The structure of Sally modules of rank one
``A complete structure theorem of Sally modules of \(\mathfrak m\)-primary ideals \(I\) in a Cohen-Macaulay local ring \((A,\mathfrak m)\) satisfying the equality \(e_1(I)=e_0(I)-\ell_A(A/I)+1\) is given, where \(e_0(I)\) and \(e_1(I)\) denote the first two Hilbert coefficients of \(I\).'' Let \((A,\mathfrak m)\) be a Cohen-Macaulay local ring of dimension \(>0\) and with infinite residue field \(k=A/ \mathfrak m\). Let \(I\) be an \(\mathfrak m\)-primary ideal of \(A\) and let \(Q\) be a minimal reduction of \(I\). The \textit{Sally module} of \(I\) with respect to \(Q\), as defined by W. V. Vasconcelos, is \(S=S_Q(I)=IA[It]/IA[Qt]\), where \(t\) is an indeterminate. The authors prove the equivalence of the following three conditions: (1) \(e_1=e_0-\ell(A/I)+1\); (2) \(\mathfrak m S=(0)\) and \(\text{rank}_B S=1\); (3) \(S\cong (X_1,X_2,\dots,X_c)B\) as graded \(T\)-modules for some integer \(0<c\leq d\), where \(X_1,\dots, X_c\) are linearly independent linear forms of the polynomial ring \(B\). Here \(T=A[Qt]\), and \(B=T/ \mathfrak m T\) (which is a polynomial ring over the residue field \(k\)). Further results are included, and connections with previous works on this topic are described in the paper.
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