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Homological algebra in bivariant $$K$$-theory and other triangulated categories. I. (English) Zbl 1234.18008
Holm, Thorsten (ed.) et al., Triangulated categories. Based on a workshop, Leeds, UK, August 2006. Cambridge: Cambridge University Press (ISBN 978-0-521-74431-7/pbk). London Mathematical Society Lecture Note Series 375, 236-289 (2010).
Some basic notions from homotopy theory and homological algebra are introduced in the setting of bivariant $$K$$-theory. This paper defines phantom maps, exact chain complexes, projective resolutions, and derived functors in bivariant $$K$$-theory and apply them to examples. The approximation of a bivariant category by an abelian category, the abelian approximation [A. Beligiannis, J. Algebra 227, No. 1, 268–361 (2000; Zbl 0964.18008)] is computed in several examples. The homological concepts introduced here then reduce to the ones in the abelian category. The derived functors are the second page in a spectral sequence which in some cases converges towards the Kasparov $$KK$$-group. This spectral sequence is a generalisation of the Adams spectral sequence in stable homotopy theory. In this paper, the simple cases, where the spectral sequence degenerates to an exact sequence are studied. It is then a generalisation of the universal coefficient theorem for $$KK_*(A,B)$$. In the sequel to this paper [R. Meyer, Tbil. Math. J. 1, 165–210 (2008; Zbl 1161.18301)], the spectral sequence is applied to examples of bivariant $$K$$-theory.
For the entire collection see [Zbl 1195.18001].

##### MSC:
 18E30 Derived categories, triangulated categories (MSC2010) 19K35 Kasparov theory ($$KK$$-theory) 46L80 $$K$$-theory and operator algebras (including cyclic theory) 55U35 Abstract and axiomatic homotopy theory in algebraic topology 18G40 Spectral sequences, hypercohomology
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