Showing minimal graded free resolutions are isomorphic I'm currently reading Rogalski's notes on noncommutative projective algebraic geometry (which can be found here) and I'm currently trying to fill out the details of Lemma 1.24 (2). The step which I don't understand is why the map $h$ is an isomorphism. I understand that there is an isomorphism $P/PA_{\ge 1} \rightarrow Q/QA_{\ge 1}$, and how we can lift this to a map $h: P \rightarrow Q$, but I can't see either injectivity or surjectivity of $h$. I feel like graded Nakayama should be relevant, but can't see why.

I want to give a second proof for $P \cong Q$ which I find more instructive. As a drawback - in contrast to the more general proof in my first answer - it requires a finiteness condition. 
Since $P$ is graded free, $P \cong \bigoplus_i A(-a_i)$ (all isomorphisms are as graded modules unless otherwise stated). For an integer $d$ let $I_{P,d} := \{i \mid a_i = d\;\}$. Hence $$P \cong \bigoplus_d \bigoplus_{I_{P,d}}A(-d)\quad,\quad Q \cong \bigoplus_d \bigoplus_{I_{Q,d}}A(-d)\tag{$\ast$}$$ Set $k := A/A_{\ge 1}$. Note that $A(-d)/A(-d)A_{\ge 1}=k(-d)$ is a graded $k$-module concentrated in the single degree $d$. Hence 
$$P/PA_{\ge 1} \cong \bigoplus_d \bigoplus_{I_{P,d}}k(-d)\quad,\quad Q/QA_{\ge 1} \cong \bigoplus_d \bigoplus_{I_{Q,d}}k(-d)$$
where the inner sum is the component of degree $d$. Since we already know 
$P/PA_{\ge 1} \cong Q/QA_{\ge 1}$, we have 
$$\bigoplus_{I_{P,d}}k(-d) \cong \bigoplus_{I_{Q,d}}k(-d)$$
as $k$-vector spaces. Now suppose $I_{P,d}$ is finite for all $d$. Then the left hand side has $k$-dimension $\#I_{P,d}$. By the isomorphism, the right hand side is also finite dimensional. Thus $I_{Q,d}$ is finite and $\#I_{P,d}=\#I_{Q,d}$. Now, $P \cong Q$ by $(\ast)$. 
