nLab non-connective algebraic K-theory

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).

Related concepts

geometric context	universal additive bivariant (preserves split exact sequences)	universal localizing bivariant (preserves all exact sequences in the middle)	universal additive invariant	universal localizing invariant
noncommutative algebraic geometry	noncommutative motives $Mot_{add}$	noncommutative motives $Mot_{loc}$	algebraic K-theory	non-connective algebraic K-theory
noncommutative topology	KK-theory	E-theory	operator 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

• Denis-Charles Cisinski, Gonçalo Tabuada, Non connective K-theory via universal invariants. Compositio

  Mathematica 147 (2011), 1281–1320 (arXiv:0903.3717)

and further expanded on in section 9 of

• Andrew Blumberg, David Gepner, Gonçalo Tabuada, A universal characterization of higher algebraic K-theory, Geometry and Topology (arXiv:1001.2282)