Valuations and higher level orders in commutative rings

Abstract

Valuation theory is one of the main tools for studying higher level orders and the reduced theory of forms over fields, see, for example [BR]. In [MW], the theory of higher level orders and reduced forms was generalized to rings with many units and many of the results for fields carried over to this setting. While it seems desirable to extend these results further, the techniques used for rings with many units will not work for general commutative rings. At the same time, there is a general theory of valuations in commutative rings (see [LM], [M], and [G]), which in [Ma] was used to study orders and the reduced theory of quadratic forms over general commutative rings. Thus it seems natural to ask if the connections between valuations and higher level orders in fields exist in commutative rings. In this paper we use valuation theory to study the space of orders and the reduced Witt ring relative to a higher level preorder in a commutative ring. As in [Ma], we first localize our ring at a multiplicative set, without changing the space of orders, in order to make the valuation theory work better. This is a standard idea from real algebraic geometry. Remarkably, many of the notions, methods, and results for fields carry over to this new setting. We define compatiblity between valuations and orders and preorders, and the ring A(T ) associated to a preorder T , which turns out to be Prufer ring as in the field case. We define the relation of dependency on the set of valuations associated to a preorder and we use this to prove a decomposition theorem for the space of orders. We can then apply this to show that, under a certain finiteness condition, the space of orders is equivalent to the space of orders of a preordered field.