# Relatively endotrivial complexes

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

Let $G$ be a finite group and $k$ a field of characteristic $p>0$. We use the
notion of projectivity relative to a $kG$-module to define multiple notions of
relatively endotrivial chain complexes, extending the definition of an
endotrivial chain complex. These definitions are motivated by Lassueur’s
analogous construction of relatively endotrivial $kG$-modules. Similar to the
study of endotrivial modules, we obtain equivalent characterizations of
relative endotriviality and corresponding h-mark homomorphisms which determine
the isomorphism class of a relatively endotrivial complex up to a torsion
element. Additionally, we study the homomorphism induced by restriction of
relatively endotrivial complexes to subgroups containing Sylow $p$-subgroups
$S$ of $G$. One result of this investigation is that the group
$\mathcal{E}_{k}(G)$ of endotrivial complexes is isomorphic to a direct
product of its torsion subgroup $\operatorname{Hom}(G,k^{\times})$ and a
fusion-stable subgroup of $\mathcal{E}_{k}(S)$.

###### Key words and phrases:

Endotrivial complex, relative projectivity, endotrivial module, restriction

###### 2010 Mathematics Subject Classification:

20J05, 20C05, 20C20

##  1\. Introduction

In her doctoral dissertation [10], Lassueur introduced the notion of
relatively endotrivial $kG$-modules. This built on the notion of relative
projectivity with respect to modules first introduced by Okuyama in [19] and
later studied by Carlson and coauthors in [5] and others. An important
consequence of Lassueur’s work was a new generalization of the Dade group of a
$p$-group to all finite groups. This was done by considering
endo-$p$-permutation modules, a generalization of endopermutation modules
first studied by Urfer in [23], that are relatively endotrivial with respect
to a certain $kG$-module. This construction coincided with a previous
generalization given by Linckelmann and Mazza in [14], who instead considered
the Dade group of a fusion system.

In this paper, we apply the ideas present in Laussuer’s work to the study of
endotrivial complexes, which were introduced by the author in [17]. Similarly
to how endotrivial modules are the invertible objects of the stable module
category, ${}_{kG}\underline{\mathbf{mod}}$, endotrivial complexes are the
invertible objects of the homotopy category $K^{b}({}_{kG}\mathbf{triv})$ of
$p$-permutation modules . Furthermore, endotrivial complexes provide examples
of splendid Rickard complexes, derived equivalences which induce “local”
derived equivalences. Broué’s abelian defect group conjecture predicts the
existence of such equivalences between a block of a group and its Brauer
correspondent.

The goal of this paper is to develop broadly the analogous theory for
“relatively” endotrivial complexes. In Lassueur’s work, there was one clear
definition choice for a relatively endotrivial module, but in our case, there
is a bit more ambiguity. What the notion of a “relatively endotrivial chain
complex” should be has multiple reasonable definitions, so we provide multiple
definitions of increasing fineness. The following is Definition 6.1.

######  Definition 1.1.

Let $C\in Ch^{b}({}_{kG}\mathbf{triv})$ and let $V$ be a $kG$-module.

  1. (a)

$C$ is weakly $V$-endotrivial if $\operatorname{End}_{k}(C)\cong
C^{*}\otimes_{k}C\simeq k[0]\oplus D$, where $D$ is a bounded chain complex of
$V$-projective $kG$-modules.

  2. (b)

$C$ is strongly $V$-endotrivial if $\operatorname{End}_{k}(C)\cong
C^{*}\otimes_{k}C\simeq k[0]\oplus D$, where $D$ is a bounded $V$-projective
$kG$-chain complex.

  3. (c)

$C$ is $V$-endosplit-trivial if $\operatorname{End}_{k}(C)\cong
C^{*}\otimes_{k}C\simeq k[0]\oplus M[0]$, where $M$ is a $V$-projective
$kG$-module.

Here, $\simeq$ denotes homotopy equivalence.

Note that we allow $V=\\{0\\}$, in which case, we recover the definition of an
endotrivial complex. Weakly and strongly $V$-endotrivial complexes are the
invertible objects of an appropriate stable homotopy categories of chain
complexes, while $V$-endosplit-trivial complexes are examples of endosplit
$p$-permutation resolutions, as introduced by Rickard in [21]. Note that we
require $C$ to consist of $p$-permutation modules. As a consequence, analogous
to how endotrivial complexes induce splendid Rickard complexes, when $V$ is
projective, weakly $V$-endotrivial complexes induce splendid stable
equivalences.

Many of the techniques used to study endotrivial complexes in [17] may be
performed in the relative setting as well. In [17], we provide a “local”
equivalent condition which determines when a chain complex $C\in
Ch^{b}({}_{kG}\mathbf{triv})$ is endotrivial. Similar characterizations are
provided here in the case of weak $V$-endotriviality and $V$-endosplit
triviality. Moreover, in these cases, we show that one may assume $V$ is a
$p$-permutation $kG$-module. As a result, the only types of relative
endotriviality arise from projectivity relative to a family of subgroups. We
say that a $kG$-module $V$ is absolutely $p$-divisible if all of its
indecomposable summands have $k$-dimension divisible by $p$. Then, the
detection theorems are as follows; these are Theorem 7.2 and Theorem 8.3.

######  Theorem 1.2.

Let $C\in Ch^{b}({}_{kG}\mathbf{triv})$, let $V$ be a $p$-permutation
$kG$-module which is absolutely $p$-divisible, and let $\mathcal{X}_{V}$ be
the set of $p$-subgroups $P$ of $G$ for which $V(P)=0$.

  1. (a)

$C$ is weakly $V$-endotrivial if and only if for all $P\in\mathcal{X}_{V}$,
$C(P)$ has nonzero homology concentrated in exactly one degree, with the
nontrivial homology having $k$-dimension one.

  2. (b)

The following are equivalent.

    1. (a)

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

    2. (b)

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

    3. (c)

$C(P)$ has nonzero homology concentrated in exactly one degree for all $P\in
s_{p}(G)$, and for the unique $i\in\mathbb{Z}$ satisfying $H_{i}(C)\neq 0$,
$H_{i}(C)$ is a $V$-endotrivial $kG$-module.

When studying endotrivial modules or chain complexes, one defines a
corresponding group parameterizing the objects with group law induced by
$\otimes_{k}$. The same applies here, though the groups of weakly
$V$-endotrivial complexes identify more complexes as equivalent than do the
groups of strongly $V$-endotrivial complexes or $V$-endosplit-trivial
complexes. One of the first landmark results in the study of endotrivial
modules, due to Puig in [20], asserts that when $G$ is a $p$-group. the
corresponding group, $T_{k}(G)$, is finitely generated abelian. We show the
same here, showing that the group of weakly $V$-endotrivial complexes and the
group of $V$-endosplit-trivial complexes are both finitely generated abelian.
This is shown using similar techniques as in [17], with a few more technical
details.

One question of interest in the study of (relatively) endotrivial modules is
the image of the homomorphism induced by restriction to a Sylow $p$-subgroup
or its normalizer. We study this question in detail here. One crucial theorem
we prove on the way is a Mackey formula describing how the Brauer construction
and additive induction commute. As a result, we are able to use the Green
correspondence for chain complexes which are “fusion stable” in the case of
restriction to normalizers. One particular result allows us to describe the
group $\mathcal{E}_{k}(G)$ of endotrivial complexes as a direct product of a
“fusion-stable” subgroup
$\mathcal{E}_{k}(S)^{\mathcal{F}}\leq\mathcal{E}_{k}(S)$, where $S$ is a Sylow
$p$-subgroup of $G$, and the torsion subgroup of $\mathcal{E}_{k}(G)$, which
is canonically isomorphic to $\operatorname{Hom}(G,k^{\times})$. These results
are from Theorem 11.7 and Corollary 11.12.

######  Theorem 1.3.

  1. (a)

Let $S\in\operatorname{Syl}_{p}(G)$. Then the group homomorphism
$\operatorname{Res}^{G}_{S}:\mathcal{E}_{k}(G)\to\mathcal{E}_{k}(S)^{\mathcal{F}}$
is surjective. Moreover, we have a split exact sequence of abelian groups

$0\to\operatorname{Hom}(G.k^{\times})\to\mathcal{E}_{k}(G)\xrightarrow{\operatorname{Res}^{G}_{S}}\mathcal{E}_{k}(S)^{\mathcal{F}}\to
0.$

  2. (b)

Let $N_{G}(S)\leq H\leq G$.
$\operatorname{Res}^{G}_{H}\mathcal{E}_{k}(G)\to\mathcal{E}_{k}(H)$ is
injective. It is an isomorphism if and only if
$\operatorname{Res}^{G}_{H}:\operatorname{Hom}(G,k^{\times})\to\operatorname{Hom}(H,k^{\times})$
is an isomorphism, or equivalently, if the Green correspondent of any
$k$-dimension one $kH$-module has $k$-dimension one.

Finally, we study the case when $V$ is projective, and consider questions
relating the weakly $V$-endotrivial, strongly $V$-endotrivial, and
$V$-endosplit-trivial complexes. We derive a number of results, including that
every strongly $V$-endotrivial complex is essentially $V$-endosplit trivial,
and that every weakly $V$-endotrivial complex may be “completed” to a
$V$-endosplit-trivial complex.

The paper is organized as follows. Sections 2 through 5 serve primarily as
preliminary information or as straightforward extensions of previous results.
Notably, most of the statements about relative projectivity with respect to
modules for modules may be easily extended for chain complexes as well. In
Section 2, we review relative projectivity for chain complexes and provide a
statement of the Green correspondence for chain complexes, which was
introduced in [6]. Section 3 mostly follows [9] in establishing properties of
relative projectivity with respect to modules, extending most statements to
relatively projective chain complexes as well. Section 4 introduces a few
relative stable categories and describes how they behave in some detail.
Section 5 recalls a few necessary properties about relatively endotrivial
modules.

Sections 6 and onwards contain most of the main results of the paper. Section
6 introduces the various definitions of endotrivial complexes, their
corresponding groups, and some basic properties. Section 7 is dedicated to
proving the equivalent local characterization of weakly $V$-endotrivial
complexes, and Section 8 is dedicated to proving the equivalent local
characterization for $V$-endosplit-trivial complexes. Section 9 introduces
h-mark homomorphisms, analogous to the h-mark homomorphism given in [17], and
proves that the group of weakly $V$-endotrivial complexes and the group of
$V$-endosplit-trivial complexes are finitely generated. Section 10 analyzes
the case of when $V$ is projective, and Section 11 studies restriction to
subgroups containing Sylow $p$-subgroups.

Notation: $G$ is always a finite group, $p$ a prime, and $k$ a field of
characteristic $p$ (not necessarily large enough for $G$). All $kG$-modules
are assumed left and finitely generated. We write ${}_{kG}\mathbf{mod}$ to
denote the category of finitely generated $kG$-modules, and
${}_{kG}\mathbf{triv}$ to denote the full subcategory consisting of
$p$-permutation modules. Given a preadditive category $\mathcal{A}$,
$Ch^{b}(\mathcal{A})$ denotes the bounded category of $\mathcal{A}$-chain
complexes and $K^{b}(\mathcal{A})$ denotes the bounded homotopy category of
$\mathcal{A}$. ${}_{kG}\underline{\mathbf{mod}}$ denotes the stable module
category of ${}_{kG}\mathbf{mod}.$ If $M,N\in\mathcal{A}$, we write $M\mid N$
to denote that $M$ is isomorphic to a direct summand of $N$. $\otimes_{k}$
denotes the tensor product of $kG$-modules or chain complexes of $kG$-modules
with diagonal $G$-action. $(-)^{*}$ denotes the $k$-dual
$\operatorname{Hom}_{k}(-,k)$ of $kG$-modules (resp. chain complexes of
$kG$-modules) which is again a $kG$-module (resp. chain complex of
$kG$-modules).

$s_{p}(G)$ denotes the set of $p$-subgroups of $G$ and
$\operatorname{Syl}_{p}(G)$ denotes the set of Sylow $p$-subgroups of $G$. If
$M\in{}_{kG}\mod$, $M[i]$ denotes the chain complex with $M$ in degree $i$ and
$0$ elsewhere. The Brauer construction at $P$ is written as a functor
$(-)(P)$. Given a group homomorphism
$\omega\in\operatorname{Hom}(G,k^{\times})$, $k_{\omega}$ denotes the
corresponding $kG$-module with $k$-dimension one, with $G$-action given by
$g\cdot m=\omega(g)m$.

##  2\. Projectivity relative to subgroups for chain complexes

We begin by reviewing the notions of relative projectivity, vertices, sources,
and the Green correspondence for chain complexes of $kG$-modules, building up
to a chain-complex theoretic Green correspondence introduced by Carlson, Wang,
and Zhang in [6]. The structural results here mirror those for $kG$-modules.
We assume familiarity with the module-theoretic versions of these concepts.

######  Definition 2.1.

A chain complex $C$ of $kG$-modules is $H$-projective if there exists some
$kH$-complex $D$ for which $C$ is isomorphic to a direct summand of
$\operatorname{Ind}^{G}_{H}D$. If $\mathcal{X}$ is a collection of subgroups
of $G$, $C$ is $\mathcal{X}$-projective if, for any indecomposable direct
summand $D$ of $C$, there exists an $H\in\mathcal{X}$ for which $D$ is
$H$-projective.

Given a set of subgroups $\mathcal{X}$ of $G$, we set
$Ch^{b}({}_{kG}\mathbf{mod},\mathcal{X})$ to be the category of
$\mathcal{X}$-projective $kG$-chain complexes. If $\mathcal{X}=\\{H\\}$, we
write $Ch^{b}({}_{kG}\mathbf{mod},H)$.

######  Remark 2.2.

  1. (a)

Recall that the categories $Ch^{b}({}_{kG}\mathbf{mod})$ and
$K^{b}({}_{kG}\mathbf{mod})$ have the Krull-Schmidt property.

  2. (b)

There is a subtle difference between $\mathcal{X}$-projectivity for chain
complexes and $\mathcal{X}$-projectivity for modules. In the module-theoretic
case, a $kG$-module is projective if and only if it is $1$-projective and
projective as $k$-module. The analogous statement does not hold for chain
complexes. For instance, the projective objects in the chain complex category
$Ch^{b}({}_{kG}\mathbf{mod})$ are split acyclic chain complexes of projective
$kG$-modules. In particular, they are additively generated by contractible
chain complexes of the form $0\to P\xrightarrow{\sim}P\to 0$ for projective
$kG$-modules $P$. On the other hand, the singleton chain complex $kG[0]$ is a
$1$-projective chain complex, but is not a projective object since it is not
acyclic.

  3. (c)

An isomorphism of modules which is natural in all components (that is, it is
induced by a natural isomorphism of functors in the appropriate module
categories) can be extended to a natural isomorphism of chain complexes.
Additionally, a natural isomorphism of bifunctors extends to a natural
isomorphism of bicomplexes, and so on. In particular, natural isomorphisms
involving internal homs and tensor products may be extended to their chain
complex counterparts, as these constructions arise from taking total complexes
of bicomplexes. For instance, the Mackey formula is a natural isomorphism, as
are all standard adjunctions involving induction and restriction.

The standard equivalent conditions for $H$-projectivity for chain complexes
are the same as the module-theoretic version.

######  Theorem 2.3.

Let $H\leq G$ and $C\in Ch({}_{kG}\mathbf{mod})$. The following are
equivalent:

  1. (a)

$C$ is $H$-projective.

  2. (b)

(Higman’s criterion) There exists $\theta\in\operatorname{End}_{kH}(C)$ such
that $\operatorname{tr}^{G}_{H}(\theta)=\operatorname{id}_{C}$.

  3. (c)

$\operatorname{End}_{kG}(C)=\operatorname{tr}^{G}_{H}(\operatorname{End}_{kH}(C))$.

  4. (d)

For every diagram ${C}$${D}$${E}$$\scriptstyle{f}$$\scriptstyle{\mu}$ with
$\mu$ surjective, the following holds. If there exists
$\lambda\in\operatorname{Hom}_{kH}(C,D)$ such that $f=\mu\circ\lambda$, then
there exists $\lambda^{\prime}\in\operatorname{Hom}_{kG}(C,D)$ such that
$f=\mu\circ\lambda^{\prime}$.

  5. (e)

If $\mu:D\to C$ is surjective and and splits in $Ch({}_{kH}\mathbf{mod})$,
then it splits in $Ch({}_{kG}\mathbf{mod}).$

  6. (f)

The surjective $kG$-complex homomorphism
$\mu:\operatorname{Ind}^{G}_{H}\operatorname{Res}^{G}_{H}C\to C$ induced by
$x\otimes m\mapsto xm$ splits.

  7. (g)

$C\mid\operatorname{Ind}^{G}_{H}\operatorname{Res}^{G}_{H}C$.

###### Proof.

This is [18, Chapter 4, Section 2, Theorem 2.2], and one may verify the proof
adapts to chain complexes. ∎

######  Definition 2.4.

An indecomposable chain complex $C$ has vertex $P$ if $P$ is minimal with
respect to the property that $C$ is $P$-projective. If $D$ is an
indecomposable complex for which $C\mid\operatorname{Ind}^{G}_{P}D$, say
$(P,D)$ is a vertex-source pair for $C$.

As in the module case, we see vertices are always conjugacy classes of
$p$-groups.

######  Proposition 2.5.

Let $C$ be a chain complex of $kG$-modules. The set of vertices of $C$ forms a
single $G$-conjugacy class of $p$-subgroups.

###### Proof.

This is [18, Chapter 4, Section 3, Theorem 3.3], and the proof adapts to chain
complexes. ∎

######  Theorem 2.6.

(Classical Green correspondence for chain complexes) Let $S$ be a Sylow
$p$-subgroup of $G$, and let $H$ be a subgroup of $G$ with $N_{G}(S)\leq H\leq
G$.

  1. (a)

Given a chain complex of $kG$-modules $C$ with vertex $S$,
$\operatorname{Res}^{G}_{H}C$ has a unique direct summand $f(C)$ with vertex
$S$. All other summands of $\operatorname{Res}^{G}_{H}C$ have vertices
contained strictly within $S$ up to conjugacy.

  2. (b)

Given a chain complex of $kH$-modules $D$ with vertex $S$,
$\operatorname{Ind}^{G}_{H}D$ has a unique direct summand $g(D)$ with vertex
$S$. All other summands of $\operatorname{Ind}^{G}_{H}D$ have vertices
contained strictly within $S$ up to conjugacy.

  3. (c)

$f$ and $g$ are inverses.

###### Proof.

This follows from the more general Green correspondence for chain complexes,
[6, Theorem 8.1], by choosing $H$ as before and $\mathfrak{B}=\\{S\\}$, again
similar to the module-theoretic case. ∎

##  3\. Relative projectivity for modules and chain complexes with respect to
modules

We assume familiarity with $p$-permutation modules, trivial source modules
(indecomposable $p$-permutation modules), and the Brauer construction. We
refer the reader to [12] or [17, Section 2] for review on those topics.

Notation: For this section, let ${}_{kG}\mathcal{C}$ denote either the
category ${}_{kG}\mathbf{mod}$ or the category $Ch^{b}({}_{kG}\mathbf{mod})$.
Both are additive, $k$-linear, Krull-Schmidt, symmetric monoidal categories
with induction, restriction, and inflation functors, as well as the Brauer
construction. If $M\in Ch^{b}({}_{kG}\mathbf{mod})$ and
$V\in{}_{kG}\mathbf{mod}$, $M\otimes_{k}V$ denotes the tensor product of chain
complexes $M\otimes_{k}V[0]$.

Some of the following section is adapted from [9, Section 2], and we extend
the previous results to the chain complex category with minimal work. Here, we
review a generalized notion of relative projectivity, first introduced by
Okuyama in [19] and later studied by Carlson and Peng in [5].

######  Definition 3.1.

Let $V$ be a $kG$-module.

  1. (a)

$M\in{}_{kG}\mathcal{C}$ is $V$-projective or projective relative to $V$ if
there exists an object $N\in{}_{kG}\mathcal{C}$ such that $M$ is isomorphic to
a direct summand of $V\otimes_{k}N$.

  2. (b)

A short exact sequence in ${}_{kG}\mathcal{C}$ $0\to
A\xrightarrow{\alpha}B\xrightarrow{\beta}C\to 0$ is $V$-split if the tensored
short exact sequence $0\to
V\otimes_{k}A\xrightarrow{\operatorname{id}\otimes\alpha}V\otimes_{k}B\xrightarrow{\operatorname{id}\otimes\beta}V\otimes_{k}C\to
0$ splits.

######  Remark 3.2.

The notion of a $V$-injective $kG$-module can be defined dually, and since
$kG$ is a symmetric algebra, the classes of $V$-projectives and $V$-injectives
coincide.

######  Definition 3.3.

We write ${}_{kG}\mathbf{mod}(V)$ for the subcategory of all $V$-projective
modules in ${}_{kG}\mathbf{mod}$. Similarly, write
$Ch^{b}({}_{kG}\mathbf{mod},V)$ for the subcategory of all $V$-projective
chain complexes in $Ch^{b}({}_{kG}\mathbf{mod})$. In generality, we write
${}_{kG}\mathcal{C}(V)$ to denote the corresponding category of $V$-projective
objects in ${}_{kG}\mathcal{C}$.

We say a module $U$ is a generator for ${}_{kG}\mathcal{C}(V)$ if and only if
${}_{kG}\mathcal{C}(U)={}_{kG}\mathcal{C}(V)$.

Similarly, if $H\leq G$, we write ${}_{kG}\mathcal{C}(H)$ for the subcategory
of ${}_{kG}\mathcal{C}$ with objects consisting of $H$-projective objects, and
if $\mathcal{X}$ is a set of subgroups of $G$, write
${}_{kG}\mathcal{C}(\mathcal{X})$ for the subcategory of ${}_{kG}\mathcal{C}$
with objects consisting of $\mathcal{X}$-projective objects.

The following list of omnibus properties will be used extensively in the
sequel.

######  Proposition 3.4.

Let $U,V$ be $kG$-modules and $M,N\in{}_{kG}\mathcal{C}$. Write
${}_{kG}\mathcal{C}(V)$ to denote the corresponding category of $V$-projective
objects in ${}_{kG}\mathcal{C}$.

  1. (a)

Any direct summand of a $V$-projective object is $V$-projective.

  2. (b)

If $U\in{}_{kG}\mathbf{mod}(V)$ (resp. $U[0]\in
Ch^{b}({}_{kG}\mathbf{mod},V)$), then
${}_{kG}\mathcal{C}(U)\subseteq{}_{kG}\mathcal{C}(V)$.

  3. (c)

If $p\nmid\dim_{k}(V)$, then ${}_{kG}\mathcal{C}(V)={}_{kG}\mathcal{C}$.

  4. (d)

${}_{kG}\mathcal{C}(V)={}_{kG}\mathcal{C}(V^{*})={}_{kG}\mathcal{C}(V^{*}\otimes_{k}V)={}_{kG}\mathcal{C}(\Omega^{n}(V))$
for $n\in\mathbb{Z}$.

  5. (e)

${}_{kG}\mathcal{C}(U\oplus
V)\cong{}_{kG}\mathcal{C}(U)\oplus{}_{kG}\mathcal{C}(V)$.

  6. (f)

${}_{kG}\mathcal{C}(U)\cap{}_{kG}\mathcal{C}(V)={}_{kG}\mathcal{C}(U\otimes_{k}V)\supseteq{}_{kG}\mathcal{C}(U)\otimes_{k}{}_{kG}\mathcal{C}(V)$.

  7. (g)

${}_{kG}\mathcal{C}(\bigoplus_{j=1}^{n}V)={}_{kG}\mathcal{C}(V)={}_{kG}\mathcal{C}(\bigotimes_{j=1}^{m}V)$
for all $m,n\in\mathbb{N}_{+}$.

  8. (h)

$M\oplus N$ is $V$-projective if and only if $M$ and $N$ are $V$-projective.

  9. (i)

Let $g\in\tilde{G}\trianglerighteq G$, then
${}^{g}({}_{kG}\mathcal{C}(V))={}_{k{}^{g}G}\mathcal{C}({}^{g}V)$.

  10. (j)

${}_{kG}\mathcal{C}(kG)\subseteq{}_{kG}\mathbf{mod}(V)$ for all $kG$-modules
$V$. Moreover, ${}_{kG}\mathcal{C}(kG)={}_{kG}\mathcal{C}(P)$ for any non-zero
projective $kG$-modules $P$.

  11. (k)

$M\in{}_{kG}\mathcal{C}(V)$ if and only if
$\operatorname{End}_{k}(M)\in{}_{kG}\mathcal{C}(V)$.

###### Proof.

All proofs can be found in [10, Proposition 2.2.2] for modules and it is
easily verified that all proofs extend to $Ch^{b}({}_{kG}\mathbf{mod})$. ∎

In general,
${}_{kG}\mathcal{C}(U)\otimes_{k}{}_{kG}\mathcal{C}(V)\not\cong{}_{kG}\mathcal{C}(U\otimes_{k}V)$
\- see [9, Remark 2.1] for a counterexample.

######  Remark 3.5.

The notion of $V$-projectivity encompasses projectivity relative to a subgroup
as well as the notion of projectivity relative to a family of subgroups. Using
Frobenius reciprocity, it is easy to show that projectivity relative to $H$ is
equivalent to projectivity relative to the $kG$-module
$k[G/H]=\operatorname{Ind}^{G}_{H}k$, and that a short exact sequence is
$H$-split if and only if it is $k[G/H]$-split. Similarly, projectivity
relative to $\mathcal{X}$ is equivalent to projectivity relative to
$\bigoplus_{H\in\mathcal{X}}k[G/H]$ and that a short exact sequence is
$\mathcal{X}$-split if and only if it is
$\bigoplus_{H\in\mathcal{X}}k[G/H]$-split if and only if it splits upon
restriction to every $H\in\mathcal{X}$.

$V$-projectivity comes equipped with corresponding equivalent definitions.

######  Theorem 3.6.

Let $V$ be a $kG$-module and $M\in{}_{kG}\mathcal{C}$. The following are
equivalent.

  1. (a)

$M$ is $V$-projective.

  2. (b)

$M\mid V^{*}\otimes_{k}V\otimes_{k}M$.

  3. (c)

$M$ is $\operatorname{End}_{k}(V)$-projective.

  4. (d)

$\operatorname{End}_{k}(M)$ is $V$-projective.

  5. (e)

For every $V$-split surjective ${}_{kG}\mathcal{C}$-morphism $\beta:B\to C$
and ${}_{kG}\mathcal{C}$-morphism $\theta:M\to C$, there exists a
${}_{kG}\mathcal{C}$-morphism $\mu:M\to B$ such that $\theta=\beta\circ\mu$.

###### Proof.

The equivalences between (a),(b), and (d) are presented in [10, Proposition
2.4.1] for modules and they extend to $Ch^{b}({}_{kG}\mathbf{mod})$. The rest
follow from Proposition 3.4. ∎

######  Proposition 3.7.

Let $H\leq G$, $P\in s_{p}(G)$, and $N\trianglelefteq G$.

  1. (a)

$\operatorname{Res}^{G}_{H}{}_{kG}\mathcal{C}(V)\subseteq{}_{kH}\mathcal{C}(\operatorname{Res}^{G}_{H}V)$.

  2. (b)

$\operatorname{Ind}^{G}_{H}{}_{kH}\mathcal{C}(V)\subseteq{}_{kG}\mathcal{C}(\operatorname{Ind}^{G}_{H}V)$.

  3. (c)

$\operatorname{Inf}^{G}_{G/N}{}_{k[G/N]}\mathcal{C}(V)\subseteq{}_{kG}\mathcal{C}(\operatorname{Inf}^{G}_{G/N}V)$.

  4. (d)

Let $\phi:G\to\tilde{G}$ be a group homomorphism and $M$ be a $V$-projective
$kG$-module.
$\operatorname{Res}_{\phi}{}_{kG}\mathcal{C}(V)\subseteq{}_{k\tilde{G}}\mathcal{C}(\operatorname{Res}_{\phi}V)$.

  5. (e)

If $V$ is a $p$-permutation $kG$-module, then
$\big{(}{}_{kG}\mathcal{C}(V)\big{)}(P)\subseteq{}_{k[N_{G}(P)/P]}\mathcal{C}(V(P))$.

###### Proof.

All of these are proven in [9, Lemma 2.1.1] for modules except (e), and each
of the proofs extends to the chain complex case.

For (e), suppose $M\mid V\otimes_{k}N$ for some $N\in{}_{kG}\mathcal{C}$. Then
$M(P)\mid(V\otimes_{k}N)(P)\cong V(P)\otimes_{k}N(P)$ by [12, Theorem 5.8.10],
since $V$ is $p$-permutation. ∎

Again, in general we do not have reverse inclusions for the above statements.
See the remark after [9, Lemma 2.1.1] for a counterexample. Next, we give some
lemmas which further describe the behavior of restriction and induction with
respect to relative projectivity.

######  Lemma 3.8.

Let $K\leq H\leq G$, $U,V$ be $kG$-modules, and $W,Z$ be $kK$-modules. The
following inclusions and equalities hold:

  1. (a)

$\operatorname{Ind}^{G}_{H}({}_{kH}\mathcal{C}(\operatorname{Res}^{G}_{H}V))\subseteq{}_{kG}\mathcal{C}(\operatorname{Ind}^{G}_{H}\operatorname{Res}^{G}_{H}V)\subseteq{}_{kG}\mathcal{C}(V)$.

If moreover, $V$ is $H$-projective, then
${}_{kG}\mathcal{C}(\operatorname{Ind}^{G}_{H}\operatorname{Res}^{G}_{H}V)={}_{kG}\mathcal{C}(V)$.

  2. (b)

If ${}_{kG}\mathcal{C}(V)={}_{kG}\mathcal{C}(U)$, then
${}_{kH}\mathcal{C}(\operatorname{Res}^{G}_{H}V)={}_{kH}\mathcal{C}(\operatorname{Res}^{G}_{H}U)$.

  3. (c)

If ${}_{kK}\mathcal{C}(W)={}_{kK}\mathcal{C}(Z)$, then
${}_{kG}\mathcal{C}(\operatorname{Ind}^{G}_{K}W)={}_{kG}\mathcal{C}(\operatorname{Ind}^{G}_{K}Z)$.

###### Proof.

All of these are proven in [9, Lemma 2.1.2] for modules, and the arguments
extend to chain complexes. ∎

######  Lemma 3.9.

Let $H\leq G$ and let $M\in{}_{kG}\mathcal{C}$. Then, the following are
equivalent:

  1. (a)

$M$ is $V$-projective.

  2. (b)

$\operatorname{Res}^{G}_{H}M$ is $\operatorname{Res}^{G}_{H}V$-projective.

  3. (c)

$\operatorname{Ind}^{G}_{H}\operatorname{Res}^{G}_{H}M$ is $V$-projective.

###### Proof.

All of these are proven in [9, Lemma 2.1.3] for modules, and the arguments
extend to chain complexes. ∎

######  Corollary 3.10.

  1. (a)

Let $S\in\operatorname{Syl}_{p}(G)$. Let $V$ and $W$ be two $kG$-modules. Then
${}_{kG}\mathcal{C}(V)={}_{kG}\mathcal{C}(W)$ if and only if
${}_{kS}\mathcal{C}(\operatorname{Res}^{G}_{S}V)={}_{kS}\mathcal{C}(\operatorname{Res}^{G}_{S}W)$.

  2. (b)

If $V$ is a trivial source module with vertex $P$, then
${}_{kG}\mathcal{C}(V)={}_{kG}\mathcal{C}(k[G/P])={}_{kG}\mathcal{C}(P)$.

  3. (c)

If $V$ is a $p$-permutation module, let $\mathcal{X}_{V}$ be the set of
$p$-subgroups $P$ of $G$ for which $V(P)\neq 0$. Then
${}_{kG}\mathcal{C}(V)={}_{kG}\mathcal{C}(\mathcal{X}_{V})$.

###### Proof.

The proof of (a) is given in [9, Corollary 2.1.4], and extends to chain
complexes.

For (b), observe that if $V$ is a trivial source module with vertex $P$, then
$V\mid k[G/P]$. Now,

$\operatorname{Res}^{G}_{S}k[G/P]\cong\bigoplus_{x\in[S\backslash
G/P]}k[S/S\cap{}^{x}P],$

and this is the complete decomposition into indecomposables, so
$\operatorname{Res}^{G}_{S}V$ must be isomorphic to a direct sum of modules of
the form $k[S/Q]$ for some subgroup $Q$ of $G$ with $Q\leq_{G}P$. If there
does not exist a direct summand of $\operatorname{Res}^{G}_{S}V$ isomorphic to
$k[S/P]$, then $V(P)=0$, implying that $V$ has vertex strictly contained in
$P$, a contradiction. Thus $\operatorname{Res}^{G}_{S}V$ has a direct summand
isomorphic to $k[S/P]$, and from the direct sum decomposition, all other
indecomposable direct summands of $\operatorname{Res}^{G}_{S}M$ have a vertex
contained in $P$.

Now, if $\mathcal{X}$ is the poset of all subgroups $Q$ for which
$\operatorname{Res}^{G}_{S}M$ has a summand isomorphic to $k[S/Q]$, then
${}_{kS}\mathcal{C}(\operatorname{Res}^{G}_{S}M)={}_{kS}\mathcal{C}(\mathcal{X})$
by Remark 3.5. Note that if $R\leq Q\leq S,$ then
${}_{kS}\mathcal{C}(R)\subseteq{}_{kS}\mathcal{C}(Q)$. Since $\mathcal{X}$ has
as its maximal elements all $S$-conjugates of $P$,
${}_{kS}\mathcal{C}(\mathcal{X})={}_{kS}\mathcal{C}(P)$. A similar argument
produces that
${}_{kS}\mathcal{C}(\operatorname{Res}^{G}_{S}k[G/P])={}_{kS}\mathcal{C}(P)$,
and the result follows by (a).

Finally, (c) follows from Proposition 3.4 and the fact that since $V$ is
$p$-permutation, $V(P)=0$ if and only if $P$ is not contained in a single
vertex of any indecomposable summand of $V$. ∎

######  Remark 3.11.

For this reason, if we assume that $V$ is a $p$-permutation module and want to
study ${}_{kG}\mathcal{C}(V)$, it suffices to assume $V$ is in fact a
permutation module, or equivalently, we can study
${}_{kG}\mathcal{C}(\mathcal{X}_{V})$, where $\mathcal{X}_{V}$ is a set of the
vertices of the indecomposable summands of $V$, or alternatively where
$\mathcal{X}_{V}$ is the set of all $p$-subgroups $P$ for which $V(P)\neq 0$.

######  Definition 3.12.

Let $V$ be a $kG$-module. Denote by ${}_{kG}\mathbf{triv}(V)$ the subcategory
of ${}_{kG}\mathbf{mod}(V)$ consisting of $p$-permutation modules, and denote
by $Ch^{b}({}_{kG}\mathbf{triv},V)$ the subcategory of
$Ch^{b}({}_{kG}\mathbf{mod},V)$ consisting of chain complexes whose components
are $p$-permutation modules. Let ${}_{kG}p\mathcal{C}(V)$ represent either
${}_{kG}\mathbf{triv}(V)$ or $Ch^{b}({}_{kG}\mathbf{triv},V)$. It is an easy
verification that all of the previous propositions hold with
${}_{kG}p\mathcal{C}(V)$ replacing ${}_{kG}\mathcal{C}(V)$.

Note that we do not necessarily assume $V$ is $p$-permutation for
${}_{kG}p\mathcal{C}(V)$. In fact, for ${}_{kG}\mathbf{triv}(V)$, we may
indeed assume $V$ is $p$-permutation, via the next proposition. As a result,
the only type of relative projectivity for ${}_{kG}\mathbf{triv}$ is
projectivity relative to a family of subgroups.

######  Proposition 3.13.

Let $V$ be a $kG$-module. Then there exists a $p$-permutation module $W$ such
that ${}_{kG}\mathbf{triv}(V)={}_{kG}\mathbf{triv}(W)$. In particular, there
exists a set of $p$-subgroups $\mathcal{X}\subseteq s_{p}(G)$ for which
${}_{kG}\mathbf{triv}(V)={}_{kG}\mathbf{triv}(\mathcal{X})$.

###### Proof.

First we show that if $M$ is a trivial source module with vertex $P\in
s_{p}(G)$ which belongs to ${}_{kG}\mathbf{triv}(V)$, then $k[G/P]$ belongs to
${}_{kG}\mathbf{triv}(V)$. If $M\mid V\otimes_{k}N$ for some $kG$-module $N$,
then
$\operatorname{Res}^{G}_{S}M\mid\operatorname{Res}^{G}_{S}(V\otimes_{k}N)$ for
$S\in\operatorname{Syl}_{p}(G)$. Since $M$ has vertex $P$,
$\operatorname{Res}^{G}_{S}M$ has a direct summand isomorphic to $k[S/P]$, and
any other nonisomorphic direct summand is isomorphic to $k[S/Q]$ for some
$Q\leq P$. Therefore $\operatorname{Res}^{G}_{S}(V\otimes_{k}N)$ has $k[S/P]$
as an isomorphic direct summand. We have
$\operatorname{Res}^{G}_{S}(V\otimes_{k}N)\cong k[S/P]\oplus N^{\prime}$, for
some $kS$-module $N$, therefore, $k[S/P]$ is
$\operatorname{Res}^{G}_{S}V$-projective. Therefore by Proposition 3.4,
${}_{kS}\mathbf{triv}(P)\subseteq{}_{kS}\mathbf{triv}(\operatorname{Res}^{G}_{S}V)$.

We have that

$\operatorname{Res}^{G}_{S}k[G/P]\cong\bigoplus_{x\in[S\backslash
G/P]}k[S/S\cap{}^{x}P],$

and for any $x\in G$, $k[S/S\cap{}^{x}P]$ is $P$-projective. Therefore,
$\operatorname{Res}^{G}_{S}k[G/P]\in{}_{kS}\mathbf{triv}(P)$, hence
$\operatorname{Res}^{G}_{S}k[G/P]\in{}_{kS}\mathbf{triv}(\operatorname{Res}^{G}_{S}V)$
as well. Equivalently, $k[G/P]$ belongs to ${}_{kG}\mathbf{mod}(V)$, as
desired.

Now, let $\mathcal{X}$ be the set of all vertices which occur for all trivial
source modules belonging to ${}_{kG}\mathbf{triv}(V)$. The previous claim and
Proposition 3.4 imply that

${}_{kG}\mathbf{triv}(V)\supset{}_{kG}\mathbf{triv}\left(\bigoplus_{P\in\mathcal{X}}k[G/P]\right)={}_{kG}\mathbf{triv}(\mathcal{X}).$

To show the other direction of set inclusion, it suffices to show that if $V$
is indecomposable with vertex $P$, then
${}_{kG}\mathbf{mod}(V)\subseteq{}_{kG}\mathbf{mod}(P)$. Suppose
$V\mid\operatorname{Ind}^{G}_{P}X$ for some $kG$-module $X$. Then for any
$M\in{}_{kG}\mathbf{mod}(V)$, we may write

$M\mid
V\otimes_{k}N\mid(\operatorname{Ind}^{G}_{P}X)\otimes_{k}N\cong\operatorname{Ind}^{G}_{P}(X\otimes_{k}\operatorname{Res}^{G}_{P}N).$

Thus $M$ is $P$-projective, and the result follows. ∎

###  3.1. Absolute $p$-divisibility

The following observation of Auslander and Carlson in [1] is crucial for the
theory of $V$-projectivity: if $V$ is a $kG$-module, then the evaluation map
or trace map $V^{*}\otimes_{k}V\to k:f\otimes v\mapsto f(v)$ is always
$V$-split, and splits when $\dim_{k}V$ is coprime to $p$.

The following arguments and definition follow from [3].

######  Theorem 3.14.

[3, 2.1] Let $k$ be an algebraically closed field of characteristic $p$
(possibly $p=0$). Let $M,N$ be finite-dimensional indecomposable $kG$-modules,
then

$k\mid M\otimes_{k}N\text{ if and only if }M\cong N^{*}\text{ and
}p\nmid\dim_{k}(N)$

Moreover, if $k$ is a direct summand of $N^{*}\otimes_{k}N$, it has
multiplicity one.

######  Proposition 3.15.

Let $V$ be a $kG$-module. The following are equivalent:

  1. (a)

The trivial $kG$-module $k$ is relatively $V$-projective.

  2. (b)

The chain complex $k[0]$ is relatively $V$-projective.

  3. (c)

$p$ does not divide the $k$-dimension of at least one of the indecomposable
direct summands of $V$.

  4. (d)

${}_{kG}\mathbf{mod}(V)={}_{kG}\mathbf{mod}.$

  5. (e)

$Ch^{b}({}_{kG}\mathbf{mod},V)=Ch^{b}({}_{kG}\mathbf{mod})$.

###### Proof.

The equivalence of (a),(c),(d) is given by [9, Lemma 2.2.2]. Now, $k[0]$ is
$V$-projective if and only if $k\mid V^{*}\otimes_{k}V$, and equivalently,
$p\nmid\dim_{k}(V)$. This shows (b) is equivalent to (c). Now, (c) implies (e)
by Proposition 3.4, and (e) implies (b) trivially, so we are done. ∎

In other words, relative $V$-projectivity is interesting if and only if the
$k$-dimensions of all indecomposable summands of $V$ are divisible by $p$.
This is analogous to the fact that if $S\in\operatorname{Syl}_{p}(G)$, all
$kG$-modules are $S$-projective.

######  Definition 3.16.

[3] A $kG$-module for which the $k$-dimensions of all indecomposable summands
of $V$ are divisible by $p$ is called absolutely $p$-divisible.

######  Lemma 3.17.

[9, Lemma 2.2.4] Let $S\in\operatorname{Syl}_{p}(G)$, and let $V$ be a
$kG$-module with vertex $Q\leq S$.

  1. (a)

For every subgroup $H\geq S$, $V$ is absolutely $p$-divisible if and only if
$\operatorname{Res}^{G}_{H}V$ is absolutely $p$-divisible.

  2. (b)

If $Q<S,$ then for every subgroup $R\leq S$ with $S\geq R>Q$, $V$ is
absolutely $p$-divisible if and only if $\operatorname{Res}^{G}_{R}V$ is.

In particular, if $V$ is a $p$-permutation module, $V$ is absolutely
$p$-divisible if and only if $V$ has no indecomposable direct summands with
vertex $S$ if and only if $V(S)=0$. Moreover, absolute $p$-divisibility is
preserved upon restriction to subgroups containing Sylow $p$-subgroups.

##  4\. $V$-stable categories

We review the stable categories corresponding to $V$-projectivity for modules
and chain complexes. These categories are analogous to the usual stable module
category, in that we factor out $V$-projective objects.

######  Definition 4.1.

Let $V$ be a $kG$-module.

  1. (a)

The $V$-stable $kG$-module category, denoted
${}_{kG}\underline{\mathbf{mod}}_{V}$, is defined as follows. The objects are
$kG$-modules, and for any two $kG$-modules $M,N$, the space of morphisms from
$M$ to $N$ is the quotient of $k$-vector spaces

$\underline{\operatorname{Hom}}^{V}_{kG}(M,N)=\operatorname{Hom}_{kG}(M,N)/\operatorname{Hom}_{kG}^{V}(M,N),$

where $\operatorname{Hom}_{kG}^{V}(M,N)$ is the subspace of
$\operatorname{Hom}_{kG}(U,V)$ consisting of morphisms which factor through a
$V$-projective module. Note this definition makes sense since
$\operatorname{Hom}_{kG}^{V}(M,N)$ is a two-sided ideal of
$\operatorname{Hom}_{kG}(M,N)$.

  2. (b)

The $V$-stable $p$-permutation $kG$-module category, denoted
${}_{kG}\underline{\mathbf{triv}}_{V}$, is defined as the full subcategory of
${}_{kG}\underline{\mathbf{mod}}_{V}$ with $p$-permutation modules as its
objects.

We have an analogous definition for chain complex categories as well.

  1. (a)

The $V$-stable category of chain complexes of $kG$-modules, denoted
$\underline{Ch}^{b}({}_{kG}\textbf{mod})_{V}$, is defined as follows. The
objects of $\underline{Ch}^{b}({}_{kG}\textbf{mod})_{V}$ are the objects of
$Ch^{b}({}_{kG}\textbf{mod})$, and for any two $C,D\in
Ch^{b}({}_{kG}\textbf{mod})$,

$\underline{\operatorname{Hom}}^{V}_{kG}(C,D)=\operatorname{Hom}_{kG}(C,D)/\operatorname{Hom}_{kG}^{V}(C,D),$

where $\operatorname{Hom}_{kG}^{V}(C,D)$ is the $k$-subspace of
$\operatorname{Hom}_{kG}(C,D)$ consisting of morphisms which factor through a
$V$-projective chain complex.

  2. (b)

The $V$-stable category of chain complexes of $p$-permutation $kG$-module,
denoted $\underline{Ch}^{b}({}_{kG}\textbf{triv})_{V}$, is defined as the full
subcategory of $\underline{Ch}^{b}({}_{kG}\textbf{mod})_{V}$ with objects
chain complexes of $p$-permutation modules.

Notation: As in the previous section, let ${}_{kG}\mathcal{C}$ refer to either
${}_{kG}\mathbf{mod}$ or $Ch^{b}({}_{kG}\mathbf{mod})$. Let
${}_{kG}\underline{\mathcal{C}}_{V}$ refer to the corresponding $V$-stable
category ${}_{kG}\underline{\mathbf{mod}}_{V}$ or
$\underline{Ch}^{b}({}_{kG}\mathbf{mod})_{V}$. For any morphism
$f\in\operatorname{Hom}_{{}_{kG}\mathcal{C}}(M,N)$, write $\underline{f}$ to
denote its image in ${}_{kG}\underline{\mathcal{C}}_{V}$. Denote the morphism
sets of ${}_{kG}\underline{\mathcal{C}}_{V}$ by
$\underline{\operatorname{Hom}}^{V}_{{}_{kG}\mathcal{C}}(M,N)$. Similarly, let
${}_{kG}p\mathcal{C}$ refer to either ${}_{kG}\mathbf{triv}$ or
$Ch^{b}({}_{kG}\mathbf{triv})$. Let ${}_{kG}p\underline{\mathcal{C}}_{V}$
refer to the corresponding $V$-stable category
${}_{kG}\underline{\mathbf{triv}}_{V}$ or
$\underline{Ch}^{b}({}_{kG}\mathbf{triv})_{V}$, and use the analogous notation
for morphisms and morphism collections, with ${}_{kG}p\mathcal{C}$ replacing
${}_{kG}\mathcal{C}$.

If $V=\\{0\\}$, we vacuously set
${}_{kG}\underline{\mathbf{mod}}_{V}={}_{kG}\mathbf{mod}$ and
${}_{kG}\underline{\mathbf{triv}}_{V}={}_{kG}\mathbf{triv}$. We also recover
$Ch^{b}({}_{kG}\textbf{mod})$ and $Ch^{b}({}_{kG}\textbf{triv})$,
respectively.

######  Proposition 4.2.

Let $M\in{}_{kG}\mathcal{C}$. The following are equivalent.

  1. (a)

$M$ is isomorphic to the zero object in ${}_{kG}\underline{\mathcal{C}}_{V}$.

  2. (b)

$\underline{\operatorname{id}}_{M}$ is the zero endomorphism of $M$ in
$\underline{\operatorname{End}}^{V}_{{}_{kG}\mathcal{C}}(M)$.

  3. (c)

$M\in{}_{kG}\mathcal{C}(V)$.

###### Proof.

(a) implies (b) is trivial since the zero object only has the zero
endomorphism. If (b) holds, then $\operatorname{id}_{M}=g\circ f$ for some
${}_{kG}\mathcal{C}$ morphisms $f:M\to P$ and $g:P\to M$, with $P$ a
$V$-projective. In particular, $g$ is split surjective, so $M$ is isomorphic
to a direct summand of $P$, hence $M$ is $V$-projective as well. Finally, (c)
implies (a) is straightforward. ∎

Note that the above proposition also holds for ${}_{kG}p\mathcal{C}$. We
previously noted that projectivity relative to modules was compatible with
standard functorial operations on groups. We translate these compatabilities
to the $V$-stable categories defined here.

######  Proposition 4.3.

Let $H\leq G$, $P\in s_{p}(G)$, and $N\trianglelefteq G$.

  1. (a)

If $V$ is a $kG$-module, $\operatorname{Res}^{G}_{H}$ induces functors
${}_{kG}\underline{\mathcal{C}}_{V}\to{}_{kH}\underline{\mathcal{C}}_{\operatorname{Res}^{G}_{H}V}$
and
${}_{kG}p\underline{\mathcal{C}}_{V}\to{}_{kH}p\underline{\mathcal{C}}_{\operatorname{Res}^{G}_{H}V}$.

  2. (b)

If $V$ is a $kH$-module, $\operatorname{Ind}^{G}_{H}$ induces functors
${}_{kH}\underline{\mathcal{C}}_{V}\to{}_{kG}\underline{\mathcal{C}}_{\operatorname{Ind}^{G}_{H}V}$
and
${}_{kH}p\underline{\mathcal{C}}_{V}\to{}_{kG}p\underline{\mathcal{C}}_{G,\operatorname{Ind}^{G}_{H}V}$.

  3. (c)

If $V$ is a $k[G/N]$-module, $\operatorname{Inf}^{G}_{G/N}$ induces functors
${}_{k[G/N]}\underline{\mathcal{C}}_{V}\to{}_{kG}\underline{\mathcal{C}}_{\operatorname{Inf}^{G}_{G/N}V}$
and
${}_{k[G/N]}p\underline{\mathcal{C}}_{V}\to{}_{kG}p\underline{\mathcal{C}}_{\operatorname{Inf}^{G}_{G/N}V}$
.

  4. (d)

If $V$ is a $kG$-module, then $-(P)$ induces a functor
${}_{kG}p\underline{\mathcal{C}}_{V}\to{}_{k[N_{G}(P)/P]}p\underline{\mathcal{C}}_{V(P)}$,
and if $V$ is $p$-permutation, $-(P)$ induces a functor
${}_{kG}\underline{\mathcal{C}}_{V}\to{}_{k[N_{G}(P)/P]}\underline{\mathcal{C}}_{V(P)}$.

###### Proof.

This follows from Lemma 3.7. ∎

In particular, if $P$ is a $p$-subgroup of $G$ for which $V(P)=0$, then $-(P)$
induces a functor
${}_{kG}p\underline{\mathcal{C}}_{V}\to{}_{k[N_{G}(P)/P]}p\mathcal{C}$. Recall
that in the usual stable module category ${}_{kG}\underline{\mathbf{mod}}$,
one can detect when a morphism factors through a relatively $k$-projective
module. We have an analogous theorem here, which in the case of $V=kG$ reduces
to the usual detection theorem.

######  Proposition 4.4.

Let $V$ be a $kG$-module and $f:M\to N$ be a ${}_{kG}\mathcal{C}$-morphism.
The following are equivalent:

  1. (a)

$f$ factors through an object in ${}_{kG}\mathcal{C}(V)$.

  2. (b)

$f$ factors through the $kG$-module homomorphism
$V^{*}\otimes_{k}V\otimes_{k}N\to N$ sending $\delta\otimes v\otimes n$ to
$\delta(v)n$ (degreewise, if necessary).

  3. (c)

$f$ factors through any $V$-split epimorphism $g:N^{\prime}\to N$.

###### Proof.

The implications (c) implies (b) implies (a) are straightforward, noting that
the homomorphism in (b) is $V$-split. Suppose (a). Then there exists a
$V$-projective object $P$ and maps $f_{1}:M\to P$ and $f_{2}:P\to N$ such that
$f=f_{2}\circ f_{1}$. Suppose $g:N^{\prime}\to N$ is a $V$-split epimorphism
as in (c). Therefore by $V$-projectivity, there exists a morphism
$g^{\prime}:P\to N^{\prime}$ such that $f_{2}=g\circ g^{\prime}$. Thus
$f=g\circ g^{\prime}\circ f_{1}$, as desired. ∎

######  Corollary 4.5.

Let $f:M\to N$ be a morphism in ${}_{kG}\mathbf{triv}$ and let $V$ be a
$kG$-module. $f$ factors through an object in ${}_{kG}\mathbf{mod}(V)$ if and
only if $f$ factors through an object in ${}_{kG}\mathbf{triv}(V)$.

###### Proof.

Recall we may replace $V$ by a $p$-permutation module by Proposition 3.13. By
the previous proposition, $f$ factors through an object in
${}_{kG}\mathbf{mod}(V)$ if and only if $f$ factors through
$V^{*}\otimes_{k}V\otimes N$, which is $p$-permutation. ∎

As a result of the above corollary, we may equivalently define
${}_{kG}p\underline{\mathcal{C}}_{V}$ as follows:
${}_{kG}p\underline{\mathcal{C}}_{V}$ has the same objects as
${}_{kG}p\mathcal{C}$, and for any two objects
$M,N\in{}_{kG}p\underline{\mathcal{C}}_{V}$, the space of morphisms from $M$
to $N$ is the quotient of $k$-vector spaces

$\underline{\operatorname{Hom}}^{V}_{{}_{kG}p\mathcal{C}}(M,N)=\operatorname{Hom}_{{}_{kG}p\mathcal{C}}(M,N)/\operatorname{Hom}_{{}_{kG}p\mathcal{C}}^{V}(M,N),$

where $\operatorname{Hom}_{{}_{kG}p\mathcal{C}}^{V}(M,N)$ is the subspace of
$\operatorname{Hom}_{{}_{kG}p\mathcal{C}}(U,V)$ consisting of morphisms which
factor through an object in ${}_{kG}p\mathcal{C}(V)$.

######  Proposition 4.6.

Let $M,N\in{}_{kG}\mathcal{C}$, and let $\varphi:M\to N$ be a morphism in
${}_{kG}\mathcal{C}$. If $\varphi$ is a $V$-split epimorphism, then the image
of $\varphi$ in $\underline{\operatorname{Hom}}_{{}_{kG}\mathcal{C}}^{V}(M,N)$
is an isomorphism if and only if $\varphi$ is a split epimorphism and
$\ker\varphi$ is a $V$-projective object.

In particular, there exists an isomorphism $M\xrightarrow[]{\sim}N$ in
${}_{kG}\underline{\mathcal{C}}_{V}$ if and only if there exist
$P,Q\in{}_{kG}\mathcal{C}(V)$ such that $M\oplus P\cong N\oplus Q$ as objects
in ${}_{kG}\mathcal{C}$.

###### Proof.

Suppose $\varphi$ is $V$-split surjective and $\underline{\varphi}$ is an
isomorphism in ${}_{kG}\underline{\mathcal{C}}_{V}$. Then there is a
${}_{kG}\mathcal{C}$ morphism $\psi:N\to M$ such that
$\underline{\varphi}\circ\underline{\psi}=\underline{\operatorname{id}}_{N}$,
or equivalently (since both categories are $k$-linear), that
$\operatorname{id}_{N}-\varphi\circ\psi$ factors through a $V$-projective
$kG$-module. Equivalently, $\operatorname{id}_{N}-\varphi\circ\psi$ factors
through any $V$-split epimorphism, in particular $\varphi$. Write

$\operatorname{id}_{N}-\varphi\circ\psi=\varphi\circ\alpha$

for some $\alpha:N\to M$. We have

$\operatorname{id}_{N}=\varphi\circ(\psi+\alpha),$

hence $\varphi$ is a split epimorphism. It follows that $\ker\psi$ is zero in
${}_{kG}\underline{\mathcal{C}}_{V}$ for $\underline{\varphi}$ to be an
isomorphism, so $\ker\varphi$ is $V$-projective. The converse of the first
statement is straightforward.

Suppose now that $\underline{\varphi}$ is an isomorphism. Define the object
$P=V^{*}\otimes_{k}V\otimes_{k}N$ and denote by $\pi:P\to V$ the canonical
morphism sending $\delta\otimes v\otimes n\mapsto\delta(v)n$ (componentwise if
necessary). Then $\varphi\oplus\pi:U\oplus P\to V$ is a $V$-split epimorphism
since $\pi$ is, and $\underline{\varphi\oplus\pi}$ is an isomorphism, as $P$
is $V$-projective, hence zero. By the first statement, $\varphi$ is a split
epimorphism with $\ker\varphi$ $V$-projective, so $M\oplus P\cong
V\oplus\ker\varphi$ as desired. The converse is straightforward. ∎

Note the above proposition also holds for ${}_{kG}p\mathcal{C}(V)$.

######  Corollary 4.7.

${}_{kG}\underline{\mathcal{C}}_{V}$ and ${}_{kG}p\underline{\mathcal{C}}_{V}$
are Krull-Schmidt categories. In particular, any isomorphism class of objects
in $\underline{\mathcal{C}},V$ and ${}_{kG}p\underline{\mathcal{C}}_{V}$ has a
unique representative which has no $V$-projective direct summands.

###### Proof.

This follows since ${}_{kG}\mathcal{C}$ is a Krull-Schmidt category, so given
any $M\in{}_{kG}\underline{\mathcal{C}}_{V}$ one may write out its unique
direct sum decomposition in $\mathcal{C}_{G}$ and strike any $V$-projective
summands to obtain a unique decomposition in
${}_{kG}\underline{\mathcal{C}}_{V}$. This follows for
${}_{kG}p\underline{\mathcal{C}}_{V}$ as well since it is a subcategory of
${}_{kG}\underline{\mathcal{C}}_{V}$. ∎

######  Remark 4.8.

In general, given a homomorphism of $kG$-modules, the image of its kernel in
the $V$-stable module category will depend on a choice of representative, thus
is not well-defined. Therefore, ${}_{kG}\underline{\mathbf{mod}}_{V}$ and
${}_{kG}\underline{\mathbf{triv}}_{V}$ are not abelian categories, however,
they are still preadditive and $k$-linear. Therefore, one may construct the
chain complex categories $Ch^{b}({}_{kG}\underline{\mathbf{mod}}_{V})$ and
$Ch^{b}({}_{kG}\underline{\mathbf{triv}}_{V})$, as well as the homotopy
categories $K^{b}({}_{kG}\underline{\mathbf{mod}}_{V})$ and
$K^{b}({}_{kG}\underline{\mathbf{triv}}_{V})$. To discuss homological
properties, we have to choose specific representatives of chain complexes, and
in general homological properties will not be well-defined for an equivalence
class of complexes.

On the other hand, since ${}_{kG}\underline{\mathbf{mod}}_{V}$ is a Krull-
Schmidt category, any isomorphism class in
$Ch^{b}({}_{kG}\underline{\mathbf{mod}}_{V})$ of complexes has a unique
representative with no $V$-projective modules in any degree. This is obtained
by taking any representative and removing all indecomposable $V$-projective
summands in each degree and corresponding homomorphisms.

######  Definition 4.9.

We define the homotopy category $\underline{K}^{b}({}_{kG}\mathbf{mod})_{V}$
of $\underline{Ch}^{b}({}_{kG}\mathbf{mod})_{V}$ in the same manner as the
non-stable setting. Two maps $f,g\in\underline{\operatorname{Hom}}_{kG}(C,D)$
are homotopic if there exists a homotopy $h:C\to D[1]$ for which
$\underline{f_{i}-g_{i}}=\underline{h_{i-1}\circ c_{i}+d_{i+1}\circ h_{i}}$ in
$\underline{Ch}^{b}({}_{kG}\mathbf{mod})_{V}$.

In this category, it is clear that both contractible complexes (in the usual
sense) and $V$-projective complexes are isomorphic to the 0 complex. It turns
out that these are all the complexes that are identified with the zero
complex.

######  Proposition 4.10.

Let $C$ be a complex of $kG$-modules which is both indecomposable in
$\underline{Ch}^{b}({}_{kG}\mathbf{mod})_{V}$ and isomorphic to the zero
complex in $\underline{K}^{b}({}_{kG}\mathbf{mod})_{V}$. Then $C\cong(0\to
M\xrightarrow{\sim}M\to 0)$ in $\underline{Ch}^{b}({}_{kG}\mathbf{mod})_{V}$
for some $kG$-module $M$.

In particular, the complexes which are isomorphic to the zero complex in
$\underline{K}^{b}({}_{kG}\mathbf{mod})_{V}$ are isomorphic to direct sums of
contractible complexes of $kG$-modules and $V$-projective complexes. Two
complexes $C,D$ for which $C\cong D$ in
$\underline{K}^{b}({}_{kG}\mathbf{mod})_{V}$ satisfy $C\oplus M\oplus N\cong
D\oplus M^{\prime}\oplus N^{\prime}$, where $M,M^{\prime}$ are contractible
chain complexes and $N,N^{\prime}$ are $V$-projective chain complexes.

###### Proof.

Let $C$ be as above, and let $i\in\mathbb{Z}$ be the maximal integer for which
$C_{i}\neq 0$, then we have a homotopy equivalence $C\simeq 0$ and therefore a
homotopy
$\underline{\operatorname{id}}=\underline{h}\circ\underline{d}+\underline{d}\circ\underline{h}$,
hence $\operatorname{id}=h\circ d+d\circ h+f$, for some endomorphism $f$ which
factors through a $V$-projective complex. In particular, we have
$\operatorname{id}_{C_{i}}=h_{i-1}\circ d_{i}+f_{i}$, with $f_{i}$ a morphism
which factors through a $V$-projective module, hence also a $V$-projective
complex by setting the module in degree $i$.

We claim $(C_{i}\to C_{i})$ is a summand of
$C\in\underline{Ch}^{b}({}_{kG}\mathbf{mod})_{V}.$ Indeed, from the homotopy,
the following setup is a well-defined sequence of morphisms which compose to
the identity in $\underline{Ch}^{b}({}_{kG}\mathbf{mod})_{V}$.

${\dots}$${0}$${C_{i}}$${C_{i}}$${0}$${\dots}$${\dots}$${0}$${C_{i}}$${C_{i-1}}$${C_{i-2}}$${\dots}$${\dots}$${0}$${C_{i}}$${C_{i}}$${0}$${\dots}$$\scriptstyle{=}$$\scriptstyle{=}$$\scriptstyle{d_{i}}$$\scriptstyle{d_{i}}$$\scriptstyle{=}$$\scriptstyle{d_{i-1}}$$\scriptstyle{h_{i-1}}$$\scriptstyle{=}$

However, $C$ was assumed to be indecomposable, hence $C=0\to C_{i}\to C_{i}\to
0$ as desired.

The final statement is proven as follows: suppose $C\cong D$ in
$\underline{K}^{b}({}_{kG}\mathbf{mod})_{V}$. Then we have a homotopy
equivalence $f:C\to D$. Let $P=C(D)[-1]$, the mapping cone of $D$, and let $p$
be the projection of $D$ onto $P$. This is a degreewise split epimorphism, and
the chain map $f\oplus p:C\oplus P\to D$ is degreewise split. Since $P$ is
contractible, we can write $C\oplus P\cong D\oplus\ker(f\oplus d)$ in
$\underline{Ch}^{b}({}_{kG}\mathbf{mod})_{V}$. It follows that $\ker(f\oplus
d)$ is contractible as well, and the result follows from Theorem 4.6. ∎

It follows that we may always take a unique up to isomorphism representative
$C$ of a class of complexes $[C]\in\underline{K}^{b}({}_{kG}\mathbf{mod})_{V}$
which has no $V$-projective or contractible summands.

##  5\. $V$-endotrivial modules

We briefly recall a few necessary facts about $V$-endotrivial modules. These
were first introduced by Lassueur in [9].

######  Definition 5.1.

Let $V$ be a $kG$-module. A $kG$-module $M$ is endotrivial relative to $V$ or
$V$-endotrivial if its $k$-endomorphism ring is the direct sum of a trivial
module and a $V$-projective module. That is, $M$ is $V$-endotrivial if and
only if

$\operatorname{End}_{k}(M)\cong M^{*}\otimes_{k}M\cong k\oplus P,\quad
P\in{}_{kG}\mathbf{mod}(V).$

Equivalently, $\operatorname{End}_{k}(M)\cong
k\in{}_{kG}\underline{\mathbf{mod}}_{V}$.

Like how endotrivial modules arise from Heller shifts, $V$-endotrivial modules
arise from relative syzygies induced by $V$-projective modules. In general,
$V$-projective covers always exist.

######  Proposition 5.2.

[9, Lemmas 3.2.1, 3.2.2, 3.3.1, 3.4.1] Let $V$ be an absolutely $p$-divisible
$kG$-module.

  1. (a)

Let $P\in{}_{kG}\mathbf{mod}(V)$ and $0\to L\to P\to N\to 0$ be a $V$-split
exact sequence. Then $N$ is $V$-endotrivial if and only if $L$ is. In
particular, if $M$ is $V$-endotrivial and $W\in{}_{kG}\mathbf{mod}(V)$, then
for every $n\in\mathbb{Z}$, $\Omega^{n}_{W}(M)$ is $V$-endotrivial.

  2. (b)

If $M,N$ are $V$-endotrivial $kG$-modules, so are $M^{*},M\otimes_{k}N$, and
$\operatorname{Hom}_{k}(M,N)$.

  3. (c)

Each $V$-endotrivial $kG$-module has a unique indecomposable $V$-endotrivial
direct summand, and all other summands are $V$-projective.

  4. (d)

If $M$ is an indecomposable $V$-endotrivial $kG$-module, then its vertex set
is $\operatorname{Syl}_{p}(G)$, and given a vertex $S$, any corresponding
$kS$-source is $\operatorname{Res}^{G}_{S}V$-endotrivial.

######  Definition 5.3.

Let $V$ be an absolutely $p$-divisible $kG$-module. The group of
$V$-endotrivial complexes $T_{V}(G)$ is defined as follows. For two
$V$-endotrivial $kG$-modules $M,N$, say $M\sim N$ if and only if $M_{0}\cong
N_{0}$, where $M_{0}$ and $N_{0}$ are their respective indecomposable
$V$-endotrivial direct summands. Equivalently, $M\sim N$ if $M\cong N$ in
${}_{kG}\underline{\mathbf{mod}}_{V}$. Let $T_{V}(G)$ denote the resulting set
of equivalence classes. In particular, any equivalence class in $T_{V}(G)$
consists of an indecomposable $V$-endotrivial module $M_{0}$, and all modules
of the form $M_{0}\oplus P$ with $P$ a $V$-projective $kG$-module.

$T_{V}(G)$ forms an abelian group under the operation

$[M]+[N]=[M\otimes_{k}N],$

with zero element $[k]$ and inverse $[M]^{-1}=[M^{*}].$

######  Remark 5.4.

If $U,V$ are absolutely $p$-divisible $kG$-modules with
${}_{kG}\mathbf{mod}(U)\subseteq{}_{kG}\mathbf{mod}(V)$, then we have an
injective group homomorphism $T_{U}(G)\to T_{V}(G)$ given by
$[M]_{U}\mapsto[M]_{V}$. Moreover, inflation, restriction, group isomorphisms,
and conjugation all induce corresponding group homomorphisms:

  * •

$\operatorname{Res}^{G}_{H}:T_{V}(G)\to T_{\operatorname{Res}^{G}_{H}V}(H)$

  * •

$\operatorname{Inf}^{G}_{G/N}:T_{V}(G/N)\to
T_{\operatorname{Inf}^{G}_{G/N}}(G)$

  * •

$\operatorname{Iso}_{f}:T_{V}(G)\to T_{\operatorname{Iso}_{f}(V)}(G^{\prime})$

  * •

$c_{g}:T_{V}(H)\to T_{{}^{g}V}({}^{g}H)$

######  Theorem 5.5.

Let $V$ be an absolutely $p$-divisible $kG$-module, and let $M$ be an
indecomposable $V$-endotrivial $kG$-module. $\operatorname{Res}^{G}_{S}M\cong
k\oplus M^{\prime}$ with
$M^{\prime}\in{}_{kS}\mathbf{mod}(\operatorname{Res}^{G}_{S}V)$ if and only if
$M$ has trivial source.

In particular, $T_{V}(G,S):=\ker\operatorname{Res}^{G}_{S}$ consists of
equivalence classes of trivial source $V$-endotrivial $kG$-modules.

###### Proof.

The forward direction is [9, Lemma 3.4.1]. We prove the reverse direction:
suppose $M$ is a trivial source $V$-endotrivial $kG$-module. Then since
$\operatorname{Res}^{G}_{S}M$ is endotrivial,
$\operatorname{Res}^{G}_{S}M\cong M^{\prime}\oplus N$, with
$N\in{}_{kG}\mathbf{mod}(\operatorname{Res}^{G}_{S}V)$, and $M^{\prime}$ an
endotrivial trivial source $kS$-module. Since $M^{\prime}$ has vertex $S$ and
is trivial source, $M^{\prime}\cong k$, as desired. ∎

In particular, $T_{V}(G,S)$ is finite, as there are finitely many trivial
source modules for any finite group $G$. In the case of $V=kG$,
$T(G,S):=T_{V}(G,S)$ has been described in detail by Balmer in [2] and by
Grodal in [8] using separate methods. It is well-known that if
$S\trianglelefteq G$,
$T(G,S)=\\{k_{\omega}\mid\omega\in\operatorname{Hom}(G,k^{\times})\\}.$ We can
extend this result to $V$-endotriviality without much difficulty.

######  Proposition 5.6.

  1. (a)

If $V$ is an absolutely $p$-divisible indecomposable $kG$-module with vertex
$P$, then ${}_{kG}\mathbf{mod}(V)\subseteq{}_{kG}\mathbf{mod}(P)$.

  2. (b)

Suppose in addition that $S\trianglelefteq G$. Then
$T_{V}(G,S)=\\{k_{\omega}\mid\omega\in\operatorname{Hom}(G,k^{\times})\\}$.

###### Proof.

  1. (a)

Suppose $V\mid\operatorname{Ind}^{G}_{P}X$ for some $kG$-module $X$. Then for
any $M\in{}_{kG}\mathbf{mod}(V)$, we may write

$M\mid
V\otimes_{k}N\mid(\operatorname{Ind}^{G}_{P}X)\otimes_{k}N\cong\operatorname{Ind}^{G}_{P}(X\otimes_{k}\operatorname{Res}^{G}_{P}N).$

Thus $M$ is $P$-projective.

  2. (b)

It suffices to assume that $V$ is indecomposable, since if there exists a
trivial source $V$-endotrivial $kG$-module $E$ with $\dim_{k}E\geq 1$,
$E\in{}_{kG}\mathbf{mod}(V^{\prime})$ for some indecomposable $V^{\prime}\mid
V$. Suppose $V$ is absolutely $p$-divisible and indecomposable, then $V$ has
non-Sylow vertex.

Suppose $M\in T_{V}(G,S)$ is an indecomposable trivial source $V$-endotrivial
module. Let $S\in\operatorname{Syl}_{p}(G)$. Then
$\operatorname{Res}^{G}_{S}M\cong k\oplus P$ with
$P\in{}_{kS}\mathbf{triv}(\operatorname{Res}^{G}_{S}V)$. Since $M$ has vertex
$S$, it is a direct summand of $k[G/S]$, hence $\operatorname{Res}^{G}_{S}M$
is a summand of

$\operatorname{Res}^{G}_{S}k[G/S]\cong\bigoplus_{x\in[G/S]}k[S/(S\cap{}^{x}S)]\cong
k^{\oplus[G:S]}.$

Since $\operatorname{Res}^{G}_{S}M$ is
$\operatorname{Res}^{G}_{S}V$-endotrivial, it must have one summand with
vertex $S$ which is necessarily $k[S/S]$, and all other terms must be
$\operatorname{Res}^{G}_{S}V$-projective. However, all indecomposable summands
of $\operatorname{Res}^{G}_{S}k[G/S]$, hence $\operatorname{Res}^{G}_{S}M$,
have vertex $S$ and are isomorphic to $k$. If $M$ has $k$-dimension greater
than one, then $\operatorname{Res}^{G}_{S}M$ necessarily decomposes into a
direct sum of trivial modules, which is not $V$-endotrivial, thus $M$ has
$k$-dimension one, as desired.

∎

We refer the reader to [9] and [10] for more details on $V$-endotrivial
modules.

##  6\. $V$-endotrivial complexes

Notation: For the rest of this paper, $V$ is an absolutely $p$-divisible
$kG$-module, and $\mathcal{X}_{V}\subseteq s_{p}(G)$ is the set of
$p$-subgroups of $G$ for which $V(P)=0$. Note that since $V$ is absolutely
$p$-divisible, $\mathcal{X}_{V}$ contains all Sylow $p$-subgroups of $G$.

######  Definition 6.1.

Let $C\in Ch^{b}({}_{kG}\mathbf{triv})$.

  1. (a)

$C$ is weakly $V$-endotrivial if $\operatorname{End}_{k}(C)\cong
C^{*}\otimes_{k}C\simeq k[0]\oplus D$, where $D$ is a bounded chain complex of
$V$-projective $kG$-modules.

  2. (b)

$C$ is strongly $V$-endotrivial if $\operatorname{End}_{k}(C)\cong
C^{*}\otimes_{k}C\simeq k[0]\oplus D$, where $D$ is a bounded $V$-projective
$kG$-chain complex.

  3. (c)

$C$ is $V$-endosplit-trivial if $\operatorname{End}_{k}(C)\cong
C^{*}\otimes_{k}C\simeq k[0]\oplus M[0]$, where $M$ is a $V$-projective
$kG$-module.

When it is useful to speak in generality, we say a chain complex is
$V$-endotrivial to refer to any one of the three definitions. If $V=0$, we say
$C$ is endotrivial, in which case all three definitions vacuously coincide.
This coincides with the definition of an endotrivial complex given in [17].

######  Remark 6.2.

It is clear that a $V$-endosplit-trivial complex is strongly $V$-endotrivial,
and a strongly $V$-endotrivial complex is weakly $V$-endotrivial. Moreover,
$V$-endosplit-trivial complexes are always examples of endosplit
$p$-permutation resolutions up to a shift in degree; we discuss endosplit
$p$-permutation resolutions in Section 8.

By Proposition 3.13, given any $V$, there exists a $p$-permutation $kG$-module
$W$ such that the classes of weakly $V$-endotrivial (resp. $V$-endosplit-
trivial) complexes and the classes of weakly $W$-endotrivial (resp.
$W$-endosplit-trivial) complexes coincide. Therefore, when considering weakly
$V$-endotrivial complexes or $V$-endosplit-trivial complexes, it suffices to
assume $V$ is $p$-permutation, and more generally, the only types of weak
$V$-endotriviality or $V$-endosplit-triviality are those which arise from
projectivity relative to subgroups.

It is not clear if the same can be said for strong $V$-endotriviality in
general, as we do not have a version of Proposition 3.13 for chain complexes.

For each version of $V$-endotriviality, we have a corresponding group
parameterizing it. These definitions will correspond to the category which
they live in - weakly $V$-endotrivial complexes are the invertible objects of
$K^{b}({}_{kG}\underline{\mathbf{triv}}_{V})$, while strongly $V$-endotrivial
complexes are the invertible objects of
$\underline{K}^{b}({}_{kG}\mathbf{triv})_{V}$, and $V$-endosplit-trivial
complexes are particular examples of strongly $V$-endotrivial complexes. We
will see later that some of these groups can be defined analogously to how
groups of (relatively) endotrivial modules are defined.

######  Definition 6.3.

  1. (a)

We define the group of weakly $V$-endotrivial chain complexes
$\mathcal{W}\mathcal{E}^{V}_{k}(G)$ to be the set of all isomorphism classes
of endotrivial complexes in the category
$K^{b}({}_{kG}\underline{\mathbf{triv}}_{V})$.
$\mathcal{W}\mathcal{E}^{V}_{k}(G)$ forms an abelian group with group addition
induced from $\otimes_{k}$ and inverse induced by taking duals.

  2. (b)

We define the group of strongly $V$-endotrivial chain complexes
$\mathcal{S}\mathcal{E}^{V}_{k}(G)$ to be the set of all isomorphism classes
of endotrivial complexes in the category
$\underline{K}^{b}({}_{kG}\mathbf{triv})_{V}$.
$\mathcal{S}\mathcal{E}^{V}_{k}(G)$ forms an abelian group with group addition
induced from $\otimes_{k}$ and inverse induced by taking duals.

  3. (c)

We define the group of $V$-endosplit-trivial chain complexes
$e\mathcal{E}^{V}_{k}(G)$ to be the set of all isomorphism classes of
endotrivial complexes in the category
$\underline{K}^{b}({}_{kG}\mathbf{triv})_{V}$. $e\mathcal{E}^{V}_{k}(G)$ forms
an abelian group with group addition induced from $\otimes_{k}$ and inverse
induced by taking duals. Moreover,
$e\mathcal{E}^{V}_{k}(G)\leq\mathcal{S}\mathcal{E}^{V}_{k}(G)$.

We write $x\mathcal{E}^{V}_{k}(G)$ to denote any of the above three groups.
For a $V$-endotrivial complex $C$, let $[C]\in x\mathcal{E}^{V}_{k}(G)$ denote
the corresponding class in the group. Set
$\mathcal{E}_{k}(G):=e\mathcal{E}^{\\{0\\}}_{k}(G)$. This definition of
$\mathcal{E}_{k}(G)$ is the same as the construction given in [17].

If $U,V$ are two $kG$-modules satisfying
${}_{kG}\mathbf{mod}(U)\subseteq{}_{kG}\mathbf{mod}(V)$, then we have a group
homomorphism $x\mathcal{E}^{U}_{k}(G)\to x\mathcal{E}^{V}_{k}(G)$. This is
injective when $x=\mathcal{S}$ or $e$, via the next lemma.

######  Proposition 6.4.

Let $U,V$ be absolutely $p$-divisible $kG$-modules (allowing $U=0$) such that
${}_{kG}\mathbf{mod}(U)\subseteq{}_{kG}\mathbf{mod}(V)$. The homomorphism
$\iota:x\mathcal{E}_{k}^{U}(G)\to x\mathcal{E}_{k}^{V}(G)$ induced by
$[C]\mapsto[C]$ is well-defined, and if $x=\mathcal{S}$ or $e$, $\iota$ is
injective.

###### Proof.

Well-definedness is straightforward since if two endotrivial chain complexes
$C_{1},C_{2}$ are isomorphic in $k^{b}({}_{kG}\underline{\mathbf{mod}}_{U})$
(resp. $\underline{K}^{b}({}_{kG}\mathbf{mod})_{U}$), they are isomorphic in
$k^{b}({}_{kG}\underline{\mathbf{mod}}_{V})$ (resp.
$\underline{K}^{b}({}_{kG}\mathbf{mod})_{V}$).

Assume $x=\mathcal{S}$ or $e$. $\ker\iota$ consists of all equivalence classes
of strongly $U$-endotrivial (resp. $U$-endosplit-trivial) complexes which,
upon being considered strongly $V$-endotrivial (resp. $V$-endosplit-trivial),
have indecomposable endotrivial summand isomorphic to $k[0]$. However, it is
straightforward to see this is simply the equivalence class $[k[0]]$, hence
$\ker\iota$ is trivial. ∎

######  Remark 6.5.

In particular, we have group homomorphisms $\iota:\mathcal{E}_{k}(G)\to
x\mathcal{E}_{k}^{V}(G)$ induced by considering an endotrivial $kG$-complex as
a $V$-endotrivial $kG$-complex. If $x=e$ or $\mathcal{S}$, this is an
inclusion. It is not, however, for $x=\mathcal{W}$. For example, if $G=C_{2}$
and $V=kC_{2}$, the endotrivial chain complex $kC_{2}\twoheadrightarrow k$
with $k$ in degree 0 is identified with the trivial weakly $V$-endotrivial
chain complex $k[0]$.

Additionally, we have a group homomorphism
$\mathcal{S}\mathcal{E}^{V}_{k}(G)\to\mathcal{W}\mathcal{E}^{V}_{k}(G)$
induced by the projection $\underline{K}^{b}({}_{kG}\mathbf{triv})_{V}\to
K^{b}({}_{kG}\underline{\mathbf{triv}}_{V})$. This homomorphism is in general
not injective. Finally, we have a homomorphism $T_{V}(G,S)\to
x\mathcal{E}_{k}^{V}(G)$ given by $[M]\mapsto[M[0]]$, which is easily checked
to be injective.

######  Proposition 6.6.

If $C$ is a weakly $V$-endotrivial (resp. strongly $V$-endotrivial,
$V$-endosplit-trivial) complex of $kG$-modules. $C$ contains a unique up to
isomorphism (in $Ch^{b}({}_{kG}\mathbf{triv})$) indecomposable weakly
$V$-endotrivial (resp. strongly $V$-endotrivial, $V$-endosplit-trivial)
complex of $kG$-modules as direct summand, and all other direct summands of
$C$ are isomorphic to the zero complex in
$K^{b}({}_{kG}\underline{\mathbf{triv}}_{V})$ (resp.
$\underline{K}^{b}({}_{kG}\mathbf{triv})_{V}$, resp.
$\underline{K}^{b}({}_{kG}\mathbf{triv})_{V}$).

###### Proof.

Let ${}_{kG}\underline{\mathcal{C}}_{V}$ denote either
$K^{b}({}_{kG}\underline{\mathbf{triv}}_{V})$ or
$\underline{K}^{b}({}_{kG}\mathbf{triv})_{V}$. Suppose $C$ is $V$-endotrivial
and is decomposable. Write $C=C_{1}\oplus C_{2}$, then

$(C_{1}\oplus C_{2})^{*}\otimes_{k}(C_{1}\oplus
C_{2})\cong\operatorname{End}_{k}(C_{1})\oplus\operatorname{End}_{k}(C_{2})\oplus(C_{1}^{*}\otimes_{k}C_{2})\oplus(C_{1}^{*}\otimes_{k}C_{2})^{*}\cong
k$

in ${}_{kG}\underline{\mathcal{C}}_{V}$. $(C_{1}\otimes_{k}C_{2}^{*})$ cannot
have direct summand isomorphic to $k$, since if it did, so would its dual.
Therefore, exactly one value of $i\in\\{1,2\\}$ satisfies
$k\mid\operatorname{End}_{k}(C_{i})$, and all other terms in the above
isomorphism are isomorphic to the zero complex in
${}_{kG}\underline{\mathcal{C}}_{V}$. Without loss of generality, assume
$i=1$. Then,

$C_{2}\cong(C_{1}\otimes_{k}C_{1}^{*})\otimes_{k}C_{2}\cong
C_{1}\otimes_{k}(C_{1}^{*}\otimes_{k}C_{2})\cong C_{1}\otimes_{k}0\cong 0.$

Thus $C_{2}$ is the zero complex in ${}_{kG}\underline{\mathcal{C}}_{V}$. It
follows that $\operatorname{End}_{k}(C_{1})\cong k$, and repeating this
argument until $C_{1}$ is indecomposable yields the result. ∎

######  Definition 6.7.

Let $C$ be a $V$-endotrivial complex. Call the unique indecomposable
$V$-endotrivial direct summand of $C$ the cap of $C$.

######  Remark 6.8.

  1. (a)

It is important to note that an element
$X\in\mathcal{W}\mathcal{E}_{k}^{V}(G)$ may have more than one representative
which is, as a chain complex of $kG$-modules, indecomposable. The previous
proposition asserts however that all such complexes are isomorphic in the
category $K^{b}({}_{kG}\underline{\mathbf{triv}}_{V})$. However, the number of
isomorphism classes of chain complexes in a single isomorphism class in
$K^{b}({}_{kG}\underline{\mathbf{triv}}_{V})$ is very large. On the other
hand, the isomorphism classes of chain complexes in a single isomorphism class
of $K^{b}({}_{kG}\underline{\mathbf{triv}}_{V})$ is more easily understood due
to Theorem 4.10.

  2. (b)

From the previous proposition, we obtain an equivalent definition of the
groups $\mathcal{S}\mathcal{E}_{k}^{V}(G)$ and $e\mathcal{E}_{k}^{V}(G)$ which
is independent of $\underline{K}^{b}({}_{kG}\mathbf{triv})_{V}$ and analogous
to the construction of the Dade group. We impose the following equivalence
relation on the set of strongly $V$-endotrivial (resp. $V$-endosplit-trivial)
complexes as follows: for two strongly $V$-endotrivial (resp. $V$-endosplit-
trivial) complexes $C,D$, we write $C\sim D$ if their unique $V$-endotrivial
summands $C_{0},D_{0}$ respectively satisfy $C_{0}\cong D_{0}$. Define the
group $\mathcal{S}\mathcal{E}_{k}^{V}(G)$ (resp. $e\mathcal{E}_{k}^{V}(G)$) to
be the group generated by these equivalences classes, with group operation
induces by $\otimes_{k}$, then it is a straightforward verification to check
that these two group definitions coincide.

Since restriction, inflation, and the Brauer construction are additive
functors and naturally commute with tensor products of $p$-permutation
modules, the following proposition is a direct consequence of Lemma 3.8 for
any definition of $V$-endotriviality.

######  Proposition 6.9.

Let $H\leq G,N\trianglelefteq G,P\in s_{p}(G)$.

  1. (a)

If $C$ is a $V$-endotrivial complex of $kG$-modules,
$\operatorname{Res}^{G}_{H}C$ is a $\operatorname{Res}^{G}_{H}V$-endotrivial
complex of $kH$-modules, and restriction induces a group homomorphism
$x\mathcal{E}_{k}^{V}(G)\to
x\mathcal{E}_{k}^{\operatorname{Res}^{G}_{H}V}(H)$.

  2. (b)

If $C$ is a $V$-endotrivial complex of $k[G/N]$-modules,
$\operatorname{Inf}^{G}_{G/N}C$ is a
$\operatorname{Inf}^{G}_{G/N}V$-endotrivial complex of $kG$-modules, and and
inflation induces an injective group homomorphism
$x\mathcal{E}_{k}^{V}(G/N)\to
x\mathcal{E}_{k}^{\operatorname{Inf}^{G}_{G/N}V}(G)$.

  3. (c)

If $C$ is a $V$-endotrivial complex of $kG$-modules, $C(P)$ is a
$V(P)$-endotrivial complex of $k[N_{G}(P)/P]$-modules for any $P\in s_{P}(G)$,
and the Brauer construction induces a group homomorphism
$x\mathcal{E}_{k}^{V}(G)\to x\mathcal{E}_{k}^{V(P)}(N_{G}(P)/P)$. In
particular, if $V(P)=0$, the Brauer construction induces a group homomorphism
$x\mathcal{E}_{k}^{V}(G)\to\mathcal{E}_{k}(N_{G}(P)/P)$.

It is clear that inflation induces an injective group homomorphism, but the
kernel and cokernel of the homomorphisms induced by restriction and the Brauer
construction are less clear. We will study restriction to subgroups containing
a Sylow $p$-subgroup in the sequel. Note that $\operatorname{Res}^{G}_{H}V$ or
$V(P)$ may not be absolutely $p$-divisible. On the other hand, inflation
always preserves absolute $p$-divisibility.

######  Proposition 6.10.

Let $V$ be an absolutely $p$-divisible $kG$-module, and let $C$ be an
indecomposable $V$-endotrivial complex. Then $C$ has vertex set
$\operatorname{Syl}_{p}(G)$.

###### Proof.

It suffices to show this for weakly $V$-endotrivial complexes. $C(S)\neq 0$
for all $S\in\operatorname{Syl}_{p}(G)$ since $V$ is not $p$-divisible,
therefore by properties of $p$-permutation modules, has components which are
not $P$-projective for any $P<S$. Observe that if a chain complex $C$ is
$H$-projective, then each of its components is $H$-projective as well. $C$
contains at least one indecompsable $p$-permutation module with vertex $S$
since $C(S)$ is nonzero (as $C(S)\otimes_{k}C(S)^{*}\simeq k$), so we are
done. ∎

Less attention will be devoted to strong $V$-endotriviality in this paper, as
this definition seems to be much more difficult to study. For instance, it is
not clear if $\mathcal{S}\mathcal{E}_{k}^{V}(G)$ is finitely generated, but we
will show in the sequel that the other two groups are, and that we have a
better understanding of their group structure.

##  7\. An equivalent definition of weak $V$-endotriviality

Recall that by Proposition 3.13, for weak $V$-endotriviality and
$V$-endosplit-triviality, we may always assume $V$ is $p$-permutation. We will
prove an important equivalent formulation of weak $V$-endotriviality, under
the assumption that $V$ is $p$-permutation.

######  Theorem 7.1.

Let $C$ be a chain complex of $p$-permutation $kG$-modules, and let
$\mathcal{X}$ be a subset of the $p$-subgroups of $G$ which is closed under
$G$-conjugation and taking subgroups. The following are equivalent:

  1. (a)

For all $P\not\in\mathcal{X}$, $C(P)$ is acyclic.

  2. (b)

There exists a chain complex $D$ with $C\simeq D$ such that for all
$i\in\mathbb{Z}$, $D_{i}$ is $\mathcal{X}$-projective or zero.

###### Proof.

See [17, Theorem 5.6]. ∎

######  Theorem 7.2.

Let $C\in Ch^{b}({}_{kG}\mathbf{triv})$, let $V$ be a $p$-permutation
$kG$-module which is absolutely $p$-divisible, and let $\mathcal{X}_{V}$ be
the set of $p$-subgroups $P$ of $G$ for which $V(P)=0$. $C$ is weakly
$V$-endotrivial if and only if for all $P\in\mathcal{X}_{V}$, $C(P)$ has
nonzero homology concentrated in one degree, with the nontrivial homology
having $k$-dimension one.

###### Proof.

Note that since $V$ is $p$-divisible, $\mathcal{X}_{V}$ contains Sylow
$p$-subgroups. First, suppose $C$ is $V$-endotrivial, i.e.
$C\otimes_{k}C^{*}\simeq k\oplus D$, for some chain complex $D$ which contains
only $V$-projective modules. Note that for any $V$-projective module $M$ and
$P\in\mathcal{X}_{V}$, $M(P)=0$. For any $p$-subgroup $P\in\mathcal{X}_{V}$,
we have

$k\cong(k\oplus D)(P)\simeq(C\otimes_{k}C^{*})(P)\cong
C(P)\otimes_{k}(C^{*})(P)\cong C(P)\otimes_{k}(C(P))^{*}.$

It follows by the Künneth formula that $C(P)$ has nonzero homology
concentrated in one degree with $k$-dimension one.

Conversely, suppose for all $p$-subgroups $P\in\mathcal{X}_{V}$, $C(P)$ has
homology concentrated in one degree with $k$-dimension one. We have the
tensor-hom adjunction

$\operatorname{Hom}_{kG}(C\otimes_{k}-,-)\cong\operatorname{Hom}_{kG}(-,\operatorname{Hom}(C,-)),$

and taking the unit we obtain an injective chain complex homomorphism
$\phi:k\to\operatorname{Hom}_{kG}(C,k[0])\otimes_{k}C\cong C^{*}\otimes_{k}C$.
Let $D$ be the mapping cone of $\phi$. For $P\in\mathcal{X}_{V}$, it follows
by assumption that $\phi(P)$ induces a homotopy equivalence
$k[0]\simeq(C^{*}\otimes_{k}C)(P)$, so in particular, $D(P)$ is contractible.
Therefore, by Theorem 7.1, $D$ is homotopy equivalent to a chain complex of
$(s_{p}(G)\setminus\mathcal{X}_{V})$-projective modules, or equivalently,
$V$-projective modules. Since $S\in\mathcal{X}_{V}$, it follows by considering
the mapping cone structure that $D$ has as a direct summand the chain complex
$k\xrightarrow{\sim}k$, implying $C\otimes_{k}C^{*}\simeq k\oplus E$, for some
bounded chain complex $E$ containing only $V$-projective modules. ∎

######  Remark 7.3.

When $V$ is projective, this theorem is analogous to a result of Rouquier,
[22, Theorem 5.6], which is as follows. Let $A$ and $B$ be block algebras with
a common defect group $D$. A complex $C$ of $p$-permutation $(A,B)$-bimodules
with diagonal vertices is a splendid stable equivalence if and only if
$eC(\Delta(P))e$ is a splendid Rickard complex for all $P\leq D$, where
$(P,e)$ is a $A$-Brauer pair.

In fact, weakly $kG$-endotrivial complexes induce splendid stable
equivalences. Denote by $\Delta G\leq G\times G$ the subgroup $\\{(g,g)\in
G\times G\mid g\in G\\}$.

######  Theorem 7.4.

Let $C$ be a weakly $kG$-endotrivial complex of $kG$-modules, which is
identified as a complex of $k\Delta G$-modules. Then
$\operatorname{Ind}^{G\times G}_{\Delta G}C$, regarded as a complex of
$(kG,kG)$-bimodules, induces a splendid stable equivalence on $kG$.

###### Proof.

The proof follows in the same manner as the series of proofs given in [17,
Section 4] that endotrivial complexes induce splendid Rickard complexes. ∎

##  8\. Endosplit $p$-permutation resolutions

In [21, Section 7], Rickard introduced the notion of an endosplit
$p$-permutation resolution (in which they are termed endosplit permutation
resolutions). $V$-endosplit-trivial complexes are, after a shift of degree,
examples of endosplit $p$-permutation resolutions, hence the naming
convention.

######  Definition 8.1.

[21, Definition 7.1] let $M$ be a $kG$-module. A chain complex of
$p$-permutation modules $C$ is said to be an endosplit $p$-permutation
resolution of $M$ if $H_{0}(C)\cong M$, $H_{i}(C)=0$ for all $i\neq 0$, and
$C^{*}\otimes_{k}C$ is split as a chain complex of $kG$-modules.

######  Remark 8.2.

  1. (a)

In particular, the Künneth formula implies
$C^{*}\otimes_{k}C\simeq(M^{*}\otimes_{k}M)[0],$ since split chain complexes
are homotopy equivalent to their homology, regarded as a chain complex with
zero differentials.

Suppose $M$ is a $V$-endotrivial module, as defined in [9], and suppose $C$ is
an endosplit $p$-permutation resolution for $M$. We have
$C^{*}\otimes_{k}C\simeq(M^{*}\otimes_{k}M)[0]\cong k\oplus P$ for some
$V$-projective $kG$-module $P$, hence $C$ is a $V$-endosplit-trivial complex.
Conversely, it is easy to see that any $V$-endosplit-trivial complex is, up to
a shift in degrees, an endosplit $p$-permutation resolution, and it follows
that $V$-endosplit-trivial complexes are equivalently shifted endosplit
$p$-permutation resolutions for $V$-endotrivial modules. Moreover, if $M$ is
one-dimensional, then $C$ is an endotrivial complex.

  2. (b)

In [21], it was proven that endosplit $p$-permutation resolutions exist for
any endopermutation $kP$-module, where $P$ is an abelian $p$-group. Using the
classification of endotrivial modules for $p$-groups, it was shown by Mazza in
[15] that if $p$ is odd, every endotrivial $kP$-module has an endosplit
$p$-permutation resolution. On the other hand, some endotrivial modules such
as the exotic endotrivial $kQ_{8}$-module does not have an endosplit
$p$-permutation resolution.

  3. (c)

Let $(K,\mathcal{O},k)$ be a $p$-modular system. [21, Proposition 7.1] implies
that every $V$-endosplit-trivial $kG$-complex $C$ has a unique lift to
$\mathcal{O}$, i.e. there exists a unique isomorphism class of chain complex
$\hat{C}$ of $p$-permutation $\mathcal{O}G$-modules such that
$k\otimes_{\mathcal{O}}\hat{C}\cong C$.

Similar to weakly $V$-endotrivial complexes, $V$-endosplit-trivial complexes
have an equivalent characterization arising via the Brauer construction. This
characterization mirrors the characterization of endotrivial complexes in [17,
Theorem 3.4].

######  Theorem 8.3.

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

  1. (a)

Let $M$ be a $kG$-module which has an endosplit $p$-permutation resolution
$C$. Then for all $P\in s_{p}(G)$, $C(P)$ has nonzero homology concentrated in
exactly one degree. Moreover, $M$ is $V$-endotrivial if and only if $C$
additionally is weakly $V$-endotrivial.

  2. (b)

Conversely, let $C$ be a chain complex which satisfies that for all $P\in
s_{p}(G)$, $C(P)$ has nontrivial homology concentrated in exactly one degree,
and $H_{0}(C)\neq 0$. Then $C$ is an endosplit $p$-permutation resolution for
$H_{0}(C)$. Furthermore, $C$ is a $V$-endosplit-trivial complex if and only if
$H_{0}(C)$ is a $V$-endotrivial module.

###### Proof.

  1. (a)

For any $P\in s_{p}(G)$,

$C(P)^{*}\otimes_{k}C(P)\cong(C^{*}\otimes_{k}C)(P)\simeq((M^{*}\otimes_{k}M)[0])(P).$

By the Künneth formula, $C(P)$ has nonzero homology in exactly one degree. For
the last statement, we have $M\otimes_{k}M^{*}\cong k\oplus N$ for some
$V$-projective $N$ if and only if $C\otimes_{k}C^{*}\simeq(k\oplus N)[0]$,
whence the result.

  2. (b)

By the Künneth formula, it follows that for any $P\in s_{p}(G)$,

$C(P)^{*}\otimes_{k}C(P)\cong(C^{*}\otimes_{k}C)(P)$

has homology concentrated in degree 0. Let $i$ (resp. $j$) denote the largest
(resp. smallest) integer for which $(C^{*}\otimes_{k}C)_{i}$ (resp.
$(C^{*}\otimes_{k}C)_{j}$) is nonzero. Observe that by considering homology,
$d_{i}(P)$ (resp. $d_{j+1}(P)$) is injective (resp. surjective) for all $P\in
s_{p}(G)$. By properties of $p$-permutation maps (see [12, Theorem 5.8.11]),
$d_{i}$ (resp. $d_{j+1}$) is split injective (resp. split surjective), so
$C^{*}\otimes_{k}C$ is homotopy equivalent to a chain complex $D$ with two
fewer nonzero terms, and with $D(P)$ having nontrivial homology concentrated
in degree zero for all $P\in s_{p}(G)$.

An inductive argument demonstrates that $C(P)^{*}\otimes_{k}C(P)\simeq M[0]$
for some $kG$-module $M$, and the Künneth formula implies $M\cong
H_{-i}(C^{*})\otimes_{k}H_{i}(C)$ for the unique $i\in\mathbb{Z}$ for which
$H_{i}(C)\neq 0$. The result follows.

∎

The following corollary is immediate.

######  Corollary 8.4.

Let $V$ be an absolutely $p$-divisible $kG$-module and $C\in
Ch^{b}({}_{kG}\mathbf{triv})$. The following are equivalent:

  1. (a)

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

  2. (b)

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

  3. (c)

$C(P)$ has nonzero homology concentrated in exactly one degree for all $P\in
s_{p}(G)$, and for the unique $i\in\mathbb{Z}$ satisfying $H_{i}(C)\neq 0$,
$H_{i}(C)$ is a $V$-endotrivial $kG$-module.

######  Remark 8.5.

As before, it follows by commutativity with tensor products that applying
restriction, inflation, or the Brauer construction to an endosplit
$p$-permutation resolution will result in an endosplit $p$-permutation
resolution. Additionally, we will see in the sequel that if $H\leq G$,
applying additive induction to an endosplit $p$-permutation resolutions which
satisfies a fusion condition results in an endosplit $p$-permutation
resolution. However, this does not preserve $V$-endosplit-triviality.

We next recall a few important results on endosplit $p$-permutation
resolutions proven by Rickard in [21]. For the first theorem, we state the
version provided in [13].

######  Theorem 8.6.

[13, 7.11.2] Let $M$ be a finitely generated $kG$-module having an endosplit
$p$-permutation resolution $C$. For any subgroup $H\leq G$, the complex
$\operatorname{Hom}_{kG}(C,C)$ has homology concentrated in degree zero,
isomorphic to $\operatorname{Hom}_{kH}(M,M)$, and there is an algebra
isomorphism

$\rho_{H}:\operatorname{Hom}_{K^{b}({}_{kH}\textbf{mod})}(C,C)\cong\operatorname{Hom}_{kH}(M,M)$

satisfying
$\operatorname{Res}^{G}_{H}\circ\rho_{G}=\rho_{H}\circ\operatorname{Res}^{G}_{H}$
and
$\operatorname{tr}^{G}_{H}\circ\rho_{H}=\rho_{G}\circ\operatorname{tr}^{G}_{H}$.
In particular, $\rho_{G}$ induces a vertex and multiplicity preserving
bijection between the sets of isomorphism classes of indecomposable direct
summands of the module $M$ and of noncontractible indecomposable direct
summands of the complex $C$.

######  Proposition 8.7.

[21, 7.3, 7.4, 7.5, 7.6]

  1. (a)

Let $N\trianglelefteq G$. If $C$ is an endosplit $p$-permutation resolution of
the $k[G/N]$-module $M$, then $\operatorname{Inf}^{G}_{G/N}M$ is an endosplit
$p$-permutation resolution of $\operatorname{Inf}^{G}_{G/N}M$.

  2. (b)

If $C,D$ are endosplit $p$-permutation resolutions of $kG$-modules $M,N$, then
$C\otimes_{k}D$ is an endosplit $p$-permutation resolution of $M\otimes_{k}N$.

  3. (c)

If $M,N$ are $kG$-modules with $M\oplus N$ having an endosplit $p$-permutation
resolution $C$, then $C=C_{M}\oplus C_{N}$, with $C_{M},C_{N}$ endosplit
$p$-permutation resolutions of $M,N$ respectively.

  4. (d)

If $M$ has an endosplit $p$-permutation resolution, any module that is a
finite direct sum of summands of the modules
$\operatorname{Ind}^{G}_{H}\operatorname{Res}^{G}_{H}M$ for all $H\leq G$ has
an endosplit $p$-permutation resolution.

One may ask the opposition question to (c): given endosplit $p$-permutation
resolutions $C,D$, when is $C\oplus D$ an endosplit $p$-permutation as well?
The answer lies in the local structure of $C$ and $D$ which we discuss in the
next section.

##  9\. h-mark homomorphisms

Similar to the study of endotrivial complexes, we have h-mark homomorphisms
which determine much of the structure of an endotrivial complex, particularly
for weakly $V$-endotrivial complexes and $V$-endosplit-trivial complexes. For
this section we assume $V$ is an absolutely $p$-divisible $kG$-module, and we
set $\mathcal{X}_{V}\subseteq s_{p}(G)$ to be the subset of all $p$-subgroups
for which $V(P)=0$.

######  Definition 9.1.

Suppose $\mathcal{X}$ is a subset of the set of subgroups of $G$ which is
closed under $G$-conjugation. $C(G,\mathcal{X})$ denotes the group of
$\mathbb{Z}$-valued class functions valued on $\mathcal{X}$, i.e. the group of
functions $f:\mathcal{X}\to\mathbb{Z}$ which are constant on conjugacy
classes. We set $C(G,p):=C(G,s_{p}(G))$.

Normally, $C(G)$ refers to the set of class functions on $G$. However, in the
context of this paper, $C(G)$ can also refer to the Brauer construction, so to
avoid confusion, we always write $C(G,p)$ to refer to the set of class
functions, even if $G$ is a $p$-group. For the scope of this paper, we are
only concerned with class functions valued on subsets of $s_{p}(G)$.

###  9.1. h-mark homomorphisms for weak and strong $V$-endotrivial complexes

The definitions in this section are motivated by the fact that if $C$ is a
weakly or strongly $V$-endotrivial complex, and if $P\in\mathcal{X}_{V}$,
$C(P)$ is an endotrivial complex.

######  Definition 9.2.

Let $V$ be an absolutely $p$-divisible $kG$-module. Let $x=\mathcal{S}$ or
$\mathcal{W}$. Recall that for any $P\in\mathcal{X}_{V}$, $C(P)$ is an
endotrivial complex.

For a class of weak or strong $V$-endotrivial complexes $[C]\in
x\mathcal{E}_{k}^{V}(G)$ and $p$-subgroup $P$ for which $V(P)=0$, denote by
$h_{C}(P)$ the degree $i\in\mathbb{Z}$ of a representative $C$ in which
$H_{i}(C(P))\neq 0$, and abusing notation, if we have
$H_{h_{C}(P))}(C(P))\cong k_{\omega}$, set
$H_{C}(P):=\omega\in\operatorname{Hom}(N_{G}(P)/P,k^{\times})$. Note this is
well-defined since $C(P)$ is an endotrivial complex. We call the values of
$h_{C}(P)$ the h-marks of $C$ at $P$, and abusively refer to $H_{C}(P)$ as the
homology of $C$ at $P$.

We have a map:

$\displaystyle\Xi:x\mathcal{E}^{V}_{k}(G)$
$\displaystyle\to\prod_{P\in\mathcal{X}_{V}}\mathbb{Z}\times\operatorname{Hom}(N_{G}(P)/P,k^{\times})$
$\displaystyle[C]$ $\displaystyle\mapsto\big{(}h_{C}(P),H_{C}(P)\big{)}_{P\in
s_{p}(G)}$

It is straightforward to verify that this map is a well-defined group
homomorphism via the Künneth formula and commutativity of the Brauer
construction with tensor products for $p$-permutation modules.

Note that for any weak or strong $V$-endotrivial complex $C$, we may regard
$h_{C}$ as a $\mathbb{Z}$-valued class function on $\mathcal{X}_{V}$. Indeed,
the Brauer construction satisfies $C({}^{g}P)={}^{g}C(P)$ for any $g\in G$ and
$P\in s_{p}(G)$. This gives another group homomorphism, which we call the
h-mark homomorphism,

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

One may easily verify that this is well-defined, that is, if
$[C_{1}]=[C_{2}]\in x\mathcal{E}_{k}^{V}(G)$, the $h_{C_{1}}=h_{C_{2}}\in
C(G,\mathcal{X}_{V})$.

We can precisely determine $\ker h_{\mathcal{W}}$ for weakly $V$-endotrivial
complexes. For strongly $V$-endotrivial complexes, determining $\ker
h_{\mathcal{S}}$ is a bit less clear, but we can discern some properties which
the elements have.

######  Lemma 9.3.

Let $C$ be a bounded complex of $p$-permutation $kG$-modules with the
following property: there exists a set $\mathcal{Y}\subset s_{p}(G)$ which
does not contain Sylow subgroups and is closed under conjugation and taking
subgroups, and which satisfies that there exists some $i\in\mathbb{Z}$ such
that for all $P\not\in\mathcal{Y}$, $\dim_{k}H_{i}(C(P))=1$ and $C(P)$ is
exact in all other degrees. $C$ is homotopy equivalent to an complex
$C^{\prime}$, with the following property: there exists an $i\in\mathbb{Z}$
such that $C^{\prime}_{i}=M\oplus N$, with $M$ $\mathcal{Y}$-projective or 0
and $N$ has Sylow vertices, and for all $j\neq i$, either $C_{j}^{\prime}=0$
or $C_{j}^{\prime}$ is $\mathcal{Y}$-projective.

###### Proof.

See [17, Lemma 5.8]. ∎

######  Corollary 9.4.

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

  1. (a)

For $h_{\mathcal{W}}:\mathcal{W}\mathcal{E}^{V}_{k}(G)\to
C(G,\mathcal{X}_{V})$, we have

$\ker h_{\mathcal{W}}\cong T_{V}(G,S).$

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

  2. (b)

For $h_{\mathcal{S}}:\mathcal{S}\mathcal{E}^{V}_{k}(G)\to
C(G,\mathcal{X}_{V})$, elements of $\ker h_{\mathcal{S}}$ have an
indecomposable representative $C$ which contains exactly one trivial source
module degree 0 with Sylow vertices, and all other module direct summands in
any degree have vertices contained in $s_{p}(G)\setminus\mathcal{X}_{V}$.

###### Proof.

Note $T_{V}(G,S)\subseteq\ker h_{\mathcal{W}}$ and $T_{V}(G,S)\subseteq\ker
h_{\mathcal{S}}$. For (a), elements of $\ker h_{\mathcal{W}}$ are
$V$-endotrivial complexes $C$ which given any $P\in\mathcal{X}_{V}$,
$\dim_{k}H_{0}(C(P))=1$ and $C(P)$ is exact in all other degrees. One may
check that any such $C$ satisfies the hypothesis of the previous lemma, with
$\mathcal{Y}=s_{p}(G)\setminus\mathcal{X}_{V}$. For weakly $V$-endotrivial
complexes, $V$-projectivity is the same as $\mathcal{Y}$-projectivity. Thus,
$C_{0}$ has an indecomposable summand with vertex $S$, and all other
indecomposable modules appearing in $C$ are $V$-projective. Therefore, $C\cong
M[0]$ for some $p$-permutation $V$-endotrivial module $M$ in
$K^{b}({}_{kG}\underline{\mathbf{triv}}_{V})$, and the result follows. The
final statement of (a) follows since $T_{V}(G,S)$ is finite (as there is a
finite number of isomorphism classes trivial source $kG$-modules), hence
torsion, whereas any complex with a nonzero h-mark cannot be torsion, as
successive powers of the complex will increase the absolute value of the
nonzero h-mark.

An analogous argument also using the previous lemma shows (b). ∎

######  Remark 9.5.

In particular, since $\ker h_{\mathcal{W}}$ is the torsion subgroup of
$\mathcal{W}\mathcal{E}^{V}_{k}(G)$, we have a split exact sequence of abelian
groups

$0\to
T_{V}(G,S)\to\mathcal{W}\mathcal{E}_{k}^{V}(G)\to\operatorname{im}h_{\mathcal{W}}\to
0.$

Unlike the situation for endotrivial complexes given in [17, Remark 3.8], this
sequence is not canonically split, one must choose a basis for the free part
of $\mathcal{W}\mathcal{E}_{k}^{V}(G)$.

######  Corollary 9.6.

Two weakly $V$-endotrivial complexes are isomorphic in
$K^{b}({}_{kG}\underline{\mathbf{triv}}_{V})$ if and only if they have the
same h-marks and local homology at all $P\in\mathcal{X}_{V}$.

###### Proof.

This follows since given two weakly $V$-endotrivial complexes which satisfy
the above, they must represent the same class in
$\mathcal{W}\mathcal{E}_{k}^{V}(G)$, whence the result. ∎

###  9.2. h-marks for endosplit $p$-permutation resolutions and $V$-endosplit-
trivial complexes

Recall that a complex of $p$-permutation $kG$-modules is a shifted endosplit
$p$-permutation resolution if and only if for all $P\in s_{p}(G)$, $C(P)$ has
nonzero homology in exactly one degree. Therefore, we have an h-mark at every
$P\in s_{p}(G)$.

######  Definition 9.7.

Let $C$ be either a $V$-endosplit-trivial complex or endosplit $p$-permutation
resolution, and let $P\in s_{p}(G)$. Denote by $h_{C}(P)$ the unique degree
$i\in\mathbb{Z}$ for which $H_{i}(C(P))\neq 0$.

If $C$ is a $V$-endosplit-trivial complex, set
$\mathcal{H}_{C}(P)=[H_{h_{C}(P)}(C(P))]\in T_{V}(N_{G}(P)/P)$. We claim this
is invariant among equivalent complexes in $e\mathcal{E}_{k}^{V}(G)$. Indeed,
suppose $[C_{1}]\sim[C_{2}]\in e\mathcal{E}_{k}^{V}(G)$. Then $C_{1}$ and
$C_{2}$ have isomorphic indecomposable $V$-endosplit-trivial direct summands,
which we denote $C_{0}$. Therefore we may write $C_{1}\cong C_{0}\oplus P_{1}$
and $C_{2}\cong C_{0}\oplus P_{2}$, with $P_{1},P_{2}$ homotopy equivalent to
a $V$-projective complex. This implies the homology of $P_{1},P_{2}$ are
$V$-projective modules. Moreover, for any $P\in s_{p}(G)$,
$h_{C_{1}}(P)=h_{C_{0}}(P)=h_{C_{2}}(P)$. Thus, for any $P\in s_{p}(G)$, we
have

$\displaystyle\mathcal{H}_{C_{1}}(P)$
$\displaystyle=[H_{h_{C_{1}}(P)}(C_{0}(P))\oplus H_{h_{C_{1}}(P)}(P_{1}(P))]$
$\displaystyle=[H_{h_{C_{0}}(P)}(C_{0}(P))]$
$\displaystyle=[H_{h_{C_{2}}(P)}(C_{0}(P))\oplus H_{h_{C_{2}}(P)}(P_{2}(P))]$
$\displaystyle=\mathcal{H}_{C_{2}}(P)\in T_{V}(N_{G}(P)/P).$

As before, we call the values of $h_{C}(P)$ the h-marks of $C$ at $P$, and
abusively refer to $\mathcal{H}_{C}(P)$ as the homology of $C$ at $P$. Note
that here, we are taking the image of homology in $T_{V}(G)$ rather than the
isomorphic image, as otherwise, the map is not well-defined. By the above
remarks, we obtain a group homomorphism:

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

We identify $T_{\emptyset}(G)=\operatorname{Hom}(G,k^{\times})$ as the group
of $k$-dimension one $kG$-modules with addition given by $\otimes_{k}$.

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

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

######  Theorem 9.8.

$\ker h_{e}\cong T_{V}(G,S)$. In particular, $\ker h_{e}$ is the torsion
subgroup of $e\mathcal{E}_{k}^{V}(G)$.

###### Proof.

Note $T_{V}(G,S)\leq\ker h_{e}$. Suppose $[C]\in\ker h_{e}$. If $C_{0}$ is the
only nonzero degree of $C$, then it follows that
$C_{0}^{*}\otimes_{k}C_{0}\cong k\oplus P$, for some $V$-projective
$kG$-module $P$, i.e. $C_{0}$ is $V$-endotrivial. Moreover, $C_{0}$ is
$p$-permutation and therefore has a unique trivial source $V$-endotrivial
direct summand, whence the result.

We show via induction on the number of nonzero terms of $C$ that $C$ is
homotopy equivalent to $M[0]$, where $M$ is a $V$-endotrivial $p$-permutation
module. The base case is $C_{0}$ being the only nonzero term, which we have
resolved.

$C$ must have nonzero terms which are either in positive or negative degree.
Suppose $C$ has nonzero terms in positive degree, the negative degree argument
follows similarly. Let $i>0$ be the greatest integer for which $C_{i}\neq 0$.
Since $h_{C}=0$, $C(P)$ is exact at $i$, so $d_{i}(P)$ is injective at all
$P\in s_{p}(G)$, hence $d_{i}$ is split injective by [12, Theorem 5.8.10].
Therefore, $C\cong C^{\prime}\oplus(C_{i}\to C_{i})$, with $C^{\prime}$
containing one less nonzero term than $C$. By the inductive hypothesis, we are
done. ∎

######  Remark 9.9.

In particular, since $\ker h_{e}$ is the torsion subgroup of
$e\mathcal{E}^{V}_{k}(G)$, we have a split exact sequence

$0\to T_{V}(G,S)\to e\mathcal{E}_{k}^{V}(G)\to\operatorname{im}h_{e}\to 0.$

Unlike the situation for endotrivial complexes, this sequence is not
canonically split, one must choose a basis for the free part of
$e\mathcal{E}_{k}^{V}(G)$. The reason for this is that in general, given any
$C\in e\mathcal{E}^{V}_{K}(V)$, $\mathcal{H}_{C}(P)\in T_{V}(N_{G}(P)/P)$ may
not have a $p$-permutation representative, and therefore we do not have a
clear well-defined map from $e\mathcal{E}_{k}^{V}(G)$ to $T_{V}(G,S)$.

######  Corollary 9.10.

Two indecomposable $V$-endosplit-trivial chain complexes are isomorphic if and
only if they have the same h-marks and isomorphic nonzero homology.

Additionally, with h-marks, we can say precisely when, given two endosplit
$p$-permutation resolutions $C,D$, $C\oplus D$ is also an endosplit
$p$-permutation resolution.

######  Theorem 9.11.

Let $C,D$ be two endosplit $p$-permutation resolutions of $kG$-modules $M,N$
respectively. $C\oplus D$ is an endosplit $p$-permutation resolution if and
only if $h_{C}=h_{D}$. If this happens, $C\oplus D$ is an endosplit
$p$-permutation resolution for $M\oplus N$.

###### Proof.

For the reverse direction, suppose $h_{C}=h_{D}$ for all $P\in s_{p}(G)$.
Then, $(C\oplus D)(P)$ also has homology concentrated in one degree for all
$P\in s_{p}(G)$, therefore $C\oplus D$ is an endosplit $p$-permutation
resolution with $h_{C}=h_{D}=h_{C\oplus D}$. For the forward direction, if
$C\oplus D$ is an endosplit $p$-permutation resolution, then $(C\oplus
D)(P)\cong C(P)\oplus D(P)$ has nonzero homology in exactly one degree for any
$P\in s_{p}(G)$. Therefore, both $C(P)$ and $D(P)$ have nonzero homology in
exactly one degree as well, with $h_{C}(P)=h_{D}(P)$. The result follows. ∎

######  Remark 9.12.

If the above scenario occurs, then the endosplit $p$-permutation resolution
$C\otimes_{k}D^{*}$ has h-marks constantly 0. By the same inductive peeling of
contractible summands as used previously,
$C\otimes_{k}D^{*}\simeq(M\otimes_{k}N^{*})[0]$, implying $M\otimes_{k}N^{*}$
is a $p$-permutation module. Therefore, the above situation can only occur if
$M\otimes_{k}N^{*}$ is $p$-permutation.

In particular, letting $N:=M$ and $D:=C$ implies that the only $kG$-modules
which can possibly have endosplit $p$-permutation resolutions are
endo-$p$-permutation $kG$-modules, that is, $kG$-modules $M$ for which
$M^{*}\otimes_{k}M\cong k\oplus P$, where $P$ is a $p$-permutation module.
These were first studied in detail by Urfer in [23].

###  9.3. Lifting for endotriviality with respect to different modules

We can use h-mark homomorphisms to study group homomorphisms between the
various groups of endotrivial complexes.

######  Remark 9.13.

Recall from Proposition 6.4 for any absolutely $p$-divisible $kG$-modules
$U,V$ such that ${}_{kG}\mathbf{mod}(U)\subseteq{}_{kG}\mathbf{mod}(V)$
(allowing $U=0$), we have a homomorphism $\iota:x\mathcal{E}_{k}^{U}(G)\to
x\mathcal{E}_{k}^{V}(G)$. In the case of $x=\mathcal{S}$ or $e$, this
homomorphism is injective. We say that a $V$-endotrivial complex $C$ lifts to
a $U$-endotrivial complex if $[C]\in x\mathcal{E}_{k}^{V}(G)$ is in the image
of the homomorphism $x\mathcal{E}_{k}^{U}(G)\to x\mathcal{E}_{k}^{V}(G)$. In
the case of $V$-endosplit-triviality, the situation is as follows. We write
$h_{U}$ to denote the h-mark homomorphism for $U$-endosplit-trivial and
$h_{V}$ for $V$-endosplit-trivials.

${0}$${T_{U}(G,S)}$${e\mathcal{E}_{k}^{U}(G)}$${C(G,p)}$${0}$${T_{V}(G,S)}$${e\mathcal{E}_{k}^{V}(G)}$${C(G,p)}$$\scriptstyle{h_{U}}$$\scriptstyle{=}$$\scriptstyle{h_{V}}$

This gives rise to a split exact sequence, since all groups present are
finitely generated abelian.

${0}$${T_{U}(G,S)}$${e\mathcal{E}_{k}^{U}(G)}$${\operatorname{im}h_{U}}$${0}$${0}$${T_{V}(G,S)}$${e\mathcal{E}_{k}^{V}(G)}$${\operatorname{im}h_{V}}$${0}$$\scriptstyle{h_{U}}$$\scriptstyle{h_{V}}$

We may identify $e\mathcal{E}_{k}^{V}(G)\cong
T_{V}(G,S)\times\operatorname{im}h_{V}\leq T_{V}(G,S)\times C(G,p)$ and
similarly for $e\mathcal{E}_{k}^{U}(G)$. These are non-canonical however;
therefore, one must be sure to verify that the choices of
$\operatorname{im}h_{U}\leq e\mathcal{E}_{k}^{U}(G)$ and
$\operatorname{im}h_{V}\leq e\mathcal{E}_{k}^{V}(G)$ are compatible, i.e.
under the inclusion $e\mathcal{E}_{k}^{U}(G)\hookrightarrow
e\mathcal{E}_{k}^{V}(G)$, we have $\iota(h_{U})\subseteq h_{V}$.

In the weakly $V$-endotrivial case, we have the following diagram. Here, there
may be more h-marks for weakly $U$-endotrivial complexes than there are for
weakly $V$-endotrivial complexes, so a projection
$\operatorname{im}h_{U}\to\operatorname{im}h_{V}$ may be necessary. The same
considerations as above must hold.

${0}$${T_{U}(G,S)}$${\mathcal{W}\mathcal{E}_{k}^{U}(G)}$${\operatorname{im}h_{U}}$${0}$${0}$${T_{V}(G,S)}$${\mathcal{W}\mathcal{E}_{k}^{V}(G)}$${\operatorname{im}h_{V}}$${0}$$\scriptstyle{h_{U}}$$\scriptstyle{\pi}$$\scriptstyle{h_{V}}$

Assuming compatible identifications as in the previous discussion, we have the
following lifting theorem which relies primarily on h-marks. The proof follows
easily from the diagrams.

######  Theorem 9.14.

Let $U,V$ be absolutely $p$-divisible $kG$-modules (allowing $U=0$) for which
${}_{kG}\mathbf{mod}(U)\subseteq{}_{kG}\mathbf{mod}(V).$

  1. (a)

If $[C]\in x\mathcal{E}_{k}^{V}(G)$ for $x=e$ (resp. $x=\mathcal{W}$) has the
h-marks of a $U$-endosplit-trivial complex (resp. a weakly $U$-endotrivial
complex), there exists a trivial source $V$-endotrivial $kG$-module $M$ for
which $M[0]\otimes_{k}C$ is isomorphic in
$\underline{K}^{b}({}_{kG}\mathbf{triv})_{V}$ (resp.
$K^{b}({}_{kG}\underline{\mathbf{triv}}_{V})$) to a $U$-endosplit-trivial
complex (resp. a weakly $U$-endotrivial complex).

  2. (b)

Under compatible identifications $e\mathcal{E}_{k}^{V}(G)\cong
T_{V}(G,S)\times\operatorname{im}h_{V}$ and $e\mathcal{E}_{k}^{U}(G)\cong
T_{V}(G,S)\times\operatorname{im}h_{U}$, $([M],(a_{P})_{P\in s_{p}(G)})\in
e\mathcal{E}_{k}^{V}(G)$ lifts to an equivalence class of $U$-endosplit-
trivial complexes if and only if $M$ has a $U$-endotrivial cap and
$(a_{P})_{P\in s_{p}(G)}$ are the h-marks of some $U$-endosplit-trivial
complex.

  3. (c)

Under compatible identifications $\mathcal{W}\mathcal{E}_{k}^{V}(G)\cong
T_{V}(G,S)\times\operatorname{im}h_{V}$ and
$\mathcal{W}\mathcal{E}_{k}^{U}(G)\cong
T_{V}(G,S)\times\operatorname{im}h_{U}$,
$([M],(a_{P})_{P\in\mathcal{X}_{V}})\in e\mathcal{E}_{k}^{V}(G)$ lifts to an
equivalence class of weakly $U$-endotrivial complexes if and only if $M$ has a
$U$-endotrivial cap and $(a_{P})_{P\in\mathcal{X}_{V}}$ coincide with the
(partial) h-marks of some weakly $U$-endotrivial complex.

One question of interest is determining in the situation of (a) when one can
say that the trivial source $V$-endotrivial module $M$ is either isomorphic to
$k$ or $U$-endotrivial. If $G$ is a $p$-group, one may remove $M$ entirely,
since the only trivial source endotrivial $kG$-module is $k$.

Additionally, we can read off via homology whether a $V$-endosplit-trivial
complex lifts to a $U$-endosplit-trivial complex.

######  Theorem 9.15.

Let $U,V$ be $kG$-modules for which
${}_{kG}\mathbf{mod}(U)\subseteq{}_{kG}\mathbf{mod}(V),$ and let $C$ be a
$V$-endosplit-trivial complex. $C$ lifts to a $U$-endosplit-trivial complex if
and only if the cap of $\mathcal{H}_{C}(1)$ is a relatively $U$-projective
$kG$-module.

###### Proof.

The forward direction is clear. Suppose the converse, then since $C$ is a
shifted endosplit $p$-permutation resolution for a representative of
$\mathcal{H}_{C}(1)$, $C$ contains as a direct summand $D$, a shifted
endosplit $p$-permutation resolution for the cap of $\mathcal{H}_{C}(1)$.
Hence $D$ is $U$-endotrivial, and moreover, has the same h-marks as $C$ by
Theorem 9.11. Since $\mathcal{H}_{C}(1)=\mathcal{H}_{D}(1)\in T_{V}(G)$ and
$C$ and $D$ have the same h-marks, the result follows. ∎

###  9.4. Lifting from weakly $V$-endotrivial to $V$-endosplit-trivial
complexes

######  Remark 9.16.

We may also consider the question of if a class of weakly $V$-endotrivial
complexes $[C]\in\mathcal{W}\mathcal{E}_{k}^{V}(G)$ has a $V$-endosplit-
trivial representative. The situation is as follows, where $\pi$ denotes
function restriction, $h_{e}$ (resp. $h_{\mathcal{W}}$) denotes the h-mark
homomorphism for $V$-endosplit-trivial complexes (resp. weakly $V$-endotrivial
complexes).

${0}$${T_{V}(G,S)}$${e\mathcal{E}_{k}^{V}(G)}$${C(G,p)}$${0}$${T_{V}(G,S)}$${\mathcal{W}\mathcal{E}_{k}^{V}(G)}$${C(G,\mathcal{X}_{V})}$$\scriptstyle{=}$$\scriptstyle{h_{e}}$$\scriptstyle{\pi}$$\scriptstyle{h_{\mathcal{W}}}$

We obtain the following homomorphism of split exact sequences.

${0}$${T_{V}(G,S)}$${e\mathcal{E}_{k}^{V}(G)}$${\operatorname{im}h_{e}}$${0}$${0}$${T_{V}(G,S)}$${\mathcal{W}\mathcal{E}_{k}^{V}(G)}$${\operatorname{im}h_{\mathcal{W}}}$${0}$$\scriptstyle{=}$$\scriptstyle{h_{e}}$$\scriptstyle{\pi}$$\scriptstyle{h_{\mathcal{W}}}$

Therefore, we can identify $e\mathcal{E}_{k}^{V}(G)\cong
T_{V}(G,S)\times\operatorname{im}h_{e}\leq T_{V}(G,S)\times C(G,p)$ and
similarly, $\mathcal{W}\mathcal{E}_{k}^{V}(G)\cong
T_{V}(G,S)\times\operatorname{im}h_{\mathcal{W}}\leq T_{V}(G,S)\times
C(G,\mathcal{X}_{V})$. These identifications are non-canonical. From this
identification, the following statement follows without much difficulty.

######  Theorem 9.17.

Let $[C]\in\mathcal{W}\mathcal{E}_{k}^{V}(G)$. $[C]$ has a $V$-endosplit-
trivial complex in its equivalence class if and only if the h-marks of $C$ are
equal to the (truncated) h-marks of an $V$-endosplit-trivial complex.

We conjecture that every $[C]\in\mathcal{W}\mathcal{E}_{k}^{V}(G)$ has a
$V$-endosplit-trivial complex in its equivalence class. We provide an example
in the next section, and a strategy for finding $V$-endosplit-trivial
complexes in an equivalence class which may extend more generally to arbitrary
cases.

##  10\. The case of $V=kG$

In this section, we study the case of $V=kG$. We set $V:=kG$ for the entire
section. In this case, $V$-endotrivial modules are endotrivial modules in the
usual sense. Therefore, $V$-endosplit-trivial complexes are shifted endosplit
$p$-permutation resolutions of endotrivial modules, Recall that from Theorem
7.4 that weakly $V$-endotrivial complexes provide examples of splendid stable
equivalences.

In fact, modulo $V$-projective chain complexes, strongly $V$-endotrivial
complexes are in fact $V$-endosplit-trivial complexes.

######  Theorem 10.1.

Let $C$ be an indecomposable strongly $V$-endotrivial complex. Then $C$ is an
$V$-endosplit-trivial complex. In particular, the inclusion
$e\mathcal{E}_{k}^{kG}(kG)\hookrightarrow\mathcal{S}\mathcal{E}_{k}^{kG}(kG)$
is an isomorphism.

###### Proof.

First, note that $V$-projective chain complexes of $kG$-modules can be
expressed as direct sums of shifts of chain complexes of the form $0\to
P\xrightarrow[]{\sim}P\to 0$ or $0\to P\to 0$, where $P$ is a projective
$kG$-module. This follows since all $k$-vector space chain complexes decompose
in this way. Therefore, $C^{*}\otimes_{k}C\simeq k\oplus D$, where $D$ is a
$V$-projective chain complex, and we may assume without loss of generality
that it has no contractible components. Therefore, $D$ is a chain complex with
zero differentials and projective homology. Moreover, since
$C^{*}\otimes_{k}C\cong C\otimes_{k}C^{*}$, it follows that $D^{*}\cong D$,
and since $D$ has zero differentials, equivalently $D_{i}^{*}\cong D_{-i}$. It
suffices to show that $H_{i}(D)=0$ for all $i\neq 0$.

We have by the Künneth formula and exactness of the $k$-dual that

$H_{0}(C^{*}\otimes_{k}C)\cong\bigoplus_{i\in\mathbb{N}_{\geq
0}}H_{-i}(C^{*})\otimes_{k}H_{i}(C)\cong\bigoplus_{i\in\mathbb{N}_{\geq
0}}H_{i}(C)^{*}\otimes_{k}H_{i}(C),$

where the direct sum has a finite number of nonzero terms. By the
characterization of $D$, $H_{0}(C^{*}\otimes_{k}C)\cong k\oplus P$ for some
projective. Therefore, by the Krull-Schmidt theorem there exists some
$i\in\mathbb{N}_{\geq 0}$ for which $H_{i}(C)$ is endotrivial, and for all
$j\neq i$, for which $H_{j}(C)\neq 0$, $H_{j}(C)\otimes_{k}H_{j}(C)^{*}$ is
projective. From 3.4 (k), it follows that $H_{j}(C)$ is projective, since
projectivity is the same as $V$-projectivity for modules.

Suppose for contradiction there exists some $j$ for which $H_{j}(C)$ is
projective. Then we have a short exact sequence of $kG$-modules

$0\to\operatorname{im}d_{j+1}\to\ker d_{j}\to H_{j}(C)\to 0,$

and projectivity implies this sequence is split. Therefore $H_{j}(C)$ is a
submodule of $\ker d_{j}$, hence a submodule of $C_{j}$. Since $kG$ is self-
injective, $H_{j}(C)$ is injective as well, hence $H_{j}(C)$ is in fact a
module direct summand of $C_{j}$. Now since $\ker
d_{j}=H_{j}(C)\oplus\operatorname{im}d_{j+1}$, in particular
$H_{j}(C)\cap\operatorname{im}d_{j+1}=\\{0\\}$ so $C$ has as direct summand
the singleton chain complex $0\to H_{j}(C)\to 0$, contradicting that $C$ is
indecomposable.

Thus there do not exist any $j$ for which $H_{j}(C)$ is projective. It follows
that $H_{i}(C)$ is nonzero for exactly one $i\in\mathbb{Z}$, hence $C$ is an
endosplit $p$-permutation resolution of an endotrivial module, i.e. a
$V$-endosplit-trivial complex. ∎

Furthermore, every weakly $V$-endotrivial complex is isomorphic in
$K^{b}({}_{kG}\underline{\mathbf{triv}}_{V})$ to a strongly $V$-endotrivial
complex, by the following process.

######  Definition 10.2.

Given any bounded chain complex $C$, one may construct a new chain complex $D$
which has nonzero homology in at most one degree $d$ as follows. Suppose
without loss of generality $C_{i}=0$ for all $i<0$ and $i>n$, and for ease of
explanation, suppose $0<d<n$. Start with $D=C$, then add a projective cover
$P_{1}$ of $C_{0}$ to $D_{1}$ as direct sum and add the corresponding cover
$P_{1}\to D_{0}$ to the differential $d_{1}$. Then $H_{0}(D)=0$. Then, if
$1\neq d$, add a projective cover $P_{2}$ of $\ker d_{1}$ to $D_{2}$ and add
the corresponding cover $P_{2}\to\ker d_{1}\hookrightarrow D_{1}$. By
construction, $H_{1}(D)=0$, and $d_{0}\circ d_{1}=0$. Repeat this process at
all $D_{i}$ for $i\\{0,\dots,d-1\\}$. Then, $H_{i}(D)=0$ for all $i<d$.
Visually, the construction is as follows:

${C_{d}}$${C_{d-1}}$${\cdots}$${C_{1}}$${C_{0}}$${P_{d}}$${P_{d-1}}$${\cdots}$${P_{1}}$

Then, take an injective hull of $C_{n},$ labeled $I_{n-1}$, and add it to
$D_{n-1}$ as direct sum and add the envelope to the differential $d_{n}$. Then
$H_{n}(D)=0$. Then if $n-1\neq d$, let $I_{n-2}$ be an injective envelope of
$D_{n-1}/\operatorname{im}d_{n}$. Add $I_{n-2}$ to $D_{n-2}$, and add the
corresponding composition $D_{n-1}\rightarrow
D_{n-1}/\operatorname{im}d_{n}\hookrightarrow I_{n-2}$ to $d_{n-1}$. By
construction, $d_{n-1}\circ d_{n}=0$ and $H_{n-1}(D)=0$. Repeat this process
for all $i\in\\{n,n-1,\dots,d+1\\}$. Then $H_{i}(D)=0$ for all $i>d$.
Visually, the end result is as follows:

${C_{n}}$${C_{n-1}}$${\cdots}$${C_{d+1}}$${C_{d}}$${C_{d-1}}$${\cdots}$${C_{1}}$${C_{0}}$${I_{n-1}}$${\cdots}$${I_{d+1}}$${I_{d}\oplus
P_{d}}$${P_{d-1}}$${\cdots}$${P_{1}}$

Call $D$ a homological concentration of $C$ at $d$. Note that $C(P)=D(P)$ for
any nontrivial $p$-subgroup $P\in s_{p}(G)$. In particular, $C\cong D$ in
$K^{b}({}_{kG}\underline{\mathbf{mod}}_{V})$.

######  Remark 10.3.

It may be possible to, in some cases, perform a local analogue of this process
as well. Suppose for instance that we wish to augment a chain complex of
$kG$-modules $C$ such that $C(P)$ has homology concentrated in one degree. The
Green correspondence gives a bijection between projective indecomposable
$k[N_{G}(P)/P]$-modules and trivial source $kG$-modules with vertex $P$.
Therefore, we can append trivial source $kG$-modules with vertex $P$ to $C$ in
such a way that the additions correspond to taking projective resolutions or
injective hulls of the $P$-local kernels and cokernels. The difficulty which
arises, however, is that while it is possible to draw the differentials at the
$P$-local level to be projective resolutions and injective hulls of
differentials of $C(P)$, there is no guarantee that the corresponding maps at
the global level result are still differentials, i.e. we may have $d^{2}\neq
0$.

######  Theorem 10.4.

The group homomorphism
$\iota:e\mathcal{E}_{k}^{V}(G)\to\mathcal{W}\mathcal{E}_{k}^{V}(G)$ is
surjective. Moreover, $\ker\iota$ is generated by representatives of truncated
projective resolutions or injective hulls of the trivial $kG$-module $k$.

###### Proof.

The surjectivity of $\iota$ follows immediately from the previous definition.
Suppose $C$ is a $V$-endosplit-trivial complex of $kG$-modules with
$[C]\in\mathcal{W}\mathcal{E}_{k}^{V}(G)$. Since the h-marks of $C$ are
trivial, it follows by Lemma 9.3 that $C$ is homotopy equivalent to a complex
consisting of only projectives, with the exception of one indecomposable
module in degree zero which is endotrivial. Moreover, since $[C]\in\ker\iota$,
this complex is stably isomorphic to the trivial endotrivial complex, so the
indecomposable endotrivial complex is the trivial complex $k[0]$. It follows
that $C$ is homotopy equivalent to a complex of projectives in all degrees
except 0, which contains only the trivial $kG$-module, and has nonzero
homology in only 1 degree since $C$ is $V$-endosplit-trivial. A standard
homological argument concludes that $C$ is homotopy equivalent to a truncated
projective resolution, as desired. ∎

######  Proposition 10.5.

Let $[C]\in\mathcal{E}_{k}^{1}(G)$ be a $V$-endosplit-trivial complex. $C$
lifts to an endotrivial complex if and only if
$\mathcal{H}_{C}(1)=[k_{\omega}]\in T_{k}(G)$ for some
$\omega\in\operatorname{Hom}(G,k^{\times})$.

###### Proof.

This is a special case of Theorem 9.15. ∎

######  Example 10.6.

  1. (a)

When $G=SD_{2^{n}}$ is a semidihedral 2-group of order at least 16, recall
from [17] that there exists a faithful endotrivial complex
$C\in\mathcal{E}_{k}(G)$ for which $h_{C}(H)=2$ for $H\leq G$, the unique (up
to conjugacy) noncentral subgroup of order 2, $h_{C}(1)=4$, and $h_{C}(K)=0$
when $K\neq 1,H$. Define the $G$-set $X:=G/H$. Recall (see for instance [4])
that $E:=\Omega(\Omega_{X})$ is an endotrivial $kG$-module which satisfies
$[E]^{2}=[k]\in T(G)$. From this, it follows that

$C_{E}:=kG\to kX\to k,$

with $k$ in degree 0, the differential $kX\to k$ the augmentation map, and the
differential $kG\to kX$ corresponding to the projective cover of $\Omega_{X}$,
is a $V$-endotrivial complex with $h_{C_{E}}(1)=2$, $h{C_{E}}(H)=1$, and
$h{C_{E}}(K)=0$ for $K\neq 1,H$. Therefore,
$[C_{E}\otimes_{k}C_{E}]=[C]\in\mathcal{E}_{k}^{1}(G)$, that is, they are
isomorphic up to homotopy equivalence and a direct sum of complexes of the
form $0\to kG\to 0$. We note this equality is not obvious, since
$\dim_{k}H_{4}(C_{E}\otimes_{k}C_{E})>1$, whereas $\dim_{k}H_{4}(C)=1$.
Therefore, it is possible for there to be torsion in the quotient
$\mathcal{E}_{k}^{1}(G)/\mathcal{E}_{k}(G)$ even when $G$ is a $p$-group.

  2. (b)

Given any $G$ which has $p$-rank greater than 1, the chain complex $C=kG\to k$
has $H_{1}(C)\cong\Omega(k)$, which is an endotrivial module. This cannot be
an endotrivial complex since $\dim_{k}\Omega(k)=|G|-1$, but is
$V$-endotrivial. Since $G$ has $p$-rank greater than 1, $C^{\otimes i}$ will
also be $V$-endotrivial but not endotrivial, thus
$\operatorname{rk}_{\mathbb{Z}}\mathcal{E}_{k}^{1}(G)$ will almost always be
strictly greater than $\operatorname{rk}_{\mathbb{Z}}\mathcal{E}_{k}(G)$.

Similarly, we may study the image of the homomorphism
$\mathcal{E}_{k}(G)\to\mathcal{W}\mathcal{E}_{k}^{V}(G)$. We have the
following commutative diagram:

${\mathcal{E}_{k}(G)}$${\mathcal{E}_{k}^{1}(G)}$${\mathcal{W}\mathcal{E}_{k}^{V}(G)}$$\scriptstyle{\iota}$

By surjectivity of $\iota$, an isomorphism class of weakly endotrivial
complexes $[C]\in\mathcal{W}\mathcal{E}_{k}^{V}(G)$ lifts to an endotrivial
complex if and only if there is a $V$-endosplit-trivial complex in $[C]$ which
lifts to an endotrivial complex. From this, we obtain the following theorem.

######  Proposition 10.7.

Let $[C]\in\mathcal{W}\mathcal{E}_{k}^{V}(G)$. $[C]$ lifts to an endotrivial
complex if and only there exists a $D\in\mathcal{E}_{k}^{1}(G)$ with
coinciding h-marks as $[C]$ and which satisfies
$[\mathcal{H}_{D}(1)]=[\Omega_{i}(k_{\omega})]$ for some $i\in\mathbb{Z}$ and
$\omega\in\operatorname{Hom}(G,k^{\times})$.

###### Proof.

The forward direction is clear, since one can simply take the endotrivial
complex belonging to $[C]$, in which case $i=0$. For the reverse direction,
suppose $D$ satisfies the above properties. If $i=0$, we are done. Otherwise,
then perform a homological concentration on $D$ with homology in degree
$h_{D}(1)-i$ \- this corresponds to taking an injective hull of
$\mathcal{H}_{D}(1)$ $i$ times if $i>0$, or a projective resolution of $H(D)$
$-i$ times if $i<0$. The resulting complex has nonzero homology isomorphic in
$T_{k}(G)$ to a $k$-dimension one module, and we are done. ∎

##  11\. Restriction to subgroups containing Sylow $p$-subgroups

For this section, fix $S\in\operatorname{Syl}_{p}(G)$, $V$ a $p$-permutation
absolutely $p$-divisible $kG$-module, and as before, $\mathcal{X}_{V}\subset
s_{p}(G)$ the set of $p$-subgroups of $G$ for which $V(P)=0$. We consider
subgroups $H\leq G$ for which $S\leq H$, and study the relationship between
$\mathcal{E}^{V}_{k}(G)$ and $\mathcal{E}_{k}^{V}(H)$ via the restriction map
$\operatorname{Res}^{G}_{H}:\mathcal{E}_{k}^{V}(G)\to\mathcal{E}_{k}^{V}(H)$.
We first note some easy properties.

######  Proposition 11.1.

Let $H\leq G$ with $S\leq H$.

  1. (a)

$C\in Ch^{b}({}_{kG}\mathbf{triv})$ is weakly $V$-endotrivial (resp.
$V$-endosplit-trivial) if and only if $\operatorname{Res}^{G}_{H}C$ is weakly
$\operatorname{Res}^{G}_{H}V$-endotrivial (resp.
$\operatorname{Res}^{G}_{H}V$-endosplit-trivial).

  2. (b)

$\operatorname{Res}^{G}_{H}:x\mathcal{E}^{V}_{k}(G)\to
x\mathcal{E}^{\operatorname{Res}^{G}_{H}V}_{k}(H)$ has kernel contained in
$\ker h$. If $N_{G}(S)\leq H\leq G$ and $x=\mathcal{W}$ or $e$, then
$\operatorname{Res}^{G}_{H}:x\mathcal{E}^{V}_{k}(G)\to
x\mathcal{E}^{V}_{k}(H)$ is injective.

  3. (c)

Let $D\in x\mathcal{E}_{k}^{\operatorname{Res}^{G}_{H}V}(H)$. If
$[D]\in\operatorname{im}(\operatorname{Res}^{G}_{H}:x\mathcal{E}^{V}_{k}(G)\to
x\mathcal{E}^{\operatorname{Res}^{G}_{H}V}_{k}(H))$, then
$\operatorname{Ind}^{G}_{H}D$ has a $V$-endotrivial complex direct summand.

  4. (d)

For every $C\in x\mathcal{E}_{k}^{V}(G)$ there exists some $D\in
x\mathcal{E}_{k}^{\operatorname{Res}^{G}_{H}V}(H)$ such that
$C\mid\operatorname{Ind}^{G}_{H}D$.

  5. (e)

For all
$D\in\operatorname{im}\operatorname{Res}^{G}_{H}:x\mathcal{E}^{V}_{k}(G)\to
x\mathcal{E}^{\operatorname{Res}^{G}_{H}V}_{k}(H)$ and $P\in s_{p}(G)$ for
which $V(P)=0$, and $g\in G$ satisfying ${}^{g}P\leq H$,
$h(D(P))=h(D({}^{g}P))$.

###### Proof.

  1. (a)

For weak endotriviality, the forward direction is clear since restriction
preserves all h-marks and $V$-endotriviality. Suppose
$\operatorname{Res}^{G}_{H}C$ is $\operatorname{Res}^{G}_{H}V$ endotrivial.
Then given any $P\leq S$ for which $(\operatorname{Res}^{G}_{H}V)(P)=0$
(equivalently, $V(P)=0$),
$(\operatorname{Res}^{G}_{H}C)(P)\cong\operatorname{Res}^{N_{G}(P)}_{N_{H}(P)}(C(P))$,
regarding the Brauer construction as a functor
${}_{kG}\mathbf{mod}\to{}_{kN_{G}(P)}\mathbf{mod}$. Since
$\operatorname{Res}^{G}_{H}C$ is $\operatorname{Res}^{G}_{H}V$-endotrivial,
$(\operatorname{Res}^{G}_{H}C)(P)$ has nonzero homology in one degree with it
having $k$-dimension one. Therefore
$\operatorname{Res}^{N_{G}(P)}_{N_{H}(P)}(C(P))$ has this property, and $C(P)$
does too. Since all $p$-subgroups of $G$ are conjugate to a subgroup of $S$,
we conclude that $C$ is endotrivial. The $V$-endosplit-trivial case follows by
a similar argument.

  2. (b)

$\operatorname{Res}^{G}_{H}$ preserves the h-marks of $C$. Therefore, if
$C\in\ker\operatorname{Res}^{G}_{H}$, all the h-marks of $C$ are 0, and
therefore $\ker\operatorname{Res}^{G}_{H}\subseteq\ker\epsilon_{x}$. The
latter claim follows from the Green correspondence for chain complexes, as in
this case, $\operatorname{Res}^{G}_{H}$ induces an injective map from the set
of isomorphism classes of trivial source $V$-endotrivial $kG$-modules to the
set of isomorphism classes of trivial source
$\operatorname{Res}^{G}_{H}V$-endotrivial $kH$-modules.

  3. (c)

By relative projectivity,
$C\mid\operatorname{Ind}^{G}_{H}\operatorname{Res}^{G}_{H}C$. Therefore, if
$D=\operatorname{Res}^{G}_{H}C$, $\operatorname{Ind}^{G}_{H}D$ has a
$V$-endotrivial direct summand, namely $C$.

  4. (d)

Set $D=\operatorname{Res}^{G}_{H}C$. Then
$C\mid\operatorname{Ind}^{G}_{H}\operatorname{Res}^{G}_{H}C$.

  5. (e)

This follows since ${}^{g}C(P)\cong C({}^{g}P)$, therefore h-marks of
$V$-endotrivial $kG$-complexes are invariant under $G$-conjugation.

∎

In the study of endotrivial modules, a common goal is to determine the image
of the restriction map $\operatorname{Res}^{G}_{H}:T_{k}(G)\to T_{k}(H)$. We
ask the same question here. First, from Proposition 11.1 (e), the following
definition makes sense.

######  Definition 11.2.

Let $S\leq H\leq G$, $W$ a $p$-permutation $kH$-module, and $C$ be a
$W$-endotrivial $kH$-complex. We say $C$ is $\mathcal{F}$-stable or fusion
stable if for all pairs of $p$-subgroups $P,Q\in s_{p}(H)$ with $P=_{G}Q$ and
$h_{C}(P),h_{C}(Q)$ defined, $h_{C}(P)=h_{C}(Q)$. Define the
$\mathcal{F}$-fusion stable subgroup
$x\mathcal{E}^{W}_{k}(H)^{\mathcal{F}}\leq x\mathcal{E}^{W}_{k}(H)$ to be the
subgroup consisting of all classes of fusion stable $W$-endotrivial complexes.

Similarly, let $C$ be an endosplit $p$-permutation resolution of $kH$-modules.
Say $C$ is $\mathcal{F}$-stable or fusion stable if for all pairs of
$p$-subgroups $P,Q\in s_{p}(G)$ with $P=_{G}Q$, $h_{C}(P)=h_{C}(Q)$, or
equivalently, $h_{C}$ is the restriction of some $f\in C(G,p)$.

######  Remark 11.3.

Note that there is a notion of fusion stability for endopermutation modules
and their corresponding endosplit $p$-permutation resolutions (see [13,
Definitions 9.9.1, 9.9.2]), which are defined over arbitrary fusion systems.
If one takes the fusion system $\mathcal{F}=\mathcal{F}_{S}(G)$, the fusion
system of the group $G$ with Sylow $p$-subgroup $S$, and sets $C$ to be an
indecomposable $V$-endosplit-trivial $kS$-complex, then if $C$ is
$\mathcal{F}$-stable as defined in the literature, then $C$ is fusion-stable
as we have defined above.

Since h-marks are preserved under conjugacy, the image of
$\operatorname{Res}^{G}_{H}:\mathcal{E}^{V}_{k}(G)\to\mathcal{E}^{\operatorname{Res}^{G}_{H}V}_{k}(H)$
is contained in $\mathcal{E}^{V}_{k}(H)^{\mathcal{F}}$. Our goal is to
determine when
$\operatorname{Res}^{G}_{H}:\mathcal{E}^{V}_{k}(G)\to\mathcal{E}^{\operatorname{Res}^{G}_{H}V}_{k}(H)^{\mathcal{F}}$
is surjective. To investigate further, we must describe how additive induction
and the Brauer construction behave together.

######  Theorem 11.4.

Let $P\in s_{p}(G)$, $H\leq G$, and $M$ be a $p$-permutation $kH$-module. We
have the following natural isomorphism:

$\left(\operatorname{Ind}^{G}_{H}M\right)(P)\cong\bigoplus_{x\in[N_{G}(P)\backslash
G/H],N_{G}(P)\cap{}^{x}H\geq_{G}P}\operatorname{Ind}^{N_{G}(P)}_{N_{G}(P)\cap{}^{x}H}\big{(}({}^{x}M)(P)\big{)}.$

###### Proof.

First, we have
$(\operatorname{Ind}^{G}_{H}M)(P)=(\operatorname{Res}^{G}_{N_{G}(P)}\operatorname{Ind}^{G}_{H}M)(P)$.
Now, applying the Mackey formula yields

$\displaystyle(\operatorname{Res}^{G}_{N_{G}(P)}\operatorname{Ind}^{G}_{H}M)(P)$
$\displaystyle\cong\left(\bigoplus_{x\in[N_{G}(P)\backslash
G/H]}\operatorname{Ind}^{N_{G}(P)}_{N_{G}(P)\cap{}^{x}H}\operatorname{Res}^{{}^{x}H}_{N_{G}(P)\cap{}^{x}H}({}^{x}M)\right)(P)$
$\displaystyle\cong\bigoplus_{x\in[N_{G}(P)\backslash
G/H]}\left(\operatorname{Ind}^{N_{G}(P)}_{N_{G}(P)\cap{}^{x}H}\operatorname{Res}^{{}^{x}H}_{N_{G}(P)\cap{}^{x}H}({}^{x}M)\right)(P)$

Given a double coset representative $x$, each indecomposable constituent of
$\operatorname{Ind}^{N_{G}(P)}_{N_{G}(P)\cap{}^{x}H}\operatorname{Res}^{{}^{x}H}_{N_{G}(P)\cap{}^{x}H}({}^{x}M)$
has a vertex contained in $N_{G}(P)\cap{}^{x}H$, so if
$\left(\operatorname{Ind}^{N_{G}(P)}_{N_{G}(P)\cap{}^{x}H}\operatorname{Res}^{{}^{x}H}_{N_{G}(P)\cap{}^{x}H}({}^{x}M)\right)(P)\neq
0$, then $N_{G}(P)\cap{}^{x}H\geq_{N_{G}(P)}P$, and since $P\trianglelefteq
N_{G}(P)$, this occurs if and only if $P\leq{}^{x}H$. Therefore,

$(\operatorname{Ind}^{G}_{H}M)(P)\cong\bigoplus_{x\in[N_{G}(P)\backslash
G/H],P\leq{}^{x}H}\left(\operatorname{Ind}^{N_{G}(P)}_{N_{G}(P)\cap{}^{x}H}\operatorname{Res}^{{}^{x}H}_{N_{G}(P)\cap{}^{x}H}({}^{x}M)\right)(P).$

We now compute the internal term when $P\leq{}^{x}H$. First, since
$P\trianglelefteq N_{G}(P)$,

$\operatorname{Ind}^{N_{G}(P)}_{N_{G}(P)\cap{}^{x}H}({}^{x}M)^{P}=\left(\operatorname{Ind}^{N_{G}(P)}_{N_{G}(P)\cap{}^{x}H}\operatorname{Res}^{{}^{x}H}_{N_{G}(P)\cap{}^{x}H}{}^{x}M\right)^{P}.$

Indeed, for any $g\in N_{G}(P)$, and $m\in{}^{x}M$, $p\cdot(g\otimes
m)=g\otimes m$ if and only if $g\otimes p^{g}m=g\otimes m$ for all $p\in P$,
and the equality follows from there. We next claim

$\sum_{Q<P}\operatorname{tr}^{P}_{Q}\left(\operatorname{Ind}^{N_{G}(P)}_{N_{G}(P)\cap{}^{x}H}\operatorname{Res}^{{}^{x}H}_{N_{G}(P)\cap{}^{x}H}{}^{x}M\right)^{Q}=\operatorname{Ind}^{N_{G}(P)}_{N_{G}(P)\cap{}^{x}H}\left(\sum_{Q<P}\operatorname{tr}^{P}_{Q}({}^{x}M)^{Q}\right).$

Indeed, for any $Q<P$, $g\otimes
m\in\left(\operatorname{Ind}^{N_{G}(P)}_{N_{G}(P)\cap{}^{x}H}\operatorname{Res}^{{}^{x}H}_{N_{G}(P)\cap{}^{x}H}{}^{x}M\right)^{Q}$
if and only if $m\in(^{x}M)^{Q^{g}}$. Now given coset representatives
$p_{1},\dots,p_{k}$ of $P/Q$, $(p_{1}+\dots+p_{k})g\otimes
m=g\otimes(p_{1}^{g}+\dots+p_{k}^{g})m$, and since $p_{1}^{g},\dots,p_{k}^{g}$
are coset representatives of $P/Q^{g}$, we have an equality of sets. Finally,
for arbitrary groups $H\leq G$ and appropriate modules $N\subseteq M$,
$\operatorname{Ind}^{G}_{H}M/N\cong\operatorname{Ind}^{G}_{H}M/\operatorname{Ind}^{G}_{H}N$
naturally by exactness of additive induction. We now compute, assuming
$P\leq{}^{x}H$ and setting $N_{{}^{x}H}(P)=N_{G}(P)\cap{}^{x}H$:

$\displaystyle\left(\operatorname{Ind}^{N_{G}(P)}_{N_{{}^{x}H}(P)}\operatorname{Res}^{{}^{x}H}_{N_{{}^{x}H}(P)}({}^{x}M)\right)(P)$
$\displaystyle=\left(\operatorname{Ind}^{N_{G}(P)}_{N_{{}^{x}H}(P)}\operatorname{Res}^{{}^{x}H}_{N_{{}^{x}H}(P)}{}^{x}M\right)^{P}/\sum_{Q<P}\operatorname{tr}^{P}_{Q}\left(\operatorname{Ind}^{N_{G}(P)}_{N_{{}^{x}H}(P)}\operatorname{Res}^{{}^{x}H}_{N_{{}^{x}H}(P)}{}^{x}M\right)^{Q}$
$\displaystyle=\operatorname{Ind}^{N_{G}(P)}_{N_{{}^{x}H}(P)}({}^{x}M)^{P}/\operatorname{Ind}^{N_{G}(P)}_{N_{{}^{x}H}(P)}\left(\sum_{Q<P}\operatorname{tr}^{P}_{Q}({}^{x}M)^{Q}\right)$
$\displaystyle\cong\operatorname{Ind}^{N_{G}(P)}_{N_{{}^{x}H(P)}}\left(({}^{x}M)^{P}/\left(\sum_{Q<P}\operatorname{tr}^{P}_{Q}({}^{x}M)^{Q}\right)\right)$
$\displaystyle=\operatorname{Ind}^{N_{G}(P)}_{N_{{}^{x}H(P)}}\big{(}({}^{x}M)(P)\big{)}$

This completes the proof of the isomorphism for modules, and since all the
isomorphisms used were natural, the result follows. ∎

As a first result of Theorem 8.6 and Theorem 11.4, we can generate endosplit
$p$-permutation resolutions for summands of induced modules which have fusion-
stable endosplit $p$-permutation resolutions.

######  Corollary 11.5.

Let $S\leq H\leq G$.

  1. (a)

Let $C\in x\mathcal{E}^{V}_{k}(H)^{\mathcal{F}}$ for $x=\mathcal{W}$ or
$\mathcal{S}$. Then for all $P\in\mathcal{X}_{V}$ ,
$\left(\operatorname{Ind}^{G}_{H}C\right)(P)$ has homology concentrated in the
degree $h_{C}({}^{g}P)$, where $g\in G$ is chosen such that ${}^{g}P\leq S$.

  2. (b)

Let $C\in e\mathcal{E}^{V}_{k}(H)^{\mathcal{F}}$. Then for all $P\in
s_{p}(G)$, $\left(\operatorname{Ind}^{G}_{H}C\right)(P)$ has homology
concentrated in the degree $h_{C}({}^{g}P)$, where $g\in G$ is chosen such
that ${}^{g}P\leq S$.

  3. (c)

If $C$ is a $\mathcal{F}$-stable endosplit $p$-permutation resolution for a
$kH$-module $M$, $\operatorname{Ind}^{G}_{H}C$ is an endosplit $p$-permutation
resolution for $\operatorname{Ind}^{G}_{H}M$, and for all indecomposable
direct summands $M\mid\operatorname{Ind}^{G}_{H}N$, there exists a
corresponding unique direct summand $C_{M}\mid\operatorname{Ind}^{G}_{H}C$
which is an endosplit $p$-permutation resolution for $M$, and for each $P\leq
S$, we have $h_{C_{M}}(P)=h_{C}(P)$.

  4. (d)

Conversely, if $C$ is an endosplit $p$-permutation resolution of a $kH$-module
which is not $\mathcal{F}$-stable, then $\operatorname{Ind}^{G}_{H}C$ is not
an endosplit $p$-permutation resolution.

###### Proof.