The kernels of completion maps and a relative form of Nakayama's lemma

Abstract

If ;4 is a commutative Noetherian ring, J an ideal of A, and E a finitely generated module over A, then the kernel of the canonical morphism from E into its J-adic completion is flJnE, the closure of (0) in the J-adic topology. A classical result of Krull shows that an element x of E is in this kernel if and only if there is a j E J with (1 -j) x = 0. A standard use of Nakayama’s lemma then shows that if J is contained in the Jacobson radical of A, such an x must necessarily by zero (cf. Bourbaki [6, III, Sect. 31). If one does not have J contained in the Jacobson radical, one has usually had to make do with calculations for specific classes of pairs (A, J) to be able to control this kernel. In this paper, I adapt the classical proof of Nakayama’s lemma to produce a relative version “modulo an additive topology.” Amongst other corollaries of this result, one can then easily show that certain conditions on J imply that fTJ”E is in the torsion submodule of E for the torsion theory associated to the additive topology. The classical case is contained in this result for the trivial additive topology, {A}. In the process of proving these results, I will have occasion to introduce several new relative versions of well known concepts. It would seem that at least some of these concepts will be useful in the study of topologised rings with properties analogous to those of local rings. I leave this possibility for later notes.