A álgebra graduada associada a uma valorização

Abstract

We begin this work with a study of the graded algebra associated to a valuation $\nu: K \longrightarrow \Gamma_{\infty}$. We verify that in the cases where the value group is free or the residue field is closed by radicals, the graded algebra is isomorphic to $K\nu \left [t ^ {\nu (R)} \right]$. Then we present the different definitions of generating sequences that appear in the literature and we verify that under certain situations they are equivalent. After that, we introduce Newton polygons and relate this object with certain properties of valuations. Finally, we apply some of the results obtained for Artin-Schreier extensions.