Vetter, Udo Linear forms on modules of projective dimension one. (English) Zbl 1058.13007 Zesz. Nauk. Uniw. Jagiell. 1255, Univ. Iagell. Acta Math. 39, 311-315 (2001). From the paper: Let \(R\) be a noetherian ring and \(M\) an \(R\)-module which has a presentation \(0\to F@>\psi>> G\to M\to 0\) with finite free \(R\)-modules \(F\) and \(G\) of rank \(m\) and \(n\). W. Bruns and M. Vetter [in: Geometric and combinatorial aspects of commutative algebra, Pap. Int. Conf. Commutative Algebra Algebraic Geometry, Messina 1999, Lect. Notes Pure Appl. Math. 217, 89–97 (2001; Zbl 0987.13006)] proved:Assume that \(r=n-m>1\) and that the first nonvanishing Fitting ideal of \(M\) has grade \(r+1\). Then the following conditions are equivalent: (1) There is a \(\chi\in M^*=\text{Hom}_R(M,R)\) such that the ideal \(\text{Im}\,\chi\) has grade \(r+1\). (2) There exists a submodule \(U\) of \(M\) with the following properties: (i) \(\text{rank}\, U=r-1\); (ii) \(U\) is reflexive, orientable, and \(U_{\mathfrak p}\) is a free direct summand of \(M_{\mathfrak p}\) for all primes \({\mathfrak p}\) of \(R\) such that \(\text{grade}\,{\mathfrak p}\leq r\).(3) \(m=1\) and \(r\) is odd. The aim of this note is to give an explicit construction of \(U\) as a submodule of the free module \(R^{{n\choose 2}-1}\). MSC: 13D02 Syzygies, resolutions, complexes and commutative rings 13D05 Homological dimension and commutative rings 13E15 Commutative rings and modules of finite generation or presentation; number of generators Keywords:free resolution; projective dimension; Fitting ideal Citations:Zbl 0987.13006 PDFBibTeX XMLCite \textit{U. Vetter}, Zesz. Nauk. Uniw. Jagiell., Univ. Iagell. Acta Math. 1255(39), 311--315 (2001; Zbl 1058.13007) Full Text: EuDML