The alpha-invariant and Gorensteinness of graded rings associated to filtrations of ideal in regular local rings

摘要

Let A be a regular local ring and let F = {F-n}(nis an element ofZ) be a filtration of ideals in A such that R (F) = +(ngreater than or equal to0) F-n is a Noetherian ring with dim R (F) = dim A + 1. Let G (F) = +(ngreater than or equal to0) F-n/Fn+1 and let a(G (F)) be the a-invariant of G (F). Then the theorem says that F-1 is a principal ideal and F-n = F-1(n) for all n is an element of Z if and only if G (F) is a Gorenstein ring and a(G (F)) = 1. Hence a(G (F)) less than or equal to-2, if G (F) is a Gorenstein ring, but the ideal F-1 is not principal. [References: 18]