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Controlled algebraic G-theory, I

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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