Local-global bounds for number of generators of projective modules over polynomial rings (Q792398)

The author proves the following theorems: (1) Let R be a commutative ring of Krull dimension d and let \(\bar R=R[X_ 1,...,X_ r]\) be the polynomial ring over R in r variables. If M is a finitely generated projective \(\bar R\)-module which is locally generated by n elements, then M can be generated by \(n(d+1)\) elements. - (2) If \(r=1\) in above, then the same conclusion holds for any finitely generated \(\bar R\)-module. For noetherian rings, these results (and better) are known. Similar results were proved by \textit{R. Weigand} and \textit{W. Vasconcelos} [Math. Z. 164, 1-7 (1978; Zbl 0362.13004)] in the non-polynomial case. The main technical tool in proving these results is a refinement of methods of \textit{R. C. Heitman} [Pac. J. Math. 62, 117-126 (1976; Zbl 0308.13014)]. ''Patch topologies'' as conceived by M. Hochster also plays a role.