Contents

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 $Mot_{add}$noncommutative motives $Mot_{loc}$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.