Stability results for projective modules over Rees algebras
Let \(R\) be a commutative Noetherian ring of Krull dimension \(d\). A classical result of \textit{J.-P. Serre} [in: Algebre Theorie Nombres, Sem. P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot 14 (1960/61), No. 2, 16 p. (1963; Zbl 0132.41301)] says that every finitely generated projective \(R\)-module \(P\) of rank \(>d\) splits off a free summand. One of the aims of this article is to provide a class of examples of commutative Noetherian rings \(R\) of dimension \(d\) such that every rank \(d\) projective \(R\)-module splits off a free summand of rank one. The authors prove the following result: Theorem 1: Let \(R\) be a commutative Noetherian domain of dimension \(d-1\) (\(d\geq 1\)), \(I\) an ideal of \(R\) and \(A=R[It]\) or \(R[It,t^{-1}]\). (i) If \(P\) is a projective \(A\)-module of rank \(d\), then \(P\overset{\sim}{\longrightarrow}Q\oplus A\) for some projective \(A\)-module \(Q\). (ii) Every finitely generated projective \(A\)-module of rank \(d\) is cancellative. For \(A=R[t]\) and for birational overrings of \(R[t]\), a similar type of result is proved by \textit{B. Plumstead} [Am. J. Math. 105, 1417--1433 (1983; Zbl 0532.13008)] and [\textit{R. A. Rao}, Patching techniques in algebra -- stability theorems for overrings of polynomial rings and extendability of quadratic modules with sufficient Witt index. Bombay University (Ph.D. Thesis) (1984)] respectively. As an application of the above theorem, the authors prove the following result: Theorem 2: Let \(R\) be a commutative Noetherian domain of dimension \(d\) and \(I\) an ideal of \(R\). Let \(M\) be a finitely generated module over \(A=R[It]\) or \(R[It,t^{-1}]\). Then \(M\) is generated by \(e(M)=\mathrm{Sup}_{\mathfrak{p}}\{\mu_{\mathfrak{p}}(M)+\dim(A/{\mathfrak{p}})\}\) elements.
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