free resolution of $\mathbb{Z}$ as an $\mathbb{Z}[x]/(x^n-1)$-module

As the tittle says, I am having a bad time thinking on the construction of a free resolution of $ \mathbb{Z}$ as an $ (\mathbb{Z}[x]/(x^n-1))$ -module.

I know that I should give an exact sequence of the form (if $ R= \mathbb{Z}[x]/(x^n-1)$ ):

$ … \longrightarrow R^{I_n} \longrightarrow …\longrightarrow R^{I_1} \longrightarrow R^{I_0} \longrightarrow \mathbb{Z} \longrightarrow 0$


$ d_i : R^{I_i} \longrightarrow R^{I_{i-1}}$ for $ i \ge 1 $

$ d_0: R^{I_0} \longrightarrow \mathbb{Z}$

$ e: \mathbb{Z} \longrightarrow 0$ .

and $ R^{I_i}$ being free $ R$ -modules.

Moreover they should satisfy the exact sequence condition: $ Im(d_i)=Ker(d_{i-1})$ .

I began thinking on the first application $ d_0: R^{I_0} \longrightarrow \mathbb{Z}$ . But I had some troubles finding it. I thought sending

$ a_0 + a_1 \bar{x} + a_2 \bar{x}^2+…+a_{n-1} \bar{x}^{n-1} \mapsto a_0$

because I need $ Im(d_0)=Ker(e) = \mathbb{Z}$ . But it gets harder to find $ d_1$ .

Then I thought:

$ a_0 + a_1 \bar{x} + a_2 \bar{x}^2+…+a_{n-1} \bar{x}^{n-1} \mapsto \sum a_i$

but again it gets hard.

Any hint/help?