Valuations and diameters of milnor K-rings

Abstract

Given a field F and a subgroup S of F × containing −1, we define a graph on F × /S associated with the relative Milnor K-ring K M * (F )/S. We prove that if the diameter of this graph is at least 4, then there exists a valuation v on F such that S is v-open. This is done by adopting to our setting a construction in a noncommutative setting due to Rapinchuk, Segev and Seitz. We study the behavior of the diameter under important K-theoretic constructions, and relate it to the elementary type conjecture. Finally, we provide an example showing that the above bound 4 is sharp.

On graphs and valuations

“…Now one defines the Milnor K-graph of F modulo N to be the undirected graph whose vertices are the non-identity elements of F × /N , and where vertices aN and bN are connected by an edge if and only if {a, b} N = 0 in K M 2 (F )/N . It follows from [E4,Lemma 2.1] that this is indeed a V-graph. The main result of [E4] is just Theorem A for this V-graph on D = F .…”

“…Namely, there are examples where the quotient D × /N supports a V-graph of diameter 3, but N is not open with respect to any non-trivial valuation (cf. [RS,Example 8.4], [E4,Example 7.2], and Examples 11.4 and 11.7).…”

“…This notion, in turn, leads to a uniform approach for constructing maps ϕ : N → Γ having certain properties resembling those of a valuation on D, where Γ is a partially ordered group (see Theorem B below). Once such a map ϕ is obtained machinery from [RS,E4] can be used to construct a valuation (see Theorem A below). Further, using ϕ and certain additional hypotheses, and expanding on machinery from [RSS, R] leads us to new openness results with respect to a finite set of valuations (and not a single valuation) for N .…”