Projecttve Modules over Laurent Polynomial Rings

Quillen's solution of Serre's problem is extended to Laurent polynomial rings. An example is given of a A[T, r~']-module P which is not extended even though A is regular and Pm is extended for all maximal ideals m of A. The object of this note is to present several comments and examples related to some problems suggested by Quillen's recent solution of Serre's problem [7]. It is an immediate consequence of Quillen's work that the analogue of Serre's problem for Laurent polynomials is also true. In other words, if G is a free abelian group and A: is a field or principal ideal domain, then all finitely generated projective &G-modules are free. This follows just as in [7, Theorem 4] once we observe that the following analogue of [7, Theorem 3] is true: If P is a finitely generated projective A[x, x~x]-module and if A(x)®A[x¡x-i]P and A(x~x) ®A[xiJC-ij P are both free, then P is free. The proof is the same as that of [7, Theorem 3]. We note that P extends over each half of the projective line over A and apply [7, Theorem 2]. This observation suggests extending the conjecture mentioned in [7] to the case of Laurent polynomial rings A[x, x~x]. By combining Quillen's methods with those of Suslin and Vaserstein one can show that the conjecture is true for projective modules of sufficiently high rank. In §2,1 will give an example to show that some such restriction is really needed for the case of Laurent rings. The example also gives a negative answer to the question raised at the end of [7]. Furthermore, it shows that the analogue of [7, Theorem 1] for Laurent rings does not hold. The conjecture of [7] would follow easily if one could extend Theorem 1 of [7] to the case of schemes. However, I will show in §3 that this extension is also false. 1. Laurent rings. The question is to decide when a finitely generated projective module P over A[x,x~x] is extended i.e., has the form F» A[x,x~x] ®A Q. A partial answer to this is given by the following result Received by the editors June 16,1976. AMS (A/05) subject classifications (1970). Primary 13C10, 16A50; Secondary 13B25, 55F25.