Math 2.0

K-theory Higher algebraic K-theory Theorem

Plus equals Q theorem

Builds on Quillen Q-constructionPlus constructionClassifying space BGLProjective moduleLoop spaceHomotopy equivalenceK0 of a ringK-theory of a symmetric monoidal category

Statement

Let RR be a ring with identity. Let 𝒫(R)\mathcal{P}(R) be the exact category of finitely generated projective left RR-modules, with the split short exact sequences. Let BQ𝒫(R)BQ\mathcal{P}(R) be the classifying space of its Q-construction. Let BGL(R)+BGL(R)^+ be the plus construction of the classifying space BGL(R)BGL(R) with respect to the commutator subgroup of GL(R)GL(R). Then there is a homotopy equivalence ΩBQ𝒫(R)≃K0(R)×BGL(R)+,\Omega BQ\mathcal{P}(R) \simeq K_0(R) \times BGL(R)^+, where Ω\Omega is the loop space and the group K0(R)K_0(R) has the discrete topology. In particular πn+1(BQ𝒫(R))≅πn(BGL(R)+)\pi_{n+1}(BQ\mathcal{P}(R)) \cong \pi_n(BGL(R)^+) for all n≥1n \ge 1, and π1(BQ𝒫(R))≅K0(R)\pi_1(BQ\mathcal{P}(R)) \cong K_0(R).

Remarks

The theorem is due to Quillen; a proof was written by Grayson. The proof passes through a third model, the category S−1SS^{-1}S. It is built from the groupoid SS of finitely generated projective modules and their isomorphisms, as in the K-theory of a symmetric monoidal category. One shows first that ΩBQ𝒫(R)≃B(S−1S)\Omega BQ\mathcal{P}(R) \simeq B(S^{-1}S), and then, by the group completion theorem, that B(S−1S)≃K0(R)×BGL(R)+B(S^{-1}S) \simeq K_0(R) \times BGL(R)^+.

At work in degree 11: the theorem gives π2(BQ𝒫(R))≅π1(BGL(R)+)=GL(R)/[GL(R),GL(R)]\pi_2(BQ\mathcal{P}(R)) \cong \pi_1(BGL(R)^+) = GL(R)/[GL(R), GL(R)], the classical group K1(R)K_1(R). So the K-groups defined by the Q-construction contain the classical K0K_0 and K1K_1.

Going deeper

For a student

The plus construction is good for computations with matrices, such as the K-theory of finite fields, and for products. The Q-construction applies to any exact category and gives the resolution, dévissage and localization theorems. The theorem lets results from one side be used on the other. For example, the localization sequence for a Dedekind domain is proved with the Q-construction, but its terms are the groups πn(BGL(R)+)\pi_n(BGL(R)^+).

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

Show the path as a graph
Classifying space of a group, concept, no evidence levelClassifying space of …conceptGeneral linear group, object, no evidence levelGeneral linear groupobjectGeometric realization, concept, no evidence levelGeometric realizationconceptGroup homology, concept, no evidence levelGroup homologyconceptHomotopy equivalence, concept, no evidence levelHomotopy equivalenceconceptK-theory of a symmetric monoidal category, concept, no evidence levelK-theory of a symmetr…conceptK0 of a ring, concept, no evidence levelK0 of a ringconceptLoop space, concept, no evidence levelLoop spaceconceptNerve of a category, concept, no evidence levelNerve of a categoryconceptPlus construction, concept, no evidence levelPlus constructionconceptProjective module, concept, no evidence levelProjective moduleconceptQuillen Q-construction, concept, no evidence levelQuillen Q-constructionconceptStable general linear group, concept, no evidence levelStable general linear…conceptClassifying space BGL, object, no evidence levelClassifying space BGLobjectPlus equals Q theorem, statement, no evidence levelPlus equals Q theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Classifying space of a group concept
  2. General linear group object
  3. Geometric realization concept
  4. Group homology concept
  5. Homotopy equivalence concept
  6. K-theory of a symmetric monoidal category concept
  7. K0 of a ring concept
  8. Loop space concept
  9. Nerve of a category concept
  10. Plus construction concept
  11. Projective module concept
  12. Quillen Q-construction concept
  13. Stable general linear group concept
  14. Classifying space BGL object
  15. Plus equals Q theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Algebraic_K-theory
About this page
Tag7X0TM18WWC
URIm2://7X0TM18WWC
Kindstatement, theorem
WritingW0 stub
MSC19D06
StatusThis copy of the site has no build data. Run m2 status 7X0TM18WWC.
AboutQuillen Q-construction, Plus construction
UsesQuillen Q-construction, Plus construction, Classifying space BGL, Projective module, Loop space, Homotopy equivalence, K0 of a ring, K-theory of a symmetric monoidal category
Changed3 November 2026

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