Some Remarks on $\delta$-Koszul Algebras

Let $A$ be a $\delta$-Koszul algebra, and let $\mathcal{K}^{\delta}(A)$ and $\mathcal{L}(A)$ denote the categories of $\delta$-Koszul modules and modules with linear presentations. Some necessary and sufficient conditions for $\mathcal{K}^{\delta}(A)=\mathcal{L}(A)$ are given. Set
$$ E(A):=\bigoplus_{i\geqslant0}\operatorname{Ext}_A^i(A_0,A_0) \qquadand\qquad \mathcal{B}(A):=\sup\{i\in \mathbb{N}\mid \operatorname{Ext}_A^i(A_0,A_0)\cap V\neq0\}, $$
where $V$ is a minimal graded generating space of $E(A)$. In the present paper, we prove that $\{\mathcal{B}(A)\mid A is \delta-Koszul\}=\mathbb{N}$. Finally, the Koszulity of the graded Hopf Galois extension of $\delta$-Koszul algebras is studied.