Hasse invariant of an algebra

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.Serre (1967) p.137

There is a commutative subfield L of D which is unramified over K, and D splits over L.Serre (1967) pp.130,138 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.Serre (1967) p.138

The Hasse invariant is thus a map defined on the Brauer group of a local field K to the divisible group Q/Z.Lorenz (2008) p.232 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,Lorenz (2008) pp.225–226 which by the Grunwald–Wang theorem and the Albert–Brauer–Hasse–Noether theorem we may take to be a cyclic algebra (L,φ,π^k) for some k mod n, where φ is the Frobenius map and π is a uniformiser.Lorenz (2008) p.226 The invariant map attaches the element k/n mod 1 to the class. This exhibits the invariant map as a homomorphism

:$\backslash underset\{L/K\}\{\backslash operatorname\{inv\}\}\; :\; \backslash operatorname\{Br\}(L/K)\; \backslash rightarrow\; \backslash mathbb\{Q\}/\backslash mathbb\{Z\}\; .$

The invariant map extends to Br(K) by representing each class by some element of Br(L/K) as above.

For a non-Archimedean local field, the invariant map is a group isomorphism.Lorenz (2008) p.233

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.Serre (1979) p.163 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.

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_v = D ⊗ K_v The extension D_v splits for all but finitely many v, so that the local invariant of D_v is almost always zero. The Brauer group Br(K) fits into an exact sequenceGille & Szamuely (2006) p.159

:$0\backslash rightarrow\; \backslash textrm\{Br\}(K)\backslash rightarrow\; \backslash bigoplus\_\{v\backslash in\; S\}\; \backslash textrm\{Br\}(K\_v)\backslash rightarrow\; \backslash mathbf\{Q\}/\backslash mathbf\{Z\}\; \backslash rightarrow\; 0,$

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.

{{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 | translator-link=Marvin Greenberg|translator-last1=Greenberg|translator-first=Marvin Jay | series=Graduate Texts in Mathematics | volume=67 | publisher=Springer-Verlag | year=1979 | isbn=0-387-90424-7 | zbl=0423.12016 }}

• {{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 (mathematics)

Category:Algebraic number theory