$ acting trivially on $S$ such that $S$ lies in the
block $X_{N^{\prime}}$ of $G/N^{\prime}$. Furthermore, $S$ lies in $X_{N}$ for
every $N\leqslant N^{\prime}$.

###### Proof.

Write $B=\varprojlim_{N}\\{X_{N},\gamma_{MN},N\in\mathcal{N}\\}$ as above and
fix $M\in\mathcal{N}$. There is a simple $k[G/M]$-module $T$ lying in $X_{M}$.
Then $T$ lies in $B$. If $N\leqslant M$ then $e_{N}T=T$, since otherwise
$e_{N}T=0$, which would imply that
$e_{M}T=e_{M}\varphi_{MN}(e_{N})T=e_{M}\cdot e_{N}T=0$. So $T$, treated as a
$k[G/N]$-module, lies in $X_{N}$.

Since $S,T$ are finite dimensional $k[[{G}]]$-modules lying in the same block
$B$, by Lemma 4.5 there is $N^{\prime}\trianglelefteq_{O}G$ with
$N^{\prime}\leq M$, such that $S,T$ lie in the same $k[G/N^{\prime}]$-block.
But $T$ lies in $X_{N^{\prime}}$, hence so does $S$. ∎

######  Proposition 4.10.

Let $G$ be a profinite group, $R$ a closed subgroup of $G$ and $B$ a block of
$G$ treated as a $k[[{G\times G}]]$-module. Then $B$ is $\delta(R)$-projective
if, and only if, $X_{N}$ is $\delta(R)(N\times N)$-projective for each
$N\trianglelefteq_{O}G$.

###### Proof.

The forward implication is easy: if $B$ is relatively $\delta(R)$-projective
then $B_{N}=B_{N\times N}$ is relatively $\delta(R)(N\times N)$-projective.
But $X_{N}$ is a direct summand of $B_{N}$, so we are done.

For the reverse implication, suppose that $X_{N}$ is $\delta(R)(N\times
N)$-projective for each $N\trianglelefteq_{O}G$.

We utilize the natural isomorphism of Lemma 2.4 to write

$B_{N}\downarrow_{{\delta(R)(N\times N)}}\uparrow^{{G\times G}}=k[G\times
G/\delta(R)(N\times N)]\widehat{\otimes}_{k}B_{N}.$

Letting $N$ vary and setting $\Psi_{MN}$ to be the tensor product of the
natural projections, we obtain an inverse system of $k[[{G\times G}]]$-modules
$\\{B_{N}\downarrow_{{\delta(R)(N\times N)}}\uparrow^{{G\times
G}},\,\Psi_{MN}\\}$, whose inverse limit is the module
$B\downarrow_{{\delta(R)}}\uparrow^{{G\times G}}$ by [18, Proposition 5.2.2].
Denoting by $\pi_{N}:B_{N}\to X_{N}$ the canonical projection, we define the
continuous $k[[{G\times G}]]$-module homomorphism

$\displaystyle q_{N}:k[G\times G/\delta(R)(N\times
N)]\widehat{\otimes}_{k}B_{N}$ $\displaystyle\to X_{N}$ $\displaystyle
z\widehat{\otimes}b$ $\displaystyle\mapsto\pi_{N}(b).$

Note that $q_{N}$ splits as a $k[[{G\times G}]]$-module homomorphism, being
the composition of split homomorphisms $B_{N}\downarrow_{{\delta(R)(N\times
N)}}\uparrow^{{G\times G}}\to X_{N}\downarrow_{{\delta(R)(N\times
N)}}\uparrow^{{G\times G}}$ and $X_{N}\downarrow_{{\delta(R)(N\times
N)}}\uparrow^{{G\times G}}\to X_{N}$ (the latter split since $X_{N}$ is
relatively $\delta(R)(N\times N)$-projective). Furthermore the $q_{N}$ yield a
map of inverse systems, as may be easily checked. The induced homomorphism
$q:B\downarrow_{{\delta(R)}}\uparrow^{{G\times
G}}\to\underleftarrow{\textnormal{lim}}\,X_{N}=B$ is thus split by Lemma 2.10,
so that $B$ is relatively $\delta(R)$-projective by Theorem 2.16. ∎

##  5 Defect Groups

###  5.1 Definition and basic properties

Recall that we treat $k[[{G}]]$ as a $G$-algebra with action from $G$ given by
conjugation. Whenever we treat a subalgebra or a quotient algebra of
$k[[{G}]]$ as a $G$-algebra, the action from $G$ is induced from this one.
Note that the center $\textnormal{Z}(k[[{G}]])$ of $k[[{G}]]$ is
$k[[{G}]]^{G}$, so block idempotents of $k[[{G}]]$ belong to $k[[{G}]]^{G}$.
Recall the definition 3.4 of $\textnormal{oTr}_{D}^{G}(k[[{G}]]^{D})$.

######  Definition 5.1.

Let $B$ be a block of a profinite group $G$ with block idempotent $e$. A
_defect group_ of $B$ is a closed subgroup $D$ of $G$ such that
$e\in\textnormal{oTr}_{D}^{G}(k[[{G}]]^{D})$ and minimal with this property.

######  Theorem 5.2.

Let $B$ be a block of $G$. A defect group of $B$ exists.

###### Proof.

Let $e$ be the block idempotent of $B$ and consider
$\mathcal{D}=\\{H\leq_{C}G\,:\,e\in\textnormal{oTr}_{H}^{G}(k[[{G}]]^{H})\\}$.
Then $\mathcal{D}$ is partially ordered by inclusion and non-empty, since
$G\in\mathcal{D}$. If $\mathcal{C}$ is a chain in $\mathcal{D}$, we assert
that $L=\bigcap\limits_{H\in\mathcal{C}}H\in\mathcal{D}$. Fix
$N\trianglelefteq_{O}G$. As $LN$ is open in $G$, $LN=HN$ for some
$H\in\mathcal{C}$ and hence $e\in\textnormal{Tr}_{LN}^{G}(k[[{G}]]^{LN})$.
Since $N$ was arbitrary, $e\in\textnormal{oTr}_{L}^{G}(k[[{G}]]^{L})$ by
definition and hence $L\in\mathcal{D}$ as claimed. The existence of a defect
group for $B$ now follows from Zorn’s Lemma. ∎

In several proofs that follow we use Tr to mean two different things (the Tr
of Definitions 2.15 and 3.1), but see Remark 3.2.

######  Proposition 5.3.

If $B$ is a block of $G$ with defect group $D$, then any finitely generated
$k[[{G}]]$-module lying in $B$ is relatively $D$-projective.

###### Proof.

Let $U$ be a finitely generated $k[[{G}]]$-module lying in $B$. For each
$N\trianglelefteq_{O}G$ we have $e\in\textnormal{Tr}_{DN}^{G}(k[[{G}]]^{DN})$
by hypothesis, so there is $a_{N}\in k[[{G}]]^{DN}$ such that
$e=\textnormal{Tr}_{DN}^{G}(a_{N})$. Denote by $\alpha_{N}:U\to U$ the
continuous $k[[{DN}]]$-module endomorphism defined by $u\mapsto a_{N}u$. One
easily checks that $\textnormal{id}_{U}=\textnormal{Tr}_{DN}^{G}(\alpha_{N})$,
which implies by Theorem 2.16 that $U$ is relatively $D$-projective. ∎

######  Proposition 5.4.

Let $B$ be a block of $G$. Then $B$ has a vertex of the form $\delta(D)$ as a
$k[[{G\times G}]]$-module, where $D$ is a pro-$p$ subgroup of $G$.

###### Proof.

By Lemma 4.6, $B$ is relatively $\delta(G)$-projective. The result now follows
since vertices are pro-$p$ subgroups of $G$ by Lemma 2.17. ∎

######  Proposition 5.5.

Let $H$ be a closed subgroup of $G$ and $B$ a block of $G$ with block
idempotent $e$. Then $e\in\textnormal{oTr}_{H}^{G}(k[[{G}]]^{H})$ if, and only
if, $B$ is $\delta(H)$-projective as a $k[[{G\times G}]]$-module.

###### Proof.

Suppose that $e\in\textnormal{oTr}_{H}^{G}(k[[{G}]]^{H})$ so that for each
$N\trianglelefteq_{O}G$, there is $a_{N}\in k[[{G}]]^{HN}$ such that
$e=\textnormal{Tr}_{HN}^{G}(a_{N})$. We claim first that
$B\downarrow_{\delta(G)(N\times N)}$ is $\delta(H)(N\times N)$-projective.
Consider the continuous $k[[{\delta(H)(N\times N)}]]$-module homomorphism
$\alpha_{N}:B\to B$ given by $y\mapsto a_{N}y$. Observe that if $R$ is a set
of left coset representatives of $HN$ in $G$, then $\\{(r,r)\,:\,r\in R\\}$ is
a set of left coset representatives of $\delta(H)(N\times N)$ in
$\delta(G)(N\times N)$. So

$\displaystyle\textnormal{Tr}_{\delta(H)(N\times N)}^{\delta(G)(N\times
N)}(\alpha_{N})(e)$ $\displaystyle=$
$\displaystyle\sum_{(r,r)\in\delta(G)(N\times N)/\delta(H)(N\times
N)}(r,r)\alpha_{N}((r^{-1},r^{-1})e)$ $\displaystyle=$
$\displaystyle\sum_{(r,r)\in\delta(G)(N\times N)/\delta(H)(N\times
N)}r\alpha_{N}(r^{-1}er))r^{-1}$ $\displaystyle=$
$\displaystyle\sum_{(r,r)\in\delta(G)(N\times N)/\delta(H)(N\times
N)}ra_{N}r^{-1}err^{-1}$ $\displaystyle=$
$\displaystyle\textnormal{Tr}_{HN}^{G}(a_{N})e$ $\displaystyle=$
$\displaystyle e.$

Thus $\textnormal{Tr}_{\delta(H)(N\times N)}^{\delta(G)(N\times
N)}(\alpha_{N})$ is the identity map on $B$ and so by Theorem 2.16,
$B\downarrow_{\delta(G)(N\times N)}$ is $\delta(H)(N\times N)$-projective. Let
$Z$ be a $k[[{\delta(H)(N\times N)}]]$-module such that

$B\downarrow_{{\delta(G)(N\times
N)}}\raisebox{0.5pt}{\,\big{|}\,}Z\uparrow^{{\delta(G)(N\times N)}}.$

Then, since $B$ is $\delta(G)(N\times N)$-projective for each
$N\trianglelefteq_{O}G$ by Lemma 4.6, we have

$\displaystyle B$ — $\displaystyle B\downarrow_{\delta(G)(N\times
N)}\uparrow^{G\times
G}\raisebox{0.5pt}{\,\big{|}\,}Z\uparrow^{\delta(G)(N\times
N)}\uparrow^{G\times G}$

which implies that $B$ is relatively $\delta(H)(N\times N)$-projective for
each $N\trianglelefteq_{O}G$, and hence is relatively $\delta(H)$-projective
by Theorem 2.16.

To show the reverse implication, suppose that $B$ is relatively
$\delta(H)$-projective. Then $B$ is relatively $(G\times HN)$-projective for
each $N\trianglelefteq_{O}G$. By Theorem 2.16, there is thus a continuous
$k[[{G\times HN}]]$-module endomorphism $\alpha_{N}$ of $B$ such that
$\textnormal{id}_{B}=\textnormal{Tr}_{G\times HN}^{G\times G}(\alpha_{N})$.
Arguing as in the finite case [13, Proposition 6.2.3] it follows that
$e\in\textnormal{Tr}_{HN}^{G}(k[[{G}]]^{HN})$, and hence
$e\in\bigcap_{N}\textnormal{Tr}_{HN}^{G}(k[[{G}]]^{HN})=\textnormal{oTr}_{H}^{G}(k[[{G}]]^{H})$,
as required. ∎

######  Proposition 5.6.

Let $D$ be a closed subgroup of $G$ and let $B$ be a block of $G$. The
following statements are equivalent.

  1. 1.

$D$ is a defect group of $B$.

  2. 2.

$\delta(D)$ is a vertex of $B$ as a $k[[{G\times G}]]$-module.

###### Proof.

This follows from Proposition 5.5. ∎

######  Proposition 5.7.

Let $G$ be a profinite group and $B$ a block of $G$. The defect groups of $B$
are a conjugacy class of pro-$p$ subgroups of $G$.

###### Proof.

If $C,D$ are defect groups of $B$ then they are pro-$p$ subgroups of $G$ and
$\delta(C),\delta(D)$ are conjugate in $G\times G$ by Proposition 5.6 and
Proposition 2.17. As with finite groups (cf. [1, p. 98]) a direct calculation
shows that this implies that $C$ and $D$ are conjugate in $G$. ∎

######  Proposition 5.8.

Let $B$ be a block of $G$ with defect group $D$ and let $P$ be a $p$-Sylow
subgroup of $G$ containing $D$. Then $D$ is open in $P$.

###### Proof.

Let $S$ be a simple $k[[{G}]]$-module lying in $B$. By Proposition 5.3, $S$
has a vertex $Q$ contained in $D$. But $Q$ is open in $P$ by Proposition 2.19,
hence so is $D$. ∎

###  5.2 Defect groups and inverse systems

We show that defect groups are well-behaved with respect to the inverse system
of finite dimensional blocks of Remark 4.8. The following fact, due to Green
in the finite case [8, Theorem 12], is rarely mentioned for finite groups, but
becomes incredibly useful for profinite groups because simple modules are
finite dimensional:

######  Proposition 5.9.

Let $G$ be a profinite group and $B$ a block of $G$ with defect group $D$.
There is a simple module $T$ lying in $B$ with vertex $D$.

###### Proof.

Recall the notation $X_{N}$ of Remark 4.8. Let $T$ be a simple
$k[[{G}]]$-module lying in $B$ with the following property: in an algebraic
closure $\overline{k}$ of $k$, the module $\overline{k}\otimes_{k}T$ has an
indecomposable direct summand of dimension $p^{r}a$ with $p$ not dividing $a$
and $r$ as small as possible among all simple $k[[{G}]]$-modules lying in $B$.
By Proposition 5.3, $T$ has a vertex $Q$ inside $D$. Using Corollary 4.9, fix
an inverse system $\mathcal{N}$ of open normal subgroups $N$ of $G$ acting
trivially on $T$ and such that $T$ lies in $X_{N}$.

Given $N\in\mathcal{N}$, $T$ has vertex $QN/N$ as a $k[G/N]$-module by Lemma
2.18. As simple $k[G/N]$-modules are simple as $k[[{G}]]$-modules, one may
follow the arguments of [10, Theorem 2.2] and [8, Theorem 12] to see that our
choice of $T$ implies that $X_{N}$ has defect group $QN/N$.

By Proposition 5.6, $X_{N}$ thus has vertex $\delta(QN/N)$ as a $k[G/N\times
G/N]$-module. We have a natural isomorphism of groups
$\delta(\frac{QN}{N})\cong\frac{\delta(Q)(N\times N)}{N\times N}$ and hence
$X_{N}$ is relatively $\delta(Q)(N\times N)$-projective as a $k[[{G\times
G}]]$-module. It now follows from Proposition 4.10 that $B$ is
$\delta(Q)$-projective and hence that $D=Q$. ∎

######  Corollary 5.10.

Let $B$ be a block of $G$ with defect group $D$. There is
$N_{0}\trianglelefteq_{O}G$ such that $X_{N}$ has defect group $DN/N$ for
every open normal subgroup $N$ of $G$ contained in $N_{0}$.

###### Proof.

By Proposition 5.9, we may consider a simple $k[[{G}]]$-module $T$ lying in
$B$ with vertex $D$. Fix $N_{0}$ such that $T$ lies in $X_{N_{0}}$ and
consider an open normal subgroup $N$ of $G$ contained in $N_{0}$. By Lemma
2.18, $T$ has vertex $DN/N$ as a $k[G/N]$-module and hence by Proposition 5.3,
there is a defect group $D^{\prime}/N$ of $X_{N}$ containing $DN/N$. But on
the other hand, since $B$ is relatively $\delta(D)$-projective as a
$k[[{G\times G}]]$-module, $X_{N}$ is relatively $\delta(DN/N)$-projective as
a $k[G/N\times G/N]$-module, and so $DN/N=D^{\prime}/N$, as required. ∎

We can now be more precise about the nature of defect groups:

######  Proposition 5.11.

Let $D$ be a defect group of the block $B$ of $G$ and let $P$ be a $p$-Sylow
subgroup of $G$ containing $D$. There is $g$ in the centralizer
$\textnormal{C}_{G}(D)$ of $D$ in $G$ such that $D=P\cap{}^{g}\\!P$.

###### Proof.

By Corollary 5.10, work within the cofinal system of $N$ for which $X_{N}$ has
defect group $DN/N$. By the finite version of this result [22, Theorem 12.3.3]
there is a non-empty closed set $C_{N}$ of $g\in\textnormal{C}_{G}(DN)$ such
that $DN=PN\cap{}^{g}\\!PN$. By compactness, the intersection $C$ of the
$C_{N}$ is non-empty and if $g\in C$ then $D=P\cap{}^{g}\\!P$. ∎

######  Remark 5.12.

Propositions 5.8 and 5.11 together say that the candidates for a defect group
may be quite limited. For instance, if $G$ has the property that the
intersection of a $p$-Sylow subgroup $P$ with any proper conjugate of $P$ has
infinite index in $P$, or if $\textnormal{C}_{G}(D)\leqslant{N}_{G}(P)$ for
every open subgroup $D$ of $P$, then every block of $G$ has defect group $P$.

######  Corollary 5.13.

Let $G$ be a profinite group and $B$ a block of $G$ with defect group $D$.
Then $D$ is the largest normal pro-$p$ subgroup of $\textnormal{N}_{G}(D)$.

###### Proof.

Given Proposition 5.11, the proof of this is just like for finite groups [22,
Corollary 12.3.4]. ∎

######  Proposition 5.14.

Let $G$ be a profinite group and $B$ a block of $G$ with block idempotent $e$.
A closed subgroup $D$ of $G$ is a defect group of $B$ if, and only if, it is
minimal among the closed subgroups $R$ of $G$ such that $|G:R|_{p}<\infty$
with the property that

$e\in\widetilde{\textnormal{Tr}}_{R}^{G}(k[[{G}]]^{R}).$

###### Proof.

When $D$ is a defect group of $B$, $|G:D|_{p}<\infty$ by Proposition 5.8, so
by Lemma 3.7, $e\in\widetilde{\textnormal{Tr}}_{D}^{G}(k[[{G}]]^{D})$. Let $R$
be an open subgroup of $D$ with the property that

$e\in\widetilde{\textnormal{Tr}}_{R}^{G}(k[[{G}]]^{R}).$

We must check that $R=D$.

By Proposition 5.9, let $T$ be a simple $k[[{G}]]$-module lying in $B$ with
vertex $D$. We work within the cofinal inverse system of $N$ for which $T$ is
in $X_{N}$ (as we may by Corollary 4.9). Fix such an $N$. By hypothesis,
$\varphi_{N}(e)=\textnormal{Tr}_{RN/N}^{G/N}(a)$ for some $a$ in
$k[G/N]^{RN/N}$. The $k[RN/N]$-module endomorphism $\beta:T\to T$ sending $x$
to $ax$ is such that
$\textnormal{id}_{T}=\textnormal{Tr}_{RN/N}^{G/N}(\beta_{N})$, and hence
$T=T_{N}$ is relatively $RN/N$-projective. Since $N$ was arbitrary, $T$ is
thus relatively $R$-projective as a $k[[{G}]]$-module by Theorem 2.16. But $R$
is a subgroup of $D$ and $D$ is a vertex of $T$, so $R=D$ as required. ∎

###  5.3 The Brauer homomorphism

We give two further characterizations of defect groups.

Consider as ever the continuous action of $G$ on $k[[{G}]]$ by conjugation.
When $D$ is a closed subgroup of $G$ and $Q$ an open subgroup of $D$, then
$\textnormal{Tr}_{Q}^{D}(k[[{G}]]^{Q})$ is an ideal of $k[[{G}]]^{D}$ by Part
1 of Lemma 3.3. Thus
$\sum\limits_{Q\lneqq_{O}D}\textnormal{Tr}_{Q}^{D}(k[[{G}]]^{Q})$ is an
abstract ideal of $k[[{G}]]^{D}$ and so its closure is a closed ideal of
$k[[{G}]]^{D}$ [21, Theorem 4.2].

######  Definition 5.15.

[19, Definition 3.2] Let $G$ be a profinite group and $D$ a closed pro-$p$
subgroup of $G$. The corresponding _Brauer quotient_ is the quotient algebra

$k[[{G}]]^{[D]}:=k[[{G}]]^{D}/\overline{\sum\limits_{Q\lneqq_{O}D}\textnormal{Tr}_{Q}^{D}(k[[{G}]]^{Q})}.$

The _Brauer homomorphism_ is the natural projection

$\textnormal{Br}_{D}:k[[{G}]]^{D}\to k[[{G}]]^{[D]}.$

######  Proposition 5.16.

Let $B$ be a block of a profinite group $G$ with block idempotent $e$. Then
$B$ has defect group $D$ if, and only if,
$e\in\textnormal{oTr}_{D}^{G}(k[[{G}]]^{D})$ and $\textnormal{Br}_{D}(e)\neq
0$.

###### Proof.

Suppose first that $B$ has defect group $D$. Then
$e\in\textnormal{oTr}_{D}^{G}(k[[{G}]]^{D})$ by definition, so we need only
check that $\textnormal{Br}_{D}(e)\neq 0$. If $\textnormal{Br}_{D}(e)=0$, then
by Rosenberg’s Lemma 4.2, $e\in\textnormal{Tr}_{Q}^{D}(k[[{G}]]^{Q})$ for some
proper open subgroup $Q$ of $D$. Fix an open normal subgroup $M$ of $G$ such
that $X_{M}$ has defect group $DM/M$ (by Corollary 5.10) and such that $QM/M$
has the same index in $DM/M$ as $Q$ has in $D$. Then $\varphi_{M}(e)$ is an
element of $\textnormal{Tr}_{QM/M}^{DM/M}(k[G/M]^{QM/M})$ so that
$\textnormal{Br}_{DM/M}(\varphi_{M}(e))=0$. But if $f$ is the block idempotent
of $X_{M}$ then $\varphi_{M}(e)f=f$, and hence $\textnormal{Br}_{DM/M}(f)=0$
since $\textnormal{Br}_{DM/M}$ is an algebra homomorphism. This contradicts
the finite version of the result [2, Lemma 6.2.4], since $X_{M}$ has defect
group $DM/M$.

Suppose now that $e\in\textnormal{oTr}_{D}^{G}(k[[{G}]]^{D})$ and
$\textnormal{Br}_{D}(e)\neq 0$. Then certainly $B$ has a defect group $R$
contained in $D$. Note that $R$ is open in $D$ by Proposition 2.19, so we can
find an open normal subgroup $M$ of $G$ such that $D\cap RM=R$. We have
$e=\textnormal{Tr}_{RM}^{G}(a)$ for some $a\in k[[{G}]]^{RM}$ by hypothesis.
Applying Mackey’s formula (Lemma 3.3),

$e=\sum_{g\in D\backslash
G/RM}\textnormal{Tr}_{D\cap{}^{g}\\!RM}^{D}({}^{g}\\!a).$

We claim that if $R$ is a proper subgroup of $D$, then so is each of the
$D\cap{}^{g}\\!RM$: if for some $g$ we had $D\cap{}^{g}\\!RM=D$, then
$R<D\leqslant{}^{g}\\!RM$, which implies that $RM\leqslant{}^{g}\\!RM$ and
hence that $g$ is in the normalizer $\textnormal{N}_{G}(RM)$. But then
$D\cap{}^{g}\\!RM=D\cap RM=R<D$, a contradiction. It follows that if $R$ is
proper in $D$, then $e$ is in the kernel of $\textnormal{Br}_{D}$, contrary to
our hypothesis. Hence $R=D$ and we are done. ∎

For the convenience of the reader we collect together the characterizations of
a defect group proved above, and provide one further characterization that
will help us to prove Brauer’s first main theorem. We need one more simple
lemma.

######  Lemma 5.17.

Let $R,D$ be closed pro-$p$ subgroups of $G$. If, for each
$N\trianglelefteq_{O}G$, $\textnormal{Br}_{D}(\textnormal{Tr}_{RN}^{G}(a))\neq
0$ for some $a\in k[[{G}]]^{RN}$, then $D$ is conjugate to a subgroup of $R$.

###### Proof.

Fix $N\trianglelefteq_{O}G$ and $a$ such that
$\textnormal{Br}_{D}(\textnormal{Tr}_{RN}^{G}(a))\neq 0$. By the Mackey
formula,

$0\neq\textnormal{Br}_{D}(\textnormal{Tr}_{RN}^{G}(a))=\sum_{g\in D\backslash
G/RM}\textnormal{Br}_{D}(\textnormal{Tr}_{D\cap{}^{g}\\!RN}^{D}({}^{g}\\!a)),$

and hence the closed set $C_{N}$ of $g\in G$ with $D\subseteq{}^{g}\\!RN$ is
non-empty. A standard compactness argument yields that
$C=\bigcap_{N\trianglelefteq_{O}G}C_{N}$ is non-empty, and $g\in C$ has the
property that $D\subseteq{}^{g}\\!R$. ∎

######  Theorem 5.18.

Let $G$ be a profinite group and $B$ a block of $G$ with block idempotent $e$.
The following are equivalent for a closed subgroup $D$ of $G$:

  1. 1.

$e\in\textnormal{oTr}_{D}^{G}(k[[{G}]]^{D})$ and $D$ is minimal with this
property,

  2. 2.

$|G:D|_{p}<\infty$, $e\in\widetilde{\textnormal{Tr}}_{D}^{G}(k[[{G}]]^{D})$
and $D$ is minimal among the closed subgroups of $G$ having these two
properties,

  3. 3.

$\delta(D)$ is a vertex of $B$ as a $k[[{G\times G}]]$-module,

  4. 4.

$e\in\textnormal{oTr}_{D}^{G}(k[[{G}]]^{D})$ and $\textnormal{Br}_{D}(e)\neq
0$.

  5. 5.

$\textnormal{Br}_{D}(e)\neq 0$ and $D$ is maximal among the pro-$p$ subgroups
of $G$ with this property.

###### Proof.

The equivalence of $1,2,3,4$ is the content of Propositions 5.6, 5.16 and
5.14. To conclude the result, we will check that $4$ and $5$ are equivalent.

Suppose that $D$ satisfies $4$ and let $R$ be a pro-$p$ subgroup of $G$
containing $D$ such that $\textnormal{Br}_{R}(e)\neq 0$. Lemma 5.17 (applied
with the roles of $R$ and $D$ interchanged) implies that $D\subseteq
R\subseteq{}^{g}\\!D$ for some $g\in G$ and hence $R=D$.

Suppose now that $D$ satisfies $5$. Let $R$ be a defect group of $B$, so that
$e\in\textnormal{oTr}_{R}^{G}(k[[{G}]]^{R})$ and $\textnormal{Br}_{R}(e)\neq
0$. By Lemma 5.17 (with $R$ and $D$ the right way round this time), $D$ is
conjugate to a subgroup of $R$, and hence by Proposition 5.7 we may as well
choose $R$ containing $D$. But $D$ is by hypothesis maximal with
$\textnormal{Br}_{D}(e)\neq 0$ and hence $D=R$, as required. ∎

##  6 Brauer’s first main theorem for profinite groups

Brauer’s first main theorem for finite groups (cf. [2, Theorem 6.2.6]) is a
natural bijection between the blocks of a finite group $G$ with defect group
$D$, and the blocks of the normalizer of $D$ in $G$ with defect group $D$. In
this section we use the results thus obtained to demonstrate our main theorem,
a version of Brauer’s first main theorem for profinite groups. We follow the
approaches of [2, §6.2] and [22, §12.6]. Given the results already obtained,
the proof is mostly similar to the finite case, with the exception of Lemma
6.3. Remark 6.9 provides a simple example that illustrates a certain subtlety
of the correspondence.

######  Lemma 6.1.

Let $D$ be a closed pro-$p$ subgroup of $G$. The composition

$k[[{\textnormal{C}_{G}(D)}]]\to k[[{G}]]^{D}\to k[[{G}]]^{[D]},$

where the first map is induced from the inclusion of the centralizer of $D$ in
$G$ into $k[[{G}]]^{D}$ and the second is the Brauer homomorphism, is an
isomorphism of $\textnormal{N}_{G}(D)$-algebras.

###### Proof.

This is [19, Lemma 3.4]. ∎

It will be convenient for us to consider $\textnormal{Br}_{D}$ as taking
values in $k[[{C_{G}(D)}]]$. Thus we will regard $\textnormal{Br}_{D}$ as the
map

$k[[{G}]]^{D}\to
k[[{G}]]^{[D]}\xrightarrow{\sim}k[[{\textnormal{C}_{G}(D)}]],$

where the second arrow is the inverse of the isomorphism of Lemma 6.1.

Lemma 6.3 below corresponds to [2, Lemma 6.2.5]. It is the principal reason we
introduce the modified trace function in Section 3 and its proof is rather
delicate. It seems to be a fundamental obstruction that the lemma cannot be
formulated without the hypothesis that $D$ is open in $P$ – this is reflected
in the failure of the correspondence when $D$ has infinite index in $P$ (see
Remark 6.9). The following auxiliary lemma is not strictly necessary for the
proof of Lemma 6.3, but without it the argument may look rather suspicious.

######  Lemma 6.2.

Let $D$ be a pro-$p$ subgroup of $G$ that is open in a $p$-Sylow subgroup of
$G$ that contains it. Let $\mathcal{N}$ denote the set of open normal
subgroups of $G$ for which $D$ is a $p$-Sylow subgroup of $DN$. The element
$|G:\textnormal{N}_{G}(DN)|_{p^{\prime}}$ of $k$ does not depend on
$N\in\mathcal{N}$.

###### Proof.

It is sufficient to prove that if $G$ is a finite group with $p$-subgroup $D$
and $N$ is a normal subgroup such that $D$ is a Sylow $p$-subgroup of $DN$,
then
$|G:\textnormal{N}_{G}(D)|_{p^{\prime}}=|G/N:\textnormal{N}_{G/N}(DN/N)|_{p^{\prime}}$
in $k$. By the orbit-stabilizer theorem,
$|G:\textnormal{N}_{G}(D)|_{p^{\prime}}$ is the $p^{\prime}$-part of the size
of the set ${}^{G}\\!D$ of $G$-conjugates of $D$ in $G$, while
$|G/N:\textnormal{N}_{G/N}(DN/N)|_{p^{\prime}}$ is the $p^{\prime}$-part of
the size of the set ${}^{G/N}\\!DN/N$ of $G/N$-conjugates of $DN/N$ in $G/N$.
Consider the surjective map $\rho:{}^{G}\\!D\to{}^{G/N}\\!DN/N$ given by
${}^{g}\\!D\mapsto{}^{gN}DN/N$. By our hypothesis on $N$, the inverse image
under $\rho$ of an element ${}^{gN}DN/N$ is precisely the set
$\textnormal{Syl}_{p}({}^{g}\\!DN)$ of Sylow $p$-subgroups of ${}^{g}\\!DN$.
The size of this set does not depend on $g$ and by Sylow’s third theorem,
$|\textnormal{Syl}_{p}({}^{g}\\!DN)|=1$ in $k$. Hence

$|{}^{G}\\!D|_{p^{\prime}}=|{}^{G/N}\\!DN/N|_{p^{\prime}}\cdot|\textnormal{Syl}_{p}(DN)|_{p^{\prime}}=|{}^{G/N}\\!DN/N|_{p^{\prime}},$

as required. ∎

######  Lemma 6.3.

Let $D$ be a pro-$p$ subgroup of $G$ that is open in a $p$-Sylow subgroup of
$G$ that contains it. Let $N$ be an open normal subgroup of $G$ such that $D$
is a $p$-Sylow subgroup of $DN$, and denote by $b$ the element
$1/|G:\textnormal{N}_{G}(DN)|_{p^{\prime}}$ of $k$. The diagram

$\textstyle{k[[{G}]]^{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{Br}_{D}}$$\scriptstyle{\widetilde{\textnormal{Tr}}_{D}^{G}}$$\textstyle{k[[{\textnormal{C}_{G}(D)}]]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b\cdot\widetilde{\textnormal{Tr}}_{D}^{\textnormal{N}_{G}(D)}}$$\textstyle{\widetilde{\textnormal{Tr}}_{D}^{G}(k[[{G}]]^{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{Br}_{D}}$$\textstyle{k[[{\textnormal{C}_{G}(D)}]]}$

commutes. In particular the image of the lower map is precisely
$\widetilde{\textnormal{Tr}}_{D}^{\textnormal{N}_{G}(D)}(k[[{\textnormal{C}_{G}(D)}]])$.

###### Proof.

We work within a cofinal set $\mathcal{N}$ of open normal subgroups $N$ of $G$
such that

$|\textnormal{N}_{G}(D)N:DN|_{p}=|\textnormal{N}_{G}(D):D|_{p}\,\hbox{ and
}\,|G:\textnormal{N}_{G}(D)N|_{p}=|G:\textnormal{N}_{G}(D)|_{p}$

so that Lemma 3.6 holds when required. These conditions imply that $D$ is a
$p$-Sylow subgroup of $DN$ and hence by Lemma 6.2, the value
$1/|G:\textnormal{N}_{G}(DN)|_{p^{\prime}}$ is equal to $b$ for every
$N\in\mathcal{N}$.

We first claim that the following diagram is well-defined and commutes:

$\textstyle{k[G/N]^{DN/N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{Br}_{DN/N}}$$\scriptstyle{\varphi_{MN}}$$\textstyle{k[G/N]^{DN/N}/\sum_{QN<DN}\textnormal{Tr}_{QN/N}^{DN/N}(k[G/N]^{DN/N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{MN}}$$\textstyle{k[\textnormal{C}_{G/N}(DN/N)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{MN}}$$\textstyle{k[G/M]^{DM/M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{Br}_{DM/M}}$$\textstyle{k[G/M]^{DM/M}/\sum_{QM<DM}\textnormal{Tr}_{QM/M}^{DM/M}(k[G/M]^{DM/M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k[\textnormal{C}_{G/M}(DM/M)]}$

wherein the symbol $\varphi_{MN}$ is abusively being used to indicate the map
induced by $\varphi_{MN}:k[G/N]\to k[G/M]$ on the corresponding subquotients.
The outer triangles commute by definition, and as long as the middle vertical
map is well-defined, then the inner squares also commute by definition. So the
only thing that needs to be checked is that the middle vertical map is well
defined, and for this we must see what happens to an element
$\textnormal{Tr}_{QN/N}^{DN/N}(a)$ when $QN$ is proper in $DN$, but $QM=DM$.
But in this case

$\varphi_{MN}(\textnormal{Tr}_{QN/N}^{DN/N}(a))=\sum_{dN\in
DN/QN}{}^{dM}\varphi_{MN}(a)=|DN:QN|\varphi_{MN}(a)=0$

because $\varphi_{MN}(a)$ is a $DM/M$-fixed point and $|DN:QN|=0$ in $k$. The
map is thus well-defined and the claim follows. In particular, the outer shape
commutes, and so
$\varphi_{MN}\textnormal{Br}_{DN/N}=\textnormal{Br}_{DM/M}\varphi_{MN}$.

Write
$k[[{G}]]^{D}=k[[{\textnormal{C}_{G}(D)}]]\oplus\overline{\sum_{Q<_{O}D}\textnormal{Tr}_{Q}^{D}(k[[{G}]]^{Q})}$.
We first claim that the second summand is sent to $0$ when passed in either
direction through the diagram of the statement of the lemma, so suppose that
$x=\textnormal{Tr}_{Q}^{D}(a)$ for $Q$ a proper open subgroup of $D$ and $a\in
k[[{G}]]^{Q}$. Note that
$\widetilde{\textnormal{Tr}}_{D}^{G}\textnormal{Tr}_{Q}^{D}(a)=\widetilde{\textnormal{Tr}}_{Q}^{G}(a)$,
as can be seen by looking at the cofinal subset of those $N$ for which $QN\cap
D=Q$. For such $N$ we have

$\textnormal{Tr}_{QN/N}^{G/N}(\varphi_{N}(a))=\sum_{g\in DN\backslash
G/QN}\textnormal{Tr}_{({}^{g}\\!QN\cap
DN)/N}^{DN/N}({}^{g}\\!\varphi_{N}(a)).$

The argument of Proposition 5.16 shows that the subgroups $({}^{g}\\!QN\cap
DN)/N$ are proper in $DN/N$, and hence

$\textnormal{Tr}_{QN/N}^{G/N}(\varphi_{N}(a))\in\textnormal{Ker}(\textnormal{Br}_{DN/N}).$

Thus $\textnormal{Br}_{D}\widetilde{\textnormal{Tr}}_{D}^{G}(x)=0$. But going
the other way round the square,
$b\cdot\widetilde{\textnormal{Tr}}_{D}^{\textnormal{N}_{G}(D)}\textnormal{Br}_{D}(x)$
is clearly $0$, so the claim follows.

We from now on assume that $x\in k[[{\textnormal{C}_{G}(D)}]]$. Denote by $s$
the element $\textnormal{Br}_{D}\widetilde{\textnormal{Tr}}_{D}^{G}(x)$ and by
$t$ the element
$\widetilde{\textnormal{Tr}}_{D}^{\textnormal{N}_{G}(D)}\textnormal{Br}_{D}(x)=\widetilde{\textnormal{Tr}}_{D}^{\textnormal{N}_{G}(D)}(x)$.
We will show that for each $M\in\mathcal{N}$, we have

$\varphi_{M}(s)=b\cdot\varphi_{M}(t),$

and hence that $s=b\cdot t$. Since $b$ does not depend on $s$ or $t$, the
lemma follows.

We first calculate $\varphi_{N}(s)$ for arbitrary $N\in\mathcal{N}$:

$\displaystyle\varphi_{N}(s)$
$\displaystyle=\varphi_{N}\textnormal{Br}_{D}\widetilde{\textnormal{Tr}}_{D}^{G}(x)$
$\displaystyle=\textnormal{Br}_{DN/N}\varphi_{N}\widetilde{\textnormal{Tr}}_{D}^{G}(x)$
$\displaystyle=\frac{1}{|G:DN|_{p^{\prime}}}\textnormal{Br}_{DN/N}\textnormal{Tr}_{DN/N}^{G/N}(\varphi_{N}(x))$
$\displaystyle=\frac{1}{|G:DN|_{p^{\prime}}}\textnormal{Br}_{DN/N}\bigg{(}\sum_{g\in
DN\backslash
G/DN}\textnormal{Tr}_{(DN\cap{}^{g}\\!DN)/N}^{DN/N}({}^{gN}\varphi_{N}(x))\bigg{)}.$

Note that $DN\cap{}^{g}\\!DN=(D\cap{}^{g}\\!DN)N$ by Dedekind’s modular law
applied to $D$ and $N\leqslant{}^{g}\\!DN$. We claim that if
$g\not\in\textnormal{N}_{G}(DN)$, then
$\textnormal{Tr}_{(DN\cap{}^{g}\\!DN)/N}^{DN/N}({}^{gN}\varphi_{N}(x))$ is
sent to $0$ by $\textnormal{Br}_{DN/N}$: observe that

$D\subseteq{}^{g}\\!DN\Longleftrightarrow g\in\textnormal{N}_{G}(DN)$

and hence for our $g\not\in\textnormal{N}_{G}(DN)$, the subgroup
$Q=D\cap{}^{g}\\!DN$ is proper in $D$. Note further that

$\left|\frac{DN}{QN}\right|=\left|\frac{DQN}{QN}\right|=\left|\frac{D}{D\cap(D\cap{}^{g}\\!DN)N}\right|=\left|\frac{D}{D\cap{}^{g}\\!DN}\right|=\left|\frac{D}{Q}\right|.$

Hence
$\textnormal{Tr}_{(DN\cap{}^{g}\\!DN)/N}^{DN/N}({}^{gN}\varphi_{N}(x))=\textnormal{Tr}_{QN/N}^{DN/N}({}^{gN}\varphi_{N}(x))$
is in the kernel of $\textnormal{Br}_{DN/N}$ as claimed, and so continuing the
above calculation we have

$\displaystyle\varphi_{N}(s)$
$\displaystyle=\frac{1}{|G:DN|_{p^{\prime}}}\textnormal{Br}_{DN/N}\bigg{(}\sum_{g\in\textnormal{N}_{G}(DN)/DN}\textnormal{Tr}_{(DN\cap{}^{g}\\!DN)/N}^{DN/N}({}^{gN}\varphi_{N}(x))\bigg{)}$
$\displaystyle=\frac{1}{|G:DN|_{p^{\prime}}}\textnormal{Br}_{DN/N}\textnormal{Tr}_{DN/N}^{\textnormal{N}_{G}(DN)/N}(\varphi_{N}(x))$
$\displaystyle=\frac{1}{|G:\textnormal{N}_{G}(DN)|_{p^{\prime}}}\textnormal{Br}_{DN/N}\widetilde{\textnormal{Tr}}_{DN/N}^{\textnormal{N}_{G}(DN)/N}(\varphi_{N}(x)).\qquad\qquad\qquad(*)$

Meanwhile

$\varphi_{N}(t)=\widetilde{\textnormal{Tr}}_{DN/N}^{\textnormal{N}_{G}(D)N/N}(\varphi_{N}(x)).$

Fix $M\in\mathcal{N}$. Since $M$ is open and
$\bigcap_{N\in\mathcal{N}}\textnormal{N}_{G}(DN)=\textnormal{N}_{G}(D)$, there
is $N\in\mathcal{N}$ contained in $M$ with the property that
$\textnormal{N}_{G}(DN)M=\textnormal{N}_{G}(D)M$. Hence

$\varphi_{MN}\left(\textnormal{Tr}_{\textnormal{N}_{G}(D)N/N}^{\textnormal{N}_{G}(DN)/N}(\varphi_{N}(t))\right)=\left|\frac{\textnormal{N}_{G}(DN)}{\textnormal{N}_{G}(D)N}\right|\varphi_{M}(t),$

or in other words,

$\varphi_{MN}\widetilde{\textnormal{Tr}}_{\textnormal{N}_{G}(D)N/N}^{\textnormal{N}_{G}(DN)/N}(\varphi_{N}(t))=\varphi_{M}(t).\qquad\qquad\qquad(**)$

We put all these calculations together to obtain the claimed relationship
between $\varphi_{M}(s)$ and $\varphi_{M}(t)$:

$\displaystyle\varphi_{M}(s)$ $\displaystyle=\varphi_{MN}\varphi_{N}(s)$
$\displaystyle=\frac{1}{|G:\textnormal{N}_{G}(DN)|_{p^{\prime}}}\cdot\varphi_{MN}\textnormal{Br}_{DN/N}\widetilde{\textnormal{Tr}}_{DN/N}^{\textnormal{N}_{G}(DN)/N}(\varphi_{N}(x))$
$\displaystyle\hbox{by }(*)$
$\displaystyle=\frac{1}{|G:\textnormal{N}_{G}(DN)|_{p^{\prime}}}\cdot\varphi_{MN}\textnormal{Br}_{DN/N}\widetilde{\textnormal{Tr}}_{\textnormal{N}_{G}(D)N/N}^{\textnormal{N}_{G}(DN)/N}\widetilde{\textnormal{Tr}}_{DN/N}^{\textnormal{N}_{G}(D)N/N}(\varphi_{N}(x))$
by Lemma 3.6
$\displaystyle=\frac{1}{|G:\textnormal{N}_{G}(DN)|_{p^{\prime}}}\cdot\textnormal{Br}_{D/M}\varphi_{MN}\widetilde{\textnormal{Tr}}_{\textnormal{N}_{G}(D)N/N}^{\textnormal{N}_{G}(DN)/N}(\varphi_{N}(t))$
$\displaystyle=\frac{1}{|G:\textnormal{N}_{G}(DN)|_{p^{\prime}}}\cdot\textnormal{Br}_{D/M}\varphi_{M}(t)$
$\displaystyle\hbox{by }(**)$ $\displaystyle=b\cdot\varphi_{M}(t).$

∎

######  Lemma 6.4.

Let $D$ be a closed normal pro-$p$ subgroup of $G$. The central idempotents of
$k[[{G}]]$ lie in $k[[{\textnormal{C}_{G}(D)}]]$.

###### Proof.

Fix an idempotent $e\in\textnormal{Z}(k[[{G}]])$. For each
$N\trianglelefteq_{O}G$, $DN/N$ is a normal $p$-subgroup of $G/N$ and
$\varphi_{N}(e)$ is a central idempotent of $k[G/N]$. By the finite version of
this result [2, Proposition 6.2.2], $\varphi_{N}(e)\in
k[\textnormal{C}_{G/N}(DN/N)]$, and hence $e\in\varprojlim
k[\textnormal{C}_{G/N}(DN/N)]=k[[{\textnormal{C}_{G}(D)}]].$ ∎

######  Lemma 6.5.

Let $D$ be a closed pro-$p$ subgroup of the profinite group $G$ and let $H$ be
a closed subgroup of $G$ such that $D\textnormal{C}_{G}(D)\leqslant
H\leqslant\textnormal{N}_{G}(D)$. Let $b$ be a block of $H$ with block
idempotent $f$ and defect group $D$. There is a unique block $B$ of $G$ with
block idempotent $e$ such that $f=f\cdot\textnormal{Br}_{D}(e)$.

###### Proof.

By hypothesis $D$ is normal in $H$, and hence $f\in
k[[{\textnormal{C}_{H}(D)}]]$ by Lemma 6.4. But again by hypothesis
$\textnormal{C}_{H}(D)=\textnormal{C}_{G}(D)$, and hence $f\in
k[[{\textnormal{C}_{G}(D)}]]$.

The Brauer homomorphism is a surjective algebra homomorphism $k[[{G}]]^{D}\to
k[[{\textnormal{C}_{G}(D)}]]$, and hence (because $\textnormal{Br}_{D}(1)=1$)
there is at least one block idempotent $e$ such that
$f=f\cdot\textnormal{Br}_{D}(e)$. If $e^{\prime}$ is another block idempotent
with this property, then

$f=f\cdot\textnormal{Br}_{D}(e)\cdot\textnormal{Br}_{D}(e^{\prime})=f\cdot\textnormal{Br}_{D}(ee^{\prime}),$

which implies that $e=e^{\prime}$, since distinct block idempotents are
orthogonal. ∎

######  Definition 6.6.

Let $D$ be a closed pro-$p$ subgroup of $G$ and $H$ a closed subgroup of $G$
with $D\textnormal{C}_{G}(D)\leq H\leq\textnormal{N}_{G}(D)$. If $b$ is a
block of $H$ with block idempotent $f$, define the _Brauer correspondent_
$b^{G}$ of $b$ to be the unique block of $G$ with block idempotent $e$ such
that $f=f\cdot\textnormal{Br}_{D}(e)$.

We may finally prove our main theorem – Brauer’s first main theorem for
profinite groups.

######  Theorem 6.7.

Let $G$ be a profinite group, $P$ a $p$-Sylow subgroup of $G$ and $D$ an open
subgroup of $P$. There is a bijective correspondence between the blocks of
$\textnormal{N}_{G}(D)$ with defect group $D$ and the blocks of $G$ with
defect group $D$, given by sending the block $b$ of $\textnormal{N}_{G}(D)$
with defect group $D$ to its Brauer correspondent $b^{G}$.

###### Proof.

Given the results already obtained, we are in a position to proceed as with
finite groups. We mostly follow [22, Theorem 12.6.4].

By Lemma 6.4, the block idempotents of $k[[{\textnormal{N}_{G}(D)}]]$ lie in
$k[[{\textnormal{C}_{\textnormal{N}_{G}(D)}(D)}]]=k[[{\textnormal{C}_{G}(D)}]]$.

Let $b$ be a block of $\textnormal{N}_{G}(D)$ with block idempotent $f$ and
defect group $D$. By Lemma 6.5 the Brauer correspondent $b^{G}$ exists. Let
$e$ be the block idempotent of $b^{G}$, so that
$f=f\cdot\textnormal{Br}_{D}(e)$ by definition. Since
$\textnormal{Br}_{D}(e)\neq 0$, $b^{G}$ has a defect group $R$ containing $D$
by Theorem 5.18. We claim that $R=D$.

The subspace $e\widetilde{\textnormal{Tr}}_{D}^{G}(k[[{G}]]^{D})$ is an ideal
of $e\textnormal{Z}(k[[{G}]])$ by Part 3 of Lemma 3.6. By Lemma 6.3 and since
$\textnormal{Br}_{D}$ is an algebra homomorphism,

$\textnormal{Br}_{D}\left(e\widetilde{\textnormal{Tr}}_{D}^{G}(k[[{G}]]^{D})\right)=\textnormal{Br}_{D}(e)\textnormal{Br}_{D}\left(\widetilde{\textnormal{Tr}}_{D}^{G}(k[[{G}]]^{D})\right)=\textnormal{Br}_{D}(e)\widetilde{\textnormal{Tr}}_{D}^{\textnormal{N}_{G}(D)}(k[[{\textnormal{C}_{G}(D)}]]),$

which contains $f$. Note that $f\not\in J(k[[{C_{G}(D)}]])$, since if it were
then $\varphi_{N}(f)$ would be a non-zero idempotent in
$J(k[[{C_{G}(D)}]]_{N})$ for some $N$, which is impossible since the radical
of a finite dimensional algebra is nilpotent. Hence
$\textnormal{Br}_{D}\left(e\widetilde{\textnormal{Tr}}_{D}^{G}(k[[{G}]]^{D})\right)\not\subseteq
J(k[[{C_{G}(D)}]])$. It follows that
$e\widetilde{\textnormal{Tr}}_{D}^{G}(k[[{G}]]^{D})\not\subseteq
J(e\textnormal{Z}(k[[{G}]]))$. But the idempotent $e$ is primitive in
$\textnormal{Z}(k[[{G}]])$ so that $e\textnormal{Z}(k[[{G}]])$ is local by
Lemma 4.2, and hence
$e\textnormal{Z}(k[[{G}]])=e\widetilde{\textnormal{Tr}}_{D}^{G}(k[[{G}]]^{D})=\widetilde{\textnormal{Tr}}_{D}^{G}(ek[[{G}]]^{D})$.
In particular $e\in\textnormal{Im}(\widetilde{\textnormal{Tr}}_{D}^{G})$,
showing that $R$ is contained in $D$ by Theorem 5.18. So $R=D$ as claimed.

For convenience denote by $\Phi$ the map from the blocks of $N_{G}(D)$ with
defect group $D$ to the blocks of $G$ with defect group $D$, sending the block
$b$ to its Brauer correspondent $b^{G}$. The argument above shows that $\Phi$
is well-defined, so it remains to check that it is bijective.

Suppose that ${b_{1}}^{G}={b_{2}}^{G}$ has block idempotent $e$. Then, letting
$f_{i}$ be the block idempotent of $b_{i}$ we have

$f_{i}=f_{i}\textnormal{Br}_{D}(e)\in\textnormal{Br}_{D}(e\textnormal{Z}(k[[{G}]])).$

But $\textnormal{Br}_{D}(e\textnormal{Z}(k[[{G}]]))$ is local and hence
$f_{1}=f_{2}$, showing that $\Phi$ is injective.

Let $B$ be a block of $G$ with defect group $D$ and block idempotent $e$, so
that $\textnormal{Br}_{D}(e)\neq 0$ and
$e\in\widetilde{\textnormal{Tr}}_{D}^{G}(k[[{G}]]^{D})$. Let $f$ be a block
idempotent of $k[[{\textnormal{N}_{G}(D)}]]$ such that
$f\cdot\textnormal{Br}_{D}(e)=f$. If $b=k[[{\textnormal{N}_{G}(D)}]]f$ has
defect group $D$, then $B=\Phi(b)$ and $\Phi$ is surjective. It thus remains
to check that $b$ has defect group $D$. By Lemma 6.3,
$\textnormal{Br}_{D}(e)=\widetilde{\textnormal{Tr}}_{D}^{\textnormal{N}_{G}(D)}(a)$
for some $a$, so that by Lemma 3.6,

$f=f\cdot\widetilde{\textnormal{Tr}}_{D}^{\textnormal{N}_{G}(D)}(a)=\widetilde{\textnormal{Tr}}_{D}^{\textnormal{N}_{G}(D)}(fa)$

and hence $b$ has a defect group $R$ contained in $D$ by Theorem 5.18. On the
other hand, $R$ is an intersection of $p$-Sylow subgroups of
$\textnormal{N}_{G}(D)$ by Proposition 5.11, while $D$ is the intersection of
all the $p$-Sylow subgroups of $\textnormal{N}_{G}(D)$ by Corollary 5.13.
Hence $D\leqslant R$ and we are done. ∎

######  Corollary 6.8.

Let $G$ be a profinite group, $P$ a $p$-Sylow subgroup of $G$, $D$ an open
subgroup of $P$ and $L$ a closed subgroup of $G$ containing
$\textnormal{N}_{G}(D)$. There is a canonical bijection between the blocks of
$G$ with defect group $D$ and the blocks of $L$ with defect group $D$.

###### Proof.

Since $\textnormal{N}_{G}(D)=\textnormal{N}_{L}(D)$, Theorem 6.7 applies for
both $G$ and $L$, yielding the result. ∎

######  Remark 6.9.

Given that a defect group is open in a $p$-Sylow subgroup of $G$ that contains
it by Proposition 5.8, Theorem 6.7 applies to arbitrary blocks of $G$. It is
never-the-less important to note that Theorem 6.7 is false when we replace the
word “open” in its statement with the word “closed”: Let $G$ be the free
pro-$p$ group of rank $2$, freely generated by $x$ and $y$. If $D$ is the
closed subgroup generated by $x$, then $D=\textnormal{N}_{G}(D)$. There is of
course a block of $k[[{\textnormal{N}_{G}(D)}]]$ with defect group $D$ (namely
$k[[{\textnormal{N}_{G}(D)}]]$ itself), but there is not a block of $k[[{G}]]$
with defect group $D$ ($k[[{G}]]$ has only one block and its defect group is
$G$).

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