Browsing by Subject "Homology theory"
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Item The discriminant algebra in cohomology(2008-08) Mallmann, Katja, 1973-; Saltman, D. J. (David J.), 1951-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.Item Pattern-equivariant cohomology of tiling spaces with rotations(2006) Rand, Betseygail; Sadun, LorenzoThis paper develops a new cohomology theory on generalized tiling spaces. This theory incorporates both the rotational geometry of the tiling space and the local pattern geometry into the structure of the cohomology groups. Our use of the local pattern geometry is a generalization of pattern-equivariant cohomology, a theory developed by Ian Putnam and Johannes Kellendonk in 2003. It was defined for tilings whose tiles appear as translates. The most general setting in tiling theory is to work with tiling spaces, with an action of a subgroup of the Euclidean group. This paper defines a new, general pattern-equivariant cohomology for tiling spaces with finite rotation groups, and proves that it is preserved under homeomorphisms which commute with the action of the group. It is conjectured here that this theory is not a topological invariant for tiling spaces with infinite rotation group.Item Regular realizations of p-groups(2008-05) Hammond, John Lockwood; Saltman, D. J. (David J.), 1951-This thesis is concerned with the Regular Inverse Galois Problem for p-groups over fields of characteristic unequal to p. Building upon results of Saltman, Dentzer characterized a class of finite groups that are automatically realized over every field, and proceeded to show that every group of order dividing p⁴ belongs to this class. We extend this result to include groups of order p⁵, provided that the base field k contains the p³-th roots of unity. The proof involves reducing to certain Brauer embedding problems defined over the rational function field k(x). Through explicit computation, we describe the cohomological obstructions to these embedding problems. Then by applying results about the Brauer group of a Dedekind domain, we show that they all possess solutions.