Mathematical Sciences: Topics in Commutative Algebra

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This project addresses three different areas. The first is concerned with various topics in connection with the construction of local noetherian rings. The principal investigator will consider formal fibers and birational extensions, questions concerning rings with low dimensional formal fibers, and a closure operation for local morphisms of local excellent rings. The second part deals with a problem on divisor class groups in rings with the approximation property while the third part is concerned with a question on local-global annihilation of local cohomology. This research is concerned with a number of questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of families of polynomial equations. One can either study the geometry of the solution set or approach problems algebraically by investigating certain functions on the solution set that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry.