Is this a projective resolution?

Question

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?

Answer 1

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

Answer 2

Let $f:k[x]\rightarrow k[x]$ and $g:k[x]\rightarrow k$, $g(f(1))=1\neq 0$, thus it is not a resolution.