A remark on Koszul complexes (Q1267319)

The authors study the Koszul complex \(K( \overline \varphi)\) induced by a linear map \(\overline \varphi: M \to R\), where \(M\) is an \(R\)-module of the form \(M=F/Rx\), with \(F\) a finite free \(R\)-module. Their aim is to extend to this case the description of the homology of \(K(\overline \varphi)\), well known when \(M\) is a free module. If \(\Phi:F\to R\) and \(\Psi: R\to F\), \(\Psi(1) =x\) are the linear maps inducing \(\overline\Phi: M\to R\), then the homology of \(K(\overline \varphi)\) turns out to be strictly linked to the degrees \(q_\Phi\) and \(q_{\Psi^*}\) of \(\Phi\) and \(\Psi^*\) (the superscript \(^*\) denotes the \(R\)-dual); in particular, it is completely determined when \(g_{\Psi^*}\) has its maximal value. As an application of their results, the authors answer some questions asked by \textit{G. Boffi} [Adv. Math. 123, No. 1, 91-103 (1996; Zbl 0861.18006)] concerning a Koszul complex linked to a bilinear form.