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]).
Similar content being viewed by others
Literatur
BASS, H.: Some problems in “classical” algebraic K-theory, Algebraic K-Theory II, Lect. Notes in Math. 342, Springer-Verlag, Berlin-Heidelberg-New York
FERRAND, D.: Les modules projectifs de type fini sur un anneau de polynomes sur un corps sont libres, Sem Bourbaki, 28 e année, 1975/76
LINDEL, H.: Wenn B ein Hauptidealring ist, so sind alle projektiven B[X,Y]-Moduln frei, Math. Ann. 222, 283–289 (1976)
LINDEL, H., LÜTKEBOHMERT, N.: Projektive Moduln über polynomialen Erweiterungen von Potenzreihenalgebren, Archiv der Math. 27, 51–54 (1977)
MOHAN KUMAR, N.: On a question of Bass and Quillen, preprint 1977
MURTHY, P.: Projective A[X]-modules, J. London Math. Soc. 41, 453–456 (1966)
QUILLEN, D.: Projective modules over polynomial rings, Invent. math. 36, 166–172 (1976)
SUSLIN, A.A.: Projektive Moduln über Polynomringen, Dokl. Akad. Nauk S.S.R. (1976), in russisch
ZARISKI, O., SAMUEL, P.: Commutative Algebra, Vol. II, van Nostrand Comp., Princeton (1958)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01180569