Let $R = k[x,y,z]$ where $k$ is considered as an $R$-module. I need to write down a projective resolution for $k$.
What I already do not understand: how is the $R$-module structure on $k$ defined in this case?
We learned projective resolutions as sequences $$ \ldots \rightarrow P_2 \rightarrow P_1 \rightarrow P_1 \rightarrow M \rightarrow 0 $$ where all the $P_j$ are projective objects.
In categorical terms, we defined a projective object $P$ as an object such that for any arrow $f : P \rightarrow X$ and any epimorphisms $e : Y \rightarrow X$, there exists a unique factorization $f = e \circ f^{'}$.
But I'm complete at loss of how to use this definition here. How does one solve these kind of problems? Any help is appreciated.