On Sally modules (Q2069817)

Let \((R, \mathfrak{m})\) be a local Noetherian ring of dimension \(d >0\) with infinite residue field. Let \(I\) be an \(\mathfrak{m}\)-primary ideal and \(J\) a reduction of \(I\). For each pair of integers \((r,k)\) with \(r \geq 0\) and \(0 \leq k \leq d\), the author shows that there exists a largest ideal \(I_k^r \subseteq I^{r+1}\) such that for \(n \gg 0\) the length \[ f(n)=\lambda\left(\frac{ I_k^r J^{n-r}+J^nI}{J^nI} \right) \] is a polynomial function of degree at most \(d-(k+1)\). The author provides structure theorems for these ideals and uses them to study the height of the associated prime ideals of the Sally module \(S_J(I)=\bigoplus_{n \geq 1} I^{n+1}/JI^n\) . (The notation chosen in the paper is a bit unfortunate, as \(I_k^r \) is not a power ideal.) All results are inspired by the original construction of \textit{K. Shah} [Trans. Am. Math. Soc. 327, No. 1, 373--384 (1991; Zbl 0738.13007)] of the coefficient ideals of an \(\mathfrak{m}\)-primary ideal.