Hasse invariant of an algebra

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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.[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]

References

  1. 1.0 1.1 Falko (2008) p.232
  2. Falko (2008) pp.225–226
  3. Falko (2008) p.226
  4. Falko (2008) p.233