Weak dimension of power series rings over valuation rings

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.