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.

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Observed surface forms (1)

Surface form	Occurrences
Milnor conjecture on quadratic forms	1

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 ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Vladimir Voevodsky → notableWork → proof of the Milnor conjecture ⓘ

Milnor K-theory → usedToFormulate → proof of the Milnor conjecture ⓘ

this entity surface form: Milnor conjecture on quadratic forms