2013/01/20 by Peter Webb, Webb, Peter
Mathematics · #16G70 (Primary) 18E30 #20C20 (Secondary) #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1301.4701
openalex publication_date 2013/01/20 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
In a k-linear triangulated category (where k is a field) we show that the\nexistence of Auslander-Reiten triangles implies that objects are determined, up\nto shift, by knowing dimensions of homomorphisms between them. In most cases\nthe objects themselves are distinguished by this information, a conclusion\nwhich was also reached under slightly different hypotheses in a theorem of\nJensen, Su and Zimmermann. The approach is to consider bilinear forms on\nGrothendieck groups which are analogous to the Green ring of a finite group.\n We specialize to the category of perfect complexes for a self-injective\nalgebra, for which the Auslander-Reiten quiver has a known shape. We\ncharacterize the position in the quiver of many kinds of perfect complexes,\nincluding those of lengths 1, 2 and 3, rigid complexes and truncated projective\nresolutions. We describe completely the quiver components which contain\nprojective modules. We obtain relationships between the homology of complexes\nat different places in the quiver, deducing that every self-injective algebra\nof radical length at least 3 has indecomposable perfect complexes with\narbitrarily large homology in any given degree. We find also that homology\nstabilizes away from the rim of the quiver. We show that when the algebra is\nsymmetric, one of the forms considered earlier is Hermitian, and this allows us\nto compute its values knowing them only on objects on the rim of the quiver.\n