Coherence and the Hilbert-syzygy identity for power series over valuation domains

Abstract

Let $V$ be a complete valuation domain of rank 1, and let $\Gamma$ be its value group. We investigate the power series ring $V[[t]]$, and finally answer the long standing question of when this ring is coherent. Specifically, we prove that it is coherent if and only if $V$ is almost maximal, and either $\Gamma =\mathbb{R}$ or $\Gamma$ is discrete. Furthermore, we demonstrate that the weak Hilbert-syzygy identity $w.dimV[[X]]=w.dimV+1$ holds if and only if $V[[X]]$ is a coherent ring.