### Ghosts in modular representation theory, by Sunil K. Chebolu, J. Daniel Christensen, and Jan Minac

A ghost over a finite group G is a map between modular representations of G
which is invisible in Tate cohomology. Motivated by the failure of the
generating hypothesis---the statement that ghosts between finite-dimensional
G-representations factor through a projective---we define the compact ghost
number of kG to be the smallest integer l such that the composition of any l
ghosts between finite-dimensional G-representations factors through a
projective. In this paper we study ghosts and the compact ghost numbers of
p-groups. We begin by showing that a weaker version of the generating
hypothesis, where the target of the ghost is fixed to be the trivial
representation k, holds for all p-groups. We do this by proving that a map
between finite-dimensional G-representations is a ghost if and only if it is a
dual ghost. We then compute the compact ghost numbers of all cyclic p-groups
and all abelian 2-groups with C_2 as a summand. We obtain bounds on the compact
ghost numbers for abelian p-groups and for all 2-groups which have a cyclic
subgroup of index 2. Using these bounds we determine the finite abelian groups
which have compact ghost number at most 2. Finally, using universal ghosts, we
establish various sets of conditions which guarantee the existence of a
non-trivial ghost out of a G-representation. Our methods involve techniques
from group theory, representation theory, triangulated category theory, and
constructions motivated from homotopy theory.

Sunil K. Chebolu <schebolu@uwo.ca>

J. Daniel Christensen <jdc@uwo.ca>

Jan Minac <minac@uwo.ca>