Sally modules of $\mathfrak{m}$-primary ideals in local rings

Abstract

Given a local Noetherian ring $(R,\mathfrak{m})$ of dimension $d>0$ and infinite residue field, we study the invariants $($dimension and multiplicity$)$ of the Sally module $S_J(I)$ of any $\mathfrak{m}$-primary ideal $I$ with respect to a minimal reduction $J$. As a by-product we obtain an estimate for the Hilbert coefficients of $\mathfrak{m}$ that generalizes a bound established by J. Elias and G. Valla in a local Cohen-Macaulay setting. We also find sharp estimates for the multiplicity of the special fiber ring $\mathcal{F}(I)$, which recover previous bounds established by C. Polini, W.V. Vasconcelos and the author in the local Cohen-Macaulay case. Great attention is also paid to Sally modules in local Buchsbaum rings.

10 pages