Hasse invariant of an algebra: Difference between revisions
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In mathematics, the '''Hasse invariant of an algebra''' is an invariant attached to a Brauer class of algebras over a field. The concept is named after [[Helmut Hasse]]. | |||
The Hasse invariant is a map from the [[Brauer group]] of a [[local field]] ''K'' to the [[divisible group]] '''Q'''/'''Z'''.<ref name=F232>Falko (2008) p.232</ref> Every class in the Brauer group is represented by a class in the Brauer group of an unramified extension of ''L''/''K'' of degree ''n'',<ref name=F2256>Falko (2008) pp.225–226</ref> which by the [[Grunwald–Wang theorem]] and the [[Albert–Brauer–Hasse–Noether theorem]] we may take to be a [[cyclic algebra]] (''L'',φ,π<sup>''k''</sup>) for some ''k'' mod ''n'', where φ is the [[Frobenius_map#Frobenius_for_local_fields|Frobenius map]] and π is a uniformiser.<ref name=F226>Falko (2008) p.226</ref> The ''invariant map'' attaches the element ''k''/''n'' mod 1 to the class and gives rise to a homomophism | |||
:<math> \underset{L/K}{\operatorname{inv}} : \operatorname{Br}(L/K) \rightarrow \mathbb{Q}/\mathbb{Z} . </math> | |||
The invariant map now extends to Br(''K'') by representing each class by some element of Br(''L''/''K'') as above.<ref name=F232/> | |||
For a non-Archimedean local field, the invariant map is a [[group isomorphism]].<ref name=F233>Falko (2008) p.233</ref> | |||
==References== | |||
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Latest revision as of 06:00, 26 August 2024
In mathematics, the Hasse invariant of an algebra is an invariant attached to a Brauer class of algebras over a field. The concept is named after Helmut Hasse.
The Hasse invariant is a map from the Brauer group of a local field K to the divisible group Q/Z.[1] Every class in the Brauer group is represented by a class in the Brauer group of an unramified extension of L/K of degree n,[2] which by the Grunwald–Wang theorem and the Albert–Brauer–Hasse–Noether theorem we may take to be a cyclic algebra (L,φ,πk) for some k mod n, where φ is the Frobenius map and π is a uniformiser.[3] The invariant map attaches the element k/n mod 1 to the class and gives rise to a homomophism
The invariant map now extends to Br(K) by representing each class by some element of Br(L/K) as above.[1]
For a non-Archimedean local field, the invariant map is a group isomorphism.[4]