Generating modules efficiently: Theorems from algebraic K-theory

Several of the fundamental theorems about algebraic K, and Kr are concerned with finding unimodular elements, that is, elements of a projective module which generate a free summand. In this paper we use the notion of a basic element (in the terminology of Swan [22]) to extend these theorems to the context of finitely generated modules. Our techniques allow a simplification and strengthening of existing results even in the projective case. Our main theorem, Theorem A, is an extension to the nonprojective case of a strong version of Serre’s famous theorem on free summands of projective modules. It has as its immediate consequences (in the projective case) Bass’s theorems about cancellation of modules and stable range of rings [ 1, Theorems 9.3 and 11.11 and the theorem of Forster and Swan on “the number of generators of a module.” In the non-projective case it implies a mild strengthening of Kronecker’s well-known result that every radical ideal in an n-dimensional noetherian ring is the radical of n + 1 elements. (If the ring is a polynomial ring, then n elements suffice [7]; this can be proved by methods similar to those of Theorem A.) Theorem A also contains the essential point of Bourbaki’s theorem [4, Theorem 4.61 that any torsion-free module over an integrally closed ring is an extension of an ideal by a free module. We also prove a theorem which gives an improvement of the Forster-Swan theorem already mentioned. The Forster-Swan theorem gives a bound (in terms of some local information) on the number of elements required to generate certain modules over a nice ring A. Our Theorem B says that, if g is the number of generators for a module M which the Forster-Swan theorem