SUMMARY ON NEW FINDINGS IN PhD. THESIS

 1.   Opening summary

-Thesis title: On the set of attached primes of local cohomology modules

-  Major: Algebra and Number Theory

- Major code: 62.46.01.04

-  PhD. Student:  Tran Do Minh Chau

-  Supervisor:    1. Assoc. Prof. Dr. Le Thi Thanh Nhan

                               2. Prof. Dr. Le Van Thuyet

-  Training institution:  College of Education, Hue University

2. The new contributions of the thesis

The thesis provided some new results on the set of the attached primes of the local cohomology modules with respect to the maximal ideal and the top local local cohomology module with respect to an arbitrary ideal.

For the local cohomology modules with respect to the maximal ideal, the thesis showed the formula passing the attached primes through  m-adic completion when the base ring is a quotient of a Gorenstein local ring. A new characterization of universally catenary local ring having all Cohen-Macaulay formal fibers was given via this formula. The thesis also gave a necessary condition for the existence of a “co-localization” functor which is compatible with every local cohomology modules with respect to the maximal ideal.

In the case of the top local local cohomology module with respect to an arbitrary ideal, the thesis described the set of the the attached primes by characterizing the prime saturation in terms of the catenary of the base ring and reducing to the case of local cohomology modules with respect to the maximal ideal. The thesis also characterized the prime saturation via the co-support and then established the associativity formula for multiplicity of this module.

Supervisor 1

 Assoc. Prof. Dr. Le Thi Thanh Nhan

 Supervisor 2

 Prof. Dr. Le Van Thuyet

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