Quadratic algebras with Ext algebras generated in two degrees

Green and Marcos (2005) [2] call a graded kk-algebra δδ-Koszul if the corresponding Yoneda algebra Ext(k,k) is finitely generated and there exists a function δ:N→Nδ:N→N such that Exti,j(k,k) is zero if j≠δ(i)j≠δ(i). For any integer m≥3m≥3 we exhibit a noncommutative quadratic δδ-Koszul algebra for which the Yoneda algebra is generated in degrees (1,1)(1,1) and (m,m+1)(m,m+1). These examples answer a question of Green and Marcos. These algebras are not Koszul but mm-Koszul (in the sense of Backelin).