We define the cohomology of a group $G$ as $H^n(G,A)=Ext^n_{\mathbb Z(G)}(\mathbb Z,A)$ for every $G$-module $A$.
To compute it, we use the standard resolution $E_\bullet$ for $\mathbb Z$ (see Lang, Algebra, page 826). But, since the choice of the (projective!) resolution should be ineffective, we could choose another resolution for $\mathbb Z$. Now, what's wrong with $$0\longrightarrow \mathbb Z\longrightarrow \mathbb Z\longrightarrow 0\quad ?$$ This should give the same result, and this seems actually somehow strange to me, so perhaps I have some confusion with definitions.
Thank you in advance for any help to clarify this.