  We consider the homotopy category of perfect complexes for a finite
dimensional self-injective algebra over a field, identifying many aspects of
perfect complexes according to their position in the Auslander-Reiten quiver.
Short complexes lie close to the rim. We characterize the position in the
quiver of complexes of lengths 1, 2 and 3, as well as rigid complexes and
truncated projective resolutions. We describe completely the quiver components
that contain projective modules (complexes of length 1). We obtain
relationships between the homology of complexes at different places in the
quiver, deducing that every self-injective algebra of radical length at least 3
has indecomposable perfect complexes with arbitrarily large homology in any
given degree. We show that homology stabilizes, in a certain sense, away from
the rim of the quiver.
