Induction is left adjoint to restriction for finite-group modules over a commutative ring

For $w\in W$, define ${\eta }_w:G\to W$ by ${\eta }_w(h):=h^{-1}\cdot w$ for $h\in H$ and ${\eta }_w(g):=0$ for $g\mathrm{\notin }H$. If $x\mathrm{\notin }H$, then $xh\mathrm{\notin }H$ for every $h\in H$, so ${\eta }_w(xh)=0=h^{-1}\cdot 0$; if $x\in H$, then ${\eta }_w(xh)=h^{-1}{x}^{-1}\cdot w=h^{-1}\cdot {\eta }_w(x)$. Thus ${\eta }_w\in {\mathrm{Ind}}_H^{G}W$.

Choose a left transversal $T$ for $G\mathrm{/}H$ with $e\in T$. If $\psi :W\to {\mathrm{Res}}_H^{G}V$ is $H$-equivariant, define $\beta (\psi )(f):={\sum }_{t\in T}t\cdot \psi (f(t))$. The sum is finite because $T$ is finite.

If $\mathrm{\Phi }:{\mathrm{Ind}}_H^{G}W\to V$ is $G$-equivariant, define $\alpha (\mathrm{\Phi })(w):=\mathrm{\Phi }({\eta }_w)$. For $h\in H$, one has ${\eta }_{h\cdot w}=h\cdot {\eta }_w$ by the action formula in [F1], so $\alpha (\mathrm{\Phi })(h\cdot w)=\mathrm{\Phi }(h\cdot {\eta }_w)=h\cdot \mathrm{\Phi }({\eta }_w)=h\cdot \alpha (\mathrm{\Phi })(w)$. Hence $\alpha (\mathrm{\Phi })\in {\mathrm{Hom}}_H(W,{\mathrm{Res}}_H^{G}V)$.

The map $\beta (\psi )$ is $G$-equivariant. Indeed, for $x\in G$ and each $t\in T$, write $x^{-1}t=t^{\mathrm{\prime }}h$ with $t^{\mathrm{\prime }}\in T$ and $h\in H$. Then $t\cdot \psi ((x\cdot f)(t))=t\cdot \psi (f(x^{-1}t))=t\cdot \psi (f(t^{\mathrm{\prime }}h))=t\cdot \psi (h^{-1}\cdot f(t^{\mathrm{\prime }}))=t{h}^{-1}\cdot \psi (f(t^{\mathrm{\prime }}))=x{t}^{\mathrm{\prime }}\cdot \psi (f(t^{\mathrm{\prime }}))$, using the covariance from [F1] and the $H$-equivariance of $\psi$. Reindexing the finite sum by $t^{\mathrm{\prime }}$ shows $\beta (\psi )(x\cdot f)=x\cdot \beta (\psi )(f)$.

For $\psi \in {\mathrm{Hom}}_H(W,{\mathrm{Res}}_H^{G}V)$, $\beta (\psi )({\eta }_w)=\psi (w)$ because ${\eta }_w(t)=0$ for $t\mathrm{\ne }e$ and ${\eta }_w(e)=w$. Hence $\alpha (\beta (\psi ))=\psi$.

For $\mathrm{\Phi }\in {\mathrm{Hom}}_G({\mathrm{Ind}}_H^{G}W,V)$ and $f\in {\mathrm{Ind}}_H^{G}W$, one has $\beta (\alpha (\mathrm{\Phi }))(f)={\sum }_{t\in T}t\cdot \alpha (\mathrm{\Phi })(f(t))={\sum }_{t\in T}t\cdot \mathrm{\Phi }({\eta }_{f(t)})=\mathrm{\Phi }({\sum }_{t\in T}t\cdot {\eta }_{f(t)})$. The function inside $\mathrm{\Phi }$ equals $f$, because at a point $th$ it takes the value $h^{-1}\cdot f(t)=f(th)$ by covariance. So $\beta (\alpha (\mathrm{\Phi }))=\mathrm{\Phi }$.

Steps 3.1 and 3.2 show that $\alpha$ and $\beta$ are inverse group isomorphisms, giving the stated adjunction. Via [F3], this is equally the usual $R[G]$-module adjunction.