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\begin{document}
\begin{frontmatter}
\title{Finiteness conditions in the stable module category}
\author{Jonathan Cornick}
\address{Department of Mathematics and Computer Science,
Queensborough Community College, CUNY
222-05 56th Avenue,
Bayside, New York 11364, USA}
\ead{jcornick@qcc.cuny.edu}
\author{Ioannis Emmanouil}
\address{Department of Mathematics, University of Athens, Athens 15784, Greece}
\ead{emmmanoui@math.uoa.gr}
\author{Peter Kropholler}
\address{Mathematical Sciences, University of Southampton,
Southampton SO17 1BJ, UK}
\ead{p.h.kropholler@soton.ac.uk}
\author{Olympia Talelli}
\address{Department of Mathematics, University of Athens, Athens 15784, Greece}
\ead{otalelli@math.uoa.gr}
\begin{abstract}
We study groups whose cohomology functors commute with filtered colimits in high dimensions.
We relate this condition to projective resolutions which exhibit some finiteness properties in high dimensions, and to the existence of Eilenberg--Mac Lane spaces with finitely many $n$-cells for all sufficiently large $n$. To that end we determine the structure of finitary Gorenstein projective modules.
The methods are inspired by representation theory and make use of the stable module category in which morphisms are defined through complete cohomology.
In order to carry out these methods we need to restrict ourselves to certain classes of hierarchically decomposable groups.
\end{abstract}
\begin{keyword}
cohomology of groups, Gorenstein projective dimension, stable module category
\end{keyword}
\end{frontmatter}
\setcounter{tocdepth}{1}
\maketitle
\tableofcontents
%\begin{abstract}
%We study groups whose cohomology functors commute with filtered colimits in high dimensions.
%We relate this condition to projective resolutions which exhibit some finiteness properties in high dimensions, and to the existence of Eilenberg--Mac Lane spaces with finitely many $n$-cells for all sufficiently large $n$.
%The methods are inspired by representation theory and make use of the stable module category in which morphisms are defined through complete cohomology.
%In order to carry out these methods we need to restrict to certain classes of hierarchically decomposable groups. Over these groups we determine the structure of finitary Gorenstein projective modules.
%\end{abstract}
\addtocounter{section}{-1}
\section{Introduction}
It follows from work of Bieri and Eckmann that a group $G$ (resp. an $R$-module $M$ over a ring $R$) is of type $FP_n$ if and only if the functors $H^i(G,\_\!\_)$ (resp. $\mbox{Ext}_R^i(M,\_\!\_)$) commute with filtered colimits (also referred to as inductive limits) of coefficient modules for all $i$ in the range $0\le i0$. If $P_{*} \longrightarrow M \longrightarrow 0$ is a projective
resolution of $M$, then $\Omega^{n}M$ is identified with the $n$-th syzygy module
of $M$, i.e.\ with the image of the map $P_{n} \longrightarrow P_{n-1}$ for all
$n \geq 1$. If $A,B,C$ are three left $R$-modules, then we identify the abelian
group $\mbox{Hom}_{R}(A,B \oplus C)$ with the direct sum
$\mbox{Hom}_{R}(A,B) \oplus \mbox{Hom}_{R}(A,C)$. In this way, an element
$(f,g) \in \mbox{Hom}_{R}(A,B) \oplus \mbox{Hom}_{R}(A,C)$ is identified
with the $R$-linear map $A \longrightarrow B \oplus C$, which is given by
$a \mapsto (f(a),g(a))$, $a \in A$. In a similar way, there is an identification
of the abelian group $\mbox{Hom}_{R}(B \oplus C,A)$ with the direct sum
$\mbox{Hom}_{R}(B,A) \oplus \mbox{Hom}_{R}(C,A)$. For any two elements
$f \in \mbox{Hom}_{R}(B,A)$ and $g \in \mbox{Hom}_{R}(C,A)$ we shall denote
by $[f,g] : B \oplus C \longrightarrow A$ the corresponding $R$-linear map
(which is given by $(b,c) \mapsto f(b) + g(c)$, $(b,c) \in B \oplus C$).
\section{Preliminary notions}
In this section, we collect certain basic notions and preliminary results
that will be used in the sequel. These involve:
\begin{enumerate}
\item
the relation between complete cohomology and the finiteness of the Gorenstein
projective dimension,
\item
certain basic results concerning finiteness conditions in the stable module
category, which are mainly related to the notions of stably compact and completely
finitary modules,
\item
a compactness result concerning poly-$\mathfrak{C}$ and hyper-$\mathfrak{C}$
modules (where $\mathfrak{C}$ is a given class of modules) and
\item
a concrete description of Rickard's algebraic homotopy colimit associated
with a class $\mathfrak{C}$ of modules.
\end{enumerate}
\vspace{0.15in}
\subsection{Complete cohomology and Gorenstein projective dimension.}
The existence of terminal (projective or injective) completions of the
ordinary Ext functors in module categories has been studied by Gedrich
and Gruenberg in \cite{Ge-Gru}. Using an approach that involves satellites and
is heavily influenced by Gedrich and Gruenberg's theory of terminal
completions, Mislin defined in \cite{M} for any ring $R$ and any left
$R$-module $M$ complete cohomology functors
$\widehat{\mbox{Ext}}^{*}_{R}(M,\_\!\_)$ and a natural transformation
${\mbox{Ext}}^{*}_{R}(M,\_\!\_) \longrightarrow
\widehat{\mbox{Ext}}^{*}_{R}(M,\_\!\_)$,
as the projective completion of the ordinary Ext functors
${\mbox{Ext}}^{*}_{R}(M,\_\!\_)$. Equivalent definitions of the complete
cohomology functors have been independently formulated by Vogel in \cite{Go}
(using a hypercohomology approach) and by Benson and Carlson in \cite{Be-Ca}
(using projective resolutions). Using the approach by Benson and Carlson,
it follows that the kernel of the canonical map
${\mbox{Hom}}_{R}(M,N) \longrightarrow \widehat{\mbox{Ext}}^{0}_{R}(M,N)$
consists of those $R$-linear maps $f : M \longrightarrow N$, which are such
that the map $\Omega^{n}f : \Omega^{n}M \longrightarrow \Omega^{n}N$ induced
by $f$ between the associated $n$-th syzygy modules factors through a
projective left $R$-module for $n \gg 0$.
There is a special case, where the complete cohomology functors may be
computed by means of complexes, as we shall now describe: A left $R$-module
$M$ is said to admit a \emph{complete resolution of coincidence index $n$} if
there exists a doubly infinite acyclic complex of projective left $R$-modules
$P_{*}$, which coincides with a projective resolution of $M$ in degrees
$\geq n$. If, in addition, $P_{*}$ remains acyclic after applying the functor
$\mbox{Hom}_{R}(\_\!\_ ,P)$ for any projective left $R$-module $P$, then
$P_{*}$ is called a \emph{complete resolution of $M$ in the strong sense}. A left
$R$-module $M$ is called \emph{Gorenstein projective} if it admits a complete
resolution in the strong sense of coincidence index $0$. The class of
Gorenstein projective left $R$-modules, which will be denoted by
${\tt GP}(R)$, contains all projective left $R$-modules, is closed under
arbitrary direct sums and has many interesting properties. In particular,
as shown by Holm in \cite{Ho}, ${\tt GP}(R)$ is closed under direct summands,
extensions and kernels of epimorphisms. As in classical homological
algebra, we say that a left $R$-module $M$ has \emph{finite Gorenstein projective
dimension} if there exist a non-negative integer $n$ and an exact sequence of left $R$-modules
\[ 0 \longrightarrow M_{n} \longrightarrow \cdots \longrightarrow M_{1}
\longrightarrow M_{0} \longrightarrow M \longrightarrow 0 , \]
such that $M_{i}$ is Gorenstein projective for all $i=0,1, \ldots, n$. In
that case, the \emph{Gorenstein projective dimension} $\mbox{Gpd}_{R}M$ of $M$ is
defined to be the least such $n$. It follows from results of Holm \cite{Ho} that
a left $R$-module $M$ has finite Gorenstein projective dimension if and only
if the module $M$ admits a complete resolution in the strong sense. We note
that the latter may be chosen to have coincidence index equal to $\mbox{Gpd}_{R}M$.
Since in the present paper we only consider complete resolutions in the strong
sense, for brevity, we shall refer to them as complete resolutions instead. It
follows from Mislin's approach to complete cohomology that the functors
$\widehat{\mbox{Ext}}^{*}_{R}(M,\_\!\_)$ may be computed, in the case where
$M$ has a complete resolution $P_{*}$, as the cohomology groups of the complex
$\mbox{Hom}_{R}(P_{*},\_\!\_)$, (cf. \cite{Co-Kr}, Theorem 1.2). In particular, if $M$
is Gorenstein projective, then the complete cohomology group
$\widehat{\mbox{Ext}}^{0}_{R}(M,N)$ may be naturally identified for any left
$R$-module $N$ with the quotient $\underline{\mbox{Hom}}_{R}(M,N)$ of the
abelian group $\mbox{Hom}_{R}(M,N)$ by the subgroup consisting of all
$R$-linear maps $f : M \longrightarrow N$ that factor through a projective
left $R$-module.
In view of the above discussion, it is important to know whether all left
$R$-modules have finite Gorenstein projective dimension, i.e.\ whether all left
$R$-modules have a complete resolution. The answer to this question involves the
cohomological invariants $\mbox{silp} \, R$ and $\mbox{spli} \, R$, which were
introduced by Gedrich and Gruenberg in \cite{Ge-Gru}. Here, $\mbox{silp} \, R$ is defined
as the supremum of the injective lengths of projective left $R$-modules, whereas
$\mbox{spli} \, R$ is the supremum of the projective lengths of injective left
$R$-modules. It follows from (\cite{Co-Kr}, Theorem 3.10) that the finiteness of these invariants
is equivalent to the assertion that all left $R$-modules have finite Gorenstein
projective dimension.
In the sequel, we shall need the following simple result.
\begin{Lemma}\label{ori_1.1}
Let $R$ be a ring and assume that $S \subseteq R$ is a subring, such that
$R$ is flat when regarded as a right $S$-module and projective when regarded
as a left $S$-module. Then, for any Gorenstein projective left $S$-module $M$
the left $R$-module $R \otimes_{S} M$ is Gorenstein projective as well.
\end{Lemma}
\vspace{-0.1in}
\begin{proof}
Let $P_{*}$ be a complete resolution in the category of left $S$-modules
having $M$ as a syzygy. Since the right $S$-module $R$ is flat, the induced
complex $R \otimes_{S} P_{*}$ is acyclic, consists of projective left
$R$-modules and has $R \otimes_{S} M$ as a syzygy. Since the left $S$-module
$R$ is projective, the restriction of any projective left $R$-module is a
projective left $S$-module. It follows readily that $R \otimes_{S} P_{*}$
is a complete resolution in the category of left $R$-modules and hence
$R \otimes_{S} M$ is a Gorenstein projective left $R$-module, as needed.
\end{proof}
In the special case where $R = \Z G$ is the integral group ring of a
group $G$, the finiteness of the Gorenstein projective dimension of all
$\Z G$-modules is known to be controlled by the finiteness of the
Gorenstein projective dimension of the trivial $\Z G$-module $\Z $.
In fact, if we define the Gorenstein cohomological dimension $\mbox{Gcd} \, G$
of $G$ by letting $\mbox{Gcd} \, G = \mbox{Gpd}_{\Z G}\Z $, then
for any $\Z G$-module $M$ there is an inequality
$\mbox{Gpd}_{\Z G}M \leq \mbox{Gcd} \, G + 1$ (cf. \cite{Bah-D-T}, Proposition
\ref{ori_2.4}(c)). As shown in [loc.cit., Theorem 2.5], the Gorenstein cohomological
dimension $\mbox{Gcd} \, G$ of $G$ is the supremum of those integers $n$,
for which there exist $\Z G$-modules $M$ and $P$, with $M$
$\Z $-free and $P$ projective, such that
$\mbox{Ext}_{\Z G}^{n}(M,P) \neq 0$.
\vspace{0.15in}
\subsection{Stably compact and completely finitary modules.}
The \emph{stable module category} of a ring $R$ is the category whose objects are
all left $R$-modules and whose morphisms $M \longrightarrow N$, where $M,N$
are two left $R$-modules, are the elements of the complete cohomology group
$\widehat{\mbox{Ext}}^{0}_{R}(M,N)$. For later use, we record the following
simple lemma.
\begin{Lemma}\label{ori_1.2}
Let $R$ be a ring. Then:
\begin{enumerate}
\item
The composition of morphisms in the stable module category of $R$ is
biadditive, i.e.\ for any three left $R$-modules $A,B,C$ it is described
by means of an additive map
\[ \widehat{\mbox{Ext}}^{0}_{R}(B,C) \otimes
\widehat{\mbox{Ext}}^{0}_{R}(A,B) \longrightarrow
\widehat{\mbox{Ext}}^{0}_{R}(A,C) . \]
\item
If $f : B \longrightarrow C$ is an $R$-linear map and
$[f] = [0] \in \widehat{\mbox{Ext}}^{0}_{R}(B,C)$, then the additive map
\[ f_{*} : \widehat{\mbox{Ext}}^{0}_{R}(A,B) \longrightarrow
\widehat{\mbox{Ext}}^{0}_{R}(A,C) \]
is trivial for any left $R$-module $A$.
\end{enumerate}
\end{Lemma}
\begin{proof}
Assertion (i) is an immediate consequence of the definitions, in view
of the biadditivity of the composition of $R$-linear maps, whereas (ii)
follows readily from (i). \end{proof}
We say that a left $R$-module $M$ is \emph{stably compact} if the functor
$\widehat{\mbox{Ext}}^{0}_{R}(M,\_\!\_)$ commutes with filtered colimits.
If the functors $\widehat{\mbox{Ext}}^{n}_{R}(M,\_\!\_)$ commute with
filtered colimits for all $n \in \Z $, then we say that $M$ is
\emph{completely finitary}. We denote by ${\tt SC}(R)$ (resp.\ ${\tt CF}(R)$)
the class of stably compact (resp.\ completely finitary) left $R$-modules.
We note that all projective left $R$-modules are stably compact, whereas
the class of completely finitary left $R$-modules is closed under direct
summands, extensions, kernels of epimorphisms, and
cokernels of monomorphisms.
If $\mathfrak{C}$ is any
class of left $R$-modules, then we define the class $\dlimit\ \mathfrak{C}$
as the class consisting of those left $R$-modules, which may be expressed as
filtered colimits of modules in $\mathfrak{C}$. We also define the class
$\mathfrak{C}^{\perp}$ as the class consisting of those left $R$-modules $N$,
which are such that $\widehat{\mbox{Ext}}^{0}_{R}(M,N) = 0$ for all
$M \in \mathfrak{C}$. It is clear that if $\mathfrak{C}, \mathfrak{D}$ are
two classes of left $R$-modules and $\mathfrak{C} \subseteq \mathfrak{D}$,
then $\mathfrak{D}^{\perp} \subseteq \mathfrak{C}^{\perp}$. In particular,
since ${\tt CF}(R) \subseteq {\tt SC}(R)$, we have an inclusion
${\tt SC}(R)^{\perp} \subseteq {\tt CF}(R)^{\perp}$.
\begin{Proposition}\label{ori_1.3}
Let $S$ be a subring of a ring $R$, such that $R$ is flat when regarded
as a right $S$-module, and consider a class $\mathfrak{C}$ of left
$S$-modules. Then, for any left $S$-module $N$ contained in
$\mathfrak{C}^{\perp} \cap \dlimit\ \mathfrak{C}$ the left
$R$-module $R \otimes_{S} N$ is contained in ${\tt SC}(R)^{\perp}$.
\end{Proposition}
\vspace{-0.1in}
\begin{proof}
Since $N \in \dlimit\ \mathfrak{C}$, we may express $N$ as the colimit of a
filtered direct system of left $S$-modules $(N_{i})_{i}$, where
$N_{i} \in \mathfrak{C}$ for all $i$; we denote by
$f_{i} : N_{i} \longrightarrow N$ the canonical maps. Having fixed the index
$i$, we note that our assumption that $N \in \mathfrak{C}^{\perp}$ implies
that the abelian group $\widehat{\mbox{Ext}}^{0}_{S}(N_{i},N)$ is trivial;
in particular, $[f_{i}] = [0] \in \widehat{\mbox{Ext}}^{0}_{S}(N_{i},N)$.
In other words, the $S$-linear map
\[ \Omega^{n}f_{i} : \Omega^{n}N_{i} \longrightarrow \Omega^{n}N , \]
which is induced by $f_{i}$, factors through a projective left $S$-module
when $n \gg 0$. Since $R$ is flat as a right $S$-module, the $R$-linear map
\[ \Omega^{n}(1 \otimes f_{i}) : \Omega^{n} (R \otimes_{S} N_{i})
\longrightarrow \Omega^{n}(R \otimes_{S} N) , \]
which is induced by
$1 \otimes f_{i} : R \otimes_{S} N_{i} \longrightarrow R \otimes_{S} N$, is
identified with the $R$-linear map
\[ 1 \otimes \Omega^{n}f_{i} : R \otimes_{S} \Omega^{n}N_{i} \longrightarrow
R \otimes_{S} \Omega^{n}N . \]
Since the latter map factors through a projective left $R$-module when $n \gg 0$,
it follows that
$[1 \otimes f_{i}] = [0] \in
\widehat{\mbox{Ext}}^{0}_{R}(R \otimes_{R} N_{i},R \otimes_{S} N)$.
Therefore, Lemma \ref{ori_1.2} implies that for any left $R$-module $M$ the additive map
\begin{equation}
(1 \otimes f_{i})_{*} : \widehat{\mbox{Ext}}^{0}_{R}(M,R \otimes_{S} N_{i})
\longrightarrow \widehat{\mbox{Ext}}^{0}_{R}(M,R \otimes_{S} N)
\end{equation}
is trivial.
In order to show that $R \otimes_{S} N \in {\tt SC}(R)^{\perp}$, assume
that the left $R$-module $M$ is stably compact. Since the left $R$-module
$R \otimes_{S} N$ is the colimit of the filtered direct system
$(R \otimes_{S} N_{i})_{i}$, we have
\[ \widehat{\mbox{Ext}}^{0}_{R}(M,R \otimes_{S} N) =
\dlimiti\ \widehat{\mbox{Ext}}^{0}_{R}(M,R \otimes_{S} N_{i}) \]
and hence the abelian group $\widehat{\mbox{Ext}}^{0}_{S}(M,R \otimes_{S} N)$
is the union of the images of the maps (1) for all $i$. The triviality of
these maps shows that $\widehat{\mbox{Ext}}^{0}_{R}(M,R \otimes_{S} N) = 0$,
as needed. \end{proof}
In the sequel, we shall use Proposition \ref{ori_1.3}, in the form of the following
corollary.
\begin{Corollary}\label{ori_1.4}
Let $S$ be a subring of a ring $R$ and assume that $R$ is flat as a right
$S$-module. If any left $S$-module can be expressed as a filtered colimit
of completely finitary left $S$-modules, then for any left $S$-module
$N \in {\tt CF}(S)^{\perp}$, the left $R$-module $R \otimes_{S} N$ is
contained in ${\tt CF}(R)^{\perp}$.
\end{Corollary}
\vspace{-0.1in}
\begin{proof}
Since ${\tt SC}(R)^{\perp} \subseteq {\tt CF}(R)^{\perp}$, this is an
immediate consequence of Proposition \ref{ori_1.3}, applied in the special case
where $\mathfrak{C} = {\tt CF}(S)$. \end{proof}
We shall also need the following property of stably compact modules.
\begin{Lemma}\label{ori_1.5}
Let $R$ be a ring and consider a left $R$-module $M$, which is stably
compact and Gorenstein projective. Then, $M$ is isomorphic to a direct
summand of the direct sum $P \oplus N$ of two left $R$-modules $P$ and
$N$, where $P$ is projective and $N$ is finitely presented.
\end{Lemma}
\vspace{-0.1in}
\begin{proof}
We may express $M$ as the colimit of a filtered direct system $(M_{i})_{i}$
of finitely presented left $R$-modules and denote by
$f_{i} : M_{i} \longrightarrow M$ the canonical maps. Since the left
$R$-module $M$ is stably compact and Gorenstein projective, the functor
$\underline{\mbox{Hom}}_{R}(M,\_\!\_)$ commutes with filtered colimits;
in particular, there is an isomorphism
\[ \underline{\mbox{Hom}}_{R}(M,M) \simeq \dlimiti\
\underline{\mbox{Hom}}_{R}(M,M_{i}) .\]
Considering the identity map $1_{M} : M \longrightarrow M$, it follows that
there exists an index $i$ and an $R$-linear map
$g_{i} : M \longrightarrow M_{i}$, such that
$[1_{M}] = [f_{i}g_{i}] \in \underline{\mbox{Hom}}_{R}(M,M)$. Then, the
endomorphism $1_{M}-f_{i}g_{i}$ of $M$ factors through a projective left
$R$-module $P$; in other words, there exist $R$-linear maps
$a : M \longrightarrow P$ and $b : P \longrightarrow M$, such that
$1_{M} - f_{i}g_{i} = ba$. The composition
\[ M \stackrel{(a,g_{i})}{\longrightarrow} P \oplus M_{i}
\stackrel{[b,f_{i}]}{\longrightarrow} M \]
is then equal to the identity map of $M$ and hence $M$ is a direct summand
of $P \oplus M_{i}$. \end{proof}
\subsection{Poly-$\mathfrak{C}$ and hyper-$\mathfrak{C}$ modules.}
Let $R$ be a ring and consider a class $\mathfrak{C}$ of left $R$-modules.
We say that a left $R$-module $E$ is a poly-$\mathfrak{C}$ module if there
exists a non-negative integer $n$ and an ascending filtration
$E_{0} \subseteq E_{1} \subseteq \ldots \subseteq E_{n}$ of $E$ by submodules,
such that $E_{0} = 0$, $E_{n} = E$ and the quotient $E_{i}/E_{i-1}$ is contained
in $\mathfrak{C}$ for all $i=1, \ldots ,n$. In other words, the class of
poly-$\mathfrak{C}$ modules consists precisely of the iterated extensions
of modules in $\mathfrak{C}$. We say that a left $R$-module $E$ is a
hyper-$\mathfrak{C}$ module if there exists an ordinal $\alpha$ and an ascending
filtration of $E$ by submodules $E_{\beta}$, which is indexed by the ordinals
$\beta \leq \alpha$, such that $E_{0} = 0$, $E_{\alpha} = E$ and
$E_{\beta}/E_{\beta -1} \in \mathfrak{C}$ (resp.\
$E_{\beta} = \bigcup_{\gamma < \beta} E_{\gamma}$) if $\beta \leq \alpha$
is a successor (resp.\ a limit) ordinal. Such an ascending chain of submodules
$(E_{\beta})_{\beta \leq \alpha}$ will be referred to as a continuous ascending
chain of submodules with sections in $\mathfrak{C}$. We note that an arbitrary
direct sum of modules contained in $\mathfrak{C}$ is a hyper-$\mathfrak{C}$
module, whereas the class of hyper-$\mathfrak{C}$ modules is closed under
extensions. In the sequel, we shall use the following compactness result.
\begin{Lemma}\label{ori_1.6}
Let $R$ be a ring and consider a class $\mathfrak{C}$ of left $R$-modules,
which is such that:
\begin{enumerate}
\item
$\mathfrak{C}$ contains all projective left $R$-modules and
\item
$\mathfrak{C}$ consists of completely finitary Gorenstein projective modules.
\end{enumerate}
We also consider two left $R$-modules $M$ and $E$ and assume that:
\begin{enumerate}
\setcounter{enumi}{2}
\item $M$ is completely finitary and Gorenstein projective and
\item $E$ is a hyper-$\mathfrak{C}$ module.
\end{enumerate}
Then, any $R$-linear map $f : M \longrightarrow E$ factors through a
poly-$\mathfrak{C}$ module $E'$.
\end{Lemma}
\vspace{-0.1in}
\begin{proof}
In view of (iv), there exists a continuous ascending chain of submodules
$(E_{\beta})_{\beta \leq \alpha}$ with sections in $\mathfrak{C}$, such
that $E=E_{\alpha}$. We shall prove by transfinite induction on the ordinal
$\beta \leq \alpha$ that any $R$-linear map $f : M \longrightarrow E_{\beta}$
factors through a poly-$\mathfrak{C}$ module $E'$ for any completely finitary
Gorenstein projective left $R$-module $M$. Since $E = E_{\alpha}$, this will
complete the proof of the lemma. We note that there is nothing to prove if
$\beta = 0$, since $E_{0}=0$. We now proceed with the inductive step:
{\em Case 1:} If $\beta$ is a limit ordinal, then the continuity of the chain
implies that $E_{\beta} = \bigcup_{\gamma < \beta} E_{\gamma}$. In view of
assumption (iii), the functor $\underline{\mbox{Hom}}_{R}(M,\_\!\_)$ commutes
with filtered colimits and hence $\underline{\mbox{Hom}}_{R}(M,E_{\beta})$ is
the colimit of the filtered direct system
$\left( \underline{\mbox{Hom}}_{R}(M,E_{\gamma}) \right) \! _{\gamma < \beta}$.
In particular, there exists an ordinal $\gamma < \beta$ and an $R$-linear map
$g : M \longrightarrow E_{\gamma}$, such that
\[ [f] = [\iota g] \in \underline{\mbox{Hom}}_{R}(M,E_{\beta}) , \]
where we denote by $\iota$ the inclusion $E_{\gamma} \hookrightarrow E_{\beta}$.
Then, $[f - \iota g] = [0] \in \underline{\mbox{Hom}}_{R}(M,E_{\beta})$ and
hence there exists a projective left $R$-module $P$ and $R$-linear maps
$a : M \longrightarrow P$ and $b : P \longrightarrow E_{\beta}$, such that
$f - \iota g = ba$. Therefore, the $R$-linear map
$f = \iota g + ba : M \longrightarrow E_{\beta}$ factors as the composition
\[ M \stackrel{(g,a)}{\longrightarrow} E_{\gamma} \oplus P
\stackrel{[\iota,b]}{\longrightarrow} E_{\beta} . \]
Invoking the induction hypothesis, we conclude that
$g : M \longrightarrow E_{\gamma}$ factors through a poly-$\mathfrak{C}$ module
$E'$ and hence $f : M \longrightarrow E_{\beta}$ factors through
$E' \oplus P$; since $P \in \mathfrak{C}$ (in view of assumption (i)), the
latter module is poly-$\mathfrak{C}$. Hence, we have proved the existence of
a factorization of $f$, as needed.
{\em Case 2a:} We now assume that $\beta$ is a successor ordinal and $f$ is
surjective. We consider the pullback $N$ of the diagram
\[ \begin{array}{ccc}
& & M \\ & & \;\; \downarrow {\scriptstyle{f}} \\
E_{\beta -1} & \stackrel{\iota}{\longrightarrow} & E_{\beta}
\end{array} \]
where $\iota$ is the inclusion $E_{\beta -1} \hookrightarrow E_{\beta}$, and
the associated commutative diagram with exact rows
\begin{equation}
\begin{array}{ccccccccc}
0 & \longrightarrow & N & \stackrel{\jmath}{\longrightarrow} & M
& \longrightarrow & M/N & \longrightarrow & 0 \\
& & {\scriptstyle{u}} \downarrow \;\;
& & {\scriptstyle{f}} \downarrow \;\;
& & {\scriptstyle{v}} \downarrow \;\; & & \\
0 & \longrightarrow & E_{\beta -1} & \stackrel{\iota}{\longrightarrow}
& E_{\beta} & \longrightarrow & E_{\beta}/E_{\beta -1}
& \longrightarrow & 0
\end{array}
\end{equation}
Since $f$ is surjective, it follows that $E_{\beta}$ is the pushout of the
diagram
\[ \begin{array}{ccc}
N & \stackrel{\jmath}{\longrightarrow} & M \\
{\scriptstyle{u}} \downarrow \;\; & & \\ E_{\beta -1}
\end{array} \]
and hence the $R$-linear map $v : M/N \longrightarrow E_{\beta}/E_{\beta -1}$
is bijective. Then, assumption (ii) implies that the left $R$-module $M/N$ is
completely finitary and Gorenstein projective; it follows that the left $R$-module
$N$ is completely finitary and Gorenstein projective as well. Hence, invoking the
induction hypothesis, we conclude that the $R$-linear map $u$ factors as the
composition
\[ N \stackrel{u_{1}}{\longrightarrow} E' \stackrel{u_{2}}{\longrightarrow}
E_{\beta -1} , \]
where $E'$ is a poly-$\mathfrak{C}$ module. We now consider the pushout $E''$
of the diagram
\[ \begin{array}{ccc}
N & \stackrel{\jmath}{\longrightarrow} & M \\
{\scriptstyle{u_{1}}} \downarrow \;\;\; & & \\ E'
\end{array} \]
which fits into a commutative diagram with exact rows
\[
\begin{array}{ccccccccc}
0 & \longrightarrow & N & \stackrel{\jmath}{\longrightarrow} & M
& \longrightarrow & M/N & \longrightarrow & 0 \\
& & {\scriptstyle{u_{1}}} \downarrow \;\;\;
& & {\scriptstyle{f_{1}}} \downarrow \;\;\;
& & {\scriptstyle{v_{1}}} \downarrow \;\;\; & & \\
0 & \longrightarrow & E' & \stackrel{\eta}{\longrightarrow} & E''
& \longrightarrow & E''/E' & \longrightarrow & 0
\end{array}
\]
Then, the $R$-linear map $v_{1}$ is bijective and hence the left $R$-module
$E''/E' \simeq M/N \simeq E_{\beta}/E_{\beta -1}$ is isomorphic with a module
in $\mathfrak{C}$. Since $E'$ is a poly-$\mathfrak{C}$ module, it follows that
$E''$ is a poly-$\mathfrak{C}$ module as well. The equalities
$f \jmath = \iota u = (\iota u_{2})u_{1}$ and the definition of $E''$ as a pushout
imply that there is a unique $R$-linear map $f_{2} : E'' \longrightarrow E_{\beta}$,
such that $f = f_{2}f_{1}$ and $\iota u_{2} = f_{2} \eta$. In this way, we obtain
a factorization of (2), viewed as a morphism of extensions, as pictured below
\[
\begin{array}{ccccccccc}
0 & \longrightarrow & N & \stackrel{\jmath}{\longrightarrow} & M
& \longrightarrow & M/N & \longrightarrow & 0 \\
& & {\scriptstyle{u_{1}}} \downarrow \;\;\;
& & {\scriptstyle{f_{1}}} \downarrow \;\;\;
& & {\scriptstyle{v_{1}}} \downarrow \;\;\; & & \\
0 & \longrightarrow & E' & \stackrel{\eta}{\longrightarrow} & E''
& \longrightarrow & E''/E' & \longrightarrow & 0 \\
& & {\scriptstyle{u_{2}}} \downarrow \;\;\;
& & {\scriptstyle{f_{2}}} \downarrow \;\;\;
& & {\scriptstyle{v_{2}}} \downarrow \;\;\; & & \\
0 & \longrightarrow & E_{\beta -1} & \stackrel{\iota}{\longrightarrow}
& E_{\beta} & \longrightarrow & E_{\beta}/E_{\beta -1}
& \longrightarrow & 0
\end{array}
\]
In particular, we have shown that the
$R$-linear map $f$ factors as the composition
\[ M \stackrel{f_{1}}{\longrightarrow} E'' \stackrel{f_{2}}{\longrightarrow}
E_{\beta} , \]
where $E''$ is a poly-$\mathfrak{C}$ module, as needed.
{\em Case 2b:} We now assume that $\beta$ is a successor ordinal and $f$ is any
$R$-linear map. Then, we may choose a projective left $R$-module $L$, such that
there exists a surjective $R$-linear map $p : L \longrightarrow E_{\beta}$, and
note that $f$ factors as the composition
\[ M \stackrel{(1_{M},0)}{\longrightarrow} M \oplus L
\stackrel{[f,p]}{\longrightarrow} E_{\beta} . \]
Since the left $R$-module $M \oplus L$ is completely finitary and Gorenstein
projective, we may apply Case 2a above and conclude that the {\em surjective}
$R$-linear map $[f,p] : M \oplus L \longrightarrow E_{\beta}$ factors through
a poly-$\mathfrak{C}$ module. It follows that this is also the case for $f$,
as needed. \end{proof}
\subsection{Rickard's algebraic homotopy colimit.}
We now turn to an important method originally inspired by techniques in algebraic topology and brought into the realm of representation theory by Rickard \cite{rickard}.
Let $R$ be a ring and fix a class $\mathfrak{C}$ of stably compact Gorenstein
projective left $R$-modules. For any left $R$-module $M$ we shall define below
a sequence $(M_{n})_{n}$ of left $R$-modules and injective $R$-linear maps
\[ M_{0} \stackrel{\eta_{0}}{\longrightarrow}
M_{1} \stackrel{\eta_{1}}{\longrightarrow} \cdots
\stackrel{\eta_{n-1}}{\longrightarrow}
M_{n} \stackrel{\eta_{n}}{\longrightarrow} \cdots , \]
in such a way that:
\begin{enumerate}
\item $M_{0} = M$,
\item For any $n \geq 0$ the following condition (which will be referred to below
as condition (ii)$_{n}$) is satisfied: If we denote by
$\iota_{n} : M \longrightarrow M_{n}$ the composition
$M_{0} \stackrel{\eta_{0}}{\longrightarrow}
M_{1} \stackrel{\eta_{1}}{\longrightarrow} \cdots
\stackrel{\eta_{n-1}}{\longrightarrow} M_{n}$,
then there exists a projective left $R$-module $Q_{n}$ and an $R$-linear map
$f_{n} : Q_{n} \longrightarrow M_{n}$, such that the map
$[\iota_{n},f_{n}] : M \oplus Q_{n} \longrightarrow M_{n}$ is surjective and
its kernel $K_{n}$ is a hyper-$\mathfrak{C}$ module.
\item If $M_{\infty} = \dlimitn\ M_{n}$, then $M_{\infty} \in \mathfrak{C}^{\perp}$.
\end{enumerate}
The definition of the above sequence proceeds by induction on $n$. We begin by letting
$M_{0}=M$. Having constructed $M_{k}$ for $k=0,1, \ldots ,n$ and embeddings
$M_{0} \stackrel{\eta_{0}}{\longrightarrow}
M_{1} \stackrel{\eta_{1}}{\longrightarrow} \cdots
\stackrel{\eta_{n-1}}{\longrightarrow} M_{n}$,
in such a way that condition (ii)$_{k}$ is satisfied for all $k=0,1, \ldots ,n$,
we proceed with the inductive step as follows: We consider the category whose objects
are the pairs $(C,f)$, where $C \in \mathfrak{C}$ and $f \in \mbox{Hom}_{R}(C,M_{n})$,
with morphisms the obvious commutative triangles. We choose a set $\Lambda_{n}$ of
objects of that category, consisting of one object from each isomorphism class therein.
We let $C_{n} = \bigoplus_{(C,f) \in \Lambda_{n}} C$ and use the maps $f$ associated
with each object $(C,f) \in \Lambda_{n}$, in order to define an $R$-linear map
$f_{n} : C_{n} \longrightarrow M_{n}$. Since $\mathfrak{C}$ consists of Gorenstein
projective modules and ${\tt GP}(R)$ is closed under arbitrary direct sums, it follows
that the left $R$-module $C_{n}$ is Gorenstein projective. In particular, there exists
a projective left $R$-module $P_{n}$ and an embedding
$\jmath_{n} : C_{n} \longrightarrow P_{n}$. We now define $M_{n+1}$ as the pushout of
the diagram
\[
\begin{array}{ccc}
C_{n} & \stackrel{\jmath_{n}}{\longrightarrow} & P_{n} \\
{\scriptstyle{f_{n}}} \downarrow \,\,\,\,\,\, & & \\
M_{n} & &
\end{array}
\]
In other words, $M_{n+1}$ fits into a commutative diagram
\[
\begin{array}{ccc}
C_{n} & \stackrel{\jmath_{n}}{\longrightarrow} & P_{n} \\
{\scriptstyle{f_{n}}} \downarrow \,\,\,\,\,\, & &
\,\,\,\,\,\, \downarrow {\scriptstyle{\varphi_{n}}} \\
M_{n} & \stackrel{\eta_{n}}{\longrightarrow} & M_{n+1} \\
\end{array}
\]
and there is a short exact sequence of left $R$-modules
\[ 0 \longrightarrow C_{n} \longrightarrow M_{n} \oplus P_{n}
\stackrel{[\eta_{n},\varphi_{n}]}{\longrightarrow} M_{n+1}
\longrightarrow 0 . \]
Since $\jmath_{n}$ is injective, it follows that $\eta_{n}$ is injective as well.
We now let $Q_{n}$ be a projective left $R$-module and
$f_{n} : Q_{n} \longrightarrow M_{n}$ an $R$-linear map, such that
$[\iota_{n},f_{n}] : M \oplus Q_{n} \longrightarrow M_{n}$ is surjective and its
kernel $K_{n}$ is a hyper-$\mathfrak{C}$ module. Then, the composition
\[ M \oplus Q_{n} \oplus P_{n}
\stackrel{[\iota_{n},f_{n}] \oplus 1_{P_{n}}}{\longrightarrow}
M_{n} \oplus P_{n}
\stackrel{[\eta_{n},\varphi_{n}]}{\longrightarrow} M_{n+1} \]
is surjective (as both of it factor maps are), coincides with
$[\eta_{n}\iota_{n},f_{n+1}] = [\iota_{n+1},f_{n+1}]$ for a suitable
$R$-linear map $f_{n+1} : Q_{n} \oplus P_{n} \longrightarrow M_{n+1}$
(in fact, $f_{n+1} = [\eta_{n}f_{n},\varphi_{n}]$) and its kernel $K_{n+1}$
is a hyper-$\mathfrak{C}$ module (being an extension of the hyper-$\mathfrak{C}$
module $\mbox{ker} \, [\eta_{n},\varphi_{n}] = C_{n}$ by the hyper-$\mathfrak{C}$
module
$\mbox{ker} \left( [\iota_{n},f_{n}] \oplus 1_{P_{n}} \right) \! =
\mbox{ker} \, [\iota_{n},f_{n}] = K_{n}$\footnote{Giver $R$-linear maps
$A \stackrel{a}{\longrightarrow} B \stackrel{b}{\longrightarrow} C$ with $a$
surjective, it is easily seen that there is a short exact sequence of left
$R$-modules
$0 \longrightarrow \mbox{ker} \, a
\longrightarrow \mbox{ker} \, ba
\stackrel{a \mid}{\longrightarrow} \mbox{ker} \, b
\longrightarrow 0$.}).
Therefore, condition (ii)$_{n+1}$ is satisfied and this completes the inductive step.
It only remains to show that $M_{\infty} = \dlimitn\ M_{n}$ satisfies condition
(iii); since $\mathfrak{C} \subseteq {\tt GP}(R)$, this amounts to showing that
$\underline{\mbox{Hom}}_{R}(C,M_{\infty}) = 0$ for any left $R$-module
$C \in \mathfrak{C}$. To that end, we fix a left $R$-module $C \in \mathfrak{C}$,
a non-negative integer $n$ and consider an $R$-linear map
$f : C \longrightarrow M_{n}$. Then, by the very definition of the map
$f_{n} : C_{n} \longrightarrow M_{n}$, we may write $f = f_{n}g$ for a suitable
$R$-linear map $g : C \longrightarrow C_{n}$. In particular, it follows that
$\eta_{n}f = \eta_{n}f_{n}g = \varphi_{n} \jmath_{n}g$ factors through the
projective left $R$-module $P_{n}$ and hence
$[\eta_{n}f] = [0] \in \underline{\mbox{Hom}}_{R}(C,M_{n+1})$. This being the
case for all $f \in \mbox{Hom}_{R}(C,M_{n})$, we conclude that
\[ \eta_{n*} : \underline{\mbox{Hom}}_{R}(C,M_{n}) \longrightarrow
\underline{\mbox{Hom}}_{R}(C,M_{n+1}) \]
is the zero map for all $n \geq 0$. Since $\mathfrak{C}$ consists of stably
compact modules and $M_{\infty} = \dlimitn\ M_{n}$, the abelian group
$\underline{\mbox{Hom}}_{R}(C,M_{\infty})$ is the direct limit of the system
\[ \underline{\mbox{Hom}}_{R}(C,M_{0}) \stackrel{\eta_{0*}}{\longrightarrow}
\underline{\mbox{Hom}}_{R}(C,M_{1}) \stackrel{\eta_{1*}}{\longrightarrow}
\cdots \stackrel{\eta_{n-1*}}{\longrightarrow}
\underline{\mbox{Hom}}_{R}(C,M_{n}) \stackrel{\eta_{n*}}{\longrightarrow}
\cdots . \]
Hence, we conclude that $\underline{\mbox{Hom}}_{R}(C,M_{\infty}) = 0$, as
needed.
\section{Orthogonal classes over hierarchically decomposable groups}
Let $\mathfrak{C}$ be a class of left $R$-modules, which contains all projective
modules and is closed under direct sums and direct summands. Then, an application
of Schanuel's lemma shows that the following two conditions are equivalent for a
left $R$-module $M$:
\begin{enumerate}
\item
There is a projective left $R$-module $P$ and a surjective $R$-linear
map $f : P \longrightarrow M$, such that $\mbox{ker} \, f \in \mathfrak{C}$.
\item
For any projective left $R$-module $P$ and any surjective $R$-linear
map $f : P \longrightarrow M$, we have $\mbox{ker} \, f \in \mathfrak{C}$.
\end{enumerate}
If these conditions are satisfied, then we say that the first syzygy module of
$M$ is contained in $\mathfrak{C}$ and write $\Omega M \in \mathfrak{C}$. We say
that $\mathfrak{C}$ is closed under syzygy modules if $\Omega M \in \mathfrak{C}$
for any left $R$-module $M \in \mathfrak{C}$.
\vspace{0.15in}
\newline
{\bf Remark 2.1.}
Let $\mathfrak{C}$ be a class of left $R$-modules, which contains all projective
modules and is closed under direct sums, direct summands and syzygy modules. We
also consider a left $R$-module $N$, which fits into an exact sequence
\[ 0 \longrightarrow N_{k} \longrightarrow \cdots
\longrightarrow N_{1} \longrightarrow N_{0}
\longrightarrow N \longrightarrow 0 , \]
where $k \geq 0$ and $N_{i} \in \mathfrak{C}^{\perp}$ for all $i=0,1, \ldots ,k$.
Then, using induction on $k$, one can easily show that $N \in \mathfrak{C}^{\perp}$
as well.
\addtocounter{Lemma}{1}
\begin{Proposition}\label{ori_Prop2.2}
Let $G$ be a group and $\mathfrak{C}$ a class consisting of stably compact
$\Z G$-modules, which contains all projective modules and is closed
under direct sums, direct summands and syzygy modules. We also consider a
class $\mathfrak{X}$ of groups, which is closed under subgroups. Then, the
following conditions are equivalent for a $\Z G$-module $N$:
\begin{enumerate}
\item $\mbox{ind}_{H}^{\, G}\mbox{res}_{H}^{G}N \in \mathfrak{C}^{\perp}$ for any
subgroup $H \subseteq G$ with $H \in \mathfrak{X}$.
\item $\mbox{ind}_{H}^{\, G}\mbox{res}_{H}^{G}N \in \mathfrak{C}^{\perp}$ for any
subgroup $H \subseteq G$ with $H \in {\scriptstyle{{\bf LH}}}\mathfrak{X}$.
\end{enumerate}
\end{Proposition}
\begin{proof}
We only have to show that (i)$\rightarrow$(ii). To that end, we let
\[ \mathfrak{H} = \{ H : \mbox{$H$ is a subgroup of $G$ and
$\mbox{ind}_{H}^{G}\mbox{res}_{H}^{G}N \in \mathfrak{C}^{\perp}$} \} . \]
Our assumption implies that $\mathfrak{H}$ contains all $\mathfrak{X}$-subgroups
of $G$. First of all, we shall prove that $\mathfrak{H}$ contains all
${\scriptstyle{{\bf H}}}\mathfrak{X}$-subgroups of $G$ as well. To that end, it
suffices to show that if $H \subseteq G$ is any subgroup for which there exists
a finite dimensional contractible $H$-CW-complex $X$ with cell stabilizers in
$\mathfrak{H}$, then we have $H \in \mathfrak{H}$. If $H \subseteq G$ is such a
subgroup, then the augmented cellular chain complex of $X$ is an exact sequence
of $\Z H$-modules
\[ 0 \longrightarrow C_{m} \longrightarrow \cdots
\longrightarrow C_{0} \longrightarrow \Z \longrightarrow 0 . \]
Tensoring the exact sequence above with $\mbox{res}_{H}^{G}N$ over $\Z$ from the right and allowing the group $G$ to act diagonally, we obtain an exact
sequence of $\Z H$-modules
\[ 0 \longrightarrow C_m\otimes\mbox{res}_{H}^{G}N
\longrightarrow \cdots
\longrightarrow C_0\otimes\mbox{res}_{H}^{G}N
\longrightarrow \mbox{res}_{H}^{G}N \longrightarrow 0 , \]
which itself yields an exact sequence of $\Z G$-modules
\[ 0 \longrightarrow
\mbox{ind}_{H}^{\, G} \! \left[ C_m\otimes\mbox{res}_{H}^{G}N
\right] \!
\longrightarrow \cdots \longrightarrow
\mbox{ind}_{H}^{\, G} \! \left[ C_0\otimes\mbox{res}_{H}^{G}N\right] \!
\longrightarrow
\mbox{ind}_{H}^{\, G}\mbox{res}_{H}^{G}N \longrightarrow 0 . \]
Since the $\Z H$-module $C_{i}$ is a permutation module with stabilizers
in $\mathfrak{H}$, it follows that the $\Z G$-module
$\mbox{ind}_{H}^{G} \! \left[ C_i\otimes\mbox{res}_{H}^{G}N \right]$ is a
direct sum of modules of the form $\mbox{ind}_{H'}^{G}\mbox{res}_{H'}^{G}N$,
where $H'$ is an $\mathfrak{H}$-subgroup of $G$. We note that the latter modules
are contained (by the very definition of $\mathfrak{H}$) in $\mathfrak{C}^{\perp}$.
Since the class $\mathfrak{C}$ consists of stably compact modules, the orthogonal
class $\mathfrak{C}^{\perp}$ is closed under arbitrary direct sums. In particular,
we conclude that
$\mbox{ind}_{H}^{G} \! \left[ C_i\otimes\mbox{res}_{H}^{G}N \otimes \right] \! \in
\mathfrak{C}^{\perp}$
for all $i=0,1, \ldots ,m$. Then, Remark 2.1 implies that
$\mbox{ind}_{H}^{G}\mbox{res}_{H}^{G}N \in \mathfrak{C}^{\perp}$ and hence
$H \in \mathfrak{H}$.
We can now complete the proof and show that all
${\scriptstyle{{\bf LH}}}\mathfrak{X}$-subgroups $H$ of $G$ are contained in
$\mathfrak{H}$. Indeed, let $H \subseteq G$ be such a subgroup; then, all
finitely generated subgroups $H'$ of $H$ are contained in
${\scriptstyle{{\bf H}}}\mathfrak{X}$. We note that the $\Z G$-module
$\mbox{ind}_{H}^{G}\mbox{res}_{H}^{G}N$ is the filtered colimit of the
$\Z G$-modules $\mbox{ind}_{H'}^{G}\mbox{res}_{H'}^{G}N$, where $H'$
runs through the collection of finitely generated subgroups of $H$. In view
of the result that we have just proved for the
${\scriptstyle{{\bf H}}}\mathfrak{X}$-subgroups of $G$, these latter
modules are all contained in $\mathfrak{C}^{\perp}$. Since $\mathfrak{C}$ consists
of stably compact modules, the orthogonal class $\mathfrak{C}^{\perp}$ is closed
under filtered colimits. It follows that
$\mbox{ind}_{H}^{G}\mbox{res}_{H}^{G}N \in \mathfrak{C}^{\perp}$ and hence
$H \in \mathfrak{H}$, as needed. \end{proof}
\begin{Definition}\label{ori_2.3}
Let $\mathfrak{Y}$ be the class consisting of those groups $G$, which are
such that any $\Z G$-module may be expressed as a filtered colimit
of completely finitary $\Z G$-modules. In other words, if $G$ is a
group then $G \in \mathfrak{Y}$ if and only if $\dlimit\ {\tt CF}(\Z G)$
is the class of all $\Z G$-modules.
\end{Definition}
Using this terminology, we may reformulate some of our earlier results. As
an example, let $G$ be a group and consider a subgroup $H \subseteq G$ with
$H \in \mathfrak{Y}$. Then, Corollary 1.4 implies that for any
$\Z H$-module $N \in {\tt CF}(\Z H)^{\perp}$, the induced
$\Z G$-module $\mbox{ind}_{H}^{G} N$ is contained in
${\tt CF}(\Z G)^{\perp}$.
\begin{Corollary}\label{ori_2.4}
Let $\mathfrak{X}$ be a subgroup-closed subclass of $\mathfrak{Y}$
and consider an ${\scriptstyle{{\bf LH}}}\mathfrak{X}$-group $G$. Then, the
following conditions are equivalent for a $\Z G$-module $N$:
\begin{enumerate}
\item $N \in {\tt CF}(\Z G)^{\perp}$ and
\item $\mbox{res}_{H}^{G}N \in {\tt CF}(\Z H)^{\perp}$ for any subgroup
$H \subseteq G$ with $H \in \mathfrak{X}$.
\end{enumerate}
\end{Corollary}
\begin{proof}
The implication (i)$\rightarrow$(ii) is an immediate consequence of the
Eckmann-Shapiro lemma for complete cohomology. We shall now prove that
(ii)$\rightarrow$(i). We note that condition (ii) and our assumption that
$\mathfrak{X} \subseteq \mathfrak{Y}$ imply (as we noted above) that
$\mbox{ind}_{H}^{G}\mbox{res}_{H}^{G}N \in {\tt CF}(\Z G)^{\perp}$
for all $\mathfrak{X}$-subgroups $H \subseteq G$. Since $G$ is an
${\scriptstyle{{\bf LH}}}\mathfrak{X}$-group, we may use Proposition 2.2
and conclude that
$N = \mbox{ind}_{G}^{G}\mbox{res}_{G}^{G}N \in {\tt CF}(\Z G)^{\perp}$,
as needed. \end{proof}
\section{Groups of finite Gorenstein cohomological dimension}
In this section, we shall obtain certain explicit results concerning completely
finitary modules over the integral group ring of certain groups. In particular,
these results include:
\begin{enumerate}
\item a description of the structure of completely finitary Gorenstein projective
modules over the integral group ring of a hierarchically decomposable group of
finite Gorenstein cohomological dimension,
\item a study of the extent to which completely finitary modules over the integral
group ring of certain hierarchically decomposable groups of finite Gorenstein
cohomological dimension have projective resolutions which are eventually of finite
type and
\item a result relating groups with eventually finitary cohomology to the existence
of models for the associated Eilenberg--Mac Lane space with finitely many $n$-cells
for all $n \gg 0$.
\end{enumerate}
\subsection{The structure of completely finitary Gorenstein projective modules.}
The Eckmann-Shapiro lemma for complete cohomology implies that induction from
a subgroup $H$ of a group $G$ maps completely finitary $\Z H$-modules
to completely finitary $\Z G$-modules. Moreover, the class of completely
finitary modules is easily seen to be closed under extensions and direct summands.
Analogous assertions are valid for the class of Gorenstein projective modules as
well. The following result shows that these operations suffice to construct in an
economic way all completely finitary Gorenstein projective $\Z G$-modules,
in the case where the group $G$ is hierarchically decomposable and has finite
Gorenstein cohomological dimension.
\begin{Theorem}\label{ori_Thm3.1}
Let $\mathfrak{X}$ be subgroup-closed subclass of the class $\mathfrak{Y}$ introduced in Definition 2.2.
We consider an ${\scriptstyle{{\bf LH}}}\mathfrak{X}$-group $G$ of finite
Gorenstein cohomological dimension and the class
\[ \mathfrak{C} = \{ \mbox{ind}_{H}^{\, G}M : \mbox{$H$ is an
$\mathfrak{X}$-subgroup of $G$ and $M \in {\tt CF}(\Z H) \cap
{\tt GP}(\Z H)$} \} . \]
Then:
(i) ${\tt CF}(\Z G)^{\perp} = \mathfrak{C}^{\perp}$ and
(ii) the class ${\tt CF}(\Z G) \cap {\tt GP}(\Z G)$ of completely
finitary Gorenstein projective $\Z G$-modules is precisely the class
consisting of the direct summands of poly-$\mathfrak{C}$ modules.
\end{Theorem}
\vspace{-0.1in}
\begin{proof}
(i) If $H$ is any subgroup of $G$ and $M$ a completely finitary $\Z H$-module,
then the Eckmann-Shapiro lemma for complete cohomology implies that
$\mbox{ind}_{H}^{G}M$ is a completely finitary $\Z G$-module. In particular,
it follows that $\mathfrak{C} \subseteq {\tt CF}(\Z G)$ and hence
${\tt CF}(\Z G)^{\perp} \subseteq \mathfrak{C}^{\perp}$. Conversely, let
$N$ be a $\Z G$-module contained in $\mathfrak{C}^{\perp}$. In order to
show that $N \in {\tt CF}(\Z G)^{\perp}$, it suffices (in view of Corollary
2.4) to show that $\mbox{res}_{H}^{G}N \in {\tt CF}(\Z H)^{\perp}$ for any
$\mathfrak{X}$-subgroup $H$ of $G$. To that end, let $H$ be an $\mathfrak{X}$-subgroup
of $G$ and consider a completely finitary $\Z H$-module $M$. Since $G$ has
finite Gorenstein cohomological dimension, its subgroup $H$ has finite Gorenstein
cohomological dimension as well (cf. \cite{Ge-Gru}, $\S $5.1(iii)). Then, the
$\Z H$-module $M$ has finite Gorenstein projective dimension and hence the
functor $\widehat{\mbox{Ext}}^{0}_{\Z H}(M,\_\!\_)$ is naturally equivalent
to the functor $\widehat{\mbox{Ext}}^{0}_{\Z H}(M',\_\!\_)$, for a suitable
completely finitary Gorenstein projective $\Z H$-module $M'$. (In fact, one
may choose $M'$ to be the $0$-th syzygy of a complete resolution of $M$.) Then, we
have
\[ \widehat{\mbox{Ext}}^{0}_{\Z H} \! \left( M,\mbox{res}_{H}^{G}N \right)
\! =
\widehat{\mbox{Ext}}^{0}_{\Z H} \! \left( M',\mbox{res}_{H}^{G}N \right)
\! =
\widehat{\mbox{Ext}}^{0}_{\Z H} \! \left( \mbox{ind}_{H}^{\, G}M',N \right)
\! = 0 . \]
In the above chain of equalities, the second one is the Eckmann-Shapiro lemma for
complete cohomology, whereas the last one follows since $N \in \mathfrak{C}^{\perp}$.
Since the group
$\widehat{\mbox{Ext}}^{0}_{\Z H} \! \left( M,\mbox{res}_{H}^{G}N \right)$ is
trivial for any $M \in {\tt CF}(\Z H)$, it follows that
$\mbox{res}_{H}^{G}N \in {\tt CF}(\Z H)^{\perp}$, as needed.
(ii) First of all, we note that $\mathfrak{C}$ is contained in the intersection
${\tt CF}(\Z G) \cap {\tt GP}(\Z G)$; as we noted above, the inclusion
$\mathfrak{C} \subseteq {\tt CF}(\Z G)$ follows from the Eckmann-Shapiro
lemma for complete cohomology, whereas Lemma \ref{ori_1.1} implies that
$\mathfrak{C} \subseteq {\tt GP}(\Z G)$. Since both classes
${\tt CF}(\Z G)$ and ${\tt GP}(\Z G)$ are closed under extensions
and direct summands, we conclude that direct summands of poly-$\mathfrak{C}$ modules
are completely finitary and Gorenstein projective. Conversely, let $M$ be a completely
finitary Gorenstein projective $\Z G$-module. We consider the $\Z G$-module
$M_{\infty}$ constructed in $\S 1.$IV and adopt the notation used therein. Since $M$
is completely finitary and Gorenstein projective, whereas
$M_{\infty} \in \mathfrak{C}^{\perp}$, it follows from (i) above that
$\underline{\mbox{Hom}}_{\Z G}(M,M_{\infty}) = 0$. Since $M$ is, of course,
stably compact and $M_{\infty} = \dlimitn\ M_{n}$, we conclude that
\[ \dlimitn\ \underline{\mbox{Hom}}_{\Z G}(M,M_{n}) =
\underline{\mbox{Hom}}_{\Z G}(M,M_{\infty}) = 0 . \]
In particular, the class of the identity map of $M=M_{0}$ maps onto zero under
the additive map
\[ \iota_{n*} : \underline{\mbox{Hom}}_{\Z G}(M,M_{0}) \longrightarrow
\underline{\mbox{Hom}}_{\Z G}(M,M_{n}) \]
for some $n \gg 0$. It follows that
$[\iota_{n}] = [0] \in \underline{\mbox{Hom}}_{\Z G}(M,M_{n})$ and hence
$\iota_{n} : M \longrightarrow M_{n}$ factors through a projective
$\Z G$-module $P$ for some $n \gg 0$, say as the composition
\[ M \stackrel{a}{\longrightarrow} P \stackrel{b}{\longrightarrow} M_{n} . \]
We now invoke condition (ii)$_{n}$ in the construction of the $M_{n}$'s in
$\S 1.$IV: There is a projective $\Z G$-module $Q_{n}$ and a
$\Z G$-linear map $f_{n} : Q_{n} \longrightarrow M_{n}$, such that
$[\iota_{n},f_{n}] : M \oplus Q_{n} \longrightarrow M_{n}$ is a surjective
$\Z G$-linear map, whose kernel $K_{n}$ is a hyper-$\mathfrak{C}$ module.
Since the latter map factors as the composition
\[ M \oplus Q_{n} \stackrel{a \oplus 1_{Q_{n}}}{\longrightarrow}
P \oplus Q_{n} \stackrel{[b,f_{n}]}{\longrightarrow} M_{n} \]
and $P \oplus Q_{n}$ is a projective $\Z G$-module, we may invoke
(\cite{Kr2}, Lemma \ref{ori_Thm3.1}) in order to conclude that the $\Z G$-module $M \oplus Q_{n}$
(and, a fortiori, the $\Z G$-module $M$) is a direct summand of
$K_{n} \oplus P \oplus Q_{n}$. We note that the regular module $\Z G$ is
obviously contained in $\mathfrak{C}$ and hence all free $\Z G$-modules
are hyper-$\mathfrak{C}$. Since $K_{n}$ is a hyper-$\mathfrak{C}$ module, we
conclude that $M$ is a direct summand of a hyper-$\mathfrak{C}$ module $E$. In
other words, there exist $\Z G$-linear maps $f : M \longrightarrow E$ and
$g : E \longrightarrow M$, such that $gf = 1_{M} : M \longrightarrow M$. We now
invoke Lemma \ref{ori_1.6} and express $f$ as the composition of two $\Z G$-linear
maps $f_{1} : M \longrightarrow E'$ and $f_{2} : E' \longrightarrow E$, where
$E'$ is a poly-$\mathfrak{C}$ module. Since the composition
\[ M \stackrel{f_{1}}{\longrightarrow} E' \stackrel{f_{2}}{\longrightarrow}
E \stackrel{g}{\longrightarrow} M \]
is the identity map of $M$, it follows that $M$ is a direct summand of the
poly-$\mathfrak{C}$ module $E'$. \end{proof}
\begin{Corollary}\label{ori_Cor3.2}
Let $\mathfrak{X}$ be a subgroup-closed subclass of $\mathfrak{Y}$.
We consider an ${\scriptstyle{{\bf LH}}}\mathfrak{X}$-group $G$ of finite
Gorenstein cohomological dimension and the class
\[ \mathfrak{C} = \{ \mbox{ind}_{H}^{\, G}M : \mbox{$H$ is an
$\mathfrak{X}$-subgroup of $G$ and $M \in {\tt CF}(\Z H) \cap
{\tt GP}(\Z H)$} \} . \]
If $P_{*} \longrightarrow M \longrightarrow 0$ is a projective resolution of a
completely finitary $\Z G$-module $M$, then the image $\Omega^{n}M$ of the
map $P_{n} \longrightarrow P_{n-1}$ is a direct summand of a poly-$\mathfrak{C}$
module for all $n \geq \mbox{Gcd} \, G + 1$.
\end{Corollary}
\vspace{-0.1in}
\begin{proof}
The Gorenstein projective dimension of $M$ is $\leq \mbox{Gcd} \, G + 1$ and
hence the image $\Omega^{n}M$ of the map $P_{n} \longrightarrow P_{n-1}$ is
Gorenstein projective for all $n \geq \mbox{Gcd} \, G + 1$. Since $M$ is
completely finitary, it is clear that $\Omega^{n}M$ is completely finitary as
well. Then, the result follows from Theorem \ref{ori_Thm3.1}(ii). \end{proof}
\subsection{Coherent group rings.}
Recall that a ring is \emph{left coherent} if and only if every finitely generated left ideal is finitely presented, or equivalently if every finitely presented module is of type $FP_{\infty}$. This latter formulation of coherence may be compared with the observation that a ring is Noetherian if and only if every \emph{finitely generated} module is of type $FP_{\infty}$; therefore this formulation is both relevant to the present article and can help to generalize results and methods from the Noetherian case.
We consider the class $\mathfrak{K}$, consisting of those groups $G$ for which
the integral group ring $\Z G$ is (left) coherent.
We shall use the
following properties of this class of groups below:
(i) $\mathfrak{K}$ is subgroup-closed.
\newline
Indeed, let $G$ be a $\mathfrak{K}$-group and consider a subgroup $H \subseteq G$.
If $I$ is a finitely generated left ideal of $\Z H$, then $\mbox{ind}_{H}^{G}I$
is a finitely generated left ideal of $\Z G$. Since $G \in \mathfrak{K}$,
the $\Z G$-module $\mbox{ind}_{H}^{G}I$ is finitely presented and hence the
functor $\mbox{Hom}_{\Z G} \! \left( \mbox{ind}_{H}^{G}I, \_\!\_ \right)$
commutes with filtered colimits of $\Z G$-modules. Since any filtered direct
system of $\Z H$-modules is a direct summand of the restriction to $H$ of a
filtered direct system of $\Z G$-modules, it follows that the functor
$\mbox{Hom}_{\Z H}(I, \_\!\_)$ commutes with filtered colimits of
$\Z H$-modules. This implies that $I$ is a finitely presented
$\Z H$-module. Since this is the case for any finitely generated left ideal
$I \subseteq \Z H$, it follows that the group ring $\Z H$ is (left)
coherent and hence $H \in \mathfrak{K}$.
(ii) $\mathfrak{K}$ is a subclass of $\mathfrak{Y}$.
\newline
Indeed, if $G \in \mathfrak{K}$, then any finitely presented $\Z G$-module
is of type $FP_{\infty}$. Since any module (over any ring) may be expressed as
a filtered colimit of finitely presented modules, we conclude that any
$\Z G$-module may be expressed (in the case where $G$ is a $\mathfrak{K}$-group)
as a filtered colimit of modules of type $FP_{\infty}$. As modules of type
$FP_{\infty}$ are completely finitary (cf.\ \cite{kr1}, $\S $4.1(ii)), it follows that
$G \in \mathfrak{Y}$.
(iii) If $G$ is a $\mathfrak{K}$-group then any completely finitary Gorenstein
projective $\Z G$-module $M$ is a direct summand of a $\Z G$-module
$M'$, which possesses a projective resolution
$P'_{*} \longrightarrow M' \longrightarrow 0$ that is finitely generated after
one step, i.e.\ which is such that the projective $\Z G$-module $P'_{n}$
is finitely generated for all $n \geq 1$.
\newline
Indeed, we know that any completely finitary Gorenstein projective
$\Z G$-module $M$ is a direct summand of the direct sum $P \oplus N$
of two $\Z G$-modules $P$ and $N$, where $P$ is projective and $N$ is
finitely presented (cf.\ Lemma \ref{ori_1.5}). As we have already noted above, our
assumption that $G$ is a $\mathfrak{K}$-group implies that the
$\Z G$-module $N$ is of type $FP_{\infty}$ and this proves the claim.
\vskip0.1in
At this point it may be worth drawing attention to at least one source of examples of coherent group rings. The following is a natural generalization of (\cite{Bi-S}, Theorem B (iii)$\implies$(i)).
\begin{Proposition}\label{coherent_examples}
Let $G$ be the fundamental group of a graph of groups in which the group ring of each edge group is Noetherian and each vertex group is contained in $\mathfrak K$. Then $G$ belongs to $\mathfrak K$.
\end{Proposition}
\begin{proof} To prove that $\Z G$ is left coherent it suffices, according to Chase's theorem, to prove that an arbitrary product of flat right $\Z G$-modules is flat (see \cite{TYLam} Theorem 4.47 for example). Let $(F_i)_i$ be a family of flat right $\Z G$-modules and let $F=\prod_iF_i$ be the correspodning direct product. We shall prove that $\mbox{Tor}^{\Z G}_1(F,M)=0$ for any left
$\Z G$-module $M$.
Let $T$ be a $G$-tree \emph{with left action of $G$} that is witness to the hypotheses, with edge set $E$ and vertex set $V$. Then there is a short exact sequence $0\to\Z E\to\Z V\to \Z\to 0$ and this yields a short exact sequence with the diagonal action of $G$ upon tensoring with $M$, namely
$0\to \Z E\otimes M\to\Z V\otimes M\to M\to 0$. From here we obtain a commutative diagram combining long exact Mayer--Vietoris sequences:
{\footnotesize
\[
\xymatrix{
\mbox{Tor}_1^{\Z G}(F, \Z V\otimes M)\ar[r]\ar[d]&
\mbox{Tor}_1^{\Z G}(F,M)\ar[r]\ar[d]&
F\otimes_{\Z G}(\Z E\otimes M)\ar[r]\ar[d]&
F\otimes_{\Z G}(\Z V\otimes M)\ar[d]\\
\prod_i\mbox{Tor}_1^{\Z G}(F_i, \Z V\otimes M)\ar[r]&
\prod_i\mbox{Tor}_1^{\Z G}(F_i,M)\ar[r]&
\prod_i\left[F_i\otimes_{\Z G}(\Z E\otimes M)\right]\ar[r]&
\prod_i\left[F_i\otimes_{\Z G}(\Z V\otimes M)\right].\\
}
\]
}
Let $V_0$ be a set of orbit representatives for the vertex set $V$. Then the top left entry of the diagram,
$\mbox{Tor}_1^{\Z G}(F, \Z V\otimes M)$ is isomorphic to
$\bigoplus_{v\in V_0}\mbox{Tor}_1^{\Z G}(F,\Z[G/G_v]\otimes M)
\cong
\bigoplus_{v\in V_0}\mbox{Tor}_1^{\Z G_v}(F,M)$ and this is zero because each group ring $\Z G_v$ is coherent. Also the bottom centre-left entries are zero because each $F_i$ is $\Z G_v$-flat. So the diagram simplifies as shown.
\[
\xymatrix{
0\ar[r]&
\mbox{Tor}_1^{\Z G}(F,M)\ar[r]&
F\otimes_{\Z G}(\Z E\otimes M)\ar[r]\ar[d]^\alpha&
F\otimes_{\Z G}(\Z V\otimes M)\ar[d]\\
&
0\ar[r]&
\prod_i\left[F_i\otimes_{\Z G}(\Z E\otimes M)\ar[r]\right]&
\prod_i\left[F_i\otimes_{\Z G}(\Z V\otimes M)\right].\\
}
\]
The desired conclusion $\mbox{Tor}_1^{\Z G}(F,M)=0$ follows from the commutativity and exactness properties of this diagram once we show that the map labelled $\alpha$ is injective.
The codomain of $\alpha$ may be expressed as
$\prod_i\left[\bigoplus_e(F_i\otimes_{\Z G_e}M)\right]$ where $e$ runs through a set of orbit representatives of edges of our tree. This in turn sits inside
$\prod_i\prod_e(F_i\otimes_{\Z G_e}M)=\prod_e\prod_i(F_i\otimes_{\Z G_e}M)$ and so $\alpha$ can be seen to be the restriction of the product over a set of orbit representatives of edges of the natural maps $F\otimes_{\Z G_e}M\to\prod_i(F_i\otimes_{\Z G_e}M)$.
Injectivity follows from the hypothesis that each $\Z G_e$ is Noetherian: it is enough to check injectivity when $M$ is replaced by a finitely generated $\Z G_e$-submodule and therefore finitely presented and in such a case one can appeal to (\cite{TYLam}, Proposition 4.44) for example.
\end{proof}
\begin{Proposition}\label{ori_Prop3.3}
Let $G$ be an ${\scriptstyle{{\bf LH}}}\mathfrak{K}$-group of finite Gorenstein
cohomological dimension. Then, any completely finitary Gorenstein projective
$\Z G$-module $M$ is a direct summand of a $\Z G$-module $M'$,
which itself has a projective resolution that is finitely generated after
one step.
\end{Proposition}
\vspace{-0.1in}
\begin{proof}
Let
\[ \mathfrak{C} = \{ \mbox{ind}_{H}^{\, G}M : \mbox{$H$ is a
$\mathfrak{K}$-subgroup of $G$ and $M \in {\tt CF}(\Z H) \cap
{\tt GP}(\Z H)$} \} . \]
Since $\mathfrak{K}$ is subgroup-closed (property (i) above) and
$\mathfrak{K} \subseteq \mathfrak{Y}$ (property (ii) above), we may invoke
Theorem \ref{ori_Thm3.1}(ii) and conclude that any completely finitary Gorenstein projective
$\Z G$-module is a direct summand of a poly-$\mathfrak{C}$ module.
Therefore, it only remains to show that any poly-$\mathfrak{C}$ module $M$ is
a direct summand of a $\Z G$-module $M'$, which itself has a projective
resolution that is finitely generated after one step. This latter claim in an
immediate consequence of the following two assertions:
(a) Any $\Z G$-module contained in $\mathfrak{C}$ is a direct summand
of a $\Z G$-module, which itself has a projective resolution that is
finitely generated after one step.
(b) The class of those $\Z G$-modules which are direct summands of other
$\Z G$-modules, which themselves have a projective resolution that is
finitely generated after one step, is extension-closed.
In order to prove assertion (a), we consider a $\mathfrak{K}$-subgroup $H$
of $G$ and a completely finitary Gorenstein projective $\Z H$-module
$M$. Then, $M$ is a direct summand of a $\Z H$-module $M'$, which itself
has a projective resolution that is finitely generated after one step (property
(iii) above). Hence, the $\Z G$-module $\mbox{ind}_{H}^{G}M$ is a direct
summand of the $\Z G$-module $\mbox{ind}_{H}^{G}M'$, which itself has a
projective resolution that is finitely generated after one step.
In order to prove assertion (b), we consider a short exact sequence of
$\Z G$-modules
\[ 0 \longrightarrow M' \stackrel{\iota}{\longrightarrow} M
\stackrel{p}{\longrightarrow} M'' \longrightarrow 0 \]
and assume that $M'$ and $M''$ are direct summands of other $\Z G$-modules,
which themselves have projective resolutions that are finitely generated after
one step. In other words, we assume that there exist $\Z G$-modules $N'$
and $N''$, such that the $\Z G$-modules $M' \oplus N'$ and $M'' \oplus N''$
have projective resolutions that are finitely generated after one step. We have
to show that $M$ is a direct summand of a $\Z G$-module, which itself has
a projective resolution that is finitely generated after one step. To that end,
we consider the short exact sequence of $\Z G$-modules
\[ 0 \longrightarrow M' \oplus N'
\stackrel{\jmath}{\longrightarrow} M \oplus N' \oplus N''
\stackrel{q}{\longrightarrow} M'' \oplus N''
\longrightarrow 0 , \]
where $\jmath(x,y) = (\iota(x),y,0)$ for all $(x,y) \in M' \oplus N'$ and
$q(w,y,z) = (p(w),z)$ for all $(w,y,z) \in M \oplus N' \oplus N''$. Applying
the horseshoe lemma to the latter short exact sequence, we conclude that the
$\Z G$-module $M \oplus N' \oplus N''$ has a projective resolution that
is finitely generated after one step. This proves the claim made for $M$ and
finishes the proof of assertion (b). \end{proof}
Let $R$ be a ring. We say that a projective resolution
$P_{*} \longrightarrow M \longrightarrow 0$ of a left $R$-module $M$ is finitely
generated after $k$ steps if the projective module $P_{n}$ is finitely generated
for all $n \geq k$.
\begin{Lemma}\label{ori_3.4}
Let $R$ be a ring and consider a short exact sequence of left $R$-modules
\[ 0 \longrightarrow M' \stackrel{\iota}{\longrightarrow} M
\stackrel{p}{\longrightarrow} M'' \longrightarrow 0 . \]
If $M'$ (resp.\ $M$) has a projective resolution which is finitely generated
after $k'$ steps (resp.\ after $k$ steps), then $M''$ has a projective resolution
which is finitely generated after $k''$ steps, where $k'' = \max \{ k'+1,k \}$.
\end{Lemma}
\vspace{-0.1in}
\begin{proof}
Let $P'_{*} \longrightarrow M' \longrightarrow 0$ and
$P_{*} \stackrel{\varepsilon}{\longrightarrow} M \longrightarrow 0$ be projective
resolutions which are finitely generated after $k'$- and $k$ steps respectively.
We consider a chain map $\iota_{*} : P'_{*} \longrightarrow P_{*}$, which lifts
$\iota$, and consider the mapping cone $P''_{*} = \mbox{cone}(\iota_{*})$. If
$\varepsilon'' : P''_{0} \longrightarrow M''$ is the composition
\[ P''_{0} = P_{0} \stackrel{\varepsilon}{\longrightarrow}
M \stackrel{p}{\longrightarrow} M'' , \]
then it is easily seen that
$P''_{*} \stackrel{\varepsilon''}{\longrightarrow} M'' \longrightarrow 0$ is a
projective resolution of $M''$. By its very definition,
$P''_{n} = P_{n} \oplus P'_{n-1}$ is finitely generated if $n \geq k$ and
$n-1 \geq k'$ and this finishes the proof. \end{proof}
\begin{Corollary}\label{ori_Cor3.5}
Let $G$ be an ${\scriptstyle{{\bf LH}}}\mathfrak{K}$-group of finite Gorenstein
cohomological dimension. Then, any completely finitary $\Z G$-module $M$
is a direct summand of a $\Z G$-module $M'$, which itself has a projective
resolution that is finitely generated after $k$ steps, where
$k = \mbox{Gpd}_{\Z G}M + 1$.
\end{Corollary}
\vspace{-0.1in}
\begin{proof}
We proceed by induction on the Gorenstein projective dimension
$\mbox{Gpd}_{\Z G}M$ of $M$. If $M$ is Gorenstein projective, the result
follows from Proposition \ref{ori_Prop3.3}. We now assume that $\mbox{Gpd}_{\Z G}M=m>0$
and consider a short exact sequence of $\Z G$-modules
\[ 0 \longrightarrow K \stackrel{\iota}{\longrightarrow} P
\stackrel{p}{\longrightarrow} M \longrightarrow 0 , \]
where $P$ is projective. Then, $K$ is completely finitary and has Gorenstein
projective dimension $m-1$. In view of the induction hypothesis, we conclude that
there exists a $\Z G$-module $L$, such that $K \oplus L$ has a projective
resolution which is finitely generated after $m$ steps. We now consider the short
exact sequence of $\Z G$-modules
\[ 0 \longrightarrow K \oplus L
\stackrel{\jmath}{\longrightarrow} P \oplus K \oplus L
\stackrel{q}{\longrightarrow} M \oplus K
\longrightarrow 0 , \]
where $\jmath(x,y) = (\iota(x),0,y)$ for all $(x,y) \in K \oplus L$ and
$q(w,x,y) = (p(w),x)$ for all $(w,x,y) \in P \oplus K \oplus L$. Since
$P \oplus K \oplus L$ has clearly a projective resolution which is finitely
generated after $\max \{ m,1 \} = m$ steps, we may invoke Lemma \ref{ori_3.4} and conclude
that $M \oplus K$ has a projective resolution which is finitely generated
after $\max \{ m+1, m \} = m+1$ steps, as needed. \end{proof}
\noindent{\bf Remark 3.7.}
Let $G$ be an ${\scriptstyle{{\bf LH}}}\mathfrak{K}$-group of finite Gorenstein
cohomological dimension. Then, the proof of Corollary \ref{ori_Cor3.5} above shows that for
any completely finitary $\Z G$-module $M$ the direct sum
$M \oplus \Omega M$ has a projective resolution that is finitely generated
after $k$ steps, where $k = \max \{ 2, \mbox{Gpd}_{\Z G}M + 1 \}$. (Using
Schanuel's lemma and Lemma 3.5, it is easily seen that this latter condition does
not depend upon the particular choice of the module $\Omega M$.)
\addtocounter{Lemma}{1}
\begin{Corollary}\label{ori_Cor3.7}
Let $G$ be an ${\scriptstyle{{\bf LH}}}\mathfrak{K}$-group of finite Gorenstein
cohomological dimension. Then, any completely finitary $\Z G$-module $M$
is a direct summand of a $\Z G$-module $M'$, which itself has a projective
resolution that is finitely generated after $k$ steps, where $k = \mbox{Gcd} \, G + 2$.
\end{Corollary}
\vspace{-0.1in}
\begin{proof}
Since $\mbox{Gpd}_{\Z G}M \leq \mbox{Gcd} \, G + 1$, the result follows
from Corollary 3.6. \end{proof}
\subsection{Groups with finitary cohomology.}
We say that a group $G$ has infinitely often finitary cohomology if there are
infinitely many non-negative integers $n$, such that the ordinary cohomology
functors $H^{n}(G,\_\!\_) = \mbox{Ext}^{n}_{\Z G}(\Z ,\_\!\_)$
commute with filtered colimits. If this latter condition holds for all but
finitely many $n$'s, we say that $G$ has eventually finitary cohomology. The
following result subsumes both Theorem A and Theorem B as stated in the Introduction.
\begin{Theorem}\label{ori_Thm3.8}
Let $G$ be an ${\scriptstyle{{\bf LH}}}\mathfrak{K}$-group of Gorenstein
cohomological dimension $\mbox{Gcd} \, G =k<\infty$. Then the following conditions are equivalent:
\begin{enumerate}
\item % i
The cohomology functors $H^{n}(G,\_\!\_)$ commute with direct limits for all $n \geq k + 3$.
\item % ii
The group $G$ has eventually finitary cohomology.
\item % iii
The group $G$ has infinitely often finitary cohomology.
\item % iv
In every projective resolution of $\Z$ over $\Z G$, the $k$th kernel is a direct summand of a $\Z G$-module which has a projective resolution that is of finite type after one step.
\item % v
Over the integral group ring of the group $G\times\Z$, the trivial module $\Z$ has
a projective resolution $P_*\twoheadrightarrow\Z$ in which $P_n$ is finitely generated for all sufficiently large $n$.
\item % vi
The group $G \times \Z $ has an Eilenberg--Mac Lane space
$K(G \times \Z ,1)$ with finitely many $n$-cells for all sufficiently large $n\ge k+2$.
\item % vii
The group $G$ has an Eilenberg--Mac Lane space $K(G,1)$, which is dominated
by a CW-complex with finitely many $n$-cells for all $n \geq k + 2$.
\item % viii
The group $G$ has an Eilenberg--Mac Lane space $K(G,1)$, which is dominated
by a CW-complex with finitely many $n$-cells for all sufficiently large $n$.
\end{enumerate}
\end{Theorem}
\begin{proof}
The implications (i)$\rightarrow$(ii)$\rightarrow$(iii), (vi)$\rightarrow$(v) and
(vii)$\rightarrow$(viii) are trivial, whereas the implications
(vi)$\rightarrow$(vii)$\rightarrow$(i) and (vi)$\rightarrow$(viii)$\rightarrow$(ii)
are valid for any group, as shown in (\cite{Ha}, \S2.2 and \S2.3). In order to show
that (iii)$\rightarrow$(v)$\rightarrow$(vi), we note that condition (iii) implies that the complete
cohomology functors
$\widehat{H}^{n}(G,\_\!\_) = \widehat{\mbox{Ext}}^{n}_{\Z G}(\Z ,\_\!\_)$
commute with filtered colimits for all $n \in \Z $ (cf.\ \cite{kr1}, \S4.1(ii)),
i.e.\ that the trivial $\Z G$-module $\Z $ is completely finitary. If
\[ 0 \longrightarrow M \longrightarrow F_{k-1} \longrightarrow \cdots
\longrightarrow F_{1} \longrightarrow F_{0} \longrightarrow \Z
\longrightarrow 0 \]
is any exact sequence of $\Z G$-modules, where $F_{i}$ is free for all
$i = 0,1, \ldots ,k-1$, then $M$ is a completely finitary Gorenstein projective
$\Z G$-module. Therefore, $M$ is a direct summand of a $\Z G$-module
$M'$, which itself has a projective resolution that is finitely generated after
one step (cf.\ Proposition \ref{ori_Prop3.3}) and this shows that (iv) holds. Now suppose that (iv) holds, so that we have a partial resolution as above.
Since we wish to think of the additive group $\Z$ as a multiplicative
group, we introduce the notation $C = \langle c \rangle$ for it, so that the group algebra $\Z C$ is a Laurent polynomial ring $\Z[c,c^{-1}]$ in one variable.
Then we may tensor the exact sequence above with the projective resolution
\[ 0 \longrightarrow \Z C \stackrel{1-c}{\longrightarrow} \Z C
\longrightarrow \Z \longrightarrow 0 \]
of the trivial $\Z C$-module $\Z $ and obtain an exact sequence
of $\Z (G \times C)$-modules
\[ \!\!\! \!\!\! \!\!\! \!\!\!\!\!\! \!\!\! \!\!\! \!\!\! \!\!\! \!\!\!\!\!\!
\!\!\! \!\!\! \!\!\! \!\!\!\!\!\! \!\!\! \!\!\! \!\!\! \!\!\! \!\!\!\!\!\!
0 \longrightarrow M \otimes \Z C
\longrightarrow (M \otimes \Z C) \oplus (F_{k-1} \otimes \Z C)
\longrightarrow \]
\[ (F_{k-1} \otimes \Z C) \oplus (F_{k-2} \otimes \Z C)
\longrightarrow \cdots
\longrightarrow F_{0} \otimes \Z C \longrightarrow \Z
\longrightarrow 0 . \]
Adding a suitable $\Z (G \times C)$-module in degrees $k$ and $k+1$, we
then obtain an exact sequence of $\Z (G \times C)$-modules
\[ \!\!\! \!\!\! \!\!\! \!\!\!\!\!\! \!\!\! \!\!\! \!\!\! \!\!\! \!\!\!\!\!\!
\!\!\! \!\!\! \!\!\! \!\!\!\!\!\! \!\!\! \!\!\! \!\!\! \!\!\! \!\!\!\!\!\!
0 \longrightarrow M' \otimes \Z C
\stackrel{f}{\longrightarrow}
(M' \otimes \Z C) \oplus (F_{k-1} \otimes \Z C)
\longrightarrow \]
\[ (F_{k-1} \otimes \Z C) \oplus (F_{k-2} \otimes \Z C)
\longrightarrow \cdots
\longrightarrow F_{0} \otimes \Z C \longrightarrow \Z
\longrightarrow 0 . \]
Since the $\Z G$-module $M'$ admits a projective resolution that is finitely
generated after one step, the argument in the proof of Lemma \ref{ori_3.4} shows that there is
a $\Z (G \times C)$-projective resolution
$Q_{*} \longrightarrow \mbox{coker} \, f \longrightarrow 0$ of $\mbox{coker} \, f$,
such that $Q_{0}$ is a free $\Z (G \times C)$-module and $Q_{n}$ is finitely
generated and free for all $n \geq 2$. By concatenation, we obtain a projective
resolution of the trivial $\Z (G \times C)$-module $\Z $
\[ \cdots \longrightarrow Q_{2} \longrightarrow Q_{1} \longrightarrow Q_{0}
\longrightarrow
(F_{k-1} \otimes \Z C) \oplus (F_{k-2} \otimes \Z C)
\longrightarrow \cdots
\longrightarrow F_{0} \otimes \Z C \longrightarrow \Z
\longrightarrow 0 , \]
where all modules with the possible exception of $Q_{1}$ are free and $Q_{n}$
is finitely generated (and free) for all $n \geq 2$. Using Eilenberg's trick,
we may add a free $\Z (G \times C)$-module to both $Q_{1}$ and $Q_{0}$
and obtain a resolution of the trivial $\Z (G \times C)$-module $\Z $
\[ \cdots \longrightarrow Q_{2} \longrightarrow Q'_{1} \longrightarrow Q'_{0}
\longrightarrow
(F_{k-1} \otimes \Z C) \oplus (F_{k-2} \otimes \Z C)
\longrightarrow \cdots
\longrightarrow F_{0} \otimes \Z C \longrightarrow \Z
\longrightarrow 0 , \]
where all modules are free and $Q_{n}$ is finitely generated (and free) for all
$n \geq 2$. Thus (v) holds.
Then, the topological arguments used in the proof of (\cite{Ha}, Theorem 2.20)
show that there is a model for the Eilenberg--Mac Lane space $K(G\times C,1) = K(G \times \Z ,1)$
with finitely many $n$-cells for all $n \geq k+2$
and we conclude that (vi) holds.
\end{proof}
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\end{document}