PUBLIC VIVA-VOCE NOTIFICATION
2.00-3:15 pm, Lecture Hall 6
On the Hilbert Functions of Graded Rings and on the F-rationality of Rees Algebra
Chennai Mathematical Institute.
Problem 1: Hilbert function of a graded ideal in a standard graded ring is an important object to study in commutative algebra. One of the fundamental question in that area is given a standard graded ring can we classify Hilbert functions of its graded ideals, more precisely given a numerical function can we check whether its a Hilbert function of an ideal; if it is then how to get an ideal. In 1927, F. Macaulay characterizes the Hilbert functions of graded ideals in polynomial ring in terms of lex ideals. Then many researchers have studied whether Macaulay type theorem holds for other standard graded rings. For projective toric rings not much is known. We will show that Macaulay type theorem holds for two projective toric rings and also discuss related results on Betti numbers.
Problem 2: Notion of F-rational rings has been introduced by F. Fedder and W. Watanabe. F-rationality is closely related to rational singularity. We will discuss some problems on F-rationality of Rees algebras and extended Rees algebras. Let (R, m) be an excellent local ring of prime characteristic, I be an m-primary ideal. We will show that R is F-rational and the Rees algebra R[It] is F-rational if and only of extended Rees algebra is so and also see some of its consequences. Only if direction was proved by Hara, Watanabe, Yoshida and they conjectured about the if direction. Next we will show if the base ring is F-rational, Proj of the Rees algebra is F-rational, and all higher sheaf cohomology of Proj of the Rees algebra vanishes then for all higher Veronese subrings of Rees algebra are F -rational. This result is motivated by a result of Hyry in case of rational singularity. We will also see a criterion of F-rationality of base ring R, given Rees algebra R[It] is F-rational.