Tight closure and primary decomposition in Commutative Algebra

Georgia State University Research Foundation, Inc., Atlanta GA

Investigators

Abstract

This project is in the field of Commutative Algebra, in which the Principal Investigator (PI) will study commutative Noetherian rings. First of all, the PI plans to study commutative Noetherian rings of prime characteristic p with a focus on the tight closure and Frobenius closure of ideals (or, more generally, submodules). In particular, the PI plans to study the existence of a uniform test exponent (for tight closure) for all modules of finite phantom projective dimension as well as the existence of a uniform test exponent (for Frobenius closure) for all ideals generated by systems of parameter. In order to better understand the structure of modules of finite phantom projective dimension, the PI, in joint work with Mel Hochster, plans to show that any such module can be `weakly embedded' into another module of finite phantom projective dimension whose structure is much easier to understand. Using this technique, the PI hopes to prove the existence of a uniform test exponent (for tight closure) for all modules of finite phantom projective dimension. The PI also plans to study certain numerical invariants, namely the F-rational signature, the F-signature and the Hilbert-Kunz multiplicity of a local ring of prime characteristic. Each of the invariants carries important information about the ring, depending on whether it is positive, zero, large or small. In particular, the notion of F-rational signature is a new invariant defined in this project, whose positivity characterizes the F-rationality of the ring. Other projects that the PI plans to carry out include the Hilbert-Kunz functions, the Embedding Theorem for modules of finite projective dimension (or, more generally, finite G-dimension), and the theory of primary decomposition concerning the linear growth property for families of modules. Commutative Algebra is a branch of mathematics, which arose from a study of solutions of a set of polynomial equations. The study of the functions over the solution sets led to the investigation of the structure of the coordinate rings. In this sense, Commutative Algebra is closely related to Algebraic Geometry. By reduction to prime characteristic p, many questions in Commutative Algebra and in Algebraic Geometry have been answers by using the tight closure theory. For example, the tight closure theory has found its applications in the studies of singularities. Scientific research in Commutative Algebra will not only deepen our understanding in the area itself, but will also benefit the related branches of Mathematics as Commutative Algebra is in constant interaction with Algebraic Geometry, Homological Algebra, Combinatorics, coding theory and Number Theory, etc. Given the importance of Mathematics in the development of modern sciences, the research shall be of importance to the overall advance of science and technology.