Abstract
We investigate (existence of) AuslanderâReiten triangles in a triangulated category in connection with torsion pairs, existence of Serre functors, representability of homological functors and realizability of injective modules. We also develop an AuslanderâReiten theory in a compactly generated triangulated category and we study the connections with the naturally associated Ziegler spectrum. Our analysis is based on the relative homological theory of purity and Brownâs Representability Theorem. Our main interest lies in the structure of AuslanderâReiten triangles in the full subcategory of compact objects. We also study the connections and the interplay between AuslanderâReiten theory, pure-semisimplicity and the finite type property, Grothendieck groups, and we give applications to derived categories of
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