This question came up as I was reading Beilinson, Ginzburg, Soergel paper Koszul Duality Patterns in Representation Theory.
Suppose that $A$ is a Koszul ring (for the definition of Koszul ring see page 2 of BGS) and let $N$ be a graded $A$-module. As I understand it should be true that $$ \operatorname{ext}_A ^{i+1} (A_0, N) = \hom_A (K^i,N)$$ where $ K^i = \ker(P^i \rightarrow P^{i-1})$.
However I haven't been able to show it. The way I have been trying to show it is by induction since in the case were $ i = 0$ maybe its possible to use the fact that $ ext^1 _A (A_0, N) = \hom_A (K^0, N)/ Im(i^*)$ where $ K^0 = \ker ( P^0 \rightarrow A_0)$ and $ i: K^0 \hookrightarrow P^0$. But in order to even start here I have to show that $ Im(i^*)$ is zero and I haven't been able to convince myself that it is.