Let $M$ be a finitely generated graded $k[x]$-module for some field $k$.
Let $$\cdots \rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow M \rightarrow 0$$
be a minimal graded projective resolution for $M$. In particular each $P_i$ is projective and hence free since $k[x]$ is a PID. I want to show that each map $P_{i+1} \rightarrow P_i$ maps each homogeneous $e\in P_{i+1}$ to some $(p_1, ... , p_r)$ where each $p_i$ is homogeneous and non-constant. The homogeneous part of this is by definition of a graded map, but the non-constant part is stumping me. I think I'm missing something obvious.
Suppose $e \mapsto (p_1, ... , p_r)$ where say $p_i$ is constant. Then since $(p_1, ... , p_r)$ is in the kernel of the next map, we have $p_i\partial(e_i) = -\sum p_j \partial(e_j)$, where $e_i$ is the $i$th standard basis. This should contradict minimality, and it does if $\{\partial(e_i)\}$ is the generating set of $P_{i-1}$ by which we defined the resolution, but is this necessarily so?