Math 2.0

K-theory Higher algebraic K-theory Theorem

Quillen's finite generation theorem

Builds on Algebraic number fieldIntegral closureAlgebraic K-theoryQuillen Q-constructionDirichlet's unit theoremBass–Milnor–Serre theoremBuilding

Statement

Let FF be a number field, and let 𝒪F\mathcal{O}_F be its ring of integers, the integral closure of ℤ\Z in FF. Then for every n≥0n \ge 0 the algebraic K-group Kn(𝒪F)K_n(\mathcal{O}_F) is a finitely generated abelian group.

Remarks

In low degrees the statement is classical. K0(𝒪F)≅ℤ⊕Cl⁡(F)K_0(\mathcal{O}_F) \cong \Z \oplus \operatorname{Cl}(F), and the ideal class group Cl⁡(F)\operatorname{Cl}(F) is finite. K1(𝒪F)≅𝒪F×K_1(\mathcal{O}_F) \cong \mathcal{O}_F^\times by the Bass–Milnor–Serre theorem, and this group is finitely generated by Dirichlet's unit theorem.

Quillen's proof works with the Q-construction of the finitely generated projective 𝒪F\mathcal{O}_F-modules, filtered by rank. He identifies the successive layers with the homology of the groups GLn(𝒪F)GL_n(\mathcal{O}_F) with coefficients in the Steinberg module, which comes from the Tits building of GLn(F)GL_n(F). The Borel–Serre compactification shows that these homology groups are finitely generated, so BQBQ has finitely generated homology. For this kind of space, finitely generated homology implies finitely generated homotopy groups.

The ranks of the groups Kn(𝒪F)K_n(\mathcal{O}_F) were then computed by Borel.

Going deeper

For a student

The theorem generalizes the finiteness of the class number and Dirichlet's unit theorem to all K-groups. It is the starting point for Borel's computation of ranks and for the conjectures relating K-groups to special values of zeta functions. The Bass conjecture asks for the same finiteness for all regular rings of finite type over ℤ\Z, and Quillen's theorem is its case of rings of integers.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

Show the path as a graph
Algebraic K-theory, concept, no evidence levelAlgebraic K-theoryconceptAlgebraic number field, concept, no evidence levelAlgebraic number fieldconceptBuilding, concept, no evidence levelBuildingconceptCongruence subgroup, concept, no evidence levelCongruence subgroupconceptFree abelian group, concept, no evidence levelFree abelian groupconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectGroup of units of a ring, concept, no evidence levelGroup of units of a r…conceptIdeal, concept, no evidence levelIdealconceptIdentity matrix, object, no evidence levelIdentity matrixobjectIndex of a subgroup, concept, no evidence levelIndex of a subgroupconceptIntegers, object, no evidence levelIntegersobjectIntegral closure, concept, no evidence levelIntegral closureconceptLattice (group), concept, no evidence levelLattice (group)conceptMatrix, definition, no evidence levelMatrixdefinitionMinkowski's theorem, statement, no evidence levelMinkowski's theoremstatementQuillen Q-construction, concept, no evidence levelQuillen Q-constructionconceptRegulator of a number field, concept, no evidence levelRegulator of a number…conceptRoot of unity, concept, no evidence levelRoot of unityconceptVector space, definition, no evidence levelVector spacedefinitionDirichlet's unit theorem, statement, no evidence levelDirichlet's unit theo…statementRing of integers, statement, no evidence levelRing of integersstatementSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial linear group, object, no evidence levelSpecial linear groupobjectBass–Milnor–Serre theorem, statement, no evidence levelBass–Milnor–Serre the…statementQuillen's finite generation theorem, statement, no evidence levelQuillen's finite gene…statement
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. Algebraic K-theory concept
  2. Algebraic number field concept
  3. Building concept
  4. Congruence subgroup concept
  5. Free abelian group concept
  6. General linear group object
  7. Group of units of a ring concept
  8. Ideal concept
  9. Identity matrix object
  10. Index of a subgroup concept
  11. Integers object
  12. Integral closure concept
  13. Lattice (group) concept
  14. Matrix definition
  15. Minkowski's theorem statement
  16. Quillen Q-construction concept
  17. Regulator of a number field concept
  18. Root of unity concept
  19. Vector space definition
  20. Dirichlet's unit theorem statement
  21. Ring of integers statement
  22. Square matrix definition
  23. Linear map definition
  24. Multilinear map definition
  25. Alternating multilinear map definition
  26. Determinant definition
  27. Special linear group object
  28. Bass–Milnor–Serre theorem statement
  29. Quillen's finite generation theorem statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Algebraic_K-theory
About this page
TagPXYP8675C2
URIm2://PXYP8675C2
Kindstatement, theorem
WritingW0 stub
MSC19D50, 11R70
StatusThis copy of the site has no build data. Run m2 status PXYP8675C2.
AboutAlgebraic K-theory
UsesAlgebraic number field, Integral closure, Algebraic K-theory, Quillen Q-construction, Dirichlet's unit theorem, Bass–Milnor–Serre theorem, Building
Changed3 November 2026

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