I need to find a free projective resolution of $\mathbb{C}[x,y,t]/(x^2+y^2-t)$ as a $\mathbb{C}[t]$-module. A projective resolution of an $R$-module $M$ is a long exact sequence $$...\rightarrow P_n\rightarrow P_{n-1}\rightarrow...\rightarrow P_0\rightarrow M\rightarrow 0.$$ Here each $P_i$ is required to be a projective $R$-module (hence the name). I require a free resolution, so the $P_i$ must be free as $C[t]$-modules. I can do simple examples, such as projective resolutions of $\mathbb{Z}/2$ as a $\mathbb{Z}$ module, i.e $$0\rightarrow \mathbb{Z} \xrightarrow{2} \mathbb{Z} \rightarrow \mathbb{Z}/2\rightarrow 0.$$
I can do slightly harder example as well, but am at a loss whenever it comes to more complex examples. It would be a great help if some one could describe a projective resolution of this $R$-module, and even better if they could give some general techniques for tackling these sorts of questions.
Note: This is not a homework question, but rather me trying to understand some algebraic geometry machinery, and this came up in the process.
$\Bbb C [x,y,t] $ is a free $\Bbb C [t] $-module, hence we have a free resolution over $\Bbb C [t] $: $$\{0\}\to\Bbb C [x,y,t]\to\Bbb C [x,y,t]\to\Bbb C [x,y,t]/\langle x^2+y^2-t\rangle\to\{0\} $$ where the first map is the multiplication by $x^2+y^2-t $, while the second is the canonical projection.