Math 2.0

Associative rings and algebras Definition

Algebraic K-theory

Builds on RingProjective moduleGrothendieck groupGeneral linear groupClassifying space of a groupHomotopy group

Definition

Let RR be a ring with identity.

  1. K0(R)K_0(R) is the Grothendieck group of the commutative monoid of isomorphism classes of finitely generated projective left RR-modules under direct sum.
  2. Let GL(R)=⋃n≥1GLn(R)GL(R) = \bigcup_{n \ge 1} GL_n(R), where GLn(R)GL_n(R) is the general linear group and GLn(R)⊆GLn+1(R)GL_n(R) \subseteq GL_{n+1}(R) through g↦(g001)g \mapsto \begin{pmatrix} g & 0 \\ 0 & 1 \end{pmatrix}. Then K1(R)=GL(R)/[GL(R),GL(R)]K_1(R) = GL(R)/[GL(R), GL(R)].
  3. For n≥1n \ge 1, Kn(R)=πn(BGL(R)+)K_n(R) = \pi_n(BGL(R)^+), the nn-th homotopy group of the plus construction of the classifying space BGL(R)BGL(R) with respect to the commutator subgroup [GL(R),GL(R)][GL(R), GL(R)].

The algebraic K-theory of RR is the sequence of abelian groups Kn(R)K_n(R), n≥0n \ge 0.

Remarks

For n=1n = 1 clause 3 gives back clause 2, since π1(BGL(R)+)=GL(R)/[GL(R),GL(R)]\pi_1(BGL(R)^+) = GL(R)/[GL(R), GL(R)]. By Whitehead's lemma, [GL(R),GL(R)][GL(R), GL(R)] is the subgroup generated by elementary matrices. Example: for a field FF, every finitely generated module is free, so K0(F)≅ℤK_0(F) \cong \Z by the dimension. The determinant gives K1(F)≅F×K_1(F) \cong F^\times, because the elementary matrices generate SL(F)SL(F). Caution: replacing projective modules by all finitely generated modules gives a different group, G0(R)G_0(R). For R=k[ε]/(ε2)R = k[\varepsilon]/(\varepsilon^2) over a field kk, K0(R)≅ℤK_0(R) \cong \Z is generated by [R][R], G0(R)≅ℤG_0(R) \cong \Z is generated by [k][k], and [R]=2[k][R] = 2[k].

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Classifying space of a group, concept, no evidence levelClassifying space of …conceptGeneral linear group, object, no evidence levelGeneral linear groupobjectGrothendieck group, concept, no evidence levelGrothendieck groupconceptHomotopy group, concept, no evidence levelHomotopy groupconceptProjective module, concept, no evidence levelProjective moduleconceptRing, concept, no evidence levelRingconceptAlgebraic K-theory, definition, no evidence levelAlgebraic K-theorydefinition
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  1. Classifying space of a group concept
  2. General linear group object
  3. Grothendieck group concept
  4. Homotopy group concept
  5. Projective module concept
  6. Ring concept
  7. Algebraic K-theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Algebraic_K-theory
About this page
Tag6SX85ZKBNJ
URIm2://6SX85ZKBNJ
Kinddefinition
WritingW0 stub
MSC16E20, 19D06
DefinesAlgebraic K-theory
UsesRing, Projective module, Grothendieck group, General linear group, Classifying space of a group, Homotopy group
Changed2 November 2026

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