The reduction exponent of socle ideals associated to parameter ideals in a Buchsbaum local ring of multiplicity two

Quasi-socle ideals and Goto numbers of parameters

“…Consequently the associated graded ring G(I) = n≥0 I n /I n+1 and the fiber cone F(I) = n≥0 I n /mI n are Cohen-Macaulay and so is the ring R(I) = n≥0 I n , if dim A ≥ 2. The first author and Sakurai [9][10][11] explored also the case where the base ring is not necessarily Cohen-Macaulay but Buchsbaum, and showed that the equality I 2 = QI (here I = Q : m) holds true for numerous parameter ideals Q in a given Buchsbaum local ring (A, m), whence G(I) is a Buchsbaum ring, provided that dim A ≥ 2 or that dim A = 1 but the multiplicity e(A) of A is not less than 2. Thus socle ideals Q : m still enjoy very good properties even in the case where the base local rings are not Cohen-Macaulay.…”

The index of reducibility of parameter ideals in low dimension

“…We thank the referee for pointing out the fascinating technique in this section, and for pointing out the recent work of Goto-Sakurai [8,9] In the course of our proof for dimension two we will need to mention several generalizations of the notion of a regular sequence. These definitions can be found in the appendix of [18], which is a good source for information concerning modules having finite local cohomologies.…”

“…Let M be a finitely-generated d-dimensional A-module with finite local cohomologies. Then we have the following inequalities: In two recent papers [8,9], Goto with H. Sakurai has returned to the study of the index of reducibility of parameter ideals in order to investigate when the equality I 2 = QI holds for a parameter ideal Q in A, where I = (Q : A m). According to earlier research of A. Corso, C. Huneke, C. Polini, and W. Vasconcelos [2][3][4], this equality holds for all parameter ideals Q in case A is a Cohen-Macaulay ring which is not regular.…”

The structure of Sally modules – towards a theory of non-Cohen–Macaulay cases

“…, a d ) be the graded maximal ideal in C , where x i , v, and a j denote the images X i , V , and Z j in C , respectively. Then C is a ddimensional graded Buchsbaum ring with depth C = d − 1 and h d−1 (C) = 1 [GSa,Theorem 4.5]. We put q = (a 1 , a 2 , .…”

“…, a d is a homogeneous system of parameters for the graded ring C . Let J = q : M. We then have J 3 = q J 2 and C ( J 2 /q J ) = 1 [GSa,Proposition 4.7]. Let A = C M , I = J A, and Q = qA.…”