= Positive ramification quotients are elementary abelian
{title2=$G_i/G_{i+1}\hookrightarrow(k_L,+),\quad i\ge1$}
For $\sigma\in G_i$, the residue of $(\sigma\pi_L-\pi_L)/\pi_L^{i+1}$ gives a homomorphism to the additive residue field with kernel $G_{i+1}$. On composing two automorphisms, their leading displacement coefficients add because each fixes residues and changes the uniformizer only in degree at least two. These successive quotients are <elementary abelian groups> in residue characteristic $p$. The filtration terminates, so the <wild inertia group> is a finite $p$-group.
Back to article page
