Bloch–Kato conjecture

E911354

mathematical conjecture statement in algebraic K-theory statement in arithmetic geometry

The Bloch–Kato conjecture is a deep statement in arithmetic geometry and K-theory that predicts an exact correspondence between Galois cohomology and Milnor K-theory, linking algebraic K-groups to field arithmetic.

Try in SPARQL Jump to: Surface forms Statements Referenced by

Observed surface forms (1)

Surface form	Occurrences
Bloch–Kato norm residue conjecture	1

Statements (48)

Predicate	Object
instanceOf	mathematical conjecture ⓘ statement in algebraic K-theory ⓘ statement in arithmetic geometry ⓘ
alsoKnownAs	Bloch–Kato norm residue isomorphism conjecture NERFINISHED ⓘ norm residue isomorphism conjecture NERFINISHED ⓘ
concerns	Milnor K-groups modulo l ⓘ fields of characteristic not equal to a fixed prime l ⓘ l-torsion in Galois cohomology ⓘ
field	Galois cohomology NERFINISHED ⓘ algebraic K-theory ⓘ arithmetic geometry ⓘ motivic cohomology ⓘ number theory ⓘ
formulatedBy	Kazuya Kato NERFINISHED ⓘ Spencer Bloch NERFINISHED ⓘ
generalizes	Milnor conjecture NERFINISHED ⓘ
implies	Milnor conjecture on quadratic forms (in suitable form) NERFINISHED ⓘ
influenced	development of motivic homotopy theory ⓘ research on special values of L-functions ⓘ study of cohomological invariants of algebraic groups ⓘ
involves	continuous Galois cohomology of absolute Galois groups ⓘ graded pieces of Milnor K-theory ⓘ
proofUses	Bloch–Ogus theory NERFINISHED ⓘ Rost motives ⓘ motivic cohomology ⓘ norm varieties ⓘ
provedBy	Charles Weibel NERFINISHED ⓘ Markus Rost NERFINISHED ⓘ Vladimir Voevodsky NERFINISHED ⓘ other collaborators in the Rost–Voevodsky program ⓘ
relatedTo	Beilinson–Lichtenbaum conjecture NERFINISHED ⓘ Bloch–Kato Selmer groups (in the context of motives) NERFINISHED ⓘ Quillen K-theory NERFINISHED ⓘ
relatesConcept	Galois cohomology ⓘ Galois representations ⓘ Milnor K-theory NERFINISHED ⓘ algebraic K-groups ⓘ field arithmetic ⓘ motivic complexes ⓘ norm residue homomorphism ⓘ étale cohomology ⓘ
states	norm residue homomorphism from Milnor K-theory modulo l to Galois cohomology is an isomorphism ⓘ
status	proved ⓘ
timePeriod	late 20th century mathematics ⓘ
topic	cohomological invariants of fields ⓘ description of Galois cohomology in terms of K-theory ⓘ exact correspondence between Milnor K-theory and Galois cohomology ⓘ norm residue isomorphism in degree n ⓘ

Referenced by (3)

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

Milnor K-theory → relatedTo → Bloch–Kato conjecture ⓘ

Milnor K-theory → usedToFormulate → Bloch–Kato conjecture ⓘ

this entity surface form: Bloch–Kato norm residue conjecture

Iwasawa theory → relatedTo → Bloch–Kato conjecture ⓘ