A Bound On The Ramification Index Of A Morphism Of Dvrs

Lemma

Let \(A\) and \(B\) be DVRs with uniformers \(\pi_A\) and \(\pi_B\) respectively and let \(\varphi:A\to B\) be an injective local morphism. Then in particular \(f(\pi_A) = u\pi_B^e\) for some integer \(e\), positive by the local assumption, and unit \(u\) in \(B\). Set \(K = \Frac A\) and \(L = \Frac B\), \(\kappa_A = A/\mathfrak m_A\) and \(\kappa_B = B/\mathfrak m_B\). Then

• If the induced extension \([L:K]\) is finite then so is \(f = [\kappa_B:\kappa_A]\)
• If \([L:K]\) is finite then \(ef\leq [L:K]\).

We call \(f\) the residue degree of \(f\) and \(e\) the ramification index.