## The discriminant algebra in cohomology

##### Abstract

Invariants of involutions on central simple algebras have been extensively studied. Many important results have been collected and extended by Knus, Merkurjev, Rost and Tignol in "The Book of Involutions" [BI]. Among those invariants are, for example, the (even) Clifford algebra for involutions of the first kind and the discriminant algebra for involutions of the second kind on an algebra of even degree. In his preprint "Triality, Cocycles, Crossed Products, Involutions, Clifford Algebras and Invariants" [S05], Saltman shows that the definition of the Clifford algebra can be generalized to Azumaya algebras and introduces a special cohomology, the so-called G-H cohomology, to describe its structure. In this dissertation, we prove analogous results about the discriminant algebra D(A; [tau]), which is the algebra of invariants under a special automorphism of order two of the [lambda]-power of an algebra A of even degree n = 2m with involution of the second kind, [tau]. In particular, we generalize its construction to the Azumaya case. We identify the exterior power algebra as defined in "Exterior Powers of Fields and Subfields" [S83] as a splitting subalgebra of the m-th [lambda]-power algebra and prove that a certain invariant subalgebra is a splitting subalgebra of the discriminant algebra. Assuming well-situatedness we show how this splitting subalgebra can be described as the fixed field of an S[subscript n] x C₂- Galois extension and that the corresponding subgroup is [Sigma] = S[subscript m] x S[subscript m] [mathematic symbol] C2. We give an explicit description of the corestriction map and define a lattice E that encodes the corestriction as being trivial. Lattice methods and cohomological tools are applied in order to define the group H²(G;E) which contains the cocycle that will describe the discriminant algebra as a crossed product. We compute this group to have order four and conjecture that it is the Klein 4-group and that the mixed element is the desired cocycle.

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