Algebras with Periodic Shifts of Ext Degrees

The so-called $\lambda$-Koszul algebra and $\lambda$-Koszul module are introduced. We give different equivalent descriptions of a $\lambda$-Koszul algebra in terms of its minimal graded projective resolution and the Yoneda $\operatorname{Ext}$-algebra $E(A)=\bigoplus_{i\ge 0}\operatorname{Ext}^i_A(\mathbb F,\mathbb F)$. The "$\lambda$-Koszulity" of a finitely generated graded module is discussed and the concepts of (strongly) weakly $\lambda$-Koszul module are introduced. Finally, we discuss the $A_{\infty}$-structure on the Yoneda $\operatorname{Ext}$-algebra of a $\lambda$-Koszul algebra.