# On endotrivial complexes and the generalized Dade group

Sam K. Miller  university of California, Santa Cruz, Department of Mathematics
<EMAIL_ADDRESS>
###### Abstract.

Let $p$ be a prime, $G$ a $p$-group, and $k$ a field of characteristic $p$. We
give a complete description of the group $\mathcal{E}_{k}(G)$ of endotrivial
complexes, identifying $\mathcal{E}_{k}(G)$ with the additive subgroup of
class functions $f:s_{p}(G)\to\mathbb{Z}$ satisfying the Borel-Smith
conditions. This is done by constructing a short exact sequence isomorphic to
a short exact sequence of rational $p$-biset functors constructed by Bouc and
Yalçin, endowing $\mathcal{E}_{k}$ with rational $p$-biset structure. We also
determine all the possible h-marks which arise in $\mathcal{E}_{k}(G)$ when
$G$ is a finite group via the induced restriction map to a Sylow $p$-subgroup.
As a consequence, we prove that every $p$-permutation autoequivalence of a
$p$-group extends to a splendid Rickard autoequivalence. Additionally, we give
a positive answer to a question of Gelvin and Yalçin in [13], showing the
kernel of the Bouc homomorphism for an arbitrary finite group $G$ is described
by class functions $f:s_{p}(G)\to\mathbb{Z}$ satisfying the oriented Artin-
Borel-Smith conditions.

###### Key words and phrases:

Endotrivial complex, endopermutation module, Dade group, Bouc homomorphism,
splendid Rickard complex, $p$-permutation equivalence

###### 2010 Mathematics Subject Classification:

20J05, 19A22, 20C05, 20C20

##  1\. Introduction

Let $G$ be a finite group and $k$ a field of characteristic $p>0$. In [17] and
[18], we previously introduced the notions of endotrivial chain complexes and
relatively endotrivial chain complexes of $p$-permutation $kG$-modules
respectively. These complexes may be considered analogues of endotrivial
$kG$-modules and relatively endotrivial $kG$-modules respectively.

In this paper, we apply much of the theory previously developed to answer
questions both about endotrivial complexes and beyond; these questions concern
class functions, the Bouc homomorphism, and derived equivalences. A key
structure in this study is the Dade group $D_{k}(G)$ of a finite group. First
introduced by Dade in [11] and [12] for the case when $G$ is a $p$-group,
$D_{k}(G)$ parameterizes the “capped” endopermutation $kG$-modules. $D_{k}(G)$
was generalized to all finite groups $G$ by Linckelmann and Mazza in [16], by
Lassueur in [15], and by Gelvin and Yalçin in [13] via separate techniques.

In Lassueur’s construction, the generalized Dade group $D_{k}(G)$
parameterizes the “strongly capped” endo-$p$-permutation $kG$-modules. We take
special interest in Lassueur’s construction, as it relies on relatively
endotrivial $kG$-modules which are endo-$p$-permutation as well. Define the
$kG$-module $V(\mathcal{F}_{G})$ by

$V(\mathcal{F}_{G}):=\bigoplus_{P\in[s_{p}(G)\setminus\operatorname{Syl}_{p}(G)]}k[G/P].$

Then, $D_{k}(G)$ is realized as a subgroup of the group
$T_{V(\mathcal{F}_{G})}(G)$ of $V(\mathcal{F}_{G})$-endotrivial $kG$-modules.
These modules are closely tied to $V(\mathcal{F}_{G})$-endosplit-trivial
complexes, as defined in [18], as a $V(\mathcal{F}_{G})$-endosplit-trivial
complex is equivalently a shifted endosplit $p$-permutation resolution (as
defined by Rickard in [20]) of a relatively $V(\mathcal{F}_{G})$-endotrivial
module. This observation gives us the following short exact sequence, which is
developed in Theorems 3.8 and 3.8.

######  Theorem 1.1.

Let $G$ be a finite group and $S$ a Sylow $p$-subgroup of $G$. We have a short
exact sequence of abelian groups

$0\to\mathcal{T}\mathcal{E}_{k}(G)\to\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)\xrightarrow[]{\mathcal{H}}D_{k}^{\Omega}(G)+T_{V(\mathcal{F}_{G})}(G,S)\to
0.$

Here $\mathcal{T}\mathcal{E}_{k}(G)\leq\mathcal{E}_{k}(G)$ is the subgroup of
homology-normalized endotrivial complexes, that is, equivalences classes of
endotrivial complexes $[C]$ with nonzero homology isomorphic to the trivial
$kG$-module $k$. $\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$ is the group of
$V(\mathcal{F}_{G})$-endosplit-trivial complexes.
$D_{k}^{\Omega}(G)+T_{V(\mathcal{F}_{G})}(G,S)\leq D_{k}(G)$ is a subgroup of
the generalized Dade group.

Therefore, we have a relationship between the groups of endotrivial complexes,
$V(\mathcal{F}_{G})$-endosplit-trivial complexes, and the generalized Dade
group. If $G$ is a $p$-group, this short exact sequence is isomorphic to a
short exact sequence constructed by Bouc and Yalçin in [8], and as a result,
we can completely determine the h-marks which arise for all endotrivial
complexes for $p$-groups. This is Theorem 4.6.

######  Theorem 1.2.

Let $G$ be a $p$-group. We have an isomorphism of short exact sequences:

${0}$${\mathcal{E}_{k}(G)}$${\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)}$${D^{\Omega}(G)}$${0}$${0}$${C_{b}(G,p)}$${C(G,p)}$${D^{\Omega}(G)}$${0}$$\scriptstyle{h}$$\scriptstyle{\mathcal{H}}$$\scriptstyle{h}$$\scriptstyle{=}$$\scriptstyle{\Psi}$

Here, $C_{b}(G,p)$ is the subgroup of $C(G,p)$ consisting of Borel-Smith
functions. The map $\Psi:C(G,p)\to D^{\Omega}(G)$ is the Bouc homomorphism
(see Definition 3.12). In particular,
$[C]\in\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$ has an endotrivial cap if and
only if its h-mark function $h_{C}$ is a Borel-Smith function.

Furthermore, the bottom short exact sequence arises from a short exact
sequence of rational $p$-biset functors, allowing us to transport biset
functor structure to $\mathcal{E}_{k}(G)$ and
$\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$. This is the topic of Section 5. In
particular, we verify that the biset structure of $\mathcal{E}_{k}(G)$ is what
one might expect. Interestingly, the induction operation does not coincide
with the usual tensor induction of chain complexes. We ask the question of
whether there is a closed-form description of the induction operator - at the
moment, it is not clear.

Though it is not possible to fully realize $\mathcal{E}_{k}(G)$ as a biset
functor for all groups, we still may extend the results of $p$-groups to the
general finite group case. Let $S\in\operatorname{Syl}_{p}(G)$. In [18], we
characterized the image of the homomorphism
$\operatorname{Res}^{G}_{S}:\mathcal{E}_{k}(G)\to\mathcal{E}_{k}(S)$ as the
subgroup of “fusion-stable” endotrivial complexes, denoted
$\mathcal{E}_{k}(S)^{\mathcal{F}}$. Then, the h-marks of endotrivial complexes
of $kG$-modules correspond to class functions which upon restriction to a
Sylow $p$-subgroup are Borel-Smith functions.

Bouc and Yalçin’s short exact sequence above determines the kernel of the Bouc
homomorphism $\Psi:C(G,p)\to D^{\Omega}(G)$ when $G$ is a $p$-group. The Bouc
homomorphism was generalized to all finite groups in [13, Theorem 1.4], and in
general, we have an inclusion $\ker\Psi\subseteq C_{b}(G,p)$. Gelvin and
Yalçin in [13] proposed an additional condition, the oriented Artin condition,
and proved that $C_{ba^{+}}(G,p)\subseteq\ker\Psi\subseteq C_{b}(G,p)$, where
$C_{ba^{+}}(G,p)$ denotes the subgroup of $C(G,p)$ consisting of class
functions satisfying the oriented Artin-Borel-Smith conditions. Furthermore,
they posed the question of whether in general, $C_{ba^{+}}(G,p)=\ker\Psi$. We
provide a positive answer to this question by rephrasing it into a question
regarding certain endotrivial complexes; this is Theorem 6.15.

######  Theorem 1.3.

$\ker\Psi=C_{ba^{+}}(G,p)$.

The proof relies on the group homomorphism $\Lambda:\mathcal{E}_{k}(G)\to
O(T(kG))$ induced by taking the Lefschetz invariant of a chain complex, which
was studied by the author in [17], and some character theory developed by
Boltje and Carman in [2]. Here, $O(T(kG))$ denotes the orthogonal unit group
of the trivial source ring.

In [17], we determined that in many cases, the homomorphism
$\Lambda:\mathcal{E}_{k}(G)\to O(T(kG))$ is not surjective for non-$p$-groups.
In particular, the image of $\Lambda$ is contained in the subgroup
$O(T(\mathbb{F}_{p}G))\leq O(T(kG))$. However, the question remained open for
$p$-groups. Using further results of Bouc and Yalçin, we determine that
indeed, $\Lambda$ is surjective for any $p$-group. This is Theorem 7.5.

######  Theorem 1.4.

Let $G$ be a $p$-group. $\Lambda:\mathcal{E}_{k}(G)\to O(T(kG))$ is
surjective.

We obtain as corollary that if $P$ is a $p$-group, every $p$-permutation
autoequivalence of a block $kP$ is obtained from a splendid Rickard
autoequivalence of $kP$. This is Theorem 7.9.

######  Theorem 1.5.

Let $G$ be a $p$-group and let $\gamma\in O(T^{\Delta}(kG,kG))$ be a
$p$-permutation autoequivalence of $kG$. There exists a splendid Rickard
complex $\Gamma$ satisfying $\Lambda(\Gamma)=\gamma.$

Notation and conventions: For the paper, we fix $p$ a prime, $k$ a field of
characteristic $p$, $G$ a finite group, $V$ an absolutely $p$-divisible
$kG$-module (possibly $0$), and $\mathcal{X}_{V}\subseteq s_{p}(G)$ the set of
$p$-subgroups of $G$ for which $V(P)=0$. Note that $\mathcal{X}_{V}$ contains
all Sylow $p$-subgroups of $G$, since $V$ is absolutely $p$-divisible. We
write $s_{p}(G)$ for the set of $p$-subgroups of $G$, $[s_{p}(G)]$ for a set
of representatives of conjugacy classes of $s_{p}(G)$, and
$\operatorname{Syl}_{p}(G)$ for the set of Sylow $p$-subgroups of $G$. We let
$S\in\operatorname{Syl}_{p}(G)$ denote an arbitrary Sylow $p$-subgroup of $G$.
Given a $kG$-module $M$, we let $M[i]$ denote the chain complex with $M$ in
degree $i$ and the zero module in all other degrees.

We refer the reader to [14] for review of projectivity relative to modules,
absolute $p$-divisibility, relatively endotrivial modules, and to [18] for
relatively stable module and chain complex categories and relatively
endotrivial chain complexes. Additionally, we refer the reader to [6] for a
complete overview of biset functors, and to [8, Section 2] for a quicker
overview of rational $p$-biset functors.

##  2\. $V$-endosplit-trivial complexes

First, we review some results regarding $V$-endosplit-trivial chain complexes
and endotrivial chain complexes.

######  Definition 2.1.

Let $C\in Ch^{b}({}_{kG}\mathbf{triv})$. Say $C$ is a $V$-endosplit-trivial
complex if

$\operatorname{End}_{k}(C)\cong C^{*}\otimes_{k}C\simeq(k\oplus M)[0],$

where $M$ is a $V$-projective $kG$-module.

If $V=0$, we say $C$ is an endotrivial complex for short. This coincides with
the definition of an endotrivial complex given in [17].

######  Proposition 2.2.

(Omnibus properties) Let $C$ be a $V$-endosplit-trivial complex. Then the
following hold:

  1. (a)

$C$ has a unique indecomposable summand $C_{0}$ which is $V$-endosplit-
trivial, and all other direct summands are $V$-projective or contractible. We
call $C_{0}$ the cap of $C$.

  2. (b)

There exists a unique $i\in\mathbb{Z}$ for which $H_{i}(C)\neq 0$.

  3. (c)

$C_{0}$ has vertex set $\operatorname{Syl}_{p}(G)$.

  4. (d)

If $H\leq G$, then $\operatorname{Res}^{G}_{H}C$ is a
$\operatorname{Res}^{G}_{H}V$-endosplit-trivial complex.

  5. (e)

If $G$ is a quotient of $\tilde{G}$, then
$\operatorname{Inf}^{\tilde{G}}_{G}C$ is a
$\operatorname{Inf}^{\tilde{G}}_{G}V$-endosplit-trivial complex.

  6. (f)

If $P\in s_{p}(G)$, then $C(P)$ is a $V(P)$-endosplit-trivial complex. In
particular, if $P\in\mathcal{X}_{V}$, i.e. $V(P)=0$, then $C(P)$ is an
endotrivial complex.

  7. (g)

If $H_{i}(C)\neq 0$, then $H_{i}(C)$ is a relatively $V$-endotrivial
$kG$-module.

###### Proof.

Proofs of all these statements can be found in [18]. ∎

######  Theorem 2.3.

[18, Corollary 8.4]

Let $C\in Ch^{b}({}_{kG}\mathbf{triv})$. The following are equivalent:

  1. (a)

$C$ is $V$-endosplit-trivial.

  2. (b)

For all $P\in s_{p}(G)$, $C(P)$ has nonzero homology concentrated in exactly
one degree, and if $P\in\mathcal{X}_{V}$, that homology has $k$-dimension one.

  3. (c)

For all $P\in s_{p}(G)$, $C(P)$ has nonzero homology concentrated in exactly
one degree $i$ and $H_{i}(C)$ is a $V$-endotrivial $kG$-module.

  4. (d)

$C$ is isomorphic to a shift of an endosplit $p$-permutation resolution of a
$V$-endotrivial $kG$-module.

In particular, $C$ is endotrivial if and only if for all $P\in s_{p}(G)$,
$C(P)$ has nonzero homology concentrated in one degree with that homology
having $k$-dimension one.

######  Definition 2.4.

We define the group $\mathcal{E}^{V}_{k}(G)$ of $V$-endosplit-trivial chain
complexes as the set of all isomorphism classes of endotrivial complexes in
the relative stable homotopy category
$\underline{K}^{b}({}_{kG}\mathbf{triv})_{V}$. $\mathcal{E}^{V}_{k}(G)$ forms
an abelian group with group addition induced from $\otimes_{k}$. Write
$[C]\in\mathcal{E}_{k}^{V}(G)$ for the equivalence class of a $V$-endosplit-
trivial complex $C$. We have $[C]^{-1}=[C^{*}]$.

We have an equivalent definition of $\mathcal{E}_{k}^{V}(G)$ as follows. Say
two $V$-endosplit-trivial complexes $C_{1},C_{2}$ are equivalent, written
$C_{1}\sim C_{2}$, if and only if $C_{1}$ and $C_{2}$ have isomorphic caps.
Set $\mathcal{E}_{k}^{V}(G)$ to be the set of equivalence classes of
$V$-endosplit-trivial complexes, with group addition induced from
$\otimes_{k}$. It follows that this definition coincides with the previous
one. See [18, Remark 6.8(b)] for details.

If $C$ is a $V$-endosplit-trivial complex, we write
$[C]\in\mathcal{E}^{V}_{k}(G)$ to denote the corresponding class of complexes
in the group. We set $\mathcal{E}_{k}(G):=\mathcal{E}^{\\{0\\}}_{k}(G)$. This
construction agrees with the definition of $\mathcal{E}_{k}(G)$ given in [17]

######  Definition 2.5.

  1. (a)

[14, Proposition 3.5.1] Let $V$ be an absolutely $p$-divisible $kG$-module.
Define an equivalence relation $\sim_{V}$ on the class of $V$-endotrivial
$kG$-modules as follows: given two $V$-endotrivial modules $M,N$, write
$M\sim_{V}N$ if and only if $M$ and $N$ have isomorphic caps, or equivalently,
if $M\cong N$ in ${}_{kG}\underline{\mathbf{mod}}_{V}$. Let $T_{V}(G)$ denote
the resulting set of equivalence classes, then $\otimes_{k}$ induces an
abelian group structure on $T_{V}(G)$.

  2. (b)

Let $S\in\operatorname{Syl}_{p}(G)$. We define $T_{V}(G,S)\leq T_{V}(G)$ as
the kernel of $\operatorname{Res}^{G}_{S}:T_{V}(G)\to
T_{\operatorname{Res}^{G}_{S}V}(S)$. Equivalently, $T_{V}(G,S)$ is the
subgroup of $T_{V}(G)$ generated by trivial source $V$-endotrivial
$kG$-modules, see [18, Theorem 5.5] for details.

######  Proposition 2.6.

[18, Proposition 6.4]

  1. (a)

If $U,V$ are two $kG$-modules satisfying
${}_{kG}Proj(U)\subseteq{}_{kG}Proj(V)$, then we have an injective group
homomorphism $\mathcal{E}^{U}_{k}(G)\to\mathcal{E}^{V}_{k}(G)$. In particular,
we have an injective group homomorphism
$\mathcal{E}_{k}(G)\hookrightarrow\mathcal{E}_{k}^{V}(G)$.

  2. (b)

We have an injective group homomorphism $T_{V}(G,S)\to\mathcal{E}_{k}^{V}(G)$
given by $[M]\mapsto[M[0]]$.

In this way, we identify both $T_{V}(G,S)$ and $\mathcal{E}_{k}(G)$ as
subgroups of $\mathcal{E}_{k}^{V}(G)$. However, the equivalence classes in
$\mathcal{E}_{k}^{V}(G)$ which are in the image of the inclusion
$\mathcal{E}_{k}(G)\hookrightarrow\mathcal{E}_{k}^{V}(G)$ will in general be
larger than their preimage in $\mathcal{E}_{k}(G)$.

######  Proposition 2.7.

[18, Theorem 9.15] Let $U,V$ be $kG$-modules for which
${}_{kG}Proj(U)\subseteq{}_{kG}Proj(V),$ and let $C$ be a $V$-endosplit-
trivial complex.
$[C]\in\operatorname{im}\iota:\mathcal{E}_{k}^{U}(G)\hookrightarrow\mathcal{E}_{k}^{V}(G)$
if and only if the cap of $\mathcal{H}_{C}(1)$ is a relatively $U$-projective
$kG$-module.

In particular,
$\operatorname{im}\iota:\mathcal{E}_{k}(G)\hookrightarrow\mathcal{E}_{k}^{V}(G)$
is the following subgroup:

$\\{[C]\in\mathcal{E}_{k}^{V}(G)\mid\mathcal{H}_{C}(1)\text{ has cap with
$k$-dimension one}\\}.$

######  Definition 2.8.

Let $C(G,p)$ denote the set of $\mathbb{Z}$-valued class functions on the set
of $p$-subgroups of $G$. We may identify $C(G,p)$ in the following way:

$C(G,p)\xrightarrow{\sim}\left(\prod_{P\in
s_{p}(G)}\mathbb{Z}\right)^{G},f\mapsto(f(P))_{P\in s_{p}(G)}.$

In particular, when $G$ is a $p$-group, $C(G,p)=B(G)^{*}$, the
$\mathbb{Z}$-dual of the Burnside ring $B(G)$, via the identification

$C(G,p)\to B(G)^{*},f\mapsto\left([G/H]\mapsto f(H)\right).$

In this situation, these groups arise from isomorphic rational $p$-biset
functors, which we describe in detail in Section 4.

It is common notation to remove the $p$ in $C(G,p)$ when $G$ is a $p$-group.
However, since $C(G)$ could also refer to the Brauer construction, we keep the
$p$-notation to avoid ambiguity.

######  Definition 2.9.

Let $C$ be a $V$-endosplit-trivial chain complex. The h-mark of $C$ at $P$,
denoted $h_{C}(P)$, is the unique integer for which $H_{h_{C}(P)}(C(P))\neq
0$. The homology of $C$ at $P$, denoted $\mathcal{H}_{C}(P)$, is
$[H_{h_{C}(P)}(C(P))]\in T_{V(P)}(N_{G}(P)/P)$. If $V=0$, we identify
$T_{0}(G)=X(G)$, the group of isomorphism classes of $k$-dimension one
$kG$-modules, with group law induced by $\otimes_{k}$. $X(G)$ may also be
identified with the group of group homomorphism
$\operatorname{Hom}(G,k^{\times})$ by sending a group homomorphism
$\omega:G\to k^{\times}$ to the $k$-dimension one $kG$-module $k_{\omega}$
defined by $g\cdot m:=\omega(g)m$ for $m\in k_{\omega},g\in G$.

Note that $h_{C}$ may be regarded as a class function
$h_{C}:s_{p}(G)\to\mathbb{Z}$, since for any $g\in G$ and $P\in s_{p}(G)$
$C({}^{g}P)\cong({}^{g}C)(P),$ see [17, Proposition 2.7(c)].

This notation for h-marks differs slightly from the notation in [17], where we
instead regard h-marks as a collection of integers indexed by $s_{p}(G)$.
However, it is easy to see these are equivalent formulations.

######  Proposition 2.10.

[18, Definition 9.7] We have a well-defined, injective group homomorphism:

$\displaystyle\Xi:\mathcal{E}_{k}^{V}(G)$ $\displaystyle\to\prod_{P\in
s_{p}(G)}\left(\mathbb{Z}\times T_{V(P)}(N_{G}(P)/P)\right)$
$\displaystyle[C]$ $\displaystyle\mapsto(h_{C}(P),\mathcal{H}_{C}(P))_{P\in
s_{p}(G)}.$

Regarding $h_{C}$ as a $\mathbb{Z}$-valued class function on $s_{p}(G)$ gives
another group homomorphism, which we call the h-mark homomorphism,

$\displaystyle h:\mathcal{E}_{k}^{V}(G)$ $\displaystyle\to C(G,p)$
$\displaystyle[C]$ $\displaystyle\mapsto h_{C}$

$\ker h\cong T_{V}(G,S)$, the torsion subgroup of $\mathcal{E}_{k}^{V}(G)$. In
particular, $\mathcal{E}_{k}^{V}(G)$ is a finitely generated abelian group.

###  2.1. The module $V(\mathcal{F}_{G})$

We next introduce a specific absolutely $p$-divisible $kG$-module which plays
a crucial role in the sequel. This module was used by Lassueur in [15] in her
construction of the generalized Dade group.

######  Definition 2.11.

We define the $kG$-module $V(\mathcal{F}_{G})$ as follows:

$V(\mathcal{F}_{G})=\bigoplus_{Q\in[s_{p}(G)\setminus\operatorname{Syl}_{p}(G)]}k[G/Q].$

$V(\mathcal{F}_{G})$ is absolutely $p$-divisible, and may be considered the
“largest” absolutely $p$-divisible module via the next theorem.

######  Theorem 2.12.

Let $V$ be an absolutely $p$-divisible $kG$-module.

  1. (a)

If $M$ is a $V$-endotrivial $kG$-module, $M$ is
$V(\mathcal{F}_{G})$-endotrivial.

  2. (b)

If $C$ is a $V$-endosplit-trivial $kG$-complex, $C$ is
$V(\mathcal{F}_{G})$-endosplit-trivial.

###### Proof.

For (a), write $V=V_{1}\oplus\cdots\oplus V_{n}$ with each $V_{i}$
indecomposable. Since $V$ is absolutely $p$-divisible, each $V_{i}$ has vertex
strictly contained in a Sylow $p$-subgroup. [18, Theorem 5.6(a)] asserts that
${}_{kG}Proj(V_{i})\subseteq{}_{kG}Proj(Q_{i})$ with $Q_{i}$ the vertex of
$V_{i}$. We have

${}_{kG}Proj(V)=\bigoplus_{i=1}^{n}{}_{kG}Proj(V_{i})\subseteq\bigoplus_{i=1}^{n}{}_{kG}Proj(Q_{i})\subseteq{}_{kG}Proj(V(\mathcal{F}_{G})),$

where the last containment follows from the fact that
${}_{kG}Proj(H)={}_{kG}Proj(k[G/H])$. Now (b) follows directly from (a) and
[18, Corollary 8.4].

∎

######  Definition 2.13.

Given any $G$-set $X$, we write $\Delta(X)$ for the $kG$-module given by the
kernel of the augmentation homomorphism,

$kX\to k,\,x\mapsto 1.$

This is referred to as the relative syzygy of $X$.

######  Theorem 2.14.

Let $V=V(\mathcal{F}_{G})$. Then $h:\mathcal{E}_{k}^{V}(G)\to C(G,p)$ is
surjective. In particular, if $G$ is a $p$-group, we have an isomorphism
$\mathcal{E}_{k}^{V}(G)\cong C(G,p)\cong B(G)^{*}$.

###### Proof.

We give an explicit set of $V$-endosplit-trivial chain complexes $C_{Q}$ for
every $Q\in[s_{p}(G)]$ such that the set $\\{h_{C_{Q}}\\}$ forms a basis of
$C(G,p)$.

If $S\in\operatorname{Syl}_{p}(G)$, we define $C_{S}=k[1]$. Otherwise, if
$Q\not\in\operatorname{Syl}_{p}(G)$, we define

$C_{Q}=0\to k[G/Q]\to k\to 0,$

with $k$ in degree 0, and the differential given by the augmentation map, that
is, the homomorphism induced by $gQ\mapsto 1\in k$. $C_{Q}$ is a
$V$-endosplit-trivial chain complex, as the augmentation map $k[G/Q]\to k$ is
$Q$-split, hence $k[G/Q]$-split. Therefore $H_{1}(C_{Q})\cong\Delta(G/Q)$ is
$k[G/Q]$-endotrivial, hence $V(\mathcal{F}_{G})$-endotrivial. It follows that

$h_{C_{Q}}(P)=\begin{cases}1&P\leq_{G}Q\\\ 0&P\not\leq_{G}Q\end{cases}.$

It follows by a standard Möbius inversion argument on the poset of
$p$-subgroups of $G$ that the set $\\{h_{C_{Q}}\\}_{Q\in[s_{p}(G)]}$ forms a
$\mathbb{Z}$-basis of $C(G,p)$, as desired. The last statement follows
immediately, since if $G$ is a $p$-group, $h$ is injective as well. ∎

######  Remark 2.15.

Explicitly, if $\\{e_{Q}\\}_{Q\in[s_{p}(G)]}$ is the idempotent basis for
$C(G,p)$, i.e. $e_{Q}(P)=1$ if $Q=_{G}P$ and $e_{Q}(P)=0$ otherwise, then

$e_{Q}=\sum_{P\in[s_{p}(G)]}\mu(P,Q)h_{C_{P}},$

where $\mu$ denotes the Möbius function for the poset $[s_{p}(G)]$.

######  Definition 2.16.

For the rest of this paper, we will always use the notation used in the
previous proof for the $V(\mathcal{F}_{G})$-endosplit-trivial complex $C_{Q}$.
We write $\omega_{Q}$ for $h_{C_{Q}}$. We will use extensively the facts that
$\\{C_{Q}\\}_{Q\in[s_{p}(G)]}$ forms a $\mathbb{Z}$-basis of a free complement
of $T_{V(\mathcal{F}_{G})}(G,S)\leq\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$
(the torsion subgroup of $\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$), and
$\\{\omega_{Q}\\}_{Q\in[s_{p}(G)]}$ forms a $\mathbb{Z}$-basis of $C(G,p)$.

Moreover, observe that when $Q\not\in\operatorname{Syl}_{p}(G)$,
$\mathcal{H}_{C_{Q}}(1)=[\Delta(G/Q)]\in T_{V(\mathcal{F}_{G})}(G)$, and for
$S\in\operatorname{Syl}_{p}(G)$, $H_{C_{S}}(1)=[k]$.

Extending notation, for a $G$-set $X$, define $\omega_{X}\in C(G,p)$ as
follows:

$\omega_{X}(P)=\begin{cases}1&X^{P}\neq\emptyset\\\
0&\text{otherwise}\end{cases}.$

In this case, $\omega_{G/Q}=\omega_{Q}$.

Using Theorem 2.14, we may obtain a construction for a $V$-endosplit-trivial
chain complex $C$ given only its h-marks, up to a twist by a unique trivial
source $V$-endotrivial $kG$-module.

######  Theorem 2.17.

Let $V$ be an absolutely $p$-divisible $kG$-module (possibly 0), and $C$ be an
indecomposable $V$-endosplit-trivial chain complex of $kG$-modules with
h-marks $h_{C}\in C(G,p)$. For $P\in[s_{p}(G)]$, set

$b_{P}:=\sum_{Q\in[s_{p}(G)]}h_{C}(Q)\mu(P,Q).$

There exists a unique trivial source $V(\mathcal{F}_{G})$-endotrivial
$kG$-module $M$ for which the following holds: $C$ is isomorphic to the unique
$V$-endosplit-trivial indecomposable direct summand of the
$V(\mathcal{F}_{G})$-endosplit-trivial chain complex

$M\otimes_{k}\bigotimes_{Q\in[s_{p}(G)]}C_{Q}^{\otimes b_{Q}},$

where $C_{Q}$ is defined in the proof of Theorem 2.14.

###### Proof.

Let $h_{C}$ denote the corresponding h-marks of $C$. Note we may regard $C$ as
a $V(\mathcal{F}_{G})$-endosplit-trivial complex as well by Theorem 2.12,
since $V$ is absolutely $p$-divisible. Then we have

$h_{C}=\sum_{Q\in[s_{p}(G)]}a_{Q}e_{Q}=\sum_{Q\in[s_{p}(G)]}a_{Q}\left(\sum_{P\in
s_{p}(G)}\mu(P,Q)\omega_{P}\right)=\sum_{P\in[s_{p}(G)]}\omega_{P}\left(\sum_{Q\in[s_{p}(G)]}a_{Q}\mu(P,Q)\right)=\sum_{P\in[s_{p}(G)]}b_{P}\omega_{P}.$

It follows that the $V(\mathcal{F}_{G})$-endosplit-trivial chain complex

$D:=\bigotimes_{Q\in[s_{p}(G)]}C_{Q}^{\otimes b_{Q}}$

has the same h-marks as $C$, and thus $D^{*}\otimes_{k}C$ has h-marks all
zero. Therefore,

$[D]^{-1}+[C]=[M[0]]\in\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G),$

for some class of $V(\mathcal{F}_{G})$-endotrivial $p$-permutation modules
$[M[0]]$. Since every class of relatively $V(\mathcal{F}_{G})$-endotrivial
modules has a unique indecomposable representative, we may take $M$ to be a
trivial source $kG$-module. Therefore, we have

$[M[0]\otimes_{k}D]=[C]\in\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G).$

Since every class of $V(\mathcal{F}_{G})$-endosolit-trivial complexes has a
unique summand with Sylow vertices, and this chain complex is also
$V(\mathcal{F}_{G})$-endosplit-trivial, $C$ is the unique indecomposable
$V(\mathcal{F}_{G})$-endosplit-trivial summand of $M[0]\otimes_{k}D$. Since
$C$ is $V(\mathcal{F}_{G})$-endosplit-trivial as well, we are done. ∎

######  Remark 2.18.

In particular, if $G$ is a $p$-group, $C$ is isomorphic to the unique
indecomposable $V(\mathcal{F}_{G})$-endosplit-trivial direct summand of

$\bigotimes_{Q\in[s_{p}(G)]}C_{Q}^{\otimes b_{Q}},$

since in this case, the only trivial source relatively endotrivial $kG$-module
is $k$. Therefore, one can reconstruct an endotrivial $kG$-complex purely from
knowing its h-marks.

##  3\. Capped endosplit $p$-permutation resolutions and the Dade group of a
finite group

In this section, we construct a short exact sequence relating $V$-endosplit-
trivial complexes, endotrivial complexes, and the generalized Dade group as
defined by Lassueur in [15]. We also determine up to equivalence which
strongly capped endo-$p$-permutation modules, as defined by Lassueur, have
endosplit $p$-permutation resolutions.

######  Definition 3.1.

  1. (a)

We say a $kG$-module $M$ is endopermutation (resp. endo-$p$-permutation) if
$M^{*}\otimes_{k}M$ is a permutation module (resp. $p$-permutation module).

  2. (b)

The following definition is due to Dade in [11]. Let $P$ be a $p$-group. We
say an endopermutation $kP$-module $M$ is capped if $M$ has a direct summand
with vertex $P$. Equivalently, $M$ is capped if $(M^{*}\otimes_{k}M)(P)$ is
nonzero, and equivalently, $M^{*}\otimes_{k}M$ has $k$ as a direct summand. If
$M_{0}$ is an indecomposable summand of $M$ with vertex $P$, we say $M_{0}$ is
the cap of $M$.

Define an equivalence relation on the set of capped endopermutation
$kP$-modules as follows: say $M\sim N$ if and only if $M\oplus N$ is an
endopermutation $kP$-module. Equivalently, $M\otimes_{k}N^{*}$ is
endopermutation. It follows by a lemma of Dade that each equivalence class has
a unique indecomposable representative. Therefore, two endopermutation
$kP$-module are equivalent if and only if they have isomorphic caps. Write
$D_{k}(P)$ for the set of equivalence classes of capped endopermutation
modules. This forms an abelian group with addition induced by $\otimes_{k}$,
and is called the Dade group.

  3. (c)

A generalization of the Dade group to non-$p$-groups was first constructed by
Linckelmann and Mazza in [16], and later described using separate techniques
by Lassueur in [15]. We give the definition in [15] as it is more closely
related to our study of $V$-endosplit-trivial complexes. An
endo-$p$-permutation $kG$-module $M$ is strongly capped if $M$ is
$V(\mathcal{F}_{G})$-endotrivial. In this case, $M$ has a unique
indecomposable strongly capped direct summand, and all other direct summands
are $V(\mathcal{F}_{G})$-projective. As before, say this unique summand is the
cap of $M$.

Define an equivalence relation on the set of strongly capped
endo-$p$-permutation modules as before: given two endo-$p$-permutation
$kG$-modules $M,N$, say $M\sim N$ if and only if $M$ and $N$ have isomorphic
caps. It follows that each equivalence class has a unique indecomposable
representative. Write $D_{k}(G)$ for the set of equivalence classes of
strongly capped endo-$p$-permutation modules. This forms an abelian group as
before with addition induced by $\otimes_{k}$, and is called the generalized
Dade group. If $G$ is a $p$-group, we recover the classical Dade group, and
for this reason, we refer to $D_{k}(G)$ as the Dade group without further
mention.

Recall that $X(G)$ denotes the set of isomorphism classes of $k$-dimension one
$kG$-modules. We have an series of inclusions

$X(G)\hookrightarrow T_{V(\mathcal{F}_{G})}(G,S)\hookrightarrow
D_{k}(G)\hookrightarrow T_{V(\mathcal{F}_{G})}(G).$

In general, these maps will not be surjective.

  4. (d)

We may extend this notion to $V$-endosplit-trivial $kG$-complexes. Say a chain
complex of $kG$-modules $C$ is capped if $C$ is a
$V(\mathcal{F}_{G})$-endosplit-trivial complex. In this case, $C$ is a shifted
endosplit $p$-permutation resolution of a strongly capped endo-$p$-permutation
$kG$-module (see [18, Remark 9.11]).

The equivalence relation used for the prior definition of $D_{k}(G)$ is
already encoded in $\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$. In this way, we
may identify $\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$ as a chain-complex
theoretic “Dade group.”

######  Definition 3.2.

  1. (a)

For a $G$-set $X$, recall $\Delta(X)$ is the kernel of the augmentation
homomorphism $kX\twoheadrightarrow k$. Define the element $\Omega_{X}\in
D_{k}(G)$ as follows:

$\Omega_{X}=\begin{cases}[\Delta(X)]&\text{if }X^{S}=\emptyset.\\\
[k]&\text{if }X^{S}\neq\emptyset.\end{cases}$

Note that this is well-defined since $\Delta(X)$ is
$V(\mathcal{F}_{G})$-endotrivial. Indeed, the chain complex
$kX\twoheadrightarrow k$ is a $V(\mathcal{F}_{G})$-endosplit-trivial complex.
If $X=G/P$ for some $P\in s_{p}(G)$, we set $\Omega_{P}:=\Omega_{G/P}$.

  2. (b)

Let $D^{\Omega}_{k}(G)\leq D_{k}(G)$ be the subgroup generated by elements of
the form $\Omega_{X}$ where $X$ runs over all $G$-sets. When $G$ is a
$p$-group, this group is independent of choice of field $k$, so we write
$D^{\Omega}(G)$ (see for instance [6, Theorem 12.9.10]).

The following theorem gives a generating set of $D_{k}^{\Omega}(G)$.

######  Proposition 3.3.

[15, Lemma 12.1] The group $D^{\Omega}_{k}(G)$ is generated by the relative
syzygies $\Omega_{G/Q}$, where $Q$ runs over the proper subgroups of $S$.

This was proven using separate techniques in [13, Proposition 5.14] as well.

######  Remark 3.4.

Let $S\in\operatorname{Syl}_{p}(G)$. For any $G$-set $X$ for which $X^{S}=0$,
$\Delta(X)$ is endo-$p$-permutation and has an endosplit $p$-permutation
resolution (if $X^{S}\neq 0$, $\Delta(X)$ is permutation). Indeed, the chain
complex $kX\twoheadrightarrow k$ suffices.

It follows that every element of $D^{\Omega}_{k}(G)$ has a representative
which has an endosplit $p$-permutation resolution. Indeed, if

$[M]=\sum_{i=1}^{j}\Omega_{Q_{i}}^{\epsilon_{i}}\in D_{k}^{\Omega}(G),$

with $\epsilon_{i}\in\\{\pm 1\\}$ and $Q_{i}$ a $p$-subgroup of $G$ which is
non-Sylow, then a corresponding endosplit $p$-permutation resolution of $[M]$
(up to some representative) is

$\bigotimes_{i=1}^{j}[C_{Q}]^{\epsilon_{i}}\in\mathcal{E}_{k}^{V\mathcal{F}(G)}(G).$

In particular, given any $[M]\in D^{\Omega}_{k}(G)$, there exists a
$V(\mathcal{F}_{G})$-endosplit-trivial complex $C$ with
$\mathcal{H}_{C}(1)=[M]\in D_{k}(G)$.

However, every element which has an endosplit $p$-permutation resolution does
not necessarily belong to an equivalence class in $D^{\Omega}_{k}(G)$. For
instance, if $G$ is not a periodic group, any class in $D_{k}(G)$ with a
representative given by a nontrivial $k$-dimension one representation has an
endosplit $p$-permutation resolution but is not an element of
$D^{\Omega}_{k}(G)$, see [15, Remark 12.2]. On the other hand, it is shown in
[15, Section 12] that in many cases,
$D^{\Omega}_{k}(G)+\Gamma(X(N_{G}(S)))=D_{k}(G)$, where $\Gamma(X(N_{G}(S)))$
is the subgroup of $D_{k}(G)$ generated by Green correspondents of
1-dimensional $k[N_{G}(S)]$-modules. By [15, Proposition 4.1],
$\Gamma(X(N_{G}(S)))$ is a well-defined subgroup of $D_{k}(G)$, that is, each
representative of each equivalence class in $\Gamma(X(N_{G}(S)))$ is
$V(\mathcal{F}_{G})$-endotrivial.

This observation may be seen as an analogue of the characterization for
$p$-groups that $D_{k}(G)=D^{\Omega}(G)$ for any $p$-group which is not
generalized quaternion. It is unknown when this equality holds in general for
non-$p$-groups. Note that the elements of $\Gamma(X(N_{G}(S)))$ have trivial
source representatives, so every element of the subgroup
$D^{\Omega}_{k}(G)+\Gamma(X(N_{G}(S)))$ has an endosplit $p$-permutation
resolution.

Finally, note that the sum $D^{\Omega}_{k}(G)+\Gamma(X(N_{G}(S)))$ may not be
direct. See the comment after [15, Theorem 12.6].

In fact, $\Gamma(X(N_{G}(S)))=T_{V(\mathcal{F}_{G})}(G,S)$ by the following
argument.

######  Proposition 3.5.

$\Gamma(X(N_{G}(S)))=T_{V(\mathcal{F}_{G})}(G,S)$.

###### Proof.

Write $\Gamma(X)$ for $\Gamma(X(N_{G}(S)))$. Since induction preserves
$p$-permutation, the Green correspondents of any trivial source
$V(\mathcal{F}_{N_{G}(S)})$-endotrivial $k[N_{G}(S)]$-modules are again
trivial source. Since every element of $\Gamma(X(N_{G}(S)))$ is trivial source
and $V(\mathcal{F}_{G})$-endotrivial, $\Gamma(X)\subseteq
T_{V(\mathcal{F}_{G})}(G,S)$. Now suppose $M$ is a trivial source
$V(\mathcal{F}_{G})$-endotrivial $kG$-module. [18, Theorem 5.6(b)] implies
that since $S\trianglelefteq N_{G}(S)$, all trivial source
$V(\mathcal{F}_{N_{G}(S)})$-endotrivial $k[N_{G}(S)]$-modules have
$k$-dimension one. Since equivalently,

$\operatorname{Res}^{N_{G}(S)}_{S}(\operatorname{Res}^{G}_{N_{G}(S)}M)=\operatorname{Res}^{G}_{S}M=k\oplus
P\text{ for some }P\in{}_{kS}Proj(V(\mathcal{F}_{S})),$

$\operatorname{Res}^{G}_{N_{G}(S)}M$ is a $p$-permutation
$V(\mathcal{F}_{N_{G}(S)})$-endotrivial module, hence its indecomposable
$V(\mathcal{F}_{N_{G}(S)})$-endotrivial summand has trivial source, and has
$k$-dimension one. Therefore, $M$ is the Green correspondent of a
$k$-dimension one $k[N_{G}(S)]$-module. Thus $\Gamma(X)\supseteq
T_{V(\mathcal{F}_{G})}(G,S)$, as desired. ∎

From this point on, we replace $\Gamma(X(N_{G}(S)))$ with
$T_{V(\mathcal{F}_{G})}(G,S)$.

######  Definition 3.6.

Denote by $\mathcal{T}\mathcal{E}_{k}(G)$ the subgroup of $\mathcal{E}_{k}(G)$
consisting of endotrivial complexes $C$ for which $\mathcal{H}_{C}(1)=[k]$. We
call this subgroup the homology-normalized subgroup of $\mathcal{E}_{k}(G)$.
$\mathcal{T}\mathcal{E}_{k}(G)$ may also be identified with the subgroup of
$\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$ consisting of equivalence classes of
complexes $C$ for which $\mathcal{H}_{C}(1)=[k]$ as well.

Since $V(\mathcal{F}_{G})$-endosplit-trivial complexes are equivalently
shifted endosplit $p$-permutation resolutions of
$V(\mathcal{F}_{G})$-endotrivial $kG$-modules, the image of $\mathcal{H}$
consists of all equivalence classes of strongly capped endo-$p$-permutation
modules which have endosplit $p$-permutation resolutions. The next theorem
characterizes precisely which strongly capped endo-$p$-permutation modules, up
to equivalence in the Dade group, have an endosplit $p$-permutation
resolution.

######  Theorem 3.7.

Let $M$ be a strongly capped endo-$p$-permutation $kG$-module. $[M]\in
D_{k}(G)$ contains a representative which has an endosplit $p$-permutation
resolution if and only if $[M]\in
D_{k}^{\Omega}(G)+T_{V(\mathcal{F}_{G})}(G,S)$.

###### Proof.

The reverse direction follows since any $\Omega_{X}\in D_{k}^{\Omega}(G)$ has
the corresponding endosplit $p$-permutation resolution $kX\twoheadrightarrow
k$ and any $[M]\in T_{V(\mathcal{F}_{G})}(G,S)$ has $M[0]$ as an endosplit
$p$-permutation resolution.

For the forward direction, suppose $M$ has an endo-$p$-permutation resolution.
Since $M$ is $V(\mathcal{F}_{G})$-endotrivial, there exists an
$V(\mathcal{F}_{G})$-endosplit-trivial complex $C$ with $\mathcal{H}(C)=[M]\in
D_{k}(G)$.

Theorem 2.14 implies that $C$ shares a cap with a tensor product of the chain
complexes $C_{Q}$, their duals, and elements of $T_{V(\mathcal{F}_{G})}(G,S)$
considered as chain complexes in degree 0. It follows that

$[M]=\mathcal{H}(C)=[M_{0}]+\sum_{i=1}^{j}\Omega_{Q_{i}}^{\pm 1}\in D_{k}(G),$

where each $Q_{i}$ is some $p$-subgroup which is non-Sylow, and $M_{0}$ is a
trivial source $V(\mathcal{F}_{G})$-endotrivial module. Thus $[M]\in
D_{k}^{\Omega}(G)+T_{V(\mathcal{F}_{G})}(G,S)$, as desired. ∎

######  Theorem 3.8.

We have a well-defined group homomorphism
$\mathcal{H}:\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)\to D_{k}(G)$ induced by
$[C]\mapsto\mathcal{H}_{C}(1)$.
$\ker\mathcal{H}=\mathcal{T}\mathcal{E}_{k}(G)$ and
$\operatorname{im}\mathcal{H}=D^{\Omega}_{k}(G)+T_{V(\mathcal{F}_{G})}(G,S)$,
so we have a short exact sequence of abelian groups

$0\to\mathcal{T}\mathcal{E}_{k}(G)\to\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)\xrightarrow[]{\mathcal{H}}D_{k}^{\Omega}(G)+T_{V(\mathcal{F}_{G})}(G,S)\to
0.$

In particular, if $G$ satisfies
$D^{\Omega}_{k}(G)+T_{V(\mathcal{F}_{G})}(G,S)=D_{k}(G)$, then $\mathcal{H}$
is surjective.

###### Proof.

$\mathcal{H}$ is a well-defined group homomorphism from Theorem 2.10 after
projection, and Theorem 2.14, which demonstrates that the image of
$\mathcal{H}$ is generated by endo-$p$-permutation $kG$-modules. In
particular, $\mathcal{E}_{k}^{V(\mathcal{F}_{G})}$ is spanned by
$\\{C_{Q}\\}_{Q\in[s_{p}(G)]}$ and $T_{V}(G,S)$, all of which have
endo-$p$-permutation homology.

Given any $V(\mathcal{F}_{G})$-endosplit-trivial complex $C$,
$\mathcal{H}(C)=[k]\in D_{k}(G)$ if and only if $\mathcal{H}(C)=[k]\in
T_{V(\mathcal{F}_{G})}(G)$. From this, it follows by definition that
$\ker\mathcal{H}=\mathcal{T}\mathcal{E}_{k}(G)$.

Now,
$\operatorname{im}\mathcal{H}=D^{\Omega}_{k}(G)+T_{V(\mathcal{F}_{G})}(G,S)$
follows directly from Theorem 3.7, since $V(\mathcal{F}_{G})$-endosplit-
trivial complexes are equivalently shifted endosplit $p$-permutation
resolutions. The existence of the short exact sequence follows immediately. ∎

######  Remark 3.9.

In particular, $\mathcal{H}:\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)\to
D_{k}(G)$ is surjective in the following cases:

  * •

$G$ is a $p$-group which is not generalized quaternion.

  * •

$G$ has cyclic Sylow $p$-subgroup.

  * •

$p$ is odd and $G$ has a normal Sylow $p$-subgroup.

  * •

$N_{G}(S)$ controls $p$-fusion of $G$.

  * •

$G=GL_{3}(\mathbb{F}_{p})$ for odd $p$.

Surjectivity when $G$ is a $p$-group which is not generalized quaternion
follows by the characterization of the Dade group (see [6, Chapter 12] or
[10]), as $D_{k}(G)$ is generated by relative syzygies. The other cases are
known examples of when
$D_{k}^{\Omega}(G)+T_{V(\mathcal{F}_{G})}(G,S)=D_{k}(G)$, see [15, Remark
12.2].

######  Corollary 3.10.

Let $M$ be an indecomposable $V(\mathcal{F}_{G})$-endotrivial $kG$-module,
that is, a cap of a strongly capped endo-$p$-permutation module. $M$ has an
endosplit $p$-permutation resolution if and only if $[M]\in
D^{\Omega}_{k}(G)+T_{V(\mathcal{F}_{G})}(G,S)$.

###### Proof.

The forward direction follows immediately from the previous theorem. For the
reverse direction, the previous theorem implies that $[M]$ has a
representative which has an endosplit $p$-permutation resolution. However, any
representative of $[M]$ can be written as $M\oplus P$, where $P$ is
$V(\mathcal{F}_{G})$-projective. If $M\oplus P$ has an endosplit
$p$-permutation resolution, $M$ does as well, and the result follows. ∎

######  Corollary 3.11.

Let $G$ be a $p$-group.
$\operatorname{rk}_{\mathbb{Z}}\mathcal{E}_{k}(G)=c(G)$, where $c(G)$ is the
number of conjugacy classes of cyclic subgroups of $G$.

###### Proof.

This follows from the short exact sequence in Theorem 3.8. It is demonstrated
in Theorem 2.14 that
$\operatorname{rk}_{\mathbb{Z}}\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$ is
equal to the number of conjugacy classes of subgroups of $G$, and the
classification of $D_{k}(G)$ for $p$-groups states that
$\operatorname{rk}_{\mathbb{Z}}D_{k}(G)$ is equal to the number of conjugacy
classes of non-cyclic subgroups of $G$, see [6, Corollary 12.9.11]. ∎

We have another homomorphism, the Bouc homomorphism, which has image in
$D^{\Omega}_{k}(G)$ and which will be critical in the sequel. This
homomorphism was defined for non-$p$-groups by Gelvin and Yalçin in [13,
Theorem 1.4]

######  Definition 3.12.

Let $G$ be an finite group. $\Psi:C(G,p)\to D^{\Omega}(G)$ is defined as
follows. For the basis of $C(G,p)$ given by
$\\{\omega_{Q}\\}_{Q\in[s_{p}(G)]}$, we $\mathbb{Z}$-linearize the assignment

$\omega_{Q}\mapsto\begin{cases}\Omega_{Q}&Q\not\in\operatorname{Syl}_{p}(G)\\\
[k]&Q\in\operatorname{Syl}_{p}(G)\end{cases}$

This homomorphism is called the Bouc homomorphism.

Although the Bouc homomorphism is defined by a $\mathbb{Z}$-linearization for
transitive $G$-sets, the definitions of $\omega_{X}$ and $\Omega_{X}$ ensure
that the Bouc homomorphism behaves as expected for all $G$-sets.

######  Proposition 3.13.

[13, Proposition 6.9] For all $G$-sets $X$, $\Psi(\omega_{X})=\Omega_{X}$.

##  4\. The case of $p$-groups

We focus on the case of $p$-groups first. For this section, unless we specify
otherwise, we assume $G$ is a $p$-group. In this case, the short exact
sequence in Theorem 3.8 can be simplified as follows:

$0\to\mathcal{E}_{k}(G)\to\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)\xrightarrow{\mathcal{H}}D^{\Omega}(G)\to
0.$

In addition, we have an isomorphism $h:\mathcal{E}_{k}^{V}(G)\cong C(G,p)$
induced by the assignment $C_{Q}\mapsto\omega_{Q}$. The picture is as follows:

${0}$${\mathcal{E}_{k}(G)}$${\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)}$${D^{\Omega}(G)}$${0}$${C(G,p)}$$\scriptstyle{\mathcal{H}}$$\scriptstyle{h}$$\scriptstyle{\Psi}$

The Bouc homomorphism is the unique homomorphism which makes the above diagram
commute. The next proposition verifies this.

######  Proposition 4.1.

If $G$ is a $p$-group, $\mathcal{H}=\Psi\circ h$.

###### Proof.

Note in this case $\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$ is a free abelian
group. It suffices to show commutativity for a $\mathbb{Z}$-basis of
$\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$. Recall a basis is given by the set

$\\{C_{Q}=k[G/Q]\to k\\}_{Q\in[s_{p}(G)]\setminus G}\cup\\{C_{G}=k[1]\\}.$

In Definition 2.16, we observed $\mathcal{H}(C_{Q})=\Omega_{G/Q}$ when $Q\neq
G$ and $\mathcal{H}(C_{G})=[k]$. For the other side, if $Q\neq G$, $(\Psi\circ
h)(C_{Q})=\Psi(\omega_{Q})=\Omega_{Q}$, and $(\Psi\circ
h)(C_{G})=\Psi(\omega_{G})=[k]$. Thus the diagram commutes. ∎

######  Remark 4.2.

In general, this equality will not hold if $G$ is not a $p$-group (and in
fact, one must enlarge the codomain to
$D^{\Omega}_{k}(G)+T_{V(\mathcal{F}_{G})}(G,S))$. For instance, for any $M\in
T_{V(\mathcal{F}_{G})}(G,S)$, we have

$\mathcal{H}(M[0])=[M]\in D^{\Omega}(G)+T_{V(\mathcal{F}_{G})}(G,S),$

but

$\Psi\circ h(M[0])=[k]\in D^{\Omega}(G)+T_{V(\mathcal{F}_{G})}(G,S).$

There is a subgroup of $\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$ for which
equality holds however, which we discuss in the sequel.

In [8], Bouc and Yalçin proved that the Bouc homomorphism can be regarded as a
surjective natural transformation of biset functors, and computed its kernel,
which we now describe.

######  Definition 4.3.

  1. (a)

The assignment $G\mapsto D^{\Omega}(G)$ for all $p$-groups $G$ defines a
rational $p$-biset functor $D^{\Omega}$. We refer the reader to [6, Section
12.6] for details on this construction.

  2. (b)

The assignment $G\mapsto C(G,p)$ (resp. $G\mapsto B(G)^{*}$) for all
$p$-groups $G$ defines isomorphic rational $p$-biset functors $C\cong B^{*}$.
Given a $(H,G)$-biset $U$, $C(U)$ is defined to be the morphism

$T_{U}:f\mapsto\left(K\mapsto\sum_{u\in K\backslash U/G}f(K^{u})\right),$

where $K\leq H$, $[K\backslash U/G]$ is a set of representatives of
$(K,G)$-orbits on $U$, and $K^{u}$ is the subgroup of $G$ defined by

$K^{u}=\\{g\in G\mid\exists k\in K,ku=ug\\}.$

######  Definition 4.4.

[22, Page 210] Let $G$ be an arbitrary finite group. Say $(T,S)$ is a section
of $G$ if $S\trianglelefteq T\leq G$. A class function on $G$ is a Borel-Smith
function if it satisfies the following conditions, called the Borel-Smith
conditions.

  * •

If $(T,S)$ is a section of $G$ such that
$T/S\cong(\mathbb{Z}/p\mathbb{Z})^{2}$, for some prime number $p$, then

$h(S)-\sum_{S<Y<T}h(Y)+ph(T)=0.$

  * •

If $(T,S)$ is a section of $G$ such that $T/S$ is cyclic of order $p$, for an
odd prime $p$, or cyclic of order 4, then $h(S)\equiv h(\hat{S})\mod 2$, where
$\hat{S}/S$ is the unique subgroup of prime order of $T/S$.

  * •

If $(T,S)$ is a section of $G$ such that $T/S$ is a quaternion group of order
$8$, then $h(S)\equiv h(\hat{S})\mod 4$, where $\hat{S}/S$ is the unique
subgroup of order 2 of $T/S$.

The Borel-Smith functions of $G$ form an additive subgroup of $C(G,p)$,
denoted by $C_{b}(G,p)$. Moreover, the assignment $G\mapsto C_{b}(G,p)$ forms
a rational $p$-biset biset subfunctor of $C$, see [8, Proposition 3.7].

We state the following theorem slightly differently than originally stated. In
the original paper, the biset functor $C$ is replaced with $B^{*}$, however
they are canonically identified.

######  Theorem 4.5.

[8, Theorem 1.2] The kernel of $\Psi:C\to D^{\Omega}$ is the biset functor
$C_{b}$ of Borel-Smith functions. Here, there is an exact sequence of
$p$-biset functors of the form

$0\to C_{b}\hookrightarrow C\twoheadrightarrow D^{\Omega}\to 0.$

Recall that since $G$ is a $p$-group,
$\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$ has a canonical basis
$\\{C_{Q}\\}_{Q\in[s_{p}(G)]}$ as described in Definition 2.16.

######  Theorem 4.6.

Let $G$ be a $p$-group. We have an isomorphism of short exact sequences:

${0}$${\mathcal{E}_{k}(G)}$${\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)}$${D^{\Omega}(G)}$${0}$${0}$${C_{b}(G,p)}$${C(G,p)}$${D^{\Omega}(G)}$${0}$$\scriptstyle{h}$$\scriptstyle{\mathcal{H}}$$\scriptstyle{h}$$\scriptstyle{=}$$\scriptstyle{\Psi}$

In particular, $[C]\in\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$ has an
endotrivial cap if and only if $h_{C}$ is a Borel-Smith function.

###### Proof.

The commutativity of the right-hand square is precisely the statement of
Proposition 4.1, and $h$ is an isomorphism by Theorem 2.14. Therefore, we have
a short exact sequence isomorphism

${0}$${\ker\mathcal{H}}$${\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)}$${D^{\Omega}(G)}$${0}$${0}$${\ker\Psi}$${C(G,p)}$${D^{\Omega}(G)}$${0}$$\scriptstyle{h}$$\scriptstyle{\mathcal{H}}$$\scriptstyle{h}$$\scriptstyle{=}$$\scriptstyle{\Psi}$

Since $\ker\mathcal{H}=\mathcal{E}_{k}(G)$ and $\ker\Psi=C_{b}(G,p)$, the
result follows. ∎

######  Corollary 4.7.

$f\in C(G,p)$ satisfies $f=h_{C}$ for some endotrivial complex $C$ if and only
if $f$ is a Borel-Smith function.

###### Proof.

This is immediate. ∎

######  Remark 4.8.

Using the construction presented in Theorem 2.17, given any Borel-Smith
function $f$, we may construct a corresponding $V(\mathcal{F}_{G})$-endosplit-
trivial complex which contains as a direct summand the unique indecomposable
endotrivial complex with h-mark function $f$.

######  Remark 4.9.

The isomorphism constructed in Theorem 4.6 factors through an injective
morphism of short exact sequences, where the intermediate short exact sequence
was constructed by Yalçin’s study of $G$-Moore complexes, see [25] for
definitions and details. Indeed, there is a unique injective group
homomorphism $\mathscr{Y}$ which sends for any $Q\in s_{p}(G)\setminus\\{G\\}$
the chain complex $C_{Q}$ to the equivalence class in $\mathcal{M}_{G}$ of the
$G$-Moore space $G/Q$, regarded as a discrete $G$-CW-complex, and sends
$C_{G}$ to the discrete $G$-Moore space with two points and trivial
$G$-action. This leads to the following morphisms of short exact sequences:

${0}$${\mathcal{E}_{k}(G)}$${\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)}$${D^{\Omega}(G)}$${0}$${0}$${\mathcal{M}_{0}(G)}$${\mathcal{M}(G)}$${D^{\Omega}(G)}$${0}$${0}$${C_{b}(G,p)}$${C(G,p)}$${D^{\Omega}(G)}$${0}$$\scriptstyle{\mathscr{Y}}$$\scriptstyle{\mathcal{H}}$$\scriptstyle{\mathscr{Y}}$$\scriptstyle{=}$$\scriptstyle{\operatorname{Dim}}$$\scriptstyle{\mathfrak{hom}}$$\scriptstyle{\operatorname{Dim}}$$\scriptstyle{=}$$\scriptstyle{\Psi}$

It follows without much difficulty that $h=\operatorname{Dim}\circ\mathscr{Y}$
and that the diagram commutes.

##  5\. Transporting rational $p$-biset functor structure

######  Remark 5.1.

The bottom row of the isomorphism of short exact sequences in Theorem 4.6
arises from the short exact sequence of rational $p$-biset functors shown in
[8, Theorem 1.2],

$0\to C_{b}\to B^{*}\twoheadrightarrow D^{\Omega}\to 0,$

where $B^{*}$ is identified with $C$. We can transport the structure of those
biset functors to the top row, thus realizing
$\mathcal{E}_{k}^{V(\mathcal{F})}$ as a rational $p$-biset functor canonically
isomorphic to the rational $p$-biset functor $C$ and $\mathcal{E}_{k}(G)$ as a
subfunctor which is canonically isomorphic to the rational $p$-biset functor
$C_{b}$, both isomorphic via the h-mark homomorphism $h$.

######  Definition 5.2.

Let $\mathcal{C}$ denote the biset category.

  1. (a)

We define the biset functor

$\mathcal{E}_{k}^{V(\mathcal{F})}:\mathcal{C}\to\mathbf{Ab}$

by the assignment $G\mapsto\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$. For any
$(H,K)$-biset $U$, the action of $U$ on
$\mathcal{E}_{k}^{V(\mathcal{F}_{K})}(K)$ is given by

$h_{H}^{-1}\circ T_{U}\circ h_{K},$

where $h_{K}:\mathcal{E}_{k}^{V(\mathcal{F}_{K})}(K)\xrightarrow{\sim}C(K,p)$
and $h_{H}:\mathcal{E}_{k}^{V(\mathcal{F}_{H})}(H)\xrightarrow{\sim}C(H,p)$
denote the h-mark isomorphisms over $K$ and $H$ respectively and $T_{U}$ is
the generalized induction defined for $C$. Since $C$ is a rational $p$-biset
functor and $\mathcal{E}_{k}^{V(\mathcal{F})}\cong C$ by construction,
$\mathcal{E}_{k}^{V(\mathcal{F})}$ is a rational $p$-biset functor.

  2. (b)

We define the biset subfunctor

$\mathcal{E}_{k}^{V(\mathcal{F})}\supseteq\mathcal{E}_{k}:\mathcal{C}\to\mathbf{Ab}$

by the assignment $G\mapsto\mathcal{E}_{k}(G)$. For any $(H,K)$-biset $U$, the
action of $U$ on $\mathcal{E}_{k}(K)$ is given by

$h_{H}^{-1}\circ T_{U}\circ h_{K},$

where $h_{K}:\mathcal{E}_{k}(K)\xrightarrow{\sim}C_{b}(K,p)$ and
$h_{H}:\mathcal{E}_{k}(H)\xrightarrow{\sim}C_{b}(H,p)$ denote the restricted
h-mark isomorphisms over $K$ and $H$ respectively and $T_{U}$ is the
generalized induction defined for $C$ and restricted to $C_{b}$. Since $C_{b}$
is a rational $p$-biset subfunctor of $C$, this construction is well-defined,
and since $C_{b}$ is a rational $p$-biset functor and $\mathcal{E}_{k}\cong
C_{b}$ by construction, $\mathcal{E}_{k}$ is a rational $p$-biset functor.

In [17, Proposition 4.3], we determined what effects restriction, inflation,
and the Brauer construction have on h-marks of endotrivial complexes. For
$\mathcal{E}_{k}$, the biset functor structure coincides with those
computations. We use this fact to show that the biset functor structure of
$\mathcal{E}_{k}$ coincides in the way we hope.

######  Theorem 5.3.

Let $H\leq G$ and $N\trianglelefteq G.$

  1. (a)

Consider the inflation biset $\text{inf}_{G/N}^{G}:={}_{G}G/N_{G/N}$. The
operation

$\text{inf}_{G/N}^{G}:\mathcal{E}_{k}(G/N)\to\mathcal{E}_{k}(G)$

coincides with the usual inflation homomorphism
$\operatorname{Inf}_{G/N}^{G}:\mathcal{E}_{k}(G/N)\to\mathcal{E}_{k}(G)$.

  2. (b)

Consider the restriction biset $\text{res}_{H}^{G}:={}_{H}G_{G}$. The
operation

$\text{res}_{H}^{G}:\mathcal{E}_{k}(H)\to\mathcal{E}_{k}(G)$

coincides with the usual restriction
$\operatorname{Res}_{H}^{G}:\mathcal{E}_{k}(H)\to\mathcal{E}_{k}(G)$.

  3. (c)

Consider the deflation biset $\text{def}_{G/N}^{G}:={}_{G/N}G/N_{G}.$ The
operation

$\text{def}_{G/N}^{G}:\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)\to\mathcal{E}_{k}^{V(\mathcal{F}_{G/N})}(G/N)$

coincides with the Brauer construction
$-(N):\mathcal{E}_{k}(G)\to\mathcal{E}_{k}(G/N)$.

###### Proof.

In [17, Proposition 4.3], the effects of restriction, inflation, and the
Brauer construction on h-marks are computed. It suffices to show that these
biset operations coincide with those computations.

  1. (a)

Let $L\leq G$ and $f\in C(G/N,p)$. We compute:

$\displaystyle(T_{\text{inf}^{G}_{G/N}}f)(L)$
$\displaystyle=\sum_{u\in[L\backslash{}_{G}G/N_{G/N}/G/N]}f(L^{u})$ We may set
$u=N\in G/N$ since $\inf^{G}_{G/N}$ is transitive as right $G/N$-set. Then,
$L^{u}=\\{gN\in G/N\mid\exists l\in L,lN=gN\\}=LN/N$. $\displaystyle=f(LN/N).$

This coincides with the description of [17, Proposition 4.3(c)], which shows
that the h-mark homomorphism of $\operatorname{Inf}^{G}_{G/N}C$ is
$h_{\operatorname{Inf}^{G}_{G/N}C}(H)=h_{C}(HN/N)$ for any $H\leq G$.

  2. (b)

Let $L\leq H$ and $f\in C(G,p)$. We compute:

$\displaystyle(T_{\text{res}^{G}_{H}}f)(L)$
$\displaystyle=\sum_{u\in[L\backslash{}_{H}G_{G}/G]}f(L^{u})$ We may set
$u=1\in G$, since $\text{res}^{G}_{H}$ is transitive as a right $G$-set. Then,
$L^{1}=\\{g\in G\mid\exists l\in L,l=g\\}$. $\displaystyle=f(G/L).$

This coincides with the description of [17, Proposition 4.3(a)], which shows
that the h-mark homomorphism of $\operatorname{Res}^{G}_{H}C$ is the
restriction $\operatorname{Res}^{G}_{H}h_{C}$.

  3. (c)

Let $L/N\leq G/N$ and $f\in C(G,p)$. We compute:

$\displaystyle(T_{\text{def}^{G}_{G/N}}f)(L/N)$
$\displaystyle=\sum_{u\in[(L/N)\backslash{}_{G/N}G/N_{G}/G]}f((L/N)^{u})$ We
may set $u=N\in G/N$ since $\text{def}^{G}_{G/N}$ is transitive on the right.
Then, $(L/N)^{N}=\\{g\in G\mid\exists lN\in L/N,lN=gN\\}=L$, since
$N\trianglelefteq L$. $\displaystyle=f(L).$

This coincides with the description of [17, Proposition 4.3(b)], which shows
the h-mark homomorphism of $C(N)$ is $h_{C(N)}(L/N)=h_{C}(L)$ for any
$L\trianglerighteq N$.

∎

######  Remark 5.4.

A difficulty in writing down an explicit formula for induction on the chain
complex level is that the induction biset ${}_{G}G_{H}$ is not transitive on
the right, and therefore there may be more than one orbit for the double coset
$L\backslash G/H$. In this case, we obtain a the formula

$(T_{\text{ind}^{G}_{H}}f)(L)=\sum_{x\in[L\backslash G/H]}f(H\cap L^{x}).$

This does not correspond to the usual tensor induction of chain complexes (see
[1, Section 4.1] for a description). In fact, tensor induction of chain
complexes does not preserve homotopy equivalence, therefore it does not
preserve endotriviality in general. See [17, Remark 5.2] for a basic example.
The fact that tensor induction of chain complexes does not preserve homotopy
equivalence is instrumental in Balmer and Gallauer’s proof that every
$kG$-module has a finite resolution by $p$-permutation $kG$-modules, see [19,
Theorem 3.1].

One way to compute how induction behaves on the chain complex level would be
to use Theorem 2.17. We pose the question of whether the induction operation
coincides with any other known operations on chain complexes, or whether the
operator has a closed-form expression.

######  Remark 5.5.

By the exact same computations, we can see what effect the restriction,
inflation, and deflation bisets have on the h-marks of a
$V(\mathcal{F}_{G})$-endosplit-trivial chain complex. These, however, do not
coincide with the usual maps $\operatorname{Res}^{G}_{H}$,
$\operatorname{Inf}^{G}_{G/N}$, or $-(N)$ on the group
$\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$. For instance, the restriction biset
$\text{res}^{G}_{H}$ induces a group homomorphism

$\text{res}^{G}_{H}:\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)\to\mathcal{E}_{k}^{V(\mathcal{F}_{H})}(H),$

but the group homomorphism induced by restriction of modules is of the form

$\operatorname{Res}^{G}_{H}:\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)\to\mathcal{E}_{k}^{\operatorname{Res}^{G}_{H}V(\mathcal{F}_{G})}(H).$

If $H$ does not contain a Sylow $p$-subgroup of $G$ as a subgroup,
${}_{kH}Proj(\operatorname{Res}^{G}_{H}V)={}_{kG}\mathbf{mod}$, and the
restriction of a $V(\mathcal{F}_{G})$-endosplit-trivial chain complex may not
be $V(\mathcal{F}_{H})$-endosplit trivial. For an example, if $G=D_{16}$, the
chain complex

$C_{H}=k[D_{16}/H]\to k,$

where $H$ is any subgroup of order $4$ satisfying $Z(D_{16})\leq H$ and the
differential is the augmentation homomorphism, satisfies

$\operatorname{Res}^{D_{16}}_{Z(D_{16})}C_{H}=k\oplus k\oplus k\oplus k\to k,$

which is not a $V(\mathcal{F}_{C_{2}})$-endosplit-trivial complex. On the
other hand, one may compute via h-marks that

$\operatorname{res}^{G}_{Z(D_{16})}C_{H}=C_{Z(D_{16})}.$

A similar issue arises for deflation, since in general,
$\big{(}V(\mathcal{F}_{G})\big{)}(N)$ is not an absolutely $p$-divisible
$kG$-module. For instance, if $G=C_{4}$, then

$V(\mathcal{F}_{G})=k[C_{4}/1]\oplus k[C_{4}/C_{2}],$

therefore

$\big{(}V(\mathcal{F}_{G})\big{)}(C_{2})\cong k\oplus k.$

Inflation has a slightly different issue which arises. In general, inflation
preserves absolute $p$-divisibility, but
${}_{kG}Proj(\operatorname{Inf}^{G}_{G/N}V(\mathcal{F}_{G/N}))\subseteq{}_{kG}Proj(V(\mathcal{F}_{G}))$.
For instance, let $G=V_{4}$ with 3 nonconjugate subgroups of order 2
$H_{1},H_{2},H_{3}$. Then,

$V(\mathcal{F}_{G/H_{1}})=k[(G/H_{1})/(H_{1}/H_{1})],\text{ and
}\operatorname{Inf}^{G}_{G/H_{1}}V(\mathcal{F}_{G/H_{1}})=G/H_{1}.$

On the other hand,

$V(\mathcal{F}_{G})=k[G/H_{1}]\oplus k[G/H_{2}]\oplus k[G/H_{3}]\oplus
k[G/1].$

Therefore, $k[G/H_{2}]$ and $k[G/H_{3}]$ are $V(\mathcal{F}_{G})$-projective
but not $\operatorname{Inf}^{G}_{G/H_{1}}V(\mathcal{F}_{G/H_{1}})$-projective.

##  6\. Application to non-$p$-groups

Using the results of the previous sections, we consider the case of $G$ not
necessarily a $p$-group. Recall that in this case, we have the short exact
sequence from Theorem 3.8,

$0\to\mathcal{T}\mathcal{E}_{k}(G)\to\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)\xrightarrow[]{\mathcal{H}}D_{k}^{\Omega}(G)+T_{V(\mathcal{F}_{G})}(G,S)\to
0.$

###  6.1. On restriction to a Sylow $p$-subgroup

Via restriction, we may transport the results of the previous section to the
case of arbitrary finite groups.

######  Definition 6.1.

Let $V$ be a (possibly zero) absolutely $p$-divisible $kG$-module and let
$H\leq G$ be a subgroup containing a Sylow $p$-subgroup $S$. We define the
fusion-stable stable subgroup
$\mathcal{E}_{k}^{V}(H)^{\mathcal{F}}\leq\mathcal{E}_{k}^{V}(H)$ as follows.

$\mathcal{E}_{k}^{V}(H)^{\mathcal{F}}=\\{[C]\in\mathcal{E}_{k}^{V}(H)\mid\text{For
all }P,Q\leq H\text{ with }P=_{G}Q,h_{C}(P)=h_{C}(Q)\\}.$

######  Theorem 6.2.

[18, Theorem 11.7]
$\operatorname{Res}^{G}_{S}:\mathcal{E}_{k}(G)\to\mathcal{E}_{k}(S)^{\mathcal{F}}$
is surjective, with kernel the torsion subgroup of $\mathcal{E}_{k}(G)$.

######  Remark 6.3.

Let $S\in\operatorname{Syl}_{p}(G)$. [18, Theorem 11.7] implies that
$\mathcal{E}_{k}(G)\cong\mathcal{E}_{k}(S)\times\operatorname{Hom}(G,k^{\times})$.
Similarly, [18, Theorem 11.17 and Theorem 11.19] imply that
$\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)\cong\mathcal{E}_{k}^{V(\mathcal{F}_{S})}(S)^{\mathcal{F}}\times
T_{V(\mathcal{F}_{G})}(G,S)$. We have a commutative diagram as follows, where
each row is exact.

${0}$${\mathcal{T}\mathcal{E}_{k}(G)}$${\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)}$${D^{\Omega}_{k}(G)+T_{V(\mathcal{F}_{G})}(G,S)}$${0}$${0}$${\mathcal{E}_{k}(S)^{\mathcal{F}}}$${\mathcal{E}_{k}^{V(\mathcal{F}_{S})}(S)^{\mathcal{F}}}$${D^{\Omega}(S)^{G-st}}$$\scriptstyle{\operatorname{Res}^{G}_{S}}$$\scriptstyle{\mathcal{H}}$$\scriptstyle{\operatorname{Res}^{G}_{S}}$$\scriptstyle{\operatorname{Res}^{G}_{S}}$$\scriptstyle{\mathcal{H}}$

Here, $D^{\Omega}(S)^{G-st}$ is the subgroup of $D^{\Omega}(S)$ consisting of
$G$-stable elements, that is, elements $[M]\in D^{\Omega}(S)$ for which

$[\operatorname{Res}^{S}_{S\cap{}^{x}S}M]=[(\operatorname{Res}^{{}^{x}S}_{S\cap{}^{x}S}\circ
c_{x})(M)]$

for all $x\in G$.

[18, Theorem 11.7] asserts that
$\operatorname{Res}^{G}_{S}:\mathcal{T}\mathcal{E}_{k}(G)\cong\mathcal{E}_{k}(S)^{\mathcal{F}}$,
and [18, Theorems 11.17 and 11.19] assert that
$\ker\operatorname{Res}^{G}_{S}:\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)\to\mathcal{E}_{k}^{V(\mathcal{F}_{S})}(S)^{\mathcal{F}}$
is the embedded subgroup $T_{V(\mathcal{F}_{G})}(G,S)$.

######  Definition 6.4.

  1. (a)

Let $G$ be an arbitrary finite group and $S\in\operatorname{Syl}_{p}(G)$. Say
a class function on $p$-groups $f\in C(G,p)$ is a Borel-Smith function on $G$
if $\operatorname{Res}^{G}_{S}f$ is a Borel-Smith function. It is easy to see
this forms a subgroup of $C(G,p)$, which we denote also by $C_{b}(G,p)$.

  2. (b)

Define $C(S,p)^{\mathcal{F}}$ (resp. $C_{b}(S,p)^{\mathcal{F}}$) as the
subgroup of $C(S,p)$ (resp. $C_{b}(S,p)$) consisting of class functions (resp.
Borel-Smith functions) $f$ which satisfy $f(H)=f({}^{g}H)$ for all $H\leq S$
and $g\in G$ satisfying ${}^{g}H\leq S$. We call this subgroup the fusion-
stable or $\mathcal{F}$-stable subgroup of $C(S,p)$ (resp. $C_{b}(S,p)$).

Observe that $\operatorname{Res}^{G}_{S}:C_{b}(G,p)\to
C_{b}(S,p)^{\mathcal{F}})$ is in fact an isomorphism, since every $p$-subgroup
of $G$ is $G$-conjugate to a $p$-subgroup of $S$.

######  Theorem 6.5.

Let $f\in C(G,p)$. $f=h_{C}$ for some endotrivial $kG$-complex $C$ if and only
if $f\in C_{b}(G,p)^{\mathcal{F}}$.

###### Proof.

Let $S\in\operatorname{Syl}_{p}(G)$. Recall that
$\mathcal{E}_{k}(G)=\mathcal{T}\mathcal{E}_{k}(G)\times\operatorname{Hom}(G,k^{\times})$.
Moreover, $\operatorname{Hom}(G,k^{\times})$ is the torsion subgroup of
$\mathcal{E}_{k}(G)$, and only the unit of $\mathcal{T}\mathcal{E}_{k}(G)$ has
trivial h-marks. Therefore, it suffices to consider complexes
$C\in\mathcal{T}\mathcal{E}_{k}(G)$. We have the following commutative
diagram:

${\mathcal{T}\mathcal{E}_{k}(G)}$${C(G,p)}$${\mathcal{E}_{k}(S)^{\mathcal{F}}}$${C(S,p)}$$\scriptstyle{h}$$\scriptstyle{\operatorname{Res}^{G}_{S}}$$\scriptstyle{\operatorname{Res}^{G}_{S}}$$\scriptstyle{h}$

[18, Theorem 11.7] asserts that
$\operatorname{Res}^{G}_{S}:\mathcal{T}\mathcal{E}_{k}(G)\to\mathcal{E}_{k}(S)^{\mathcal{F}}$
is an isomorphism. Moreover, it is straightforward from the definition of
$\mathcal{E}_{k}(S)^{\mathcal{F}}$ that the image of
$h:\mathcal{E}_{k}(S)^{\mathcal{F}}\to C_{b}(S,p)$ is
$C_{b}(S,p)^{\mathcal{F}}$. Since $h$ is injective, we have
$\mathcal{T}\mathcal{E}_{k}(G)\cong C_{b}(S,p)^{\mathcal{F}}$.

Notice that $\operatorname{Res}^{G}_{S}:C(G,p)\to C(S,p)$ is injective, since
every $p$-subgroup of $G$ is conjugate to a $p$-subgroup of $S$. Since the
image of the lower arrows lies in $C_{b}(S,p)^{\mathcal{F}}$, it follows by
commutativity of the diagram that the image of
$h:\mathcal{T}\mathcal{E}_{k}(G)\to C(G,p)$ is the subgroup $C_{b}(G,p)\cong
C_{b}(S,p)^{\mathcal{F}}$. Since $h$ is injective, we conclude
$h:\mathcal{T}\mathcal{E}_{k}(G)\to C_{b}(G,p)\cong C_{b}(S,p)^{\mathcal{F}}$
is an isomorphism. The result follows. ∎

Therefore, we may identify endotrivial complexes of $kG$-modules Borel-Smith
functions defined on $G$. These can also be viewed as Borel-Smith functions on
$S$ which satisfy a fusion stability condition over the fusion system
$\mathcal{F}_{S}(G)$.

######  Remark 6.6.

From [15, Theorem 7.3], we have that $\operatorname{Res}^{G}_{S}:D_{k}(G)\to
D_{k}(S)^{G-st}$ is surjective, with kernel $T_{V(\mathcal{F}_{G})}(G,S)$.
Since the restriction of a relative syzygy is again a relative syzygy, we may
ask if the map

$\operatorname{Res}^{G}_{S}:D^{\Omega}_{k}(G)\to D^{\Omega}(S)^{G-st}$

is surjective as well. It follows from the diagram in Remark 6.3 that
$\operatorname{Res}^{G}_{S}:D^{\Omega}_{k}(G)\to D^{\Omega}(S)^{G-st}$ is
surjective if and only if

$\mathcal{H}:\mathcal{E}_{k}^{V(\mathcal{F}_{S})}(S)^{\mathcal{F}}\to
D^{\Omega}(S)^{G-st}$

is surjective.

If both maps are surjective, we have a homomorphism of short exact sequences

${0}$${\mathcal{T}\mathcal{E}_{k}(G)}$${\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)}$${D^{\Omega}_{k}(G)+T_{V(\mathcal{F}_{G})}(G,S)}$${0}$${0}$${\mathcal{E}_{k}(S)^{\mathcal{F}}}$${\mathcal{E}_{k}^{V(\mathcal{F}_{S})}(S)^{\mathcal{F}}}$${D^{\Omega}(S)^{G-st}}$${0}$$\scriptstyle{\operatorname{Res}^{G}_{S}}$$\scriptstyle{\mathcal{H}}$$\scriptstyle{\operatorname{Res}^{G}_{S}}$$\scriptstyle{\operatorname{Res}^{G}_{S}}$$\scriptstyle{\mathcal{H}}$

In particular, we can derive an explicit basis for $D^{\Omega}(S)^{G-st}$ from
the image of $\mathcal{H}$. Furthermore, if this occurs and $S$ is not
generalized quaternion (in particular, $D^{\Omega}(S)=D_{k}(S)$), we have a
homomorphism of short exact sequences

${0}$${T_{V(\mathcal{F}_{G})}(G,S)}$${D^{\Omega}_{k}(G)+T_{V(\mathcal{F}_{G})}(G,S)}$${D^{\Omega}(S)^{G-st}}$${0}$${0}$${T_{V(\mathcal{F}_{G})}(G,S)}$${D_{k}(G)}$${D_{k}(S)^{G-st}}$${0}$$\scriptstyle{\operatorname{Res}^{G}_{S}}$$\scriptstyle{\operatorname{Res}^{G}_{S}}$

By the five lemma, we have an isomorphism $D_{k}(G)\cong D^{\Omega}_{k}(G)$.
We conclude the following.

######  Theorem 6.7.

Suppose $S\in\operatorname{Syl}_{p}(G)$ is not generalized quaternion. The
following are equivalent.

  1. (a)

$\operatorname{Res}^{G}_{S}:D_{k}^{\Omega}(G)\to D(S)^{G-st}$ is surjective.

  2. (b)

$\mathcal{H}:\mathcal{E}_{k}^{V(\mathcal{F}_{S})}(S)^{\mathcal{F}}\to
D^{\Omega}(S)^{G-st}$ is surjective.

  3. (c)

$D_{k}^{\Omega}(G)+T_{V(\mathcal{F}_{G})}(G,S)=D_{k}(G)$.

  4. (d)

$D^{\Omega}(S)^{G-st}$ has the following generating set:

$\left\\{\Omega_{P}+\sum_{Q\in[s_{p}(G)],Q=_{G}P,Q\neq P}\Omega_{Q}\mid
P\in[s_{p}(G)\setminus\operatorname{Syl}_{p}(G)]\right\\}$

###### Proof.

The equivalence of (a) and (b) follows from diagram 6.3. (a) implies (c)
follows from the previous remark, and (c) implies (a) follows from the
surjective group homomorphism $\operatorname{Res}^{G}_{S}:D_{k}(G)\to
D(S)^{G-st}.$

(b) implies (d) follows since the generating set is precisely the image of
$\mathcal{H}$ when restricted to $\mathcal{E}^{V(\mathcal{F}_{G})}_{k}(G)$.
(d) implies (b) is similarly straightforward, since in this case, for each
$\Omega_{P}+\sum\Omega_{Q}$, the preimage $[C_{P}]\cdot\sum[C_{Q}]$ is fusion-
stable, and the corresponding set of chain complexes with $C_{S}$ added, forms
a spanning set for $\mathcal{E}^{V(\mathcal{F}_{G})}_{k}(G)^{\mathcal{F}}$. ∎

###  6.2. The kernel of the Bouc homomorphism for non-$p$-groups

######  Remark 6.8.

Theorem 6.5 gives us all possible h-marks which arise from endotrivial
complexes, and Theorem 2.17 gives a recipe for reconstructing an endotrivial
complex from its h-marks, up to a twist by a strongly capped indecomposable
endo-$p$-permutation $kG$-module. In the $p$-group case, this information
tells us how to reconstruct a complex, since the only strongly capped
indecomposable endo-$p$-permutation $kG$-module is $k$. However, in the
non-$p$-group case, the situation is less clear.

Recall in Theorem 2.17, the setup was as follows: for $P\in[s_{p}(G)]$, we set

$b_{P}:=\sum_{Q\in[s_{p}(G)]}h_{C}(Q)\mu(P,Q).$

Then, the chain complex

$D:=\bigotimes_{Q\in[s_{p}(G)]}C_{Q}^{b_{Q}}$

is $V(\mathcal{F}_{G})$-endosplit-trivial. One goal is to determine when $D$
has an endotrivial summand, and if not, a $M\in T_{V(\mathcal{F}_{G})}(G,S)$
for which $M[0]\otimes_{k}D$ has an endotrivial summand. An even stronger
question we may ask is whether
$D\in\ker\mathcal{H}:\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)\to
D^{\Omega}_{k}(G)+T_{V(\mathcal{F}_{G})}(G,S).$ Note that $D$ is generated by
a product of chain complexes representing relative syzygies. This motivates
the following definition.

######  Definition 6.9.

Define the subgroup
${}^{\Omega}\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)\leq\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$
by

${}^{\Omega}\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G):=\langle[C_{P}]\mid P\in
s_{p}(G)\rangle.$

Define the subgroup
$\mathcal{E}_{k}^{\Omega}(G)\leq\mathcal{E}_{k}(G)\leq\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$
by

$\mathcal{E}_{k}^{\Omega}(G):={}^{\Omega}\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)\cap\mathcal{E}_{k}(G).$

Finally, define
$\mathcal{T}\mathcal{E}_{k}^{\Omega}(G):=\mathcal{T}\mathcal{E}_{k}(G)\cap\mathcal{E}_{k}^{\Omega}(G)$.
It is straightforward that

$\mathcal{H}^{\Omega}:{}^{\Omega}\mathcal{E}_{k}^{V(\mathcal{F}_{G})}\to
D_{k}^{\Omega}(G)$

is a well-defined surjective group homomorphism, and moreover,
$\ker\mathcal{H}^{\Omega}=\mathcal{T}\mathcal{E}_{k}^{\Omega}(G)$. We have a
short exact sequence

$0\to\mathcal{T}\mathcal{E}_{k}^{\Omega}(G)\hookrightarrow{}^{\Omega}\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)\twoheadrightarrow
D^{\Omega}(G)\to 0,$

where $\mathcal{H}^{\Omega}$ is the surjection.

Rephrasing the previous remark, we wish to determine
$\mathcal{E}_{k}^{\Omega}(G)$ or $\mathcal{T}\mathcal{E}_{k}^{\Omega}(G)$. One
way we may do so is by returning to the Bouc homomorphism $\Psi:C(G,p)\to
D^{\Omega}(G)$, in the more general case of non-$p$-groups. In this case, we
no longer necessarily have an isomorphism $C_{b}(G,p)\cong\ker\Psi$.

Gelvin and Yalçin describe in [13] constraints for $\ker\Psi$.

######  Definition 6.10.

[13, Definitions 9.6, 9.7] A function $f\in C(G,p)$ satisfies the oriented
Artin condition if for any distinct prime numbers $p$ and $q$, consider
$L\triangleleft K\triangleleft H\leq N_{G}(L)$ subgroups of $G$ such that $K$
is a cyclic $p$-group, $K/L\cong\mathbb{Z}/p$, and $H/K\cong\mathbb{Z}/q^{r}$.
Then $f(L)\equiv f(K)\mod 2q^{r-l}$, where $H/K$ acts on $K/L$ with kernel of
order $q^{l}$.

A oriented Artin-Borel-Smith function is a superclass function that satisfies
the Borel-Smith conditions and the oriented Artin condition. The subgroup of
$C(G,p)$ consisting of oriented Artin-Borel-Smith functions is denoted by
$C_{ba^{+}}(G,p)$.

Note that if $p=2$, then any Borel-Smith function vacuously satisfies the
Artin condition, since there is no nontrivial automorphism of
$K/L\cong\mathbb{Z}/2$.

######  Theorem 6.11.

[13, Theorem 9.10] Let $G$ be a finite group and $\Psi:C(G,p)\to
D^{\Omega}(G)$ be the Bouc homomorphism. Then

$C_{ba^{+}}(G,p)\subseteq\ker\Psi\subseteq C_{b}(G,p).$

In particular, if $p=2$, $\ker\Psi=C_{b}(G,p)$.

In [13], Gelvin and Yalçin asked if in general, $C_{ba^{+}}(G,p)=\ker\Psi$;
they note that they could find no counterexamples to the claim. Moreover, a
nontrivial example was provided in which $C_{ba^{+}}(G,p)=\ker\Psi\subset
C_{b}(G,p)$.

######  Lemma 6.12.

[13, Lemma 9.11] Let $Q\trianglelefteq G$ be a cyclic subgroup of $G$ of order
$p$ such that $G/Q\cong\mathbb{Z}/q^{r}$. Suppose that $G/Q$ acts on $Q$ with
kernel order $q^{l}$. Then $D^{\Omega}(G)\cong\mathbb{Z}/2q^{r-l}$ and the
equality $\ker\Psi=C_{ba^{+}}(G,p)$ holds.

The next proposition gives us an equivalent formulation of $\ker\Psi$.

######  Proposition 6.13.

We have an isomorphism of short exact sequences:

${0}$${\mathcal{T}\mathcal{E}_{k}^{\Omega}(G)}$${{}^{\Omega}\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)}$${D^{\Omega}(G)}$${0}$${0}$${\ker\Psi}$${C(G,p)}$${D^{\Omega}(G)}$${0}$$\scriptstyle{h}$$\scriptstyle{\mathcal{H}^{\Omega}}$$\scriptstyle{h}$$\scriptstyle{=}$$\scriptstyle{\Psi}$

###### Proof.

First, note that $\mathcal{H}^{\Omega}=\Psi\circ h$ by a similar argument as
used in the proof of Proposition 4.1. In particular,
${}^{\Omega}\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$ is free abelian with
basis given by elements of the form $[C_{P}]$ for $P\in[s_{p}(G)]$. The left-
hand square commutes simply since the injections are both inclusions of
kernels. The image of this basis of
${}^{\Omega}\mathcal{E}_{k}^{V(\mathcal{F}_{G})}(G)$ under $h$ is precisely
the basis of $C(G,p)$ given by elements of the form $\omega_{P}$. Since
$[C_{p}]\mapsto\omega_{p}$, $h$ is an isomorphism. Since the top and bottom
injective homomorphisms are both subgroup inclusion,
$h:\mathcal{T}\mathcal{E}_{k}^{\Omega}(G)\to\ker\Psi$ is also an isomorphism.
∎

In fact, we may use the identification given in Proposition 6.13 to completely
determine $\ker\Psi$, giving a positive answer to Gelvin and Yalçin’s
question. To do this, we briefly recall a character-theoretic result of Boltje
and Carman regarding the orthogonal unit group of the trivial source ring
$O(T(kG))$, and the relation of $O(T(kG))$ to $\mathcal{E}_{k}(G)$ studied in
[17].

######  Theorem 6.14.

[2, Theorem C] Let $S$ be a Sylow $p$-subgroup of $G$ and let
$\mathcal{F}:=\mathcal{F}_{S}(G)$ be the associated fusion system on $S$. One
has a direct product decomposition

$O(T(kG))\cong B(\mathcal{F})^{\times}\times\left(\prod_{P\in
s_{p}(G)}\operatorname{Hom}(N_{G}(P)/P,k^{\times})\right)^{\prime}$

where the second factor is defined as the set of all tuples

$(\varphi_{P})\in\left(\prod_{P\in
s_{p}(G)}\operatorname{Hom}(N_{G}(P)/P,k^{\times})\right)^{G}$

satisfying

$\varphi_{P}(xP)=\varphi_{P\langle x_{p}\rangle}(xP\langle x_{p}\rangle)$

for all $P\in s_{p}(G)$ and $x\in N_{G}(P)$.

Here, $x_{p}$ denotes the $p$-part of $x$. That is, given any $x\in G$,
$x=x_{p}x_{p^{\prime}}=x_{p^{\prime}}x_{p}$, where $x_{p}$ has order a power
of $p$ and $x_{p^{\prime}}$ has $p^{\prime}$ order.

Taking the Lefschetz invariant of an endotrivial complex induces a group
homomorphism

$\Lambda:\mathcal{E}_{k}(G)\to O(T(kG)),[C]\mapsto\Lambda(C).$

Moreover, given an endotrivial complex $C$, the tuple $(\varphi_{P})$ in [2,
Theorem C] arising in the identification of $\Lambda(C)\in O(T(kG))$
corresponds to the local homology of $C$ via

$\mathcal{H}_{C}(P)=[k_{\varphi_{P}}]\in D^{\Omega}(N_{G}(P)/P).$

See [17, Proposition 4.5] for details.

We are now ready to prove Gelvin and Yalçin’s conjecture with endotrivial
complex machinery. In fact, we may do so by reducing to the case they prove in
[13, Lemma 9.11], then apply some character theory.

######  Theorem 6.15.

$\ker\Psi=C_{ba^{+}}(G,p)$.

###### Proof.

It suffices to show $C_{ba^{+}}(G,p)\supseteq\ker\Psi$. Suppose
$[C]\in\mathcal{T}\mathcal{E}_{k}^{\Omega}(G)$ and let $C$ be the unique
indecomposable representative of $[C]$. By Proposition 6.13, it suffices to
show that $h_{C}$ satisfies the oriented Artin-Borel-Smith conditions. For any
collection of subgroups $L\triangleleft K\triangleleft H\leq N_{G}(L)$ as
specified in the oriented Artin conditions, we will show $h_{C}(L)\equiv
h_{C}(K)\mod 2q^{r-l}$. Note that $r$ and $l$ are local conditions, only
depending on $K/L$ and $H/K$.

Let $D=\operatorname{Res}^{N_{G}(L)/L}_{H/L}C(L)$. Then

$h_{D}(K/L)=h_{C}(K)\text{ and }h_{D}(L/L)=h_{C}(L)$

by [17, Proposition 4.3]. Moreover, the values of $r$ and $l$ do not change.
Therefore, it suffices to show $h_{D}$ satisfies $h_{D}(L/L)\equiv
h_{D}(K/L)\mod 2q^{r-l}$. Notice that $H/L$ satisfies the conditions of [13,
Lemma 9.11], with $K/L$ as the subgroup $Q$ in the lemma. Therefore, by [13,
Lemma 9.11], if $h_{D}\in\ker\Psi$, the congruence holds, and we are done.
This occurs if and only if $\mathcal{H}_{D}(1)=[k]\in D^{\Omega}(H/K)$ by
Proposition 6.13. Note that

$\operatorname{Res}^{N_{G}(L)/L}_{H/L}\mathcal{H}_{C}(L)=\mathcal{H}_{D}(1)\in
D^{\Omega}(H/K).$

On the other hand,

$\operatorname{Res}^{N_{G}(L)/L}_{H/L}\mathcal{H}_{C}(L)=[k_{\varphi_{L}|_{H/L}}],$

where $\varphi_{L}$ is the character corresponding to $u$ from the
decomposition in [2, Theorem C] Thus, it suffices to show that for any $x\in
H$, $\varphi_{L}(xL)=1.$

Since $h_{C}\in\ker\Psi$, $\varphi_{1}$ is the trivial character. By [2,
Theorem C], we have that for any $x\in H$,

$1=\varphi_{1}(x)=\varphi_{\langle x_{p}\rangle}(x\langle
x_{p}\rangle)=\varphi_{\langle x_{p}\rangle}(x_{p^{\prime}}\langle
x_{p}\rangle).$

We claim that for every coset $xL$ of $H/L$, there exists a $y\in H$ for which
$\varphi_{L}(yL)=\varphi_{L}(xL)$ and $y_{p}$ is a generator of $L$, where
$y=y_{p}y_{p^{\prime}}$ is the unique decomposition of $y$ into $p$\- and
$p^{\prime}$-parts. Indeed, if $\langle x_{p}\rangle=L$, $y=x$ suffices. If
not, then since $K$ is a $p$-group, $x_{p}\in K$, and there exists some $k\in
K$ for which $x_{p}k$ is a generator of $L\leq K$. Let $y=xk$. Then by
construction, $\langle y_{p}\rangle=\langle x_{p}k\rangle=L$. Now, since
$\varphi_{L}$ is a linear Brauer character and $k$ is a $p$-element of $H$,

$\varphi_{L}(yL)=\varphi_{L}(xkL)=\varphi_{L}(x_{p^{\prime}}L)=\varphi_{L}(xL),$

as desired. Therefore, for every coset $xL\in H/L$, we have

$1=\varphi_{1}(y)=\varphi_{L}(yL)=\varphi_{L}(xL),$

thus $\varphi_{L}$ is the trivial character, and we are done.

∎

######  Corollary 6.16.

Let $f\in C_{b}(G,p)$. For $P\in[s_{p}(G)]$, set

$b_{P}:=\sum_{Q\in[s_{p}(G)]}f(Q)\mu(P,Q).$

The $V(\mathcal{F}_{G})$-endosplit-trivial chain complex

$C=\bigotimes_{Q\in[s_{p}(G)]}C_{Q}^{\otimes b_{Q}}$

has an endotrivial cap $C_{0}$ with $\mathcal{H}(C_{0})=k$ if and only if
$f\in C_{ba^{+}}(G,p)$.

###### Proof.

For the reverse direction, it follows from the proof of Theorem 2.17 that
$h_{C}=f$. From Proposition 6.13, it follows that
$C\in\mathcal{T}\mathcal{E}_{k}^{\Omega}(G)$, and therefore its cap $C_{0}$
has indecomposable homology isomorphic to $k$, as desired. For the forward
direction, if $f\not\in C_{ba^{+}}(G,p)$, then $[C]\not\in\ker\mathcal{H}$ by
the previous theorem and the isomorphism in Proposition 6.13, and the result
follows. ∎

It remains unclear what occurs if $f\in C_{b}(G,p)$ but $f\not\in
C_{ba^{+}}(G,p)$. One question of interest which remains is when $C$ as
constructed in the previous corollary or as in Theorem 2.17 has an endotrivial
cap which does not belong to $\ker\mathcal{H}$.

##  7\. Lifting virtual modules to invertible complexes for $p$-groups

In this section, we use results from [8] to prove that the Lefschetz
homomorphism,

$\Lambda:\mathcal{E}_{k}(G)\to
O(T(kG)),\,[C]\mapsto\Lambda(C):=\sum_{i\in\mathbb{Z}}(-1)^{i}[C_{i}],$

is surjective when $G$ is a $p$-group, giving a positive answer for a question
posed by the author in [17]. As a corollary, we show that every
$p$-permutation autoequivalence of a $p$-group (as defined by Boltje and
Perepelitsky in [4]) is the Lefschetz invariant of some splendid Rickard
autoequivalence of the group algebra.

###  7.1. Surjectivity of the Lefschetz homomorphism

######  Definition 7.1.

For any finite group $G$, the trivial source ring, $T(kG)$ is the Grothendieck
group of ${}_{kG}\mathbf{triv}$. It is a free $\mathbb{Z}$-module with basis
given by representatives of trivial source $kG$-modules, and every element
$X=T(kG)$ can be written $X=[M]-[N]$ for some $p$-permutation $kG$-modules
$M,N$. $T(kG)$ forms a commutative ring with multiplication induced by
$\otimes_{k}$.

$(-)^{*}$ induces an involution on $T(kG)$. The orthogonal unit group
$O(T(kG))$ is the subgroup of the unit group $T(kG)^{\times}$ consisting of
units $u$ for which $u^{-1}=u^{*}$.

If $G$ is a $p$-group, every permutation $kG$-module is indecomposable. In
this case, we have a canonical ring isomorphism

$B(G)\xrightarrow{\sim}T(kG),[X]\mapsto[kX].$

Moreover, $(-)^{*}$ is the identity map on $T(kG)$, therefore
$B(G)^{\times}\cong T(kG)^{\times}\cong O(T(kG))$.

######  Remark 7.2.

$B(G)^{\times}$ is famously difficult to describe for all finite groups. In
fact, by an argument of tom Dieck in [21], the statement “if $G$ has odd
order, $B(G)^{\times}\cong C_{2}$” is equivalent to the Feit-Thompson theorem.

However, in the case for which $G$ is a $p$-group, Yalçin in [24] gave a
complete generating set of $B(G)^{\times}$, which Bouc later refined to a
basis in [5] by realizing $B^{\times}$ as a rational $p$-biset functor. Note
the only interesting case is when $p=2$. The biset operations on $B^{\times}$
are described by Bouc’s generalized tensor induction. For details, we refer
the reader to [6, Section 11.2], but they will not be necessary for the scope
of this paper.

For any finite group $G$, $B(G)$ is be characterized similarly to how
endotrivial complexes are via h-marks. For a $G$-set $X$, its mark at $H$ is
the integer $|X^{H}|$. The $\mathbb{Z}$-linearization of this assignment
yields the mark homomorphism.

$m:B(G)\to B(G)^{*},X\mapsto(f_{X}:G/H\mapsto|X|^{H}).$

The mark homomorphism is injective and full-rank, so
$\mathbb{Q}\otimes_{\mathbb{Z}}m:\mathbb{Q}\otimes_{\mathbb{Z}}B(G)\to\mathbb{Q}\otimes_{\mathbb{Z}}B(G)^{*}$
is an isomorphism. However, it is rarely surjective.

Finally, it is easy to see the image of $B(G)^{\times}$ consists of functions
$f\in B(G)^{*}$ which take values only in $\pm 1$.

There are two key insights for the proof of this statement. For these next
statements, note $G$ is a $p$-group, so $B(G)^{*}\cong C(G.p)$.

######  Proposition 7.3.

Let $G$ be a $p$-group. The following diagram commutes, where $\phi$ is the
exponential map, $\phi(f)(K)=(-1)^{f(K)}$.

${\mathcal{E}_{k}(G)}$${C_{b}(G,p)}$${(B(G)^{*})^{\times}}$${O(T(kG))}$${B(G)^{\times}}$$\scriptstyle{h}$$\scriptstyle{\Lambda}$$\scriptstyle{\phi}$$\scriptstyle{\cong}$$\scriptstyle{m}$

###### Proof.

This is a reformulation of [17, Proposition 4.6]. ∎

######  Proposition 7.4.

Let $G$ be a $p$-group. Then $\phi(C_{b}(G,p))=m(B(G)^{\times})$.

###### Proof.

This is a reformulation of [8, Proposition 5.1], which uses a result of
Tornehave in [23]. ∎

As a result, we obtain a surjective homomorphism $\gamma:C_{b}(G)\to
B(G)^{\times}$ for which the following diagram commutes.

${\mathcal{E}_{k}(G)}$${C_{b}(G,p)}$${(B(G)^{*})^{\times}}$${O(T(kG))}$${B(G)^{\times}}$$\scriptstyle{h}$$\scriptstyle{\Lambda}$$\scriptstyle{\gamma}$$\scriptstyle{\phi}$$\scriptstyle{\cong}$$\scriptstyle{m}$

######  Theorem 7.5.

Let $G$ be a $p$-group. $\Lambda:\mathcal{E}_{k}(G)\to O(T(kG))$ is
surjective.

###### Proof.

This follows from Figure 7.1. Since $h$ is an isomorphism and $O(T(kG))\cong
B(G)^{\times}$, surjectivity of $\gamma:C_{b}(G,p)\to B(G)^{\times}$ implies
$\Lambda$ is surjective as well. ∎

###  7.2. Every $p$-permutation autoequivalence of a $p$-group lifts to a
splendid Rickard complex

Next, we use Theorem 7.5 to prove that for any every $p$-permutation
autoequivalence of a $p$-group $G$, $\gamma\in O(T^{\Delta}(kG,kG)$, there
exists a splendid Rickard equivalence $\Gamma\in
Ch^{b}({}_{kG}\mathbf{triv}_{kG})$ for which $\Lambda(\Gamma)=\gamma$.

We refer the reader to [4] or [9] for exposition on $p$-permutation
equivalences and splendid Rickard complexes.

######  Definition 7.6.

For any $\phi\in\operatorname{Aut}(G)$, define $\Delta_{\phi}(G)\leq G\times
G$ to be the subgroup defined by

$\Delta_{\phi}(G):=\\{(\phi(g),g)\in G\times G\mid g\in G\\}.$

Define $\Delta_{\phi}(G)^{op}\leq G\times G$ to be the subgroup defined by

$\Delta_{\phi}(G)^{op}:=\\{(g,\phi(g))\in G\times G\mid g\in G\\}.$

Note that $\Delta_{\phi}(G)^{op}\cong\Delta_{\phi^{-1}}(G)$. We have obvious
group homomorphisms

$G\cong\Delta_{\phi}(G),\,g\mapsto(\phi(g),g),$

and

$G\cong\Delta_{\phi}(G)^{op},\,g\mapsto(g,\phi(g)).$

In this way we identify $kG$-modules with $k\Delta_{\phi}(G)$-modules and
$k\Delta_{\phi}(G)^{op}$-modules.

######  Proposition 7.7.

Let $G$ be a $p$-group and let $\gamma\in O(T^{\Delta}(kG,kG))$ be a
$p$-permutation equivalence. Then there exists a orthogonal unit $u\in
O(T(k[\Delta_{\phi}G]))\cong O(T(kG))$ and group automorphism
$\phi:G\xrightarrow{\sim}G$ such that $\gamma=\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}G}(u)$.

###### Proof.

This follows from [3, Theorem 1.1(e)], however we provide a mostly self-
contained proof as well.

$\mathcal{B}\mathcal{P}(\gamma)$ has a maximal $\gamma$-Brauer pair
$\omega=(\Delta(G,\phi,G),e\otimes e^{*})$ by [4, Theorem 10.11], where
$\phi:G\to G$ is a group automorphism and $e$ is the unique block of $Z(G)$,
and all other maximal elements of $\mathcal{B}\mathcal{P}(\gamma)$ are
$G\times G$-conjugate to $\omega$, every other element of
$\mathcal{B}\mathcal{P}(\gamma)$ is $G\times G$-conjugate to some
$\omega^{\prime}$ satisfying $\omega^{\prime}\leq\omega$. Therefore, every
trivial source $(kG,kG)$-bimodule $M$ appearing in $\gamma$ has a maximal
$M$-Brauer pair $\omega^{\prime}\in\mathcal{B}\mathcal{P}(M)$ satisfying
$\omega^{\prime}\leq\omega$. In particular, there exists a subgroup $P\leq G$
for which $M$ has $\Delta_{\phi}(P)$ as a vertex. Therefore, we may write

$\gamma=\left(\sum_{i=1}^{a}[\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}P_{i}}k]\right)-\left(\sum_{i=1}^{b}[\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}Q_{i}}k]\right),$

for some $a,b\in\mathbb{N}$ and each $P_{i},Q_{j}$ a subgroup of $G$. Since
for any $P\leq G$,

$\operatorname{Ind}^{G\times G}_{\Delta_{\phi}P}=\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}G}\circ\operatorname{Ind}^{\Delta_{\phi}G}_{\Delta_{\phi}P},$

the above equality factors as

$\gamma=\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}G}\left(\sum_{i=1}^{a}[\operatorname{Ind}^{\Delta_{\phi}G}_{\Delta_{\phi}P_{i}}k]-\sum_{i=1}^{b}[\operatorname{Ind}^{\Delta_{\phi}G}_{\Delta_{\phi}Q_{i}}k]\right)=:\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}G}u,$

with $u\in T(k[\Delta_{\phi}G])$. It suffices to show $u\in
O(T(k[\Delta_{\phi}G]))$. From here, we identify $T(kG)$ with
$T(k[\Delta_{\phi}(G)])$ in the obvious way.

We have that

$\left(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)}u\right)\cdot_{kG}\left(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}G}u\right)^{*}=kG\in O(T^{\Delta}(kG,kG)).$

Since $u\in T(kG)$ and $G$ is a $p$-group, $u$ is a sum of transitive
indecomposable virtual permutation modules, hence $\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}G}u$ is a sum of transitive indecomposable virtual
permutation modules with twisted diagonal vertices. It is a straightforward
verification that for any $P\leq G$ and $\varphi\in\operatorname{Aut}(G)$,

$\left(\operatorname{Ind}^{G\times
G}_{\Delta_{\varphi}(P)}k\right)^{*}\cong\operatorname{Ind}^{G\times
G}_{\Delta_{\varphi}(P)^{op}}k,$

as $(kG,kG)$-bimodules. Therefore,

$\left(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)}u\right)^{*}=\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)^{op}}u\in T^{\Delta}(kG,kG).$

It follows from Bouc’s extended tensor product formula [7, Theorem 1.1] that
for any $H\leq G$ and $kH$-modules $M,N$ that

$\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(H)}M\otimes_{kG}\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(H)^{op}}N\cong\operatorname{Ind}^{G\times G}_{\Delta
H}(M\otimes_{k}N),$

where on the left, $M$ is regarded as a $k\Delta_{\phi}H$-module and $N$ is
regarded as a $k\Delta_{\phi}(H)^{op}$-module. Moreover, this isomorphism is
natural in both arguments. Therefore, we have the following chain of
equalities.

$kG=\left(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)}u\right)\cdot_{kG}\left(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)}u\right)^{*}=\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)}u\cdot_{kG}\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)^{op}}u\cong\operatorname{Ind}^{G\times G}_{\Delta
G}(u\cdot_{k}u)\in O(T^{\Delta}(kG,kG)).$

$\operatorname{Ind}^{G\times G}_{\Delta G}$ induces a split injective group
homomorphism $T(kG)\to T^{\Delta}(kG,kG)$ with retraction induced by taking
$1\times G$-fixed points, see [17, Lemma 4.14]. Therefore, we have

$k=(kG)^{1\times G}=\operatorname{Ind}^{G\times G}_{\Delta
G}(u\cdot_{k}u)^{1\times G}=u\cdot_{k}u\in T(kG).$

Since $u$ contains only permutation modules, as $G$ is a $p$-group, $u$ is
self-dual, and we conclude $u\in O(T(kG))$ as desired. ∎

Therefore, every $p$-permutation autoequivalence of a $p$-group is induced
twisted diagonally from an orthogonal unit. Given a $p$-permutation
equivalence $\gamma=\operatorname{Ind}^{G\times G}_{\Delta_{\phi}G}u$ with
$u\in O(T(kG))$, the obvious choice of corresponding splendid Rickard complex
$\Gamma$ which satisfies $\Lambda(\Gamma)=\gamma$ should be
$\Gamma=\operatorname{Ind}^{G\times G}_{\Delta_{\phi}G}C$, where $C$ is some
endotrivial complex for which $\Lambda(C)=u$. Indeed,

$\Lambda\left(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}G}C\right)=\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}G}u=\gamma\in O(T^{\Delta}(kG,kG)).$

######  Lemma 7.8.

Let $M$ be a $kG$-module. We have a natural isomorphism

$(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)}M)^{*}\cong(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)^{op}}M)^{*},$

as $(kG,kG)$-bimodules, where the dual induced on the left-hand side arises
from the left $kG$-module $k$-dual, and the dual on the right-hand side arises
from the bimodule $k$-dual.

###### Proof.

We first note the bimodule structure on the left- and right-hand sides of the
proposed isomorphism.

On the left, for $a,b\in G$, $(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)}M)^{*}$, has actions defined by:

$\displaystyle a\cdot f\big{(}(g_{1},g_{2})\otimes m\big{)}\cdot b$
$\displaystyle=(a,b^{-1})\cdot f\big{(}(g_{1},g_{2})\otimes m\big{)}$
$\displaystyle=f\big{(}(a^{-1},b)(g_{1},g_{2})\otimes m\big{)}$
$\displaystyle=f\big{(}(a^{-1}g_{1},bg_{2})\otimes m\big{)},$

and on the right, $(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)^{op}}M)^{*}$ corresponds to the bottom left composite,
that is, it has actions defined by:

$\displaystyle a\cdot f\big{(}(g_{1},g_{2})\otimes m\big{)}\cdot b$
$\displaystyle=f\big{(}b\cdot\big{(}(g_{1},g_{2})\otimes m\big{)}\cdot
a\big{)}$ $\displaystyle=f\big{(}(bg_{1},a^{-1}g_{2})\otimes m\big{)}.$

We define an isomorphism $\psi:(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)}M)^{*}\cong(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)^{op}}M)^{*}$ as follows:

$\displaystyle\psi:(\operatorname{Ind}^{G\times G}_{\Delta_{\phi}(G)}M)^{*}$
$\displaystyle\to(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)^{op}}M)^{*}$ $\displaystyle f$
$\displaystyle\mapsto\big{(}(g_{1},g_{2})\otimes m\mapsto
f((g_{2},g_{1})\otimes m)\big{)}$

Here, $(g_{2},g_{1})\otimes m$ as above is considered an element in
$\operatorname{Ind}^{G\times G}_{\Delta_{\phi}(G)}M)^{*}$. We first check
well-definedness of the mapping with respect to the tensor product. Let
$f\in(\operatorname{Ind}^{G\times G}_{\Delta_{\phi}(G)}M)^{*}$ and
$(g_{1},g_{2})\otimes m\in\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)^{op}}M$. We have for any $g\in G$ that

$(g_{1},g_{2})\otimes m=(g_{1}g^{-1},g_{2}\phi(g^{-1}))\otimes gm.$

Then for any $f\in(\operatorname{Ind}^{G\times G}_{\Delta_{\phi}(G)}M)^{*}$,

$\displaystyle\psi(f)((g_{1},g_{2})\otimes m)$
$\displaystyle=\psi(f)((g_{1}g^{-1},g_{2}\phi(g^{-1}))\otimes gm)$
$\displaystyle=f((g_{2}\phi(g^{-1}),(g_{1}g^{-1})\otimes gm)$
$\displaystyle=f((g_{2},g_{1})\otimes m)$
$\displaystyle=\psi(f)((g_{1},g_{2})\otimes m)$

Thus the map is well-defined. We next check that it is a $(kG,kG)$-bimodule
homomorphism. Let $a,b\in G$.

$\displaystyle\psi(a\cdot f\cdot b)((g_{1},g_{2})\otimes m)$
$\displaystyle=\psi(f((a^{-1},b)\cdot-))((g_{1},g_{2})\otimes m)$
$\displaystyle=f((a^{-1}g_{2},bg_{1})\otimes m)$
$\displaystyle=\psi(f)((bg_{1},a^{-1}g_{2})\otimes m)$
$\displaystyle=(a\cdot\psi(f)\cdot b)((g_{1},g_{2})\otimes m)$

Therefore, $\psi$ is a $(kG,kG)$-bimodule homomorphism. It is apparent $\psi$
is an isomorphism, as its inverse is constructed similarly. Finally we must
check naturality. This follows from commutativity of the following diagram,
where $f:M\to N$ is a $kG$-module homomorphism.

${(k[G\times G]\otimes_{k\Delta_{\phi}(G)}M)^{*}}$${(k[G\times
G]\otimes_{k\Delta_{\phi}(G)^{op}}M)^{*}}$${(k[G\times
G]\otimes_{k\Delta_{\phi}(G)}N)^{*}}$${(k[G\times
G]\otimes_{k\Delta_{\phi}(G)^{op}}N)^{*}}$$\scriptstyle{\psi_{M}}$$\scriptstyle{\psi_{N}}$$\scriptstyle{(\operatorname{id}\otimes
f)^{*}}$$\scriptstyle{(\operatorname{id}\otimes f)^{*}}$

∎

######  Theorem 7.9.

Let $G$ be a $p$-group and let $\gamma\in O(T^{\Delta}(kG,kG))$ be a
$p$-permutation autoequivalence of $kG$. There exists a splendid Rickard
complex $\Gamma$ satisfying $\Lambda(\Gamma)=\gamma.$

###### Proof.

By Proposition 7.7, we may write $\gamma=\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)}u$ for some $\phi\in\operatorname{Aut}(G)$ and $u\in
O(T(kG))\cong O(T(k[\Delta_{\phi}(G)]))$, After identifying $O(T(kG))$ with
$B(G)^{\times}$, by 7.5, there exists an endotrivial complex $C$ satisfying
$\Lambda(C)=u$. Moreover, by additivity of induction,

$\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)}C=\gamma=\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)}u.$

It suffices to show $\operatorname{Ind}^{G\times G}_{\Delta_{\phi}(G)}C$ is a
splendid Rickard autoequivalence of $kG$. We have:

$\displaystyle(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)}C)\otimes_{kG}(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)}C)^{*}$ $\displaystyle\cong(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)}C)\otimes_{kG}(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)^{op}}C)^{*}$ The above isomorphism is the identification
in Lemma 7.8. $\displaystyle\cong(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)}C)\otimes_{kG}(\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)^{op}}C^{*})$ This follows from the standard natural
$kG$-module isomorphism
$\operatorname{Ind}^{G}_{H}M^{*}\cong(\operatorname{Ind}^{G}_{H}M)^{*}$.
$\displaystyle\cong\operatorname{Ind}^{G\times G}_{\Delta
G}(C\otimes_{k}C^{*})$ This follows from the isomorphism used in the proof of
Proposition 7.7. $\displaystyle\cong kG$

Thus by [20, Theorem 2.1], $\operatorname{Ind}^{G\times
G}_{\Delta_{\phi}(G)}C$ is a splendid Rickard complex. ∎

######  Corollary 7.10.

Let $P$ be a $p$-group, $G$ an arbitrary finite group, and suppose $B$ is a
block of $kG$ which is $p$-permutation equivalent to $kP$. Then either all
$p$-permutation equivalences for $kP$ and $B$ lift to splendid equivalences
between $kP$ and $B$, or none do.

###### Proof.

Suppose there exists at least one $p$-permutation equivalence $\gamma\in
O(T^{\Delta}(B,kP))$ for which there exists a splendid Rickard complex
$\Gamma$ for $B$ and $kP$ satisfying $\Lambda(\Gamma)=\gamma$. Then given any
other $\gamma^{\prime}\in O(T^{\Delta}(B,kP))$,

$\gamma^{*}\cdot_{B}\gamma^{\prime}\in O(T^{\Delta}(kP,kP)).$

By Theorem 7.9, there exists a splendid autoequivalence $X$ of $kP$ such that

$\Lambda(X)=\gamma^{*}\cdot_{B}\gamma^{\prime}.$

Therefore,

$\Lambda(\Gamma\otimes_{kB}X)=\Lambda(\Gamma)\cdot_{B}\Lambda(X)=\gamma\cdot_{kP}\gamma^{*}\cdot_{B}\cdot\gamma^{\prime}=[B]\cdot_{B}\gamma^{\prime}=\gamma^{\prime}.$

Thus $\gamma^{\prime}$ lifts to a splendid Rickard complex as well, and the
result follows. ∎

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