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, ${\rm gr}{\nu^*}(S)$, is a finite ${\rm gr}{\nu^*}(R)$-module, where $S$ is a normal local ring which lies over a normal local ring $R$ and $\nu^*$ is a valuation of the quotient field of $S$ which dominates $S$. We obtain a very general result in equicharacteristic zero in Theorem 1.5. We also obtain general results for unramified extensions of excellent local rings in Proposition 1.7.