## Tbilisi Mathematical Journal

- Tbilisi Math. J.
- Volume 1 (2008), 165-210.

### Homological algebra in bivariant K-theory and other triangulated categories. II

#### Abstract

We use homological ideals in triangulated categories to get a sufficient criterion for a pair of subcategories in a triangulated category to be complementary. We apply this criterion to construct the Baum-Connes assembly map for locally compact groups and torsion-free discrete quantum groups. Our methods are related to the abstract version of the Adams spectral sequence by Brinkmann and Christensen.

#### Note

Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen.

#### Article information

**Source**

Tbilisi Math. J., Volume 1 (2008), 165-210.

**Dates**

Received: 17 July 2008

Revised: 4 December 2008

Accepted: 15 December 2008

First available in Project Euclid: 12 June 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.tbilisi/1528768828

**Digital Object Identifier**

doi:10.32513/tbilisi/1528768828

**Mathematical Reviews number (MathSciNet)**

MR2563811

**Zentralblatt MATH identifier**

1161.18301

**Subjects**

Primary: 18E30: Derived categories, triangulated categories

Secondary: 19K35: Kasparov theory ($KK$-theory) [See also 58J22] 46L80: $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] 55U35: Abstract and axiomatic homotopy theory

**Keywords**

triangulated category Baum–Connes conjecture phantom map spectral sequence

#### Citation

Meyer, Ralf. Homological algebra in bivariant K-theory and other triangulated categories. II. Tbilisi Math. J. 1 (2008), 165--210. doi:10.32513/tbilisi/1528768828. https://projecteuclid.org/euclid.tbilisi/1528768828