I-adic topology

In commutative algebra, the mathematical study of commutative rings, adic topologies are a family of topologies on the underlying set of a module, generalizing the p-adic topologies on the integers. Let R be a commutative ring and M an R-module. Then each ideal a of R determines a topology on M called the a-adic topology, characterized by the pseudometric The family is a basis for this topology. With respect to the topology, the module operations of addition and scalar multiplication are continuous, so that M becomes a topological module. However, M need not be Hausdorff; it is Hausdorff if and only ifso that d becomes a genuine metric. Related to the usual terminology in topology, where a Hausdorff space is also called separated, in that case, the a-adic topology is called separated. By Krull's intersection theorem, if R is a Noetherian ring which is an integral domain or a local ring, it holds that for any proper ideal a of R. Thus under these conditions, for any proper ideal a of R and any R-module M, the a-adic topology on M is separated. For a submodule N of M, the canonical homomorphism to M/N induces a quotient topology which coincides with the a-adic topology. The analogous result is not necessarily true for the submodule N itself: the subspace topology need not be the a-adic topology. However, the two topologies coincide when R is Noetherian and M finitely generated. This follows from the Artin-Rees lemma. Completion (algebra) When M is Hausdorff, M can be completed as a metric space; the resulting space is denoted by and has the module structure obtained by extending the module operations by continuity. It is also the same as (or canonically isomorphic to): where the right-hand side is an inverse limit of quotient modules under natural projection. For example, let be a polynomial ring over a field k and a = (x1, ..., xn) the (unique) homogeneous maximal ideal. Then , the formal power series ring over k in n variables. As a consequence of the above, the a-adic closure of a submodule is This closure coincides with N whenever R is a-adically complete and M is finitely generated.