\title{Virtual projectivity and the stable module category}
\author{David J. Benson}
\address{Institute of Mathematics, Fraser Noble Building, Univeristy
  of Aberdeen, Aberdeen AB24 3UE, United Kingdom}
\author{Jon F. Carlson}
\address{Department of Mathematics, University of Georgia, Athens GA
  30602, USA}

\subjclass[2010]{20C20, 20J06}
\keywords{Finite groups, modular representation theory, virtual
  projectivity, stable module category, nearly null map}

\begin{document}

\begin{abstract}
We introduce the concept of a nearly null map in the stable module
category, and relate it to the notion of virtual projectivity. We show
that the trivial module $k$ is virtually $M$-projective if and only if
the map $f\colon N \to k$ in the triangle 
$N \xrightarrow{f} k \to M \otimes M^*$ is nearly null. If these
conditions hold, then $M$ generates the stable module category
in fewer than $\ell$ steps, where $\ell$ is the radical length of the
group algebra. We give examples to show that if $M$ generates the
stable module category then $k$ is not necessarily virtually
$M$-projective. We also give examples to show that for a given group, 
the degree of virtual projectivity of $k$ with respect to a module $M$
is not bounded. Finally, we develop a theory of virtual $M$-variety of
a module $X$. This is a subset of the spectrum of the cohomology ring
which controls virtual projectivity of $X$ with respect to $M$.
\end{abstract}