Koszul and generalized Koszul properties for noncommutative graded algebras
| dc.contributor.author | Phan, Christopher Lee, 1980- | |
| dc.date.accessioned | 2010-05-15T00:13:21Z | |
| dc.date.available | 2010-05-15T00:13:21Z | |
| dc.date.issued | 2009-06 | |
| dc.description | 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. | en_US |
| dc.description.abstract | 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. | en_US |
| dc.description.sponsorship | Committee in charge: Brad Shelton, Chairperson, Mathematics; Victor Ostrik, Member, Mathematics; Christopher Phillips, Member, Mathematics; Sergey Yuzvinsky, Member, Mathematics; Van Kolpin, Outside Member, Economics | en_US |
| dc.identifier.uri | https://hdl.handle.net/1794/10367 | |
| dc.language.iso | en_US | en_US |
| dc.publisher | University of Oregon | en_US |
| dc.relation.ispartofseries | University of Oregon theses, Dept. of Mathematics, Ph. D., 2009; | |
| dc.subject | Koszul properties | en_US |
| dc.subject | Noncommutative graded algebras | en_US |
| dc.subject | Yoneda algebra | en_US |
| dc.subject | Grobner bases | en_US |
| dc.subject | Homological algebra | en_US |
| dc.subject | Mathematics | en_US |
| dc.subject | Algebra, Homological | |
| dc.subject | Algebra, Yoneda | |
| dc.subject | Koszul algebras | |
| dc.title | Koszul and generalized Koszul properties for noncommutative graded algebras | en_US |
| dc.type | Thesis | en_US |
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