Executive Summary : | The main goal of the proposed project is to study multiplicities of non-Noetherian filtrations of ideals and properties of the graded algebras associated to the filtrations. Multiplicities of ideals in a local ring is a numerical invariant from which one can read off many important properties of the ideals and the ring. The multiplicity of an m-primary ideal I in a local ring (R,m) was first introduced by Hilbert and generalized by Samuel. Since then the multiplicities of ideals have been studied extensively and become an active area of research in Commutative Algebra. The existence of multiplicities of non-Noetherian filtrations of ideals was proved by several authors with some extra assumptions on the base ring. Recently, Cutkosky provided necessary and sufficient conditions for the existence of multiplicities of non-Noetherian filtrations of ideals. Although relatively new, the recent research and discoveries related to multiplicities of non-Noetherian filtrations of ideals have become a central component of research in Commutative Algebra and have a great impact on other areas of Mathematics, e.g., Algebraic Geometry and Graph Theory. There are several classical results related to Noetherian filtrations of ideals that are yet to be explored for non-Noetherian filtrations of ideals. Investigate those classical results for non-Noethrian filtrations of ideals is one of the primary goals of this project. Properties like Castelnuovo-Mumford regularity, Cohen-Macaulayness, Gorensteiness, of the graded algebras associated to an ideal have always been of considerable interest and an active area of research in Commutative Algebra. In contrast to that, most questions related to these properties for the multigraded algebras associated to multigraded filtrations of ideals are still widely open and a very limited amount of research work has been done so far. The proposed research will lead to a better understanding of the filtrations (not necessarily Noetherian) of ideals. I expect that the outcome of my research will shed some light on our understanding of the properties of graded algebras associated to filtrations of ideals. |