Algebraic K-theory

Algebraic K-theoryMathworldPlanetmath is a series of functorsMathworldPlanetmath on the category of rings. Broadly speaking, it classifies ring invariants, i.e. ring properties that are Morita invariant.

The functor K0

Let R be a ring and denote by M(R) the algebraic direct limitMathworldPlanetmath of matrix algebras Mn(R) under the embeddingsPlanetmathPlanetmathPlanetmath Mn(R)Mn+1(R):a(a000). The zeroth K-group of R, K0(R), is the Grothendieck group (abelian groupMathworldPlanetmath of formal differencesPlanetmathPlanetmath) of idempotentsMathworldPlanetmathPlanetmath in M(R) up to similarity transformations. Let pMm(R) and qMn(R) be two idempotents. The sum of their equivalence classesMathworldPlanetmathPlanetmath [p] and [q] is the equivalence class of their direct sumMathworldPlanetmathPlanetmathPlanetmath: [p]+[q]=[pq] where pq=diag(p,q)Mm+n(R). Equivalently, one can work with finitely generated projective modules over R.

The functor K1

Denote by GL(R) the direct limit of general linear groupsMathworldPlanetmath GLn(R) under the embeddings GLn(R)GLn+1(R):g(g001). Give GL(R) the direct limit topology, i.e. a subset U of GL(R) is open if and only if UGLn(R) is an open subset of GLn(R), for all n. The first K-group of R, K1(R), is the abelianisation of GL(R), i.e.


Note that this is the same as K1(R)=H1(GL(R),), the first group homology group (with integer coefficients).

The functor K2

Let En(R) be the elementary subgroupMathworldPlanetmathPlanetmath of GLn(R). That is, the group generated by the elementary n×n matrices eij(r), rR, where eij(r) is the matrix with ones on the diagonals, the value r in row i, column j and zeros elsewhere. Denote by E(R) the direct limit of the En(R) using the construction above (note E(R) is a subgroup of GL(R)). The second K-group of R, K2(R), is the second group homology group (with integer coefficients) of E(R),


Higher K-functors

Higher K-groups are defined using the Quillen plus construction,

Knalg(R)=πn(BGL(R)+), (1)

where BGL(R) is the classifying spacePlanetmathPlanetmath of GL(R).

Rough sketch of suspension:

ΣR=ΣR (2)

where Σ=C/J. The cone, C, is the set of infinite matrices with integral coefficients that have a finite number of non-trivial elements on each row and column. The ideal J consists of those matrices that have only finitely many non-trivial coefficients.

Ki(R)Ki+1(ΣR) (3)

Algebraic K-theory has a product structure,

Ki(R)Kj(S)Ki+j(RS). (4)


  • 1 H. Inassaridze, Algebraic K-theory. Kluwer Academic Publishers, 1994.
  • 2 Jean-Louis Loday, Cyclic Homology. Springer-Verlag, 1992.
Title Algebraic K-theory
Canonical name AlgebraicKtheory
Date of creation 2013-03-22 13:31:32
Last modified on 2013-03-22 13:31:32
Owner mhale (572)
Last modified by mhale (572)
Numerical id 10
Author mhale (572)
Entry type Topic
Classification msc 19-00
Classification msc 18F25
Related topic KTheory
Related topic GrothendieckGroup
Related topic StableIsomorphism