Complete Intersection in Overrings of a Certain One-Dimensional Gorenstein Graded Local Ring☆

Let A be a Gorenstein local ring with the maximal ideal and let I be an -primary ideal in A. Then, following [GIW], we say that I is a good ideal in A if I contains a parameter ideal Q in A as a reduction and the associated graded ring G I = ⊕n≥0 In/In+1 of I is a Gorenstein ring with a G I = 1 − dimA, where a G I = 1 − dim A, where a G I denotes the a-invariant of G I [GW, Definition (3.1.4)]. In [GIW] Goto et al. intensively studied good ideals in a given Gorenstein local ring and among many other results they gave in the case where dimA = 1 a striking correspondence theorem between the set A of good ideals in A and the set A of certain overrings of A. To state their result explicitly let us assume that dimA = 1 and let B denote the total quotient ring of A. Let A be the set of Gorenstein A-algebras C of B which are module-finite extensions of A but C = A. Then we have the following.