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\begin{document}

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\begin{abstract}
We develop the theory of relative projectivity and the triangulated category
for the module category of a group algebra. We produce an algebraic object
that plays a similar role to the derived category in relation to the stable
module category, and prove an analogue of a result of Rickard's in  the
ordinary stable category. 
\end{abstract}

\section{The distinction from the usual stable category}
 Although a lot of the behaviour of relative stable categories mimics that of
 the ordinary stable category, there are results that fail to pass through. We
 study one of these, homotopy colimits, in another section, but let us offer
 one non-example in the finite dimensional case.

If $M$ is a (non-zero) module with no projective summands and if for all
simple modules $S$ $\underline{\Hom}_{kG}(M,S) = 0$ then $M$ is projective.

Proof: Let $S$ be any module in the top of $M$, then the surjection factors
through the projective cover $P(S)$ of $S$. Since the top of $P(S)$ contains
exactly one module, the map from $M$ to it must be a surjection, hence $P(S)$
is a summand of $M$. $S$ was arbitrary, and we are done $\Box$


There is no corresponding result for the relatively stable category since
there is no a priori reason for the top of the relative projective cover to
contain only one module. And even if this were the case, then there is no
reason for the projective cover to be a summand since the map might not factor
as a $W/F$ split map as the following example demonstrates.

 Let $G$ be the vier-gruppe, $C_2 \times C_2$, and let $C_2 \times 1 \cong
 H\leq G$, then consider the relatively projective modules induced from
 $H$. In this case we have a complete description of the $G$ modules:



% in here is an example of how to do xypic, there are others later too.


\begin{enumerate}
\item The trivial module, $k$
\item The free module $kG$
 \[\xymatrix{  
     & k \ar[dl]^{\overline{g}} \ar[dr]^{\overline{h}} &  \\
k \ar[dr]^{\overline{h}}  &      & k \ar[dl]^{\overline{g}} \\
          & k   &     }\]
\item The vee-modules
 \[\xymatrix{
     k \ar[dr] &   & k \ar[dl] \ar[dr] &   & k \ar[dl] \ar[dr] & {\ldots}\\
               & k &                    & k &     & k {\ldots} }\]
\item The em-modules
 \[\xymatrix{
      & k \ar[dr] \ar[dl]&   & k \ar[dl] \ar[dr] &   & k \ar[dl] \ar[dr] & {\ldots}\\
     k&          & k &                    & k &     & k {\ldots} }\]

\item the other ones
\[\xymatrix{
k \ar[d] \ar[dr] & k \ar[dl] \ar [d] \\
k & k }\]


\end{enumerate}





where the generators of $C_2\times C_2$ are denoted, by abuse of notation,
$g=(g,e)$ and $h=(e,h)$, so that $g$ generates $H$, and the lines indicate
the actions of $\overline{g}=1+g$ and $\overline{h}=1+h$. We label the
vee-module of dimension $n$ as $V_n$, and simlilarly we use $M_n$ for the
em-module of dimension $n$.

There are exactly two indecomposable $kH$ modules, which are induced to the
2-dimensional vee-module, $V_2$ and the free $kG$ module. Any relatively
projective module is a direct sum of copies of these.

In this case the natural inclusion $k\to V_3$ factors as the natural
inclusions $k \to V_2 \to V_3$ and we see that the lemma we started the
section with has no analogue in the relatively stable category.



Remark: we can characterize this in terms of perpendicular categories,
something we shall return to in the investigation of homotopy colimits. Let
$\mathcal{S}$ be the smallest full triangulated subcategory containing the
simple modules and closed under direct sums and summands in
$\underline{\text{mod}}_W$. If $W=kG$ then $\mathcal{S}$ generates the  
 (relatively) stable module category. However, if $W$ is not projective then it
does not necessarily generate. In this case we define the (triangulated
subcategory) $\mathcal{S}^{\perp}$ as $\{T \in \underline{\text{mod}}_W |
\text{Hom}_W(S,T)=0 \ \forall S \in \mathcal{S}\}$
 

\subsection{Relatively Projective Resolutions}


 We will continue to use example of the module category of a group, but we
 will adopt the notation of the adjoint functors, as the results apply in this
 larger framework, ie there is no harm in thinking $W\otimes_k?$, whenever we
 see $F$. Recall that we had a functor $F:A \to B$ with left and
 right adjoints $G_L$ and $G_R$, and that the $F$-projectives were summands of
 objects of the form $G_L(X)$ \&c. In order to simplify the presentation we will
 make the assumption that $G_L = G_R=G$ rather than that the class of $F$
 projectives and injectives coincides. Hence $F$ and $G$ are a pair of adjoint
 functors. 






\end{document}