I am trying hard to understand the general definition of Koszul algebras and I follow Uli Kraehmer's notes.
I got stuck at the very beginning of the definition of Koszul algebras. Look at Proposition 11 and Example 9 (with the quantum plane).
For Prop. 11, could someone please detail the proof? Why does it follow that $A$ is generated, as an algebra, by $A_1$? And further, why does it follow afterwards that the relations are quadratic?
For Example 9, with the quantum plane, $A=k_q[x,y]$: I understood that $(x,y)$ are algebra generators for $A$ and hence $P_1=A^2$. But why can we conclude from the subsequent computations that $\ker\phi_1$ is $A(-qy,x)$? And furthermore, why is $b_2=1$ and not 2, given the fact that we have two generators for $\ker\phi_1$?
And for the general definition of a Koszul ring, what does it mean that the term $P_i$ in the resolution of $k$ is generated in degree i?
Please be as explicit and elementary as you can be. It would be great if you could also point me to some more elementary expositions and examples.
I also tried to read Polishchuk and Positselski's Quadratic Algebras, BGS and most of the "famous" literature on this topic, but I simply can't understand these basic facts.
Thank you very much.
Why does it follow that $A$ is generated, as an algebra, by $A_1$? And further, why does it follow afterwards that the relations are quadratic?
This is not easy to answer in a few lines IMO, depending on how you approach to Kosul theory, there is no point to copy-and-paste a page from another literature to answer your question. I myself prefer BGS, and so to guide your reading, all of these can be found in BGS Chapter 2 (Section 2.1 - 2.10 is enough).
And for the general definition of a Koszul ring, what does it mean that the term $P_i$ in the resolution of $k$ is generated in degree $i$?
This is again can be found in BGS, but to start you off, I can extract the relevant part:
Suppose we work with left modules. For a positively graded ring $A=\oplus_{n\geq 0} A^n$, let $k=A^0$ and take a (graded) projective resolution of $k$: $$ \cdots \to P_i \to \cdots \to P_1 \to P_0 \to k \to 0 $$ $A$ is Koszul if $P_i$ is generated in degree $i$, that is, $P_i = \oplus_{j\in\mathbb{Z}} P_i^j$ (decomposition into graded piece) satisfies $P_i = AP_i^i$ (hence the name). Note that as the resolution is graded, it is immediate (regardless of Koszulity) that the graded pieces of $P_i$ lives in degree $j\geq i$, so Koszulity guarantees that the lowest graded piece determine the whole term.