Definition 3

With the notation of Definition 2, the subgroup of $G^z(p')$ consisting of those $\sigma$ such that the endomorphism $z \mapsto \sigma.z$ of $A'/p'$ is the identity is called the inertia group of $p'$ (with respect to $G$) and denoted by $G^T(p')$ (or $G^T$). The ring of elements of $A'$ invariant under $G^T(p')$ is called the inertia ring of $p'$ (with respect to $G$) and is denoted by $A^T(p')$ (or $A^T$) (†).

If $A$ is the subring of $A'$ consisting of the invariants of $G$, clearly

It follows from Definitions 2 and 3 that, for all $p \in G$,

If, for all $\sigma \in G^z(p')$, $\bar{\sigma}$ is the automorphism $z \mapsto \sigma.z$ of $A'/p'$, $\sigma \mapsto \bar{\sigma}$ is a homomorphism (called canonical) of $G^z$ to the group $\Gamma_0$ of automorphisms of

(*) The letter $Z$ is the initial of the German word "Zerlegung" which means "decomposition." (†) The letter T is the initial of the German word "Tragheit" which means "inertia."

A'/p' leaving invariant the elements of $A^{\mathbf{Z}}/(p' \cap A^{\mathbf{Z}})$ (canonically identified with a subring of $A'/p'$) and by definition $\mathscr{G}^{\Gamma}(p')$ is the kernel of this canonical homomorphism; $\mathscr{G}^{\Gamma}$ is therefore a normal subgroup of $\mathscr{G}^{\mathbf{Z}}$. If $k'$ is the field of fractions of $A'/p'$, every automorphism of $A'/p'$ can be extended uniquely to an automorphism of $k'$, so that $\sigma \mapsto \bar{\sigma}$ can be considered as a homomorphism from $\mathscr{G}^{\mathbf{Z}}(p')$ to the group of automorphisms of $k'$. Note finally that, since $\mathscr{G}^{\Gamma}$ is normal in $\mathscr{G}^{\mathbf{Z}}$, $A^{\Gamma}$ is stable under $\mathscr{G}^{\mathbf{Z}}$.