The Picard Group of an Incidence Ring of a Finite Preordered Set Over a Field
Description:... Although the history of algebra dates back almost three thousand years, much of modern algebra grew out of a desire to solve many number theoretical questions posed as recently as the sixteenth and seventeenth centuries. Many of these questions, including Fermat's Last Theorem, fascinated mathematicians such as Kummer and Dedekind, prompting them to formalize modern algebra by defining and studying the classic algebraic constructs. Dedekind, for example, developed the notion of an ideal to generalize the ideal numbers Kummer investigated. Commutative ring theorists in turn developed the ideal class group to measure the distance between a Dedekind domain and a principal ideal domain. In this thesis, we are particularly interested in studying yet a further generalization of the ideal class group--the Picard group. To further connect this paper to the work of number theorists, we also consider the incidence algebra, which Rota introduced to generalize the Mobius inversion formula. We combine the study of these two classical constructs from number theory, and we completely investigate the Picard group of an incidence algebra of a finite preordered set over a field. In the process, we are able to formulate a structure theorem for the automorphism group of such an algebra and to solve a question pertaining to invariance under Morita equivalence.
Show description