[Paper Review] Auslander-Reiten theory for simply connected differential graded algebras

Karsten Schmidt|ArXiv.org|Jan 4, 2008

Algebraic structures and combinatorial models32 references9 citations

TL;DR

This paper establishes that the Auslander-Reiten quiver of a simply connected differential graded algebra over a field has finitely many components if and only if the algebra is quasi-isomorphic to the cohomology algebra of a sphere. The author constructs infinite families of modules, each in a distinct component, showing that such algebras—beyond the sphere case—must have uncountably many components, depending on the cohomology dimension.

ABSTRACT

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 family for all n.

Motivation & Objective

• To determine the conditions under which the Auslander-Reiten quiver of a simply connected differential graded algebra has finitely many components.
• To extend Jørgensen’s earlier results on the Auslander-Reiten quiver of Poincaré duality spaces to general simply connected dg algebras.
• To characterize the structure of Auslander-Reiten components in terms of the cohomology dimension of the dg algebra.
• To demonstrate that only the cohomology of a d-dimensional sphere yields finitely many components, with all other cases having infinitely many.

Proposed method

• Utilizes the derived category of a differential graded algebra as the ambient triangulated category for Auslander-Reiten theory.
• Applies Jørgensen’s construction of the Auslander-Reiten quiver for simply connected Poincaré duality spaces to the setting of dg algebras.
• Employs minimal semi-free resolutions to model the dg algebra and analyze its module category.
• Constructs discrete or n-parameter families of indecomposable modules, each lying in a distinct Auslander-Reiten component.
• Uses an amplitude inequality to constrain the structure of components and rule out finitely many components beyond the sphere case.
• Leverages rational homotopy theory and minimal Sullivan models to relate topological properties to the algebraic structure of the dg algebra.

Experimental results

Research questions

• RQ1 When does the Auslander-Reiten quiver of a simply connected differential graded algebra have finitely many components?
• RQ2 What is the structure of the Auslander-Reiten components for dg algebras that are not quasi-isomorphic to the cohomology of a sphere?
• RQ3 How do the cohomology dimensions of a simply connected dg algebra influence the number and type of Auslander-Reiten components?
• RQ4 Can one construct infinite families of modules, each in a different component, for non-spherical dg algebras?
• RQ5 What is the topological significance of the number of Auslander-Reiten components in the derived category of a simply connected space?

Key findings

• The only simply connected differential graded algebra with finitely many Auslander-Reiten components is the cohomology algebra of a d-dimensional sphere.
• For all other simply connected dg algebras, the Auslander-Reiten quiver contains infinitely many components.
• The author constructs, for each cohomology dimension, either a discrete family or an n-parameter family of indecomposable modules, each in a distinct component.
• The number of components is uncountable when the cohomology dimension is greater than one, and the families are parameterized by arbitrary n for any n ≥ 1.
• The components of the Auslander-Reiten quiver are all of the form ℤA∞, generalizing Jørgensen’s result for Poincaré duality spaces.
• The construction shows that the sphere is the unique case where the Auslander-Reiten quiver is a finite union of ℤA∞-components, with d−1 components for a d-sphere.