TheN *-metric completion of regular rings

Abstract

We have two main objectives in this paper. Let R be a unit regular ring (the standard reference is [G1]); R admits a pseudo-metric topology, and the completion, denoted S, is called the N*-completion of R. We study the relations between R and S, complementary to a recent article of Goodearl on N*-complete regular rings. The rings R, S are representable as rings of sections of a sheaf-like object, and we clarify and extend this notion. The collection of pseudo-rank functions of R (terms not defined here, will be found in [G1]), denoted IP(R), is a compact convex set, with extreme boundary denoted either OelP(R) or E. We define N* :R--*[0, 1] via N*(r) =sup{P(r)lPslP(R)}. Then N* induces a pseudo-metric topology on R by means of d(r, s)= N*(r-s). This is a metric precisely when N*(r)=0 implies r = 0 ("R is N*-torsion-free"; in [GH2, p. 208], "R has a Hausdorff family of pseudorank functions"); we shall consider only such rings. Chapter 1 deals with properties of the N*-completion of a general (unit regular) ring R, called S. Our main result here (which takes the bulk of the chapter to establish) is that S possesses no new pseudo-rank functions (i.e., the natural map IP(S)~IP(R) is an affine homeomorphism). (This was claimed in [H1, Proposition 15] but the injectivity part of the proof has a gap.) It follows that S is complete in its intrinsic N*-metric and so the recent results of Goodearl I-G2] apply to S. When the appropriate identifications are made, the sup-norm completion of the image of Ko(R) in AfflP(R) [the affine continuous functions on IP(R)] is just Ko(S ). Some consequences of these results include: (i) If K is a metrizable Choquet simplex, there is an N*-complete regular ring S such that Ko(S ) is order-isomorphic to Aft(K). (ii) If(G, u) is an unperforated interpolation group with order unit (see [EHS]), then the sup-norm closure of its image in its natural representation as a group of affine functions on a simplex, is an interpolation group.