Morphisms in the infinite radical of the bounded derived category
Auslander-Reiten theory, the theory of irreducible morphisms and almost split sequences, plays a fundamental role in the theory of Artin algebras. In recent years, this theory has been extended to different categories, such as the derived category of bounded complexes in an abelian category. The authors study sufficient conditions for a morphism in the category of complexes of fixed size \(\mathbf{C}_n (\text{proj} A)\), with \(n \geq 2\), and an Artin algebra \(A\), to belong to the infinite radical of \(K^{-,b} (\text{proj}\: A)\) (Theorem A, page 430). They also consider a Nakayama gentle algebra \(A\), whose ordinary quiver is an oriented cycle, proving that the irreducible morphisms in \(\mathbf{C}_n (\text{proj} \: A)\) are irreducible in \(K^{-,b} (\text{proj} \:A)\) or belong to the infinite radical of \(K^{-,b} (\text{proj} \:A)\) (Theorem B, page 431). The authors characterize the irreducible morphisms between indecomposable complexes in \(\mathbf{C}_n (\text{proj} \:A)\) that belong to the infinite radical of \(K^{-,b} (\text{proj} \:A)\) for a Nakayama gentle algebra \(A\) that is not self-injective (Theorem C, page 431). For the case of a Nakayama gentle self-injective algebra, they characterize the irreducible morphisms between indecomposable complexes in \(\mathbf{C}_n (\text{proj} \:A)\) that belong to the infinite radical of the bounded homotopic category in terms of their left and right degrees (Theorem D, page 432).
scientific article; zbMATH DE number 7396024
scientific article