K-theory of Gelfand rings

Abstract

The origin of Gelfand rings comes from [9] where the Jacobson topology and the weak topology are compared. The equivalence of these topologies defines a regular Banach algebra. One of the interests of these rings resides in the fact that we have an equivalence of categories between vector bundles over a compact manifold and finitely generated projective modules over C( M), the ring of continuous real functions on M [17]. These rings have been studied by R. Bkouche (soft rings [3]) C.J. Mulvey (Gelfand rings [15]) and S. Teleman (harmonic rings [19]). Firstly we study these rings geometrically (by sheaves of modules (Theorem 2.5)) and then introduce the Čech covering dimension of their maximal spectrums. This allows us to study the stable rank of such a ring A (Theorem 6.1), the nilpotence of the nilideal of K 0( A) - The Grothendieck group of the category of finitely generated projective A-modules - (Theorem 9.3), and an upper limit on the maximal number of generators of a finitely generated A-module as a function of the afore-mentioned dimension (Theorem 4.4). Moreover theorems of stability are established for the group K 0( A), depending on the stable rank (Theorems 8.1 and 8.2). They can be compared to those for vector bundles over a finite dimensional paracompact space [18]. Thus there is an analogy between finitely generated projective modules over Gelfand rings and Čech dimension, and finitely generated projective modules over noetherian rings and Krull dimension.