Let \((A,\mathfrak m,k)\) be a \(d\)-dimensional Cohen-Macaulay local ring with maximal ideal \(\mathfrak m\) and infinite residue field \(k\). Consider an \(\mathfrak m\)-primary ideal \(I\) of \(A\). The Hilbert polynomial giving the length of \(A/I^{n+1}\) for large \(n\) can be written in the form \[ e_0\binom {n+d}d-e_1\binom{n+d-1}{d-1}+\cdots+(-1)^d e_d. \] Northcott proved that \(e_1\geq e_0-\lambda\geq 0\), where \(\lambda\) denotes the length of \(A/I\). Huneke and Ooishi independently investigated the extremal situation where \(e_1=e_0-\lambda\). It was shown that this equality holds if and only if the reduction number of \(I\) is equal to \(1\) (that is, \(I^2=QI\) for some minimal reduction \(Q\) of \(I\)). When these conditions hold, \(e_i=0\) for \(i\geq 2\), the associated graded ring of \(I\) is Cohen-Macaulay, and \(I^2=QI\) for every minimal reduction \(Q\) of \(I\). Sally considered the ideals \(I\) which satisfy \(e_1=e_0-\lambda+1\) and \(e_2\neq 0\) if \(d\geq 2\). She proved that that in this case, \(I^2/QI\) has length one and \(I^3=QI^2\) for every minimal reduction \(Q\) of \(I\) (in particular, the reduction number of \(I\) is two), and the associated graded ring of \(I\) has depth at least \(d-1\).
Vasconcelos introduced the Sally module \(S=S_Q(I)\); and thereby reformulated and extended Sally’s work. If \(Q\) is a minimal reduction of \(I\), then the Sally module of \(I\) with respect to \(Q\) is \(I\mathcal R(I)/I\mathcal R(Q)\), where \(\mathcal R(I)=A[It]\) and \(\mathcal R(Q)=A[Qt]\) are the Rees algebras of \(I\) and \(Q\), respectively.
The present paper characterizes ideals which satisfy \(e_1=e_0-\lambda+1\) without imposing the condition that \(e_2\neq 0\). The authors resolve \(S_Q(I)\) as a module over \(\mathcal R(Q)\), and, when \(d=2\), they give a complete structure Theorem for \(S_Q(I)\) as a \(\mathcal R(Q)\)-module.