The Witt Ring of a Curve with Good Reduction over a Non-dyadic Local Field

Abstract

In this work, we present a generalization to varieties and sheaves of the fundamental ideal of the Witt ring of a field by defining a sheaf of fundamental ideals $\tilde{I}$ and a sheaf of Witt rings $\tilde{W}$ in the obvious way. The Milnor conjecture then relates the associated graded of $\tilde{W}$ to Milnor K-theory and so allows the classical invariants of a bilinear space over a field to be extended to our setting using \'etale cohomology. As an application of these results, we calculate the Witt ring of a smooth curve with good reduction over a non-dyadic local field.