A note on D. Quillen’s paper: “Projective modules over polynomial rings” (Invent. Math. 36 (1976), 167–171)

Roitman, Moshe

doi:10.1090/S0002-9939-1977-0444638-1

A note on D. Quillen’s paper: “Projective modules over polynomial rings” (Invent. Math. 36 (1976), 167–171)
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by Moshe Roitman

Proc. Amer. Math. Soc. 64 (1977), 231-232

DOI: https://doi.org/10.1090/S0002-9939-1977-0444638-1

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Abstract:

We give a simplified proof to the following theorem due to D. Quillen: if A is a commutative noetherian ring of global dimension $\leqslant 1$, then finitely generated projective modules over $A[{T_1}, \ldots ,{T_n}]$ are extended from A. We prove also that if A is a commutative noetherian ring of global dimension d, then finitely generated projective modules of rank $> d$ over $A[{T_1}, \ldots ,{T_n}]$ are extended from A.

References

Daniel Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976), 167–171. MR 427303, DOI 10.1007/BF01390008
H. Bass, $K$-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 5–60. MR 174604, DOI 10.1007/BF02684689
Hyman Bass, Libération des modules projectifs sur certains anneaux de polynômes, Séminaire Bourbaki, 26ème année (1973/1974), Lecture Notes in Math., Vol. 431, Springer, Berlin, 1975, pp. Exp. No. 448, pp. 228–354 (French). MR 0472826