proof of the Milnor conjecture

E858913

mathematical proof result in Galois cohomology result in algebraic K-theory

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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (5)

Label	Occurrences
Milnor conjecture	2
Milnor conjecture on quadratic forms	1
Milnor conjecture on quadratic forms (in suitable form)	1
The Milnor conjecture	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]

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
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 ⓘ linked to: Witt group of quadratic forms fields with finite 2-cohomological dimension ⓘ quadratic forms ⓘ
contributedToAward	Vladimir Voevodsky Fields Medal 2002 ⓘ linked to: Vladimir Voevodsky
establishes	isomorphism between graded pieces of the Witt ring and Milnor K-theory mod 2 ⓘ norm residue isomorphism theorem in degree 2 ⓘ
field	Galois cohomology ⓘ algebraic K-theory ⓘ linked to: K-theory
generalizedBy	proof of the Bloch–Kato conjecture ⓘ
hasAuthor	Vladimir Voevodsky ⓘ
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 ⓘ linked to: Diophantine geometry higher K-theory ⓘ modern algebraic topology ⓘ
involves	Galois cohomology groups ⓘ Galois symbol ⓘ Milnor K-groups ⓘ linked to: Milnor K-theory norm residue homomorphism ⓘ
language	English ⓘ
ledTo	development of motivic homotopy theory ⓘ new methods in algebraic geometry ⓘ
proves	Milnor conjecture ⓘ linked to: proof of the Milnor conjecture
provesFor	fields of characteristic not equal to 2 ⓘ
publishedIn	Annals of Mathematics ⓘ
recognizedBy	Fields Medal ⓘ
relatedArea	motivic integration ⓘ triangulated categories of motives ⓘ
relatedConjecture	Bloch–Kato conjecture ⓘ
relates	Milnor K-theory ⓘ étale cohomology ⓘ
status	accepted ⓘ
titleOfMainPaper	The Milnor conjecture ⓘ linked to: proof of the Milnor conjecture
uses	A1-homotopy theory ⓘ motivic cohomology ⓘ motivic homotopy theory ⓘ
usesTechnique	homotopy-theoretic methods in algebraic geometry ⓘ simplicial sheaves ⓘ spectral sequences in motivic cohomology ⓘ
yearAnnounced	1996 ⓘ
yearPublished	1997 ⓘ

How these facts were elicited