Homological algebra in bivariant K-theory and other triangulated categories

Abstract

Bivariant (equivariant) K-theory is the standard setting for non-commutative topology. We may carry over various techniques from homotopy theory and homological algebra to this setting. Here we do this for some basic notions from homological algebra: phantom maps, exact chain complexes, projective resolutions, and derived functors. We introduce these notions and apply them to examples from bivariant K-theory. An important observation of Beligiannis is that we can approximate our category by an Abelian category in a canonical way, such that our homological concepts reduce to the corresponding ones in this Abelian category. We compute this Abelian approximation in several interesting examples, where it turns out to be very concrete and tractable. The derived functors comprise the second tableau of a spectral sequence that, in favourable cases, converges towards Kasparov groups and other interesting objects. This mechanism is the common basis for many different spectral sequences. Here we only discuss the very simple 1-dimensional case, where the spectral sequences reduce to short exact sequences.

only some minor corrections

References in corpus (4)

• C*-Algebras over Topological Spaces: The Bootstrap Class
• Categorical aspects of bivariant K-theory
• Dualities in equivariant Kasparov theory
• Coactions of Hopf-$C^{\ast }$-algebras and equivariant $E$-theory

Cited by in corpus (24)

• The Baum-Connes conjecture for free orthogonal quantum groups
• C*-Algebras over Topological Spaces: The Bootstrap Class
• E-Theory for C*-algebras over topological spaces
• Tensor triangular geometry and KK-theory
• Homology and K-theory of dynamical systems. I. torsion-free ample groupoids
• Groups with Spanier-Whitehead duality
• Gorenstein homological algebra and universal coefficient theorems
• A covariant Stinespring theorem
• Localizing subcategories in the Bootstrap category of separable C*-algebras
• Equivariant Fredholm modules for the full quantum flag manifold of $S{U}_q(3)$
• KK-theory of circle actions with the Rokhlin property
• The heart of the Banach spaces
• On the classification of group actions on C*-algebras up to equivariant KK-equivalence
• Categorical approach to the Baum-Connes conjecture for étale groupoids
• Classification of ${\mathcal{O}}_{\infty }$-stable $C^{\ast }$-algebras
• Localization techniques in circle-equivariant KK-theory
• A remark on twists and the notion of torsion-free discrete quantum groups
• The K-theory of free quantum groups
• A note on a certain Baum--Connes map for inverse semigroups
• Relative stable categories and birationality
• Independent resolutions for totally disconnected dynamical systems I: Algebraic case
• Dynamic asymptotic dimension and Matui's HK conjecture
• Compact Lie group actions with continuous Rokhlin property
• Classification of certain continuous fields of Kirchberg algebras