Author: Peter Webb
email: webb@math.umn.edu
Address:
School of Mathematics
University of Minnesota
Minneapolis, MN 55455, USA

Title:
Consequences of the Existence of Auslander-Reiten Triangles with Applications to  
Perfect Complexes for Self-Injective Algebras

Abstract:
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.

Preprint January 2013