Marshall’s and Milnor’s Conjectures for Preordered von Neumann Regular Rings
نویسندگان
چکیده
The aim of this paper is to prove that, if R is a commutative regular ring in which 2 is a unit, then the reduced theory of quadratic forms with invertible coefficients in R, modulo a proper preorder T , satisfies Marshall’s signature conjecture and Milnor’s Witt ring conjecture (for precise statements, see Section 1 below). For that purpose we use the theory of special groups (abbreviated SG), presented in [DM2] (see also Section 2 of [DM1]), and the K-theory of those structures, developed in [DM3] and [DM6]. To a pair 〈R, T 〉 as above, we associate a reduced special group (RSG), GT (R) = R×/T× (R× = units of R). A result from [DM5] (Thm. 3.16, pp. 17-18) shows that, under these conditions —in fact, even under considerably more general conditions— GT (R) faithfully reflects the reduced theory of quadratic forms modulo T , over free R-modules.
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