PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 97, Number 4, August 1986 ON STABLY EXTENDED PROJECTIVE MODULES OVER POLYNOMIAL RINGS MOSHEROITMAN ABSTRACT. We prove here that if A is a commutative noetherian ring of Krull dimension d and of finite characteristic prime to d!, then stably extended projective A[XX,..., Xn]-modules of rank > d/2 + 1 are extended from A. We denote by A a commutative ring with unit. Ur{A) is the set of unimodular rows of length r over A. As in [3, §5, p. 34], given u, v in Ur{A) and a subgroup G of GLr(A), we write u ~g v if there exists g in G such that v = ug. We abbreviate the notations u ~glt(A) v to u ~ v and u ~Er(A) V to u ~b v. For u,v in Ur{A) we denote by u <->GLr(A)v or simply u <-> v the property: u ~ (1,0,... ,0) if and only if v~ (1,0,..., 0). If tp: A —► B is a canonical ring homomorphism (such as A —> As, A —» A//, where S is a multiplicative subset and / is an ideal of A) and o £ A, we denote <p(a) — a. If f(X) is a polynomial in A[X\, we denote its leading coefficient by /(/). As usual A(X) denotes the localization of A[X] at the set of monic polynomials. If S is a multiplicative subset of A and f{X) £ A[X], we say that f(X) is unitary in As[X] if f(X) is unitary in Ag[X], that is, 1(f) = us for some s £ S and u invertible in A. We recall that a finitely generated projective module P over R = A[X\, • • •, Xn] is called stably extended from A if there exists a finitely generated .R-projective module Q extended from A such that P 0 Q is extended from A or, equivalently, if there exists m > 0 such that P © Rm is extended from A. Lemma l (cf. [12, Corollary 2]). Let (x0,...,xr) £ Ur+i(A), r > 2, and let t be an element of A which is invertible mod (Axç>+ ■ ■ ■ + Axr_2). Then (xo,...,xr) ~e (x0,...,t2xr). PROOF. Let 5Z¿=o x»y» - *• Tnen °y Í8' Lemma 1] we have (Xo,...,Xr-2,Xr-l,Xr) ~E (%0, ■ ■ ■ , %r-2, Vr-l, Vr) ~E (x0, . . . ,Xr-2,tXr-X,tXr). Let tt' = 1 mod (Axo + ■ ■ • 4- Axr_2). By Whitehead's lemma we have (Xo,.. . ,Xr-2,tXr-l,tXr) ~B (xo, . .., Xr_2, t'tXr-1, Í Xr ) ~ß (xo,...,Ir_2,Xr_i,i2Ir). D LEMMA 2 (CF. E.G., [1, THÉORÈME 1]). Let S be a multiplicative subset of A, such that As is noetherian of finite Krull dimension d. Let (5n,...,3r) £ Received by the editors December 3, 1984 and, in revised form, May 6, 1985. 1980 Mathematics Subject Classification. Primary 13C10, 13F20. ©1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page 585 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use