Weak dimension of power series rings over valuation rings

Abstract

We examine the power series ring $R\left[\right[X\left]\right]$ over a valuation ring R of rank 1, with proper, dense value group. We give a counterexample to Hilbert's syzygy theorem for $R\left[\right[X\left]\right]$, i.e. an $R\left[\right[X\left]\right]$-module C that is flat over R and has flat dimension at least 2 over $R\left[\right[X\left]\right]$, contradicting a previously published result. The key ingredient in our construction is an exploration of the valuation theory of $R\left[\right[X\left]\right]$. We also use this theory to give a new proof that $R\left[\right[X\left]\right]$ is not a coherent ring, a fact which is essential in our construction of the module C.