The Effect of Topological Properties of Commutative Algebraic Rings on Local Cohomology Modules

This research investigates the effect of topological properties of commutative algebraic rings on local cohomology modules. The main goal is to analyze in detail the relationship between Zariski topology, localization, and topological stability with the structure and properties of local cohomology modules. This research uses an analytical method and investigates the subject by reviewing the literature and basic concepts, theoretical analysis, mathematical modeling and formulation, and data and results analysis .Mathematical tools such as long cohomology sequences and fractional functions are employed. The results show that the Zariski topology and localization of rings have a direct impact on local cohomology modules. Also, the topological stability of rings under various algebraic transformations helps to preserve and analyze the properties of local cohomology modules. The findings of this research can contribute to the development of theoretical knowledge in the field of commutative algebra and algebraic geometry and lead to practical applications in various fields such as number theory and computer science.