I read in many places that the "usual" free resolution of $\mathbb{Z}$ as a trivial $\mathbb{Z}[\mathbb{Z}]$-module is given as follows
$$0 \to \mathbb{Z}[\mathbb{Z}] \xrightarrow{\partial} \mathbb{Z}[\mathbb{Z}] \xrightarrow{\epsilon} \mathbb{Z} \to 0.$$
without specifying the maps $\partial $ and $ \epsilon$. I want to understand these maps. What is the image of a Laurent Polynomial $P(t)=\displaystyle\sum_{k\in \mathbb Z}a_kt^k$. I assume that the augmentation map $\epsilon$ sends $P(t)$ to the sum of the coefficients $a_k$ and this makes sens because only a finite number of the coefficients are not zero.
Let me write $\mathbb{Z}[\mathbb{Z}]$ as $R=\mathbb{Z}[t,t^{-1}]$. Note that as an $R$-module, $\mathbb{Z}$ is just the quotient by the ideal $(t-1)$, since $t$ acts on $\mathbb{Z}$ by multiplication by $1$ and if you set $t=1$ then any Laurent polynomial reduces to an integer. So as you say, $\epsilon$ maps a Laurent polynomial to the sum of its coefficients (i.e., what you get by evaluation at $t=1$). For $\partial$, just note that the kernel of $\epsilon$ is the ideal $(t-1)$, so you can take $\partial$ to be the map given by multiplication by $t-1$.