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Concept
I-adic topology
Summary
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.
Official source
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