Abstract:
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
$$
E(A):=\bigoplus_{i\geqslant0}\operatorname{Ext}_A^i(A_0,A_0)
\qquad\text{and}\qquad \mathcal{B}(A):=\sup\{i\in \mathbb{N}\mid \operatorname{Ext}_A^i(A_0,A_0)\cap V\neq0\},
$$
where $V$ is a minimal graded generating space of $E(A)$. In the present paper, we prove that
$\{\mathcal{B}(A)\mid A\text{ is }\delta\text{-Koszul}\}=\mathbb{N}$. Finally, the Koszulity of the graded Hopf Galois extension of $\delta$-Koszul algebras is studied.
Keywords:$\delta$-Koszul algebra, $\delta$-Koszul module, graded Hopf–Galois extension,
module with linear presentation.
