Abstract
We are concerned with a question of H. Bass and D. Quillen which asks whether the following is true:
If A is a regular noetherian ring, then every finitely generated projective module P over a polynomial extension A[T] of A is extended from A. (cf. [1], § 4, (IX) and [7]).
We give an affirmative answer, either if (i) A equals a ring of fractions of a polynomial ring over a regular noetherian ring B with dim B≦2, or if (ii) A equals a ring of fractions of a polynomial extension of a power series ring over a complete regular local ring B with dim B≦2.
(ii) implies the case that A is an unramified complete regular local ring. This generalizes the result in [4], which has been proved independently in [5]. (i) spezializes to the known theorem of Quillen and Suslin if A=B (cf. [2]).
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Literatur
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Lindel, H. Projektive Moduln über Polynomringen A[T1,...,Tm] mit einem regulären Grundring A. Manuscripta Math 23, 143–154 (1978). https://doi.org/10.1007/BF01180569
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DOI: https://doi.org/10.1007/BF01180569