Controlled algebraic G-theory, I

Gunnar Carlsson and Boris Goldfarb

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.

Journal of Homotopy and Related Structures, Vol. 6(2011), No. 1, pp. 119-159