Seminar in Algebraic Geometry*, Summer 2010

webpage last modified: 6/22/10

May 25th, 1:30-2:30 (MDD 312): "Algebraic sets, the Zariski topology, and (quasi-)affine varieties," Chris Ryan (graduate student at UL). Synopsis: the definition of affine n-space and algebraic set; how algebraic sets form the closed sets of the Zariski topology on An; the definition of affine variety and quasi-affine variety; a few examples and extra details along the way.
May 28th, 11:00-11:50 (MDD 312): "Finitely generated and cogenerated modules," Chris. Synopsis: Let R be an associative ring with identity and let M be a left R-module. Then M is finitely generated if and only if for every set A of submodules of M that spans M, there exists a finite subset F of A whose elements also span M. The concept of being finitely generated has an important dual. An R-module M is finitely cogenerated if and only if for every set A of submodules of M with zero intersection, there exists a finite subset of A with zero intersection. An R-module is left noetherian if and only if every submodule is finitely generated. On the other hand, a module is left artinian if and only if every factor module is finitely cogenerated. For semisimple modules, these four concepts are all equivalent.
June 17th, 1:30-2:25 (MDD 312): "Baer rings and bounded operators on a Hilbert space," Chris. Synopsis: A Baer ring is a ring R with unity such that the right annihilator of any subset of R is generated by an idempotent. The condition that annihilators be generated by idempotents is left-right symmetric under the assumption that the ring has a multiplicative identity. However, for rings without unity, this condition is not necessarily symmetric: for an example of this, we considered the ring of 2 x 2 matrices over a field with zeros in the second row. Any ring of linear operators on a vector space is Baer, as is any ring of bounded operators on a Hilbert space.
* This seminar is partly supported by a grant from the Louisiana Board of Regents Support Fund.
organized by Daniel (Davis)