Sally Modules of 𝔪-Primary Ideals in Local Rings

Abstract

Given any local Noetherian ring (R, 𝔪), we study invariants, such as the dimension and multiplicity, of the Sally module S J (I) of any 𝔪-primary ideal I with respect to a minimal reduction J. As a by-product we obtain an estimate for the Hilbert coefficients of 𝔪 that generalizes a bound established by Elias and Valla in a local Cohen–Macaulay setting. We also find sharp estimates for the multiplicity of the special fiber ring ℱ(I) of I, which recover previous bounds established by Polini, Vasconcelos, and the author in the local Cohen–Macaulay case. A particular attention is also paid to Sally modules in local Buchsbaum rings.