We establish an inequality between the dimensions of the endomorphism and extension spaces of the indecomposable modules in generalized standard almost cyclic coherent components of the Auslander-Reiten quivers of finite dimensional algebras over an arbitrary base field. As an application we provide a homological characterization, involving the Euler quadratic form, of the tame algebras with separating families of almost cyclic coherent Auslander-Reiten components.
"On the indecomposable modules in almost cyclic coherent Auslander-Reiten components." J. Math. Soc. Japan 63 (4) 1121 - 1154, October, 2011. https://doi.org/10.2969/jmsj/06341121