Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, Jun. 2003. / Includes bibliographical references (p. 25-26). Also available on the Internet.

In this thesis, we present a detailed exposition of Koszul algebras and Koszul duality. We begin with an overview of the required concepts of graded algebras and homological algebra. We then give a precise treatment of Koszul and quadratic algebras, together with their dualities. We fill in some arguments that are omitted in the literature and work out a number of examples in full detail to illustrate the abstract concepts.

A new connection between combinatorics and noncommutative algebra is established by relating a certain class of directed graphs to noncommutative Koszul algebras. The directed graphs in this class are called full graphs and are defined by a set of criteria on the edges. The structural properties of full graphs are studied as they relate to the edge criteria. A method is introduced for generating a Koszul algebra Lambda from a full graph G. The properties of Lambda are examined as they relate to the structure of G, with special attention being given to the construction of a projective resolution of certain semisimple Lambda-modules based on the structural properties of G. The characteristics of the Koszul algebra Lambda that is derived from the product of two full graphs G' and G' are studied as they relate to the properties of the Koszul algebras Lambda' and Lambda' derived from G' and G'. / Ph. D.

x, 68 p. : ill. (some col.) / Motivated by the Adams spectral sequence for computing stable homotopy groups, Priddy defined a class of algebras called Koszul algebras with nice homological properties. Many important algebras arising naturally in mathematics are Koszul, and the Koszul property is often tied to important structure in the settings which produced the algebras. However, the strong defining conditions for a Koszul algebra imply that such algebras must be quadratic. A very natural generalization of Koszul algebras called K 2 algebras was recently introduced by Cassidy and Shelton. Unlike other generalizations of the Koszul property, the class of K 2 algebras is closed under many standard operations in ring theory. The class of K 2 algebras includes Artin-Schelter regular algebras of global dimension 4 on three linear generators as well as graded complete intersections. Our work comprises two distinct projects. Each project was motivated by an aspect of the theory of Koszul algebras which we regard as sufficiently powerful or fundamental to warrant an interpretation for K 2 algebras. A very useful theorem due to Backelin and Fröberg states that if A is a Koszul algebra and I is a quadratic ideal of A which is Koszul as a left A -module, then the factor algebra A/I is a Koszul algebra. We prove that if A is Koszul algebra and A I is a K 2 module, then A/I is a K 2 algebra provided A/I acts trivially on Ext A ( A/I,k ). As an application of our theorem, we show that the class of sequentially Cohen-Macaulay Stanley-Reisner rings are K 2 algebras and we give examples that suggest the class of K 2 Stanley-Reisner rings is actually much larger. Another important recent development in ring theory is the use of A ∞ -algebras. One can characterize Koszul algebras as those graded algebras whose Yoneda algebra admits only trivial A ∞ -structure. We show that, in contrast to the situation for Koszul algebras, vanishing of higher A ∞ -structure on the Yoneda algebra of a K 2 algebra need not be determined in any obvious way by the degrees of defining relations. We also demonstrate that obvious patterns of vanishing among higher multiplications cannot detect the K 2 property. This dissertation includes previously unpublished co-authored material. / Committee in charge: Dr. Brad Shelton, Chair; Dr. Victor Ostrik, Member; Dr. Nicholas Proudfoot, Member; Dr. Arkady Vaintrob, Member; Dr. David Boush, Outside Member

Orientador: Adriano Adrega de Moura / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T09:20:36Z (GMT). No. of bitstreams: 1 Ferreira_GilmardeSousa_M.pdf: 480832 bytes, checksum: 42db64084fe99183a4fb95f90cd9a833 (MD5) Previous issue date: 2012 / Resumo: Nessa dissertação estudamos certas categorias de módulos graduados para uma classe de álgebras de Lie que inclui as álgebras de correntes. Em particular, estudamos diversas propriedades homológicas dessas categorias tais como resoluções projetivas e o espaço de extensões entre seus objetos simples. Em certas situações, os resultados levam ao estabelecimento de um relacionamento com álgebras de Koszul. O estudo é baseado em artigos recentes de Vyjayanthi Chari e seus co-autores / Abstract: In this dissertation, we study certain categories of graded modules for a class of Lie algebras which include current algebras. In particular, we study several homological properties of these categories such as projective resolutions and the space of extensions between two given simple objects. Under certain conditions, these results establish a relationship with Koszul algebras. The study is based on recent papers by Vyjayanthi Chari and her co-authors / Mestrado / Matematica / Mestre em Matemática

Graded ideals over the polynomial ring are studied deeply with a huge of methods and results. Over the exterior algebra, there are not much known about the structures of minimal graded resolutions, Gröbner fans of graded ideals or the Koszul property of algebras defined by graded ideals. We study componentwise linearity, linear resolutions of graded ideals as well as universally, initially and strongly Koszul properties of graded algebras defined by a graded ideals over the exterior algebra. After that, we apply our results to Orlik-Solomon ideals of hyperplane arrangements and show in which way the exterior algebra is useful in the study of related combinatorial objects.

xi, 95 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We investigate some homological properties of graded algebras. If A is an R -algebra, then E (A) := Ext A ( R, R ) is an R-algebra under the cup product and is called the Yoneda algebra. (In most cases, we assume R is a field.) A well-known and widely-studied condition on E(A) is the Koszul property. We study a class of deformations of Koszul algebras that arises from the study of equivariant cohomology and algebraic groups and show that under certain circumstances these deformations are Poincaré-Birkhoff-Witt deformations. Some of our results involve the [Special characters omitted] property, recently introduced by Cassidy and Shelton, which is a generalization of the Koszul property. While a Koszul algebra must be quadratic, a [Special characters omitted] algebra may have its ideal of relations generated in different degrees. We study the structure of the Yoneda algebra corresponding to a monomial [Special characters omitted.] algebra and provide an example of a monomial [Special characters omitted] algebra whose Yoneda algebra is not also [Special characters omitted]. This example illustrates the difficulty of finding a [Special characters omitted] analogue of the classical theory of Koszul duality. It is well-known that Poincaré-Birkhoff-Witt algebras are Koszul. We find a [Special characters omitted] analogue of this theory. If V is a finite-dimensional vector space with an ordered basis, and A := [Special characters omitted] (V)/I is a connected-graded algebra, we can place a filtration F on A as well as E (A). We show there is a bigraded algebra embedding Λ: gr F E (A) [Special characters omitted] E (gr F A ). If I has a Gröbner basis meeting certain conditions and gr F A is [Special characters omitted], then Λ can be used to show that A is also [Special characters omitted]. This dissertation contains both previously published and co-authored materials. / Committee in charge: Brad Shelton, Chairperson, Mathematics; Victor Ostrik, Member, Mathematics; Christopher Phillips, Member, Mathematics; Sergey Yuzvinsky, Member, Mathematics; Van Kolpin, Outside Member, Economics

Neste trabalho estudamos algumas características das álgebras de Koszul, como por exemplo, a maneira como elas se relacionam com suas respectivas álgebras de Yoneda. Descrevemos a álgebra de Yoneda de uma álgebra monomial e como aplicação construímos uma família de álgebras: as chamadas homologicamente auto-duais. Uma álgebra de Koszul pode ser definida a partir da existência de resoluções lineares dos módulos simples. Por isso faz-se necessário a dedicação de parte de nossa atenção ao estudo destas resoluções. Além disso, achamos interessante estudar métodos para a construção de resoluções projetivas de módulos sobre quocientes de álgebras de caminhos. Para tal construção usamos essencialmente a teoria de bases de Gröbner não comutativas. Finalmente, para o caso de módulos lineares sobre álgebras de Koszul, veremos que é possível modicar essa construção de modo que a resolução resultante seja linear. / In this work we study some features of Koszul algebras as, for example, the way that they are related with their Yoneda algebras. We describe the Yoneda algebra of a monomial algebra and as an application we construct a family of algebras: the so called homologically self-dual algebras. A Koszul algebra can be dened as an algebra for which there are linear resolutions of their simple modules. Because of this we dedicate part of our attention to the study of projective resolutions. The study of methods for the construction of projectives resolutions of modules over quotients of path algebras, has an of interest its own. For the study of projective resolutions we used the theory of noncommutative, Gröbner bases. Finally, for the case of linear modules over Koszul algebras, we will see that it is possible to modify the general construction described here, so that the resulting resolution is linear.

Neste trabalho estudamos algumas características das álgebras de Koszul, como por exemplo, a maneira como elas se relacionam com suas respectivas álgebras de Yoneda. Descrevemos a álgebra de Yoneda de uma álgebra monomial e como aplicação construímos uma família de álgebras: as chamadas homologicamente auto-duais. Uma álgebra de Koszul pode ser definida a partir da existência de resoluções lineares dos módulos simples. Por isso faz-se necessário a dedicação de parte de nossa atenção ao estudo destas resoluções. Além disso, achamos interessante estudar métodos para a construção de resoluções projetivas de módulos sobre quocientes de álgebras de caminhos. Para tal construção usamos essencialmente a teoria de bases de Gröbner não comutativas. Finalmente, para o caso de módulos lineares sobre álgebras de Koszul, veremos que é possível modicar essa construção de modo que a resolução resultante seja linear. / In this work we study some features of Koszul algebras as, for example, the way that they are related with their Yoneda algebras. We describe the Yoneda algebra of a monomial algebra and as an application we construct a family of algebras: the so called homologically self-dual algebras. A Koszul algebra can be dened as an algebra for which there are linear resolutions of their simple modules. Because of this we dedicate part of our attention to the study of projective resolutions. The study of methods for the construction of projectives resolutions of modules over quotients of path algebras, has an of interest its own. For the study of projective resolutions we used the theory of noncommutative, Gröbner bases. Finally, for the case of linear modules over Koszul algebras, we will see that it is possible to modify the general construction described here, so that the resulting resolution is linear.