Inertia elements versus Frobenius elements (Q711576)

For a Galois extension of fields \(\tilde{K}|K\) let \(\mathfrak{Rm}(\tilde{K})\) be the set of the ramification elements in \(\text{Gal}(\tilde{K}|K)\), \(\mathfrak{In}(\tilde{K})\) the set of the inertia elements, \(\mathfrak{In.tm}(\tilde{K})\) the set of the tame inertia elements. The author proves that \(\mathfrak{Rm}(\tilde{K})\), \(\mathfrak{In}(\tilde{K})\), \(\mathfrak{In.tm}(\tilde{K})\) are closed in \(\text{Gal}(\tilde{K}|K)\). If \(K\) is finitely generated, then the set \(\mathfrak{In.tm.div}(\tilde{K})\) of the divisorial inertia elements is dense in \(\mathfrak{In.tm}(\tilde{K})\). If \(K|k\) is a function field, then \(\mathfrak{In}(\tilde{K}|k)\) and \(\mathfrak{In.tm}(\tilde{K}|k)\) are closed subsets of \(\text{Gal}(\tilde{K}|K)\), and \(\mathfrak{In.tm.div}(\tilde{K}|k)\) is dense in \(\mathfrak{In.tm}(\tilde{K}|k)\).