Unimodular elements in projective modules
Finitely generated modules over Noetherian commutative rings of finite Krull dimension are considered. An approach to establish the existence of unimodular elements in modules over positively graded rings and certain localizations of these rings are developed. In the case of a projective module \(P\) over the Laurent polynomial ring \(A = R [X_1, \dots, X_n\), \(Y_1^{\pm 1}, \dots, Y_m^{\pm 1}]\), the author obtains results corresponding to some results by \textit{S. M. Bhatwadekar}, \textit{H. Lindel} and \textit{R. A. Rao} [Invent. Math. 81, 189-203 (1985; Zbl 0604.13007)] without using the Quillen-Suslin theorem on projective modules over polynomial rings. The transitivity of the action of the group of elementary transformations on the set of unimodular elements \(\text{Um} (A \oplus P)\) in the case \(\text{rank} P \geq \max (2, \dim R + 1)\) is proved. A direct proof of the corresponding result by \textit{A. Wiemers} [``Eigenschaften projektiver Moduln über Laurent-Polynomringen und diskreten Hodge-Algebren'', Diss. (Münster 1989; Zbl 0716.13007)] concerning large projective modules over discrete Hodge algebras \(R[X_1, \dots, X_n]/I\), where \(I\) is generated by monomials or \(I = 0\), is given.
scientific article; zbMATH DE number 763195
scientific article