Controlled algebraic G-theory, I

TL;DR

This paper develops a geometric control framework for algebraic G-theory, extending classical relations between G-theory and K-theory to new exact categories influenced by large-scale geometric properties.

Contribution

It introduces exact categories of modules filtered by metric space subsets, relating their algebraic G-theory to bounded K-theory and applying this to the G-theoretic Novikov conjecture.

Findings

Established nonconnective bounded excision in the new setting

Related algebraic G-theory to geometric modules of Pedersen and Weibel

Applied results to strengthen the G-theoretic Novikov conjecture

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.