Controlled algebraic G-theory, I

Abstract

This paper extends the notion of geometric control in algebraic K-theory from additive categories with split exact sequences to other exact structures. In particular, we construct exact categories of modules over a Noetherian ring filtered by subsets of a metric space and sensitive to the large scale properties of the space. The algebraic K-theory of these categories is related to the bounded K-theory of geometric modules of Pedersen and Weibel the way G-theory is classically related to K-theory. We recover familiar results in the new setting, including the nonconnective bounded excision and equivariant properties. We apply the results to the G-theoretic Novikov conjecture which is shown to be stronger than the usual K-theoretic conjecture.