proof of the Milnor conjecture
E858913
The proof of the Milnor conjecture is Vladimir Voevodsky’s landmark result in algebraic K-theory and Galois cohomology that established a deep connection between Milnor K-theory and étale cohomology, earning him the Fields Medal.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Milnor conjecture on quadratic forms | 1 |
| proof of the Milnor conjecture canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10388472 — 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: proof of the Milnor conjecture Context triple: [Vladimir Voevodsky, notableWork, proof of the Milnor conjecture]
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A.
Milnor K-theory
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.
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B.
Standard Conjectures on Algebraic Cycles
The Standard Conjectures on Algebraic Cycles are a set of deep, still unproven hypotheses in algebraic geometry that aim to provide a foundational theory of algebraic cycles and their cohomological properties, underpinning much of the modern theory of motives.
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C.
Introduction to Algebraic K-Theory
Introduction to Algebraic K-Theory is a foundational graduate-level textbook by John Milnor that systematically develops the basic concepts and techniques of algebraic K-theory in a concise and influential style.
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D.
Beilinson conjectures
Beilinson conjectures are a set of deep conjectures in arithmetic geometry that relate special values of L-functions to algebraic K-theory and motivic cohomology, generalizing phenomena seen in cases like the Birch and Swinnerton-Dyer conjecture.
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E.
Hirzebruch signature theorem
The Hirzebruch signature theorem is a fundamental result in differential topology that expresses the signature of a smooth, compact, oriented 4k-dimensional manifold as a polynomial in its Pontryagin classes.
- 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: proof of the Milnor conjecture Target entity description: The proof of the Milnor conjecture is Vladimir Voevodsky’s landmark result in algebraic K-theory and Galois cohomology that established a deep connection between Milnor K-theory and étale cohomology, earning him the Fields Medal.
-
A.
Milnor K-theory
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.
-
B.
Standard Conjectures on Algebraic Cycles
The Standard Conjectures on Algebraic Cycles are a set of deep, still unproven hypotheses in algebraic geometry that aim to provide a foundational theory of algebraic cycles and their cohomological properties, underpinning much of the modern theory of motives.
-
C.
Introduction to Algebraic K-Theory
Introduction to Algebraic K-Theory is a foundational graduate-level textbook by John Milnor that systematically develops the basic concepts and techniques of algebraic K-theory in a concise and influential style.
-
D.
Beilinson conjectures
Beilinson conjectures are a set of deep conjectures in arithmetic geometry that relate special values of L-functions to algebraic K-theory and motivic cohomology, generalizing phenomena seen in cases like the Birch and Swinnerton-Dyer conjecture.
-
E.
Hirzebruch signature theorem
The Hirzebruch signature theorem is a fundamental result in differential topology that expresses the signature of a smooth, compact, oriented 4k-dimensional manifold as a polynomial in its Pontryagin classes.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical proof
ⓘ
result in Galois cohomology ⓘ result in algebraic K-theory ⓘ |
| basedOn | ideas of John Milnor ⓘ |
| concerns |
Witt ring of a field
NERFINISHED
ⓘ
fields with finite 2-cohomological dimension ⓘ quadratic forms ⓘ |
| contributedToAward | Vladimir Voevodsky Fields Medal 2002 NERFINISHED ⓘ |
| establishes |
isomorphism between graded pieces of the Witt ring and Milnor K-theory mod 2
ⓘ
norm residue isomorphism theorem in degree 2 NERFINISHED ⓘ |
| field |
Galois cohomology
ⓘ
algebraic K-theory NERFINISHED ⓘ |
| generalizedBy | proof of the Bloch–Kato conjecture ⓘ |
| hasAuthor | Vladimir Voevodsky NERFINISHED ⓘ |
| hasConsequence |
description of Galois cohomology in terms of K-theory
ⓘ
new invariants of quadratic forms ⓘ |
| implies | isomorphism between Milnor K-theory mod 2 and Galois cohomology with Z/2Z coefficients ⓘ |
| influenced |
arithmetic geometry
NERFINISHED
ⓘ
higher K-theory ⓘ modern algebraic topology ⓘ |
| involves |
Galois cohomology groups
ⓘ
Galois symbol NERFINISHED ⓘ Milnor K-groups NERFINISHED ⓘ norm residue homomorphism ⓘ |
| language | English ⓘ |
| ledTo |
development of motivic homotopy theory
ⓘ
new methods in algebraic geometry ⓘ |
| proves | Milnor conjecture NERFINISHED ⓘ |
| provesFor | fields of characteristic not equal to 2 ⓘ |
| publishedIn | Annals of Mathematics NERFINISHED ⓘ |
| recognizedBy | Fields Medal NERFINISHED ⓘ |
| relatedArea |
motivic integration
ⓘ
triangulated categories of motives ⓘ |
| relatedConjecture | Bloch–Kato conjecture NERFINISHED ⓘ |
| relates |
Milnor K-theory
NERFINISHED
ⓘ
étale cohomology ⓘ |
| status | accepted ⓘ |
| titleOfMainPaper | The Milnor conjecture NERFINISHED ⓘ |
| uses |
A1-homotopy theory
ⓘ
motivic cohomology ⓘ motivic homotopy theory NERFINISHED ⓘ |
| usesTechnique |
homotopy-theoretic methods in algebraic geometry
ⓘ
simplicial sheaves ⓘ spectral sequences in motivic cohomology ⓘ |
| yearAnnounced | 1996 ⓘ |
| yearPublished | 1997 ⓘ |
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Input
Subject: proof of the Milnor conjecture Description of subject: The proof of the Milnor conjecture is Vladimir Voevodsky’s landmark result in algebraic K-theory and Galois cohomology that established a deep connection between Milnor K-theory and étale cohomology, earning him the Fields Medal.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Milnor conjecture on quadratic forms