Theorem 2

Let G be a finite group, and let L be a commutative K-algebra of finite degree endowed with an action of G given by an injective homomorphism $\lambda$ from G to Aut$_K(L)$. The following properties are then equivalent:

(i) There exist a subgroup H of G, a Galois extension E of K of finite degree, an isomorphism from H to Gal(E$/K$), and an isomorphism of $(K,G)$-algebras from L to Coind$^G_H(E)$.

(ii) The algebra L is étale, and $\mathscr{H}$ is a homogeneous principal G-set (I, §5, No. 6, p. 60, Definition 7).

(iii) There exists an isomorphism of $(\Omega ,G)$-algebras $\psi : L_{(\Omega )}\rightarrow \mathscr{F}(G,\Omega )$; in other words, for every $g\in G$, the automorphism $\psi \circ \lambda (g)_{(\Omega )}\circ \psi^{-1}$ of $\Omega^G$ is equal to the automorphism

(iv) The algebra L is reduced, and L is a free K[G]-module of rank 1.

(v) The algebra L is reduced, we have Card(G) = [L : K], and K is the subring of L of elements invariant under the action of G.

(vi) The algebra L is reduced, the group G acts transitively on the set of maximal ideals of L, and, for every maximal ideal $\mathfrak{m}$ of L, the stabilizer $G_{\mathfrak{m}}$ of $\mathfrak{m}$ in G acts faithfully on $L/\mathfrak{m}$ and admits K as subfield of invariants.

(i)$\Rightarrow$(ii): Let E be a Galois extension of K of finite degree and $\tau$ an isomorphism from H to Aut$_K(E)$. Let S be a system of representatives of the right cosets of G modulo H. The K-algebra F = Coind$^G_H(E)$ is isomorphic to $\mathscr{F}(S,E)$ (Lemma 7, a)); it is therefore étale. We denote by $\lambda_F$ the action of G on F. Furthermore, let $\psi$ be a K-algebra homomorphism from E to Ω, and let $\chi_0$ be the homomorphism $f\mapsto \psi (f(e))$ from F to Ω. Let $g\in G$ be such that we have $\chi_0\circ \lambda_F(g) =\chi_0$; since $\psi$ is injective, we then have $f(g) =f(e)$ for every $f\in F$. In view of Lemma 7, a), we have $g\in H$, and by the formula

which holds for every $h\in H$, this can only happen if $g=e$. On the other hand, by Lemma 7, b) and Theorem 3 of V, §10, No. 6, p. 66, we have

[F : K] = (G : H)[E : K] = (G : H) Card(H) = Card(G).

The set $\mathscr{K}$ of K-homomorphisms from F to Ω has cardinal [F : K] because F is étale (V, §6, No. 5, p. 32, Proposition 4), so Card($\mathscr{K}$) $=$ Card(G). Since the stabilizer of $\chi_0$ in G is equal to $\{e\}$ by the above, $\mathscr{K}$ is a homogenous principal G-set.

(ii)$\Rightarrow$(iii): Suppose that L is étale and that $\mathscr{H}$ is a homogenous principal G-set. By Lemma 10, the $(\Omega$, G)-algebras $L_{(\Omega )}$ and $\mathscr{F}(\mathscr{H},\Omega )$ are isomorphic. Since $\mathscr{H}$ is a homogenous principal G-set, the $(\Omega$, G)-algebras $\mathscr{F}(\mathscr{H},\Omega )$ and $\mathscr{F}(G,\Omega )$ are isomorphic.

(iii)$\Rightarrow$(iv): Suppose that property (iii) holds. Then $L_{(\Omega )}$ is a free module of rank 1 over the algebra Ω[G]; the latter can be canonically identified with $K[G]_{(\Omega )}$. We then apply Theorem 3 of VIII, p. 37.

The implication (iv)$\Rightarrow$(v) is immediate.

Let us prove the implication (v)$\Rightarrow$(vi). The algebra L is reduced. By Lemma 8, c), the group G acts transitively on the set $\mathscr{S}$ of maximal ideals of L. Let $\mathfrak{m}$ be an element of $\mathscr{S}$. By Lemma 9, since $\bigcap_{\mathfrak{n}\in\mathscr{S}}\mathfrak{n}=\{0\}$, the algebra L is isomorphic to the algebra Coind$^G_{G_{\mathfrak{m}}}(L/\mathfrak{m})$. The algebra of invariants of $G_{\mathfrak{m}}$ in $L/\mathfrak{m}$ therefore coincides with K by Lemma 7, c). Hence the homomorphism $\lambda_{\mathfrak{m}}$ from $G_{\mathfrak{m}}$ to Gal((L$/\mathfrak{m}$)$/K$) is surjective. By Lemma 7, we moreover have

Card(G) $= [L : K] = (G : G_{\mathfrak{m}})[L/\mathfrak{m}: K]$. So Card(G$_{\mathfrak{m}}$) $= [L/\mathfrak{m}: K]$, and the homomorphism $\lambda_{\mathfrak{m}}$ is injective.

It remains to prove the implication (vi)$\Rightarrow$(i). Let $\mathfrak{m}$ be a maximal ideal of L. By Lemma 9, the algebra L is isomorphic to the algebra Coind$^G_{G_{\mathfrak{m}}}(L/\mathfrak{m})$ as a $(K$, G)-algebra. Since $G_{\mathfrak{m}}$ acts faithfully in $L/\mathfrak{m}$ and admits K as subfield of invariants, the group homomorphism $\lambda_{\mathfrak{m}}$ defines an isomorphism from $G_{\mathfrak{m}}$ to Gal((L$/\mathfrak{m}$)$/K$).