Hasse invariant of an algebra

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.

There is a commutative subfield L of D which is unramified over K, and D splits over L. 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.

The Hasse invariant is thus a map defined on the Brauer group of a local field K to the divisible group Q/Z. 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, 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. The invariant map attaches the element k/n mod 1 to the class. This exhibits the invariant map as a homomorphism

$\underset{L/K}{\mathrm{inv}}:\mathrm{Br}(L/K)\to \mathbb{\mathbb{Q}}/\mathbb{\mathbb{Z}}.$

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.

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. 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.

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_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 sequence

$0\to \text{<mrow data-mjx-texclass="ORD">Br</mrow>}(K)\to \underset{v\in S}{⨁}\text{<mrow data-mjx-texclass="ORD">Br</mrow>}(K_v)\to \mathbf{𝐐}/\mathbf{𝐙}\to 0,$

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.

References

• Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. 
• Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. pp. 231–238. ISBN 978-0-387-72487-4. 
• Serre, Jean-Pierre (1967). "VI. Local class field theory". in Cassels, J.W.S.; Fröhlich, A.. 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. London: Academic Press. pp. 128–161. 
• Serre, Jean-Pierre (1979). Local Fields. Graduate Texts in Mathematics. 67. Springer-Verlag. ISBN 0-387-90424-7. 

Further reading

• Shatz, Stephen S. (1972). Profinite groups, arithmetic, and geometry. Annals of Mathematics Studies. 67. Princeton, NJ: Princeton University Press. ISBN 0-691-08017-8.