On a question of Quillen

Authors:
S. M. Bhatwadekar and R. A. Rao

Journal:
Trans. Amer. Math. Soc. **279** (1983), 801-810

MSC:
Primary 13C10; Secondary 13H05, 18G05

DOI:
https://doi.org/10.1090/S0002-9947-1983-0709584-1

MathSciNet review:
709584

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Abstract: Let be a regular local ring, and a regular parameter of . Quillen asked whether every projective -module is free. We settle this question when is a regular local ring of an affine algebra over a field . Further, if has *infinite* residue field, we show that projective modules over Laurent polynomial extensions of are also free.

**[**S. M. Bhatwadekar and A. Roy,**BR**]*Inversion of monic polynomials and existence of unimodular elements*(preprint). MR**701360 (85i:13016)****[**H. Lindel,**L**]*On a question of Bass, Quillen and Suslin concerning projective modules over polynomial rings*, Invent. Math.**65**(1981), 319-323. MR**641133 (83g:13009)****[**N. Mohan Kumar,**Mo**]*On a question of Bass-Quillen*, J. Indian Math. Soc.**43**(1979), 13-18. MR**681997 (84h:13011)****[**B. S. Nashier,**N**]*Efficient generation of ideals in polynomial rings*(preprint). MR**725083 (85f:13010)****[**D. Quillen,**Q**]*Projective modules over polynomial rings*, Invent. Math.**36**(1976), 167-171. MR**0427303 (55:337)****[**R. A. Rao,**R**]*Stability theorems for overrings of polynomial rings*, II, J. Algebra**78**(1982), 337-444. MR**680370 (84m:13010)****[**A. Roy,**Ry**]*Application of patching diagrams to some questions about projective modules*, J. Pure Appl. Algebra**24**(1982), 313-319. MR**656854 (83i:13009)****[**A. A. Suslin,**Su**]*Projective modules over a polynomial ring are free*, Soviet Math. Dokl.**17**(1976), 1160-1164 (English translation).**[**R. G. Swan,**Sw**]*Projective modules over Laurent polynomial rings*, Trans. Amer. Math. Soc.**237**(1978), 111-120. MR**0469906 (57:9686)**

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DOI:
https://doi.org/10.1090/S0002-9947-1983-0709584-1

Article copyright:
© Copyright 1983
American Mathematical Society