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In mathematics, the '''Hasse invariant of an algebra''' is an invariant attached to a [[Brauer class]] of [[Algebra over a field|algebras over a field]]. The concept is named after [[Helmut Hasse]]. The invariant plays a role in [[local class field theory]]. | |||
==Local fields== | |||
Let ''K'' be a [[local field]] with valuation ''v'' and ''D'' a ''K''-algebra. We may assume ''D'' is a [[division algebra]] with centre ''K'' of degree ''n''. The valuation ''v'' can be extended to ''D'', for example by extending it compatibly to each commutative subfield of ''D'': the value group of this valuation is (1/''n'')'''Z'''.<ref name=S67137>Serre (1967) p.137</ref> | |||
There is a commutative subfield ''L'' of ''D'' which is unramified over ''K'', and ''D'' splits over ''L''.<ref name=S671308>Serre (1967) pp.130,138</ref> The field ''L'' is not unique but all such extensions are conjugate by the [[Skolem–Noether theorem]], which further shows that any automorphism of ''L'' is induced by a conjugation in ''D''. Take γ in ''D'' such that conjugation by γ induces the Frobenius automorphism of ''L''/''K'' and let ''v''(γ) = ''k''/''n''. Then ''k''/''n'' modulo 1 is the Hasse invariant of ''D''. It depends only on the Brauer class of ''D''.<ref name=S67138>Serre (1967) p.138</ref> | |||
The Hasse invariant is thus a map defined on the [[Brauer group]] of a [[local field]] ''K'' to the [[divisible group]] '''Q'''/'''Z'''.<ref name=S67138/><ref name=L232>Lorenz (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=L2256>Lorenz (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=L226>Lorenz (2008) p.226</ref> The invariant map attaches the element ''k''/''n'' mod 1 to the class. This exhibits the invariant map as a homomophism | |||
:<math> \underset{L/K}{\operatorname{inv}} : \operatorname{Br}(L/K) \rightarrow \mathbb{Q}/\mathbb{Z} . </math> | |||
The invariant map extends to Br(''K'') by representing each class by some element of Br(''L''/''K'') as above.<ref name=S67138/><ref name=L232/> | |||
For a non-Archimedean local field, the invariant map is a [[group isomorphism]].<ref name=S67138/><ref name=L233>Lorenz (2008) p.233</ref> | |||
In the case of the field '''R''' of [[real number]]s, there are two Brauer classes, represented by the algebra '''R''' itself and the [[quaternion]] algebra '''H'''.<ref name=S163>Serre (1979) p.163</ref> It is convenient to assign invariant zero to the class of '''R''' and invariant 1/2 modulo 1 to the quaternion class. | |||
In the case of the field '''C''' of complex numbers, the only Brauer class is the trivial one, with invariant zero.<ref name=GS159/> | |||
==Global fields== | |||
For a global field ''K'', given a central simple algebra ''D'' over ''K'' then for each valuation ''v'' of ''K'' we can consider the extension of scalars ''D''<sub>''v''</sub> = ''D'' ⊗ ''K''<sub>''v''</sub> The extension ''D''<sub>''v''</sub> splits for all but finitely many ''v'', so that the ''local invariant'' of ''D''<sub>''v''</sub> is almost always zero. The Brauer group Br(''K'') fits into an [[exact sequence]]<ref name=S163/><ref name=GS159>Gille & Szamuely (2006) p.159</ref> | |||
:<math> 0\rightarrow \textrm{Br}(K)\rightarrow \bigoplus_{v\in S} \textrm{Br}(K_v)\rightarrow \mathbf{Q}/\mathbf{Z} \rightarrow 0,</math> | |||
where ''S'' is the set of all valuations of ''K'' and the right arrow is the sum of the local invariants. The injectivity of the left arrow is the content of the [[Albert–Brauer–Hasse–Noether theorem]]. Exactness in the middle term is a deep fact from [[global class field theory]]. | |||
==References== | |||
{{reflist}} | |||
* {{cite book | last1=Gille | first1=Philippe | last2=Szamuely | first2=Tamás | title=Central simple algebras and Galois cohomology | series=Cambridge Studies in Advanced Mathematics | volume=101 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86103-9 | zbl=1137.12001 }} | |||
* {{cite book | first=Falko | last=Lorenz | title=Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics | year=2008 | publisher=Springer | isbn=978-0-387-72487-4 | pages=231–238 | zbl=1130.12001 }} | |||
* {{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | chapter=VI. Local class field theory | pages=128–161 | editor1-last=Cassels | editor1-first=J.W.S. | editor1-link=J. W. S. Cassels | editor2-last=Fröhlich | editor2-first=A. | editor2-link=Albrecht Fröhlich | title=Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union | location=London | publisher=Academic Press | year=1967 | zbl=0153.07403 }} | |||
* {{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=[[Local Fields (book)|Local Fields]] | others=Translated from the French by Marvin Jay Greenberg | series=[[Graduate Texts in Mathematics]] | volume=67 | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90424-7 | zbl=0423.12016 }} | |||
==Further reading== | |||
* {{cite book | last=Shatz | first=Stephen S. | title=Profinite groups, arithmetic, and geometry | series=Annals of Mathematics Studies | volume=67 | location=Princeton, NJ | publisher=[[Princeton University Press]] | year=1972 | isbn=0-691-08017-8 | zbl=0236.12002 | mr=0347778 }} | |||
[[Category:Field theory]] | |||
[[Category:Algebraic number theory]] | |||
Latest revision as of 13:14, 23 June 2013
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 invariant plays a role in local class field theory.
Local fields
Let K be a local field with valuation v and D a K-algebra. We may assume D is a division algebra with centre K of degree n. The valuation v can be extended to D, for example by extending it compatibly to each commutative subfield of D: the value group of this valuation is (1/n)Z.[1]
There is a commutative subfield L of D which is unramified over K, and D splits over L.[2] The field L is not unique but all such extensions are conjugate by the Skolem–Noether theorem, which further shows that any automorphism of L is induced by a conjugation in D. Take γ in D such that conjugation by γ induces the Frobenius automorphism of L/K and let v(γ) = k/n. Then k/n modulo 1 is the Hasse invariant of D. It depends only on the Brauer class of D.[3]
The Hasse invariant is thus a map defined on the Brauer group of a local field K to the divisible group Q/Z.[3][4] 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,[5] 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.[6] The invariant map attaches the element k/n mod 1 to the class. This exhibits the invariant map as a homomophism
The invariant map extends to Br(K) by representing each class by some element of Br(L/K) as above.[3][4]
For a non-Archimedean local field, the invariant map is a group isomorphism.[3][7]
In the case of the field R of real numbers, there are two Brauer classes, represented by the algebra R itself and the quaternion algebra H.[8] It is convenient to assign invariant zero to the class of R and invariant 1/2 modulo 1 to the quaternion class.
In the case of the field C of complex numbers, the only Brauer class is the trivial one, with invariant zero.[9]
Global fields
For a global field K, given a central simple algebra D over K then for each valuation v of K we can consider the extension of scalars Dv = D ⊗ Kv The extension Dv splits for all but finitely many v, so that the local invariant of Dv is almost always zero. The Brauer group Br(K) fits into an exact sequence[8][9]
where S is the set of all valuations of K and the right arrow is the sum of the local invariants. The injectivity of the left arrow is the content of the Albert–Brauer–Hasse–Noether theorem. Exactness in the middle term is a deep fact from global class field theory.
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
Further reading
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
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