Families of Auslander–Reiten components for simply connected differential graded algebras

Abstract

Peter Jørgensen introduced the Auslander-Reiten quiver of a simply connected Poincaré duality space. He showed that its components are of the form Z A ∞ and that the Auslander-Reiten quiver of a d-dimensional sphere consists of d − 1 such components. We show that this is essentially 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.

The Hochschild cohomology ring of the singular cochain algebra of a space

“…As for global nature of singular cochains, Jørgensen has investigated the derived category of the singular cochain on a Poincaré duality space by applying Auslander-Reiten theory. In particular, the Auslander-Reiten quiver of the full subcategory consisting of compact objects is determined in [24] and [25], see also [40]. Such the result brings us to the study of topological spaces with categorical representation theory.…”

Homological properties of cochain Differential Graded algebras

“…The resulting cochain Auslander-Buchsbaum Equality and Gap Theorem, established in Theorems 4.7 and 4.11 below, are new. A cochain Amplitude Inequality can be found already in [10,Proposition 3.11]; the proof of our Theorem 4.4 works by different methods.…”

Upper and Lower Bounds of the (co)chain Type Level of a Space

“…By applying Auslander-Reiten theory for derived categories [17] [18], Jørgensen [23] [24] has clarified the structure of the Auslander-Reiten quiver of the full subcategory D c (C * (B; K)) of compact objects of D(C * (B; K)) provided the space B is Gorenstein at K in the sense of Félix, Halperin and Thomas [11]. In fact, the result [24, Theorem 0.1] tells us that each component of the quiver is of the form ZA ∞ ; see also [23] and [44]. Depending on the detailed information of the quiver of D c (C * (S d ; K)), the computation of the level of an appropriate space over the sphere S d is performed in [35].…”

“…Depending on the detailed information of the quiver of D c (C * (S d ; K)), the computation of the level of an appropriate space over the sphere S d is performed in [35]. In particular, we see that the level of a space X over S d is less than or equal to an integer l if and only if the DG module C * (X; K) over C * (S d ; K) is made up of molecules lying between the lth horizontal line and the bottom one of the quiver; see [44,Proposition 6.6], [35,Examples 5.2 and 5.3] and [23,Theorem 8.13].…”