An abstract approach to higher level forms and rigidity

Abstract

We suggest an abstract framework for Becker & Rosenberg's reduced theory of higher level forms and Witt rings over a field. along the lines of Marshall's Spaces of Orderings treatment of the reduced theory of quadratic forms. We study rigidity in this higher level context, and generalize results of Berman. Cordes and Ware. Specifically, let T be a preorder in a field K . and set B T ( K )= { x ϵ K \{0}: at least one of x and - x is not T -rigid}. We show that B T ,( K ) is a subgroup of ḳ = K \{0). Under a flniteness assumption we identify B(K) in terms of the smallest valuation ring A K which is compatible with T . If B T (Ḳ)¦ K in this setting it follows that K is “equivalent w.r.t. T -forms” to a generalized power series field over the residue field k T of A T.