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