Shapiro's lemma for the trivial subgroup and acyclicity of free comodules

Over a $k$-algebra $R$, Yoneda gives $\mathrm{Nat}(G_R,V_R^{\mathrm{a}})=V{\otimes }_{k}R{\otimes }_R{A}_R=V{\otimes }_{k}A{\otimes }_{k}R$. The identifications are natural under base change. Right translation of a regular function corresponds to ${\mathrm{id}}_V\otimes \mathrm{\Delta }$, so they identify ${\mathrm{I}\mathrm{n}\mathrm{d}}^G(V)$ with the additive functor of the free comodule $V\otimes A$.

A degree-$n$ cochain with induced coefficients is a regular $V$-valued function $f(g_1,\dots ,g_n)(x)$ on $G^{n+1}$. Make the invertible change of variables $F(x_0,\dots ,x_n)=f(x_0^{-1}{x}_1,x_1^{-1}{x}_2,\dots ,x_{n-1}^{-1}{x}_n)(x_0)\mathrm{.}$ Its inverse sets $x_0=x$ and $x_i=x{g}_1\cdots g_i$. Under these maps the induced-coefficient coboundary becomes $\delta F={\sum }_{i=0}^{n+1}(-1{)}^{i}F(x_0,\dots ,\widehat{x_i},\dots ,x_{n+1})$: the first term uses right translation of the function argument, the middle terms multiply adjacent differences, and the last deletes the last point.

For $n\ge 1$ define $sF(x_0,\dots ,x_{n-1})=F(e,x_0,\dots ,x_{n-1})$. This is regular and natural. The alternating omission formula gives $s\delta +\delta s=\mathrm{id}$: the omission of the inserted identity in $s\delta$ gives $F$, while every remaining term is the corresponding term of $\delta s$ with opposite sign. Thus if $\delta F=0$, then $F=\delta (sF)$, so all cohomology in degrees $n\ge 1$ vanishes. Transporting through step 1.2 proves (a).

The natural isomorphism of step 1.1 identifies the complexes for induced coefficients and the free comodule, so their cohomology agrees. Step 2.1 therefore proves $H^n(G,V\otimes A)=0$ for every $n\ge 1$, which is (b).