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The use of graphs in algebraic studies is ubiquitous, whether the graphs be finite or infinite, directed or undirected. Green and Zacharia have characterized finite generation of the cohomology rings of monomial algebras, and thereafter G. Davis determined a finite criteria for such generation in the case of cycle algebras. Herein, we describe the construction of a finite directed graph upon which criteria can be established to determine finite generation of the cohomology ring of in-spoked cycle" algebras, a class of algebras that includes cycle algebras. We then show the further usefulness of this constructed graph by studying other monomial algebras, including d-Koszul monomial algebras and a new class of monomial algebras which we term "left/right-symmetric" algebras. / Ph. D.

The goal of this paper is to explore the first cohomology group of groups G that are not necessarily finitely generated. Our focus is on l^p-cohomology, 1 leq p leq infty, and what results regarding finitely generated groups change when G is infinitely generated. In particular, for abelian groups and locally finite groups, the l^p-cohomology is non-zero when G is countable, but vanishes when G has sufficient cardinality. We then show that the l^infty-cohomology remains unchanged for many classes of groups, before looking at several results regarding the injectivity of induced maps from embeddings of G-modules. We present several new results for countable groups, and discuss which results fail to hold in the general uncountable case. Lastly, we present results regarding reduced cohomology, including a useful lemma extending vanishing results for finitely generated groups to the infinitely generated case. / Ph. D. / The goal of this paper is to use a technique that originated in algebraic topology to study the properties of a structure called a group. Groups are collections of objects that interact with each other through an operation that obeys certain properties. Groups arise when considering many different mathematical questions, and they were first studied when looking at the different symmetries an object can have. Classifying the different properties of a group is an active area of mathematical research. We seek to do this by looking at collections of maps from a particular group to the real or complex numbers, then studying how the group shifts these functions.

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Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-02-26Bitstream added on 2014-06-13T20:47:30Z : No. of bitstreams: 1 santos_mv_me_sjrp.pdf: 471794 bytes, checksum: eb8010c830dbd94ac9f17418379b492f (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo principal deste trabalho é apresentar em detalhes um estudo sobre dois critérios para G-coincidências de aplicações de um espaço particular X em um CW complexo, onde G é um grupo finito. No primeiro critério G é o grupo cíclico de ordem p, com p um primo ímpar e X é uma esfera de dimensão ímpar. No segundo critério, que estende o primeiro, G é um grupo finito qualquer e X é um CW complexo com o mesmo tipo de homotopia de uma esfera de dimensão ímpar. Para o estudo desses critérios foram necessários alguns resultados da teoria de cohomologia de grupos finitos com ênfase em grupos com cohomologia periódica segundo a teoria de cohomologia de Tate. / The main objective of htis work is to present in details a study about two criteria for G-coincidences of maps from a particular spaca X into a CW-complex, where G is a finite group. In the first criterion G is the cyclic group of order p, with p an odd prime and X is an odd dimensional sphere. In the second criterion, wich extends the firt, G is any finite group and X is a CW-complex with the same type of homotopy of an odd dimensional sphere. For the study of those criteria were needed some results from the theory of cohomology of finite groups with emphasis on groups with periodic cohomology according to the Tate cohomology theory.

Orientador: Maria Gorete Carreira Andrade / Banca: Edivaldo Lopes dos Santos / Banca: Ermínia de Lourdes Campello Fanti / Resumo: O objetivo principal deste trabalho é apresentar em detalhes um estudo sobre dois critérios para G-coincidências de aplicações de um espaço particular X em um CW complexo, onde G é um grupo finito. No primeiro critério G é o grupo cíclico de ordem p, com p um primo ímpar e X é uma esfera de dimensão ímpar. No segundo critério, que estende o primeiro, G é um grupo finito qualquer e X é um CW complexo com o mesmo tipo de homotopia de uma esfera de dimensão ímpar. Para o estudo desses critérios foram necessários alguns resultados da teoria de cohomologia de grupos finitos com ênfase em grupos com cohomologia periódica segundo a teoria de cohomologia de Tate. / Abstract: The main objective of htis work is to present in details a study about two criteria for G-coincidences of maps from a particular spaca X into a CW-complex, where G is a finite group. In the first criterion G is the cyclic group of order p, with p an odd prime and X is an odd dimensional sphere. In the second criterion, wich extends the firt, G is any finite group and X is a CW-complex with the same type of homotopy of an odd dimensional sphere. For the study of those criteria were needed some results from the theory of cohomology of finite groups with emphasis on groups with periodic cohomology according to the Tate cohomology theory. / Mestre

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A concrete description of Hochschild cohomology is the first step toward exploring associative deformations of algebras. In this dissertation, deformation theory, geometry, combinatorics, invariant theory, representation theory, and homological algebra merge in an investigation of Hochschild cohomology of skew group algebras arising from complex reflection groups. Given a linear action of a finite group on a finite dimensional vector space, the skew group algebra under consideration is the semi-direct product of the group with a polynomial ring on the vector space. Each representation of a group defines a different skew group algebra, which may have its own interesting deformations. In this work, we explicitly describe all graded Hecke algebras arising as deformations of the skew group algebra of any finite group acting by the regular representation. We then focus on rank two exceptional complex reflection groups acting by any irreducible representation. We consider in-depth the reflection representation and a nonfaithful rotation representation. Alongside our study of cohomology for the rotation representation, we develop techniques valid for arbitrary finite groups acting by a representation with a central kernel. Additionally, we consider combinatorial questions about reflection length and codimension orderings on complex reflection groups. We give algorithms using character theory to compute reflection length, atoms, and poset relations. Using a mixture of theory, explicit examples, and calculations using the software GAP, we show that Coxeter groups and the infinite family G(m,1,n) are the only irreducible complex reflection groups for which the reflection length and codimension orders coincide. We describe the atoms in the codimension order for the groups G(m,p,n). For arbitrary finite groups, we show that the codimension atoms are contained in the support of every generating set for cohomology, thus yielding information about the degrees of generators for cohomology.

A version of the classical Lefschetz fixed point formula is proved for the cohomology of the cone of a cochain mapping of elliptic complexes. As a particular case we show a Lefschetz formula for the relative de Rham cohomology.

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