Finite generation of extensions of associated graded rings along a valuation

Abstract

In this paper we consider the question of when the associated graded ring along a valuation, gr ν ∗ ( S ) , is a finite gr ν ∗ ( R ) -module, where S is a normal local ring which lies over a normal local ring R and ν ∗ is a valuation of the quotient field of S which dominates S. We begin by discussing some examples and results, allowing us to refine the conditions under which finite generation can hold. We must impose the condition that the extension of valuations is defectless and performs a birational extension of R along the valuation to obtain finite generation (replacing S with the local ring of the quotient field of S determined by the valuation which lies over the extension of R). With these assumptions, we have that finite generation holds, when R is a two-dimensional excellent local ring. Our main result (in Theorem 1.5) is to show that for an arbitrary valuation in an algebraic function field over an arbitrary field of characteristic zero, after a birational extension along the valuation, we always have finite generation (all finite extensions of valued fields are defectless in characteristic zero). This generalizes an earlier result, in [Cutkosky, Math. Proc. Cambridge Soc. 106 (2016) 233–255], showing that finite generation holds (after a birational extension) with the additional assumptions that ν has rank 1 and has an algebraically closed residue field. There are essential difficulties in removing these assumptions, which are addressed in the proof in this paper. We obtain general results for unramified extensions of excellent local rings in Proposition 1.7, showing that after blowing up, the extension of associated graded rings is finitely generated of an extremely simple form. This proposition plays an essential role in the Proof of Theorem 1.5.