Families of Auslander-Reiten components for simply connected differential graded algebras. (Q846879)

Let \(\Bbbk\) be a field, \(X\) a simply connected topological space and \(C^*(X,\Bbbk)\) the corresponding singular cochain differential graded algebra. By \textit{P. Jørgensen} [Comment. Math. Helv. 79, No. 1, 160-182 (2004; Zbl 1053.55010)], \(X\) has Poincaré duality if and only if the compact derived categories of both left and right DG modules over \(C^*(X,\Bbbk)\) have Auslander-Reiten triangles. In the latter case one can define the Auslander-Reiten quiver for \(X\) and it was shown by Jørgensen that all components are of the form \(\mathbb ZA_\infty\). The present paper addresses the questions about the number of these components and their suitable parameterization. The main result is as follows: Theorem. Let \(A\) be a simply connected Gorenstein differential graded algebra of finite type. Then the Auslander-Reiten quiver of \(\mathbf D^c(A)\) has finitely many components if and only if \(\dim_\Bbbk\mathbf H^*A=2\). In the latter case the number of components equals \(\sup\{i\mid\mathbf H^iA\neq 0\}-1\). Moreover, if \(\dim_\Bbbk\mathbf H^iA\geq 2\) for some \(i\), then there is an \(n\)-parameter family of Auslander-Reiten components for each \(n\in\mathbb N\), in fact, there are objects, each lying in different components, that can be parameterized by \(\mathbb P^1(\Bbbk)^n\).