nLab
non-connective algebraic K-theory

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Motivic cohomology

Contents

Idea

Non-connective algebraic K-theory, sometimes called Bass K-theory, is a variant of algebraic K-theory with non-trivial homotopy groups/cohomology groups in negative degrees.

This universally arises as the hom-spectra in the (∞,1)-category of noncommutative motives.

A review and further discussion is in (Blumberg-Gepner-Tabuada 10, section 9).

geometric contextuniversal additive bivariant (preserves split exact sequences)universal localizing bivariant (preserves all exact sequences in the middle)universal additive invariantuniversal localizing invariant
noncommutative algebraic geometrynoncommutative motives Mot addMot_{add}noncommutative motives Mot locMot_{loc}algebraic K-theorynon-connective algebraic K-theory
noncommutative topologyKK-theoryE-theoryoperator K-theory

References

  • R. W. Thomason, Thomas Trobaugh, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, 1990, 247-435.

The characterization of non-connective algebraic K-theory via noncommutative motives is due to

and further expanded on in section 9 of

Last revised on January 5, 2015 at 17:36:34. See the history of this page for a list of all contributions to it.