Auslander Reiten Theory Research Articles

AUSLANDER–REITEN TRIANGLES AND A THEOREM OF ZIMMERMANN

A classical theorem of Zimmermann describes the relation between almost split sequences in the category of finitely presented modules and those in the category of all modules over some fixed ring. An analogue of Auslander–Reiten triangles in triangulated categories is proved in this paper. This is used to explain the relation between different existence results for Auslander–Reiten triangles, which are based either on Brown’s representability theorem, or on the existence of Serre functors. This paper is a continuation of [14], where Brown representability was introduced as a foundation for a general Auslander–Reiten theory in triangulated categories. More recently, Auslander–Reiten triangles have played an important role in work of Reiten and van den Bergh [18] and Jorgensen [11]. In [18], the existence of Auslander–Reiten triangles is translated into the existence of a Serre functor in the sense of Bondal and Kapranov [4]. This is used for a classification of the noetherian hereditary abelian categories satisfying Serre duality. In [11], the singular cochain differential graded algebra of a simply connected space is studied. The existence of

Auslander-Reiten Triangles, Ziegler Spectra and Gorenstein Rings

We investigate (existence of) Auslander–Reiten 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 Auslander–Reiten 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 Brown’s Representability Theorem. Our main interest lies in the structure of Auslander–Reiten triangles in the full subcategory of compact objects. We also study the connections and the interplay between Auslander–Reiten theory, pure-semisimplicity and the finite type property, Grothendieck groups, and we give applications to derived categories of

Symmetry and duality on n-Gorenstein rings

We will study homological properties of noetherian rings. As a bridge between the n-Gorenstein property and the dominant dimension, we will introduce the dominant numbers of rings and prove their left–right symmetry (Section 1.1). The ( l, n)-condition (Section 2.2) on selfinjective resolutions provides a useful tool on our study. Moreover, we will give two dualities on n-Gorenstein rings (Sections 1.2, 1.3), where one of them is closely related to the Auslander–Reiten theory (Section 1.3.1).

Auslander–Reiten Theory via Brown Representability Dedicated to Professor Daniel Quillen on his Sixtieth Birthday

Auslander–Reiten Theory via Brown Representability Dedicated to Professor Daniel Quillen on his Sixtieth Birthday