Multiplicity of the special fiber of blowups

Abstract

Abstract. Let (R, m) be a Noetherian local ring and let I be an m-primary ideal. In this paper we give sharp bounds on the multiplicity of the special fiber ring F of I in terms of other well-known invariants of I. A special attention is then paid in studying when equality holds in these bounds, with a particular interest in the unmixedness or, better, the Cohen-Macaulayness of F.

Sally Modules of 𝔪-Primary Ideals in Local Rings

“…To be more specific, in Theorem 3.4 we show that f 0 (I) ≤ e 1 (I) − e 0 (I) − e 1 (J) + λ(R/I) + µ(I) − d + 1, thus generalizing a previous result obtained jointly with C. Polini and W.V. Vasconcelos [1]. If in addition R is a local Buchsbaum ring, in Theorem 3.5 we show that f 0 (I) ≤ e 1 (I) + I(R) − e 1 (J) + 1, where I(R) is the Buchsbaum invariant of R introduced by J. Stückrad and W. Vogel [14].…”

“…Vasconcelos [17]. We also show that if equality holds in the latter estimate then the ideal I has minimal multiplicity in the sense of S. Goto (see [1] for a similar statement). Still in a Buchsbaum setting, we show in Proposition 3.3 that the Sally module of an ideal I containing H 0 m (R) has either dimension d or 0.…”

“…More recently, in a joint work with C. Polini and W.V. Vasconcelos [1], we have used the Sally module as the means to obtain information on the multiplicity of the special fiber ring F(I), on the unmixedness (or even Cohen-Macaulayness) of F(I), and, ultimately, on the reduction number r of I. Our investigation has been prompted by the lack of knowledge of the properties of Sally modules in non Cohen-Macaulay settings.…”

On the structure of the fiber cone of ideals with analytic spread one

“…. ) has been done by Corso, Ghezzi, Polini and Ulrich [4], Corso, Polini and Vasconcelos [3], T. Cortadellas [5], D'Cruz and Puthenpurakal [11], Heinzer and Kim [18], Heinzer, Kim and Ulrich [19], Jayanthan and Verma [26,27], Jayanthan, Puthenpurakal and Verma [28], or D.Q. Viêt [38] and others.…”

Multiplicity and Cohen-Macaulayness of Fiber Cones of Good Filtrations

“…These problems have concerned many authors over the years. Using different approaches, the authors investigated the Cohen-Macaulayness and other properties of fiber cones F m (I ) and F J (F ) (see, for instance, [1][2][3][4][5][6][7]9,11,12,20]). Using weak-(FC)-sequences of ideals in local rings, the author of [20] characterized the multiplicity and the CohenMacaulayness of F m (I ) in terms of minimal reductions of ideals.…”