Quillen K-theory
E912805
Quillen K-theory is a sophisticated algebraic K-theory framework defined via higher K-groups of exact or Waldhausen categories, providing deep invariants of rings, schemes, and topological spaces that generalize and extend earlier constructions such as Milnor K-theory.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic K-theory
ⓘ
mathematical theory ⓘ |
| appliesTo |
Waldhausen categories
NERFINISHED
ⓘ
exact categories ⓘ rings ⓘ schemes ⓘ topological spaces ⓘ |
| characteristicProperty |
excision in K-theory
ⓘ
homotopy invariance in suitable settings ⓘ localization sequences ⓘ long exact sequences of K-groups ⓘ |
| defines | higher K-groups ⓘ |
| definesGroup |
K0
ⓘ
K1 NERFINISHED ⓘ K2 ⓘ Kn for n≥0 ⓘ |
| extends |
Bass K-theory
NERFINISHED
ⓘ
Grothendieck group K0 NERFINISHED ⓘ |
| field |
algebraic geometry
ⓘ
algebraic topology ⓘ category theory ⓘ homological algebra ⓘ |
| framework | homotopy-theoretic ⓘ |
| generalizes | Milnor K-theory NERFINISHED ⓘ |
| hasApplication |
Bloch–Kato conjecture
NERFINISHED
ⓘ
Lichtenbaum–Quillen conjecture NERFINISHED ⓘ algebraic cycles ⓘ higher regulators ⓘ motivic cohomology ⓘ number theory ⓘ topological cyclic homology ⓘ |
| influenced |
Waldhausen K-theory
NERFINISHED
ⓘ
higher category theory ⓘ motivic homotopy theory NERFINISHED ⓘ |
| introducedBy | Daniel Quillen NERFINISHED ⓘ |
| introducedInYear | 1972 ⓘ |
| namedAfter | Daniel Quillen NERFINISHED ⓘ |
| relatedConcept |
Waldhausen K-theory
NERFINISHED
ⓘ
algebraic K-theory of rings ⓘ algebraic K-theory of schemes ⓘ |
| studiesInvariant |
exact sequences
ⓘ
projective modules ⓘ vector bundles ⓘ |
| usedTo |
define K-theory of Waldhausen categories
ⓘ
define K-theory of exact categories ⓘ |
| usesConstruction |
Q-construction
NERFINISHED
ⓘ
plus-construction ⓘ |
| usesTool |
classifying spaces
ⓘ
homotopy groups ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.