Commutative Rings Graded by Abelian Groups

Authors

Brian P. Johnson, University of Nebraska-LincolnFollow

Date of this Version

Summer 8-2012

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Thomas Marley. Lincoln, Nebraska: August, 2012

Copyright (c) 2012 Brian P. Johnson

Abstract

Rings graded by Z and Z^d play a central role in algebraic geometry and commutative algebra, and the purpose of this thesis is to consider rings graded by any abelian group. A commutative ring is graded by an abelian group if the ring has a direct sum decomposition by additive subgroups of the ring indexed over the group, with the additional condition that multiplication in the ring is compatible with the group operation. In this thesis, we develop a theory of graded rings by defining analogues of familiar properties---such as chain conditions, dimension, and Cohen-Macaulayness. We then study the preservation of these properties when passing to gradings induced by quotients of the grading group.

Adviser: Thomas Marley