Phantom maps and purity in modular representation theory, I.
D. J. Benson and G. Ph. Gnacadja

\begin{abstract}
Let $k$ be a field and $G$ a finite group. 
By analogy with the theory of phantom maps in topology,
a map $f:M \ra N$ between
$kG$-modules is said to be phantom if its restriction to every finitely
generated submodule of $M$ factors through a projective module. 
We investigate the relationships between the theory of phantom maps, 
the algebraic theory of purity, and Rickard's idempotent modules.
In general, adding one to
the pure global dimension of $kG$ gives an upper bound for the
number of phantoms we need to compose to get a map which factors
through a projective module. However, this bound is not sharp. For 
example, for the group $\Z/4\times\Z/2$ in characteristic two, 
the composite of $6$ phantom maps always factors through a projective
module, whereas the pure global dimension of the group algebra can
be arbitrarily large.
\end{abstract}