Graded Bourbaki ideals of graded modules (Q2664639)

If \(R\) is a normal domain and \(M\) is a finitely generated torsion-free module, then a Bourbaki sequence exists. That is, there is a free submodule \(F\) of \(M\) such that \(\frac{M}{F}\) is isomorphic to an ideal of \(R\). This is a well-known Theorem of [\textit{N. Bourbaki}, Elements of mathematics. Commutative algebra. EParis: Hermann; Reading, Mass.: Addison-Wesley Publishing Company. (1972; Zbl 0279.13001), Chapter VII, Sect. 4,9, Theorem 6]. The paper under review is devoted to studying graded Bourbaki ideals in the above existence guaranteed conditions. In section 2, the graded version of Bourbaki's theorem is established. It is shown that over factorial ring there exist Bourbaki ideals with grade at least \(2\). The authors introduce ``Bourbaki number'' in section \(3\). Section \(4\) is the main section of the paper. The authors provide criteria to determine when a monomorphism \(\varphi : F\rightarrow M\) fits in a Bourbaki sequence and how one can compute its Bourbaki ideal, explicitly [Theorems 4.2, 4.5, 4.7]. In these criteria, two cases are mainly considered \begin{itemize} \item The case where \(M\) has finite projective dimension, and \item the case \(M\) is a reflexive module. \end{itemize} In both cases, the Bourbaki sequence is determined by codimension of certain minor ideals. As a consequence, it is shown that Bourbaki sequences of a reflexive graded module M, over a Cohen Macaulay standard graded normal domain, are indexed by a Zariski open set (Theorem 4.4). Section 5 and 6 are applications of the main theorems to Koszul cycles of the residue field over the polynomial ring \(K[x_1, \dots, x_n]\). In particular, Bourbaki ideals of \(Z_{n-1}\) and \(Z_{n-2}\) are square-free monomial ideals.