We show that the odd-indexed derived Witt groups of a semi-local ring with trivial involution vanish. We show that this is wrong when the involution is not trivial and we provide examples.
Abstract. Let F be the function field of a p-adic curve. Let D be a central simple algebra over F of period n and λ ∈ F * . We show that if n is coprime to p and D · (λ) = 0 in H 3 (F, µ ⊗2 n ), then λ is a reduced norm. This leads to a Hasse principle for the SL 1 (D), namely an element λ ∈ F * is a reduced norm from D if and only if it is a reduced norm locally at all discrete valuations of F .
In this paper we show that over any field K of characteristic different from 2, the Maslov index gives rise to a 2-cocycle on the stable symplectic group with values in the Witt group. We also show that this cocycle admits a natural reduction to I 2 (K) and that the induced natural homomorphism from K 2 Sp(K) → I 2 (K) is indeed the homomorphism given by the symplectic symbol {x, y} mapping to the Pfister form 1, −x ⊗ 1, −y .
Mathematics Subject Classifications (1991): 11E81, 19CXX
For a central simple algebra with an orthogonal involution (A, σ) over a field k of characteristic different from 2, we relate the multipliers of similitudes of (A, σ) with the Clifford algebra C(A, σ). We also give a complete description of the group of multipliers of similitudes when deg A ≤ 6 or when the virtual cohomological dimension of k is at most 2.
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