Hilbert functions and Sally modules (Q1360878)

Let \((R,m)\) be a local ring of positive Krull dimension, \(I\) an \(m\)-primary ideal, and \(J\) a reduction of \(I\). The Sally module of \(I\) with respect to \(J\) is \(S_JI= \bigoplus_{n\geq 1}I^{n+1}/JI^n\). In this paper the Sally module is studied with the aim of getting information on the Hilbert function \(H_I(n)\) and the associated graded ring \(\text{gr}_I(R)\) of \(R\). In case \(S_J(I)\) can be well understood, this information can be transferred to \(\text{gr}_I(R)\). Here is an example of a result from the paper: If \(R\) is CM and \(J\) a minimal reduction of \(I\) with reduction number \(r\), then the first Hilbert coefficient of \(S_J(I)\) is \(s_0\leq \sum_{n=1}^{r-1} \lambda(I^{n+1}/JI^n)\). If there is equality, then \(S_J(I)\) is CM, all Hilbert coefficients \(s_j\) of \(S_j(I)\) can be determined, \(\text{depth(gr}_I (R))\geq \dim(R)- 1\), and all Hilbert coefficients of \(\text{gr}_I(R)\) are determined.