The Auslander-Reiten quiver of a Poincare duality space

Auslander-Reiten triangles and quivers over topological spaces

In this paper, Auslander-Reiten triangles are introduced into algebraic topology, and it is proved that their existence characterizes Poincare duality spaces. Invariants in the form of quivers are also introduced, and Auslander-Reiten triangles and quivers over spheres are computed. The quiver over the d-dimensional sphere turns out to consist of d-1 components, each isomorphic to ZA_{\infty}. So quivers are sufficiently sensitive invariants to tell spheres of different dimension apart.

Auslander-Reiten theory for simply connected differential graded algebras

Peter Jorgensen introduced the Auslander-Reiten quiver of a simply connected Poincare duality space. He showed that its components are of the form ZA_infty and that the Auslander-Reiten quiver of a d-dimensional sphere consists of d-1 such components. In this thesis we show that this is the only case where finitely many components appear. More precisely, we construct families of modules, where for each family, each module lies in a different component. Depending on the cohomology dimensions of the differential graded algebras which appear, this is either a discrete family or an n-parameter fam…

Auslander-Reiten sequences for homotopists and arithmeticians

We introduce Auslander-Reiten sequences for group algebras and give several recent applications. The first part of the paper is devoted to some fundamental problems in Tate cohomology which are motivated by homotopy theory. In the second part of the paper we interpret Auslander-Reiten sequences in the context of Galois theory and connect them to some important arithmetic objects.

Auslander-Reiten sequences, Brown-Comenetz duality, and the $K(n)$-local generating hypothesis

In this paper, we construct a version of Auslander-Reiten sequences for the $K(n)$-local stable homotopy category. In particular, the role of the Auslander-Reiten translation is played by the local Brown-Comenetz duality functor. As an application, we produce counterexamples to the $K(n)$-local generating hypothesis for all heights $n>0$ and all primes. Furthermore, our methods apply to other triangulated categories, as for example the derived category of quasi-coherent sheaves on a smooth projective scheme.

Simplicial structures in higher Auslander-Reiten theory

We develop a novel combinatorial perspective on the higher Auslander algebras of type $\mathbb{A}$, a family of algebras arising in the context of Iyama's higher Auslander-Reiten theory. This approach reveals interesting simplicial structures hidden within the representation theory of these algebras and establishes direct connections to Eilenberg-MacLane spaces and higher-dimensional versions of Waldhausen's $\operatorname{S}_\bullet$-construction in algebraic $K$-theory. As an application of our techniques we provide a generalisation of the higher reflection functors of Iyama and Oppermann to…

通过相容 Auslander-Reiten 图研究抽象表示论

Abstract representation theory via coherent Auslander-Reiten diagrams

本文提出利用 Auslander-Reiten 图研究抽象稳定同伦理论上的箭图表示，通过构造网格 $\infty$-范畴证明了表示的抽象等价性，并导出了经典表示论关键函子的抽象版本及谱 Picard 群的泛自等价。