Study of Blowing-up rings

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The purpose of this research is to study the following problems: (1) Find practical criteria for Rees algebras and associated graded rings of ideals to be Gorenstein rings. (2) Develop the theory of good ideals in Gorenstein local rings. As for the second problem we have developed a satisfactory new theory. Let (A, m) be a Gorenstein local ring of dimension d and let I be an m-primary ideal in A. Then we say that I is a good ideal in A, if the associated graded ring G(I) of I is a Gorenstein ring and a(G(I)) = 1-d. Although good ideals are the best ones next to the parameter ideals in A, we lacked deep investigations of the ideals of this kind. The project has performed a basic theory of them. While the project proceeded, unexpected results about integrally closed ideals were discovered, which led us to the other research on the reduction numbers of certain In-primary ideals in Buchsbaurn, or more generally, FLC local rings. The results generalize those on Cohen-Macaulay local rings that were given by Corso-Polini-Huneke-Vasconcelos. We especially summarized these investigations into the following three papers:[1] S.Goto and H.Sakurai, The equality $1^2 = Q1$ in Buchsbaum rings, Rendiconti del Seminario Matematico dell'Universit di Padova 110(2003),25-56[2] S.Goto and H.Sakurai, The reduction exponent of socle ideals, associated to parameter ideals in a Buchsbaum local ring of multiplicity two, J. Math. Soc. Japan (to appear)[3] S.Goto and H.Sakurai, When does the equality $1^2=Q1$ hold true in Buchsbaum rings?, Pieprint 2003