I'm recently reading the Milne's Etale Cohomology (1980 Princeton University Press). When I read the part of etale cohomology vs flat cohomology in page 115, I have trouble understanding the first step in the proof. Let $X$ be affine scheme, $X_0$ be closed sub-scheme of $X$ defined by an ideal $\mathfrak{I}$ of square zero, $G$ be smooth (quasi-projective) commutative group scheme over $X$. Define sheaf $\mathcal{N}$ (or just functor) by $Y\mapsto\ker(G(Y)\rightarrow G(Y_0))$ where $Y_0=X_0\times_XY$. $X'$ is another affine $X$-scheme which is finite and faithfully flat over $X$.
Milne tried to show cech cohomology of $\mathcal{N}$ for small finite flat topology is trivial, alternatively, $\check{H}(X'/X,\mathcal{N})=0$ which is crucial in Step 2 in the proof. His strategy is to claim $\mathcal{N}$ is representable by quasi-coherent $\mathcal{O}_X$-Module (say, $\mathcal{M}=\mathcal{Hom}_{\mathcal{O}_X}(e^*(\Omega_{G/X}),\mathfrak{I})$ where $e$ is neutral section $X\rightarrow G$) and apply Prop. 3.7 in page 114.
The question arises in the following way : I know $\mathcal N$ is a torsor of $\mathcal M$ and therefore, they coincide as sets. When we calculate cech cohomology, group structure have to be involved, however, group structure of $\mathcal{N}$ is induced from commutative group $G$ while group structure of $\mathcal{M}$ is induced from modules. How can we say the cech cohomology of $\mathcal{M}$ is trivial will imply triviality of cech cohomology of $\mathcal{N}$?
I found the similar question arised in the proof of Grothendieck's Le Groupe de Brauer III : Exemples et Complements, page 177. Their references are all just telling the coincidence of sets, which confuse me for a long while. Any help will be welcomed.
As you say, $\mathcal{N}$ is a torsor for $\mathcal{M}$. But $\mathcal{N}$ has a canonical point, namely the identity element, and hence is canonically isomorphic to $\mathcal{M}$.