Projective modules over real affine algebras
For any field k, all projective modules over the polynomial ring \(R=k[X_ 1,...,X_ d]\) are free, by the celebrated theorem of Quillen and Suslin. On the other hand, if \(d\geq 3\) and \(f\in R\), then all projective modules over the ring \(k[X_ 1,...,X_ d,1/f]\) need not be free [see example 1 in \textit{R. G. Swan's} paper in Trans. Am. Math. Soc. 105, 264-277 (1962; Zbl 0109.416)]. It is proved that if \(k={\mathbb{R}}\) and f is strictly positive then all projective modules over \({\mathbb{R}}[X_ 1,X_ 2,X_ 3,1/f]\) are free. This fact is a consequence of the following theorem: Let R be a real closed field and A an affine domain over R of dimension d, smooth in codimension one. Let \(f\in A\) be a function without any R- rational zeros and P a projective module of rank \(\geq d-1\) over \(A_ f\). If \(P\oplus A_ f\) is free then P is extended from A.
freeness of projective modules over polynomial rings
scientific article; zbMATH DE number 4139899
scientific article