Milnor K-theory
E265518
Milnor K-theory is an algebraic K-theory constructed from fields using tensor powers of their multiplicative groups modulo Steinberg relations, playing a central role in modern algebraic geometry and number theory.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Milnor K-theory canonical | 2 |
| Bloch group | 1 |
| Milnor K-groups | 1 |
| Milnor conjecture in algebraic K-theory | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2418317 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Milnor K-theory Context triple: [John Milnor, notableWork, Milnor K-theory]
-
A.
K-theory
K-theory is a branch of algebraic topology and algebraic geometry that studies vector bundles and generalized cohomology theories using algebraic and categorical methods.
-
B.
“K-Theory” (book with Friedrich Hirzebruch and others)
“K-Theory” is a foundational mathematical monograph co-authored by Michael Atiyah, Friedrich Hirzebruch, and others that systematically develops topological K-theory and its applications in geometry and topology.
-
C.
Grothendieck group
The Grothendieck group is an algebraic construction that formally turns a commutative monoid (often arising from isomorphism classes of objects like vector bundles or modules) into an abelian group, playing a central role in K-theory and modern algebraic geometry.
-
D.
Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
-
E.
Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Milnor K-theory Target entity description: Milnor K-theory is an algebraic K-theory constructed from fields using tensor powers of their multiplicative groups modulo Steinberg relations, playing a central role in modern algebraic geometry and number theory.
-
A.
K-theory
K-theory is a branch of algebraic topology and algebraic geometry that studies vector bundles and generalized cohomology theories using algebraic and categorical methods.
-
B.
“K-Theory” (book with Friedrich Hirzebruch and others)
“K-Theory” is a foundational mathematical monograph co-authored by Michael Atiyah, Friedrich Hirzebruch, and others that systematically develops topological K-theory and its applications in geometry and topology.
-
C.
Grothendieck group
The Grothendieck group is an algebraic construction that formally turns a commutative monoid (often arising from isomorphism classes of objects like vector bundles or modules) into an abelian group, playing a central role in K-theory and modern algebraic geometry.
-
D.
Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
-
E.
Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic K-theory
ⓘ
mathematical theory ⓘ |
| appearsIn | Milnor’s work on algebraic K-theory of fields ⓘ |
| appliesTo | fields ⓘ |
| approximates | higher algebraic K-theory of fields ⓘ |
| basedOn | tensor powers of multiplicative groups of fields ⓘ |
| construction | graded ring K_*^M(F) ⓘ |
| definedUsing |
symbols {a_1,…,a_n} with a_i in F^×
ⓘ
tensor algebra on F^× ⓘ |
| defines |
Milnor K-theory
self-linksurface differs
ⓘ
surface form:
Milnor K-groups
|
| degreeOnePart | K_1^M(F) ≅ F^× ⓘ |
| degreeZeroPart | K_0^M(F) ≅ ℤ ⓘ |
| fieldOfStudy |
algebra
ⓘ
algebraic K-theory ⓘ algebraic geometry ⓘ number theory ⓘ |
| generalizes | classical K_1 of a field ⓘ |
| hasComponent |
Milnor K-group K_0^M(F)
ⓘ
Milnor K-group K_1^M(F) ⓘ Milnor K-group K_n^M(F) ⓘ |
| hasGradedPiece | K_n^M(F) in degree n ⓘ |
| hasProperty |
functorial in the field F
ⓘ
multiplicative graded-commutative structure ⓘ |
| hasRelation | K_n^M(F)/ℓ ≅ H^n(G_F, μ_ℓ^{⊗ n}) for suitable fields and primes ℓ ⓘ |
| hasSymbolNotation | {a_1,…,a_n} for elements of K_n^M(F) ⓘ |
| influenced |
Voevodsky’s work on motives
ⓘ
development of motivic homotopy theory ⓘ |
| introducedBy | John Milnor ⓘ |
| introducedIn | 1960s ⓘ |
| isGradedBy | nonnegative integers ⓘ |
| namedAfter | John Milnor ⓘ |
| playsRoleIn | proof of the norm residue isomorphism theorem ⓘ |
| quotientedBy | Steinberg relation {a,1−a}=0 for a,1−a≠0 ⓘ |
| relatedTo |
Bloch–Kato conjecture
ⓘ
Galois cohomology ⓘ Quillen K-theory ⓘ motivic cohomology ⓘ |
| satisfiesRelation |
{a_1,…,a_i,…,a_j,…,a_n} = −{a_1,…,a_j,…,a_i,…,a_n}
ⓘ
{ab,c_2,…,c_n}={a,c_2,…,c_n}+{b,c_2,…,c_n} ⓘ |
| usedIn |
arithmetic geometry
ⓘ
class field theory ⓘ higher local class field theory ⓘ study of central simple algebras ⓘ study of quadratic forms ⓘ |
| usedToFormulate |
Bloch–Kato conjecture
ⓘ
surface form:
Bloch–Kato norm residue conjecture
proof of the Milnor conjecture ⓘ
surface form:
Milnor conjecture on quadratic forms
|
| usesConcept | Steinberg relations ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Milnor K-theory Description of subject: Milnor K-theory is an algebraic K-theory constructed from fields using tensor powers of their multiplicative groups modulo Steinberg relations, playing a central role in modern algebraic geometry and number theory.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Milnor conjecture in algebraic K-theory
this entity surface form:
Bloch group
this entity surface form:
Milnor K-groups