A note on D. Quillen's paper: ``Projective modules over polynomial rings'' (Invent. Math. 36 (1976), 167-171)

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.