Discriminants of involutions on Henselian division algebras

Decomposition of involutions on inertially split division algebras

“…We can apply this result to S and to I to get that the types of σ and σ * are equal to the type of σ| C . Also, applying it to I = (C, G, (ω, f )), we see that the type of σ is equal to the type of σ| C , which is equal to the type of σ| C , by [3,§1,Prop. 3].…”

“…We thus extend Dherte's results to the context of inertially split algebras over a Henselian valued field with residue characteristic not 2. We remark that in [3,Thm. 5] it is proved that if D is a division algebra of exponent 2 over a Henselian valued field F (with char(F ) = 2) and if D has an involution σ of the first kind with σ = id on D, then D can be decomposed as D = S ⊗ F T with S inertially split and T totally ramified, and each of S and T are stable under σ.…”

“…Since Q is stable under σ and σ is trivial on Z, we see that Proof. By [3,§1,Prop. 4], there is an involution σ on J with σ = τ .…”

“…3 We have disc(σ * ) = disc(σ) = i(disc(σ)). If C = Z, then these are also equal to N Z/F (disc(σ| C )).…”

“…In particular, by [6,Lemma 5.14], there is a natural but not unique decomposition of S, up to similarity, as S ∼ I ⊗ F N , where I is similar to an inertial division algebra and N is a nicely semiramified division algebra (see [6,Secs. 2,3] for definitions). For the Malcev-Neumann algebra S(A/K/F ) mentioned above, by viewing f ∈ Z 2 (G, K * ), the cocycle f factors as f = fc with c a symmetric cocycle having values in F * , and this factorization gives a similarity relation G, c), which is a special case of the decomposition of [6].…”