  In a k-linear triangulated category (where k is a field) we show that the
existence of Auslander-Reiten triangles implies that objects are determined, up
to shift, by knowing dimensions of homomorphisms between them. In most cases
the objects themselves are distinguished by this information, a conclusion
which was also reached under slightly different hypotheses in a theorem of
Jensen, Su and Zimmermann. The approach is to consider bilinear forms on
Grothendieck groups which are analogous to the Green ring of a finite group.
  We specialize to the category of perfect complexes for a self-injective
algebra, for which the Auslander-Reiten quiver has a known shape. We
characterize the position in the quiver of many kinds of perfect complexes,
including those of lengths 1, 2 and 3, rigid complexes and truncated projective
resolutions. We describe completely the quiver components which contain
projective modules. 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 find also that homology
stabilizes away from the rim of the quiver. We show that when the algebra is
symmetric, one of the forms considered earlier is Hermitian, and this allows us
to compute its values knowing them only on objects on the rim of the quiver.
