Controlled algebraic \(G\)-theory. II. (English) Zbl 1362.19002
Tillmann, Ulrike (ed.) et al., Algebraic topology: applications and new directions. Stanford symposium on algebraic topology: applications and new directions, Stanford University, Stanford, CA, USA, July 23–27, 2012. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9474-3/pbk; 978-1-4704-1855-7/ebook). Contemporary Mathematics 620, 111-132 (2014).
Summary: There are two established ways to introduce geometric control in the category of free modules—the bounded control and the continuous control at infinity. Both types of control can be generalized to arbitrary modules over a noetherian ring and applied to study algebraic \(K\)-theory of infinite groups. This was accomplished for bounded control in part I of the present paper [G. Carlsson and B. Goldfarb, J. Homotopy Relat. Struct. 6, No. 1, 119–159 (2011; Zbl 1278.19002)] and the subsequent work of G. Carlsson and the first author, in the context of spaces of finite asymptotic dimension. This part II develops the theory of filtered modules over a proper metric space with a good compactification. It is applicable in particular to CAT(0) groups which do not necessarily have finite asymptotic dimension.
For the entire collection see [Zbl 1294.00030].
For the entire collection see [Zbl 1294.00030].
MSC:
| 19D35 | Negative \(K\)-theory, NK and Nil |
| 18E10 | Abelian categories, Grothendieck categories |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 18E35 | Localization of categories, calculus of fractions |
| 18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |
