Is this a projective resolution? Let $k$ be a field, $R=k[x]$. Is this a projective resolution of $k$ over $R$?
$$0\to k[x]\to k[x]\to k\to 0$$
where the left map is $x\mapsto x-1$ and the right map is $x\mapsto 1$ ?
If not, what is a projective resolution in this case?

The projective resolution of $k$ over $k[x]$ is this:
$$0\longrightarrow k[x]\xrightarrow{\times (x-1)}\begin{aligned}[t]k[x]&\longrightarrow k\\\scriptstyle P(x)&\longmapsto \scriptstyle P(1)\end{aligned}\!\!\!\!\longrightarrow 0$$
