Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1979. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Vita. / Bibliography: leaves 131-133. / Ph.D.

We develop some basic homological theory of hopfological algebra as defined by Khovanov. A simplicial bar resolution for an arbitrary hopfological module is constructed, and some derived analogue of Morita theory is established. We also discuss about some special classes of examples that appear naturally in categorification.

We calculate the second and third integral homology of arbitrary finite rank Coxeter groups. The first of these calculations refines a theorem of Howlett, the second is entirely new. We then prove that families of Artin monoids, which have the braid monoid as a submonoid, satisfy homological stability. When the K(π,1) conjecture holds this gives a homological stability result for the associated families of Artin groups. In particular, we recover a classic result of Arnol'd.

by Au Pat Nien. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 86-88). / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Valuations --- p.4 / Chapter 2.1 --- Preliminaries --- p.4 / Chapter 2.2 --- Complete Fields --- p.6 / Chapter 2.3 --- Unramified Extension of Complete Field --- p.10 / Chapter 2.4 --- Local Fields --- p.12 / Chapter 3 --- Ramification Groups and Hasse-Herbrand Function --- p.16 / Chapter 3.1 --- Ramification Groups --- p.16 / Chapter 3.2 --- "The Quotients Gi/Gi+1, i ≥ 0" --- p.17 / Chapter 3.3 --- The Hasse-Herbrand function --- p.19 / Chapter 4 --- The Norm Map --- p.21 / Chapter 4.1 --- Lemmas --- p.21 / Chapter 4.2 --- The Norm Map on the Residue Field of a Totally Ramified Extension of Prime Degree --- p.22 / Chapter 4.3 --- Extension of the Perfect Residue Field in a Totally Ramified Extension --- p.26 / Chapter 4.4 --- The Norm Map on Finite Separable Extension of Knr with K Perfect --- p.28 / Chapter 5 --- Cohomology of Finite Groups --- p.30 / Chapter 5.1 --- Preliminaries --- p.30 / Chapter 5.2 --- Mappings of Cohomology Groups --- p.32 / Chapter 5.2.1 --- Restriction and Inflation --- p.32 / Chapter 5.2.2 --- Corestriction --- p.34 / Chapter 5.3 --- Cup Product --- p.34 / Chapter 5.4 --- Cohomology Groups of Low Dimensions --- p.35 / Chapter 5.5 --- Some Results of Group Cohomology --- p.43 / Chapter 6 --- The Brauer Group of a Field --- p.57 / Chapter 7 --- The Norm Residue Map --- p.60 / Chapter 7.1 --- Determination of the Brauer Group of a Local Field --- p.60 / Chapter 7.2 --- Canonical Class --- p.62 / Chapter 7.3 --- The Reciprocity Law --- p.64 / Chapter 8 --- The Local Symbol --- p.74 / Chapter 8.1 --- Definition --- p.74 / Chapter 8.2 --- The Hilbert Symbol --- p.74 / Chapter 8.3 --- The Differential of the Formal Power Series --- p.76 / Chapter 8.4 --- The Artin-Schreier Symbol --- p.78 / Chapter 9 --- Characterization of a Norm Group --- p.81 / Bibliography

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1999. / Includes bibliographical references. Also available on the Internet.

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. / text

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