Remark 1

Let G be a group, H a subgroup of G, and N a subgroup of H. Let E be a $(K$, N)-algebra. Let $\alpha$ be an element of $\mathscr{F}_H(G,\mathscr{F}_N(H,E))$. We have the relation

for every $g\in G,h\in H$, and $n\in N$ and the relations

for all $h, h'\in H$ and every $g\in G$. Consequently, $\alpha (ng)(e) =\alpha (g)(n) =$ $n(\alpha (g)(e))$ for every $g\in G$ and $n\in N$. We can therefore consider the mapping

defined by the relation $\psi (\alpha )(g) =\alpha (g)(e)$ for $\alpha$ in $\mathscr{F}_H(G,\mathscr{F}_N(H,E))$ and $g$ in G. The mapping $\psi$ is an algebra isomorphism from Coind$^G_H$(Coind$^H_N(E)$) to Coind$^G_N(E)$ whose inverse sends an element $\beta$ from $\mathscr{F}_N(G,E)$ to the mapping $\alpha$ defined by the relation $\alpha (g)(h) =\beta (hg)$ for $g\in G$ and $h\in H$.

We now suppose given a finite group G and a reduced (V, §6, No. 6, p. 32) commutative K-algebra L of finite degree endowed with an action of G given by a homomorphism $\lambda$ from G to Aut$_K(L)$. For $x\in L$ and $g\in G$, we denote by $g\cdot x$ the transform of $x$ under the automorphism $\lambda (g)$ of L. Let $\mathscr{S}$ be the set of maximal ideals of L; we denote by $g\cdot \mathfrak{m}$ the transform of an element $\mathfrak{m}$ of $\mathscr{S}$ under the automorphism $\lambda (g)$ of L. It is an element of $\mathscr{S}$. For every $\mathfrak{m}$ in $\mathscr{S}$, the field $L/\mathfrak{m}$ is a finite extension of K. We write $\pi_{\mathfrak{m}}: L\rightarrow L/\mathfrak{m}$ for the projection and denote by $G_{\mathfrak{m}}$ the stabilizer of $\mathfrak{m}$ in G, that is, the set of $g\in G$ such that $g\cdot \mathfrak{m}=\mathfrak{m}$. The K-algebra $L/\mathfrak{m}$ is endowed with an action of $G_{\mathfrak{m}}$ through the homomorphism $\lambda_{\mathfrak{m}}$ from $G_{\mathfrak{m}}$ to Aut$_K(L/\mathfrak{m})$ that sends an element $h$ of $G_{\mathfrak{m}}$ to the automorphism of $L/\mathfrak{m}$ deduced from $\lambda (h)$ when passing to the quotients.

Let $\mathscr{O}$ be the set of orbits of G in $\mathscr{S}$. Given an orbit $\sigma \in \mathscr{O}$, set $\mathfrak{a}_{\sigma}=\bigcap_{\mathfrak{m}\in\sigma}\mathfrak{m}$ and $L_{\sigma}= L/\mathfrak{a}_{\sigma}$. Since $\mathfrak{a}_{\sigma}$ is invariant under G, by passing to the quotients, the action of G on L defines a homomorphism $\lambda_{\sigma}$ from G to Aut$_K(L_{\sigma})$. Finally, denote by $\pi_{\sigma}$ the canonical mapping from L to $L_{\sigma}$.