Finiteness conditions on the Ext-algebra of a cycle algebra

Let A be a finite-dimensional algebra given by quiver and monomial relations. In [E.L. Green, D. Zacharia, Manuscripta Math. 85 (1994) 11–23] we see that the Ext-algebra of A is finitely generated only if all the Ext-algebras of certain cycle algebras overlying A are finitely generated. Here a cycle algebra Λ is a finite-dimensional algebra given by quiver and monomial relations where the quiver is an oriented cycle. The main result of this paper gives necessary and sufficient conditions for the Ext-algebra of such a Λ to be finitely generated; this is achieved by defining a computable invariant of Λ, the smo-tube. We also give necessary and sufficient conditions for the Ext-algebra of Λ to be Noetherian.