I-adic topology
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Article ImagesIn 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 ๐ of R determines a topology on M called the ๐-adic topology, characterized by the pseudometric ย The family ย is a basis for this topology.[1]
An ๐-adic topology is a linear topology (a topology generated by some submodules).
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 ifย so 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 ๐-adic topology is called separated.[1]
By Krull's intersection theorem, if R is a Noetherian ring which is an integral domain or a local ring, it holds that ย for any proper ideal ๐ of R. Thus under these conditions, for any proper ideal ๐ of R and any R-module M, the ๐-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 ๐-adic topology. The analogous result is not necessarily true for the submodule N itself: the subspace topology need not be the ๐-adic topology. However, the two topologies coincide when R is Noetherian and M finitely generated. This follows from the Artin-Rees lemma.[2]
When M is Hausdorff, M can be completed as a metric space; the resulting space is denoted by ย and has the module structure obtained by extending the module operations by continuity. It is also the same as (or canonically isomorphic to): ย where the right-hand side is an inverse limit of quotient modules under natural projection.[3]
For example, let ย be a polynomial ring over a field k and ๐ = (x1, ..., xn) the (unique) homogeneous maximal ideal. Then ย , the formal power series ring over k in n variables.[4]
The ๐-adic closure of a submodule ย is ย [5] This closure coincides with N whenever R is ๐-adically complete and M is finitely generated.[6]
R is called Zariski with respect to ๐ if every ideal in R is ๐-adically closed. There is a characterization:
- R is Zariski with respect to ๐ if and only if ๐ is contained in the Jacobson radical of R.
In particular a Noetherian local ring is Zariski with respect to the maximal ideal.[7]
- ^ a b Singh 2011, p.ย 147.
- ^ Singh 2011, p.ย 148.
- ^ Singh 2011, pp.ย 148โ151.
- ^ Singh 2011, problemย 8.16.
- ^ Singh 2011, problemย 8.4.
- ^ Singh 2011, problemย 8.8
- ^ Atiyah & MacDonald 1969, p.ย 114, exercise 6.
- Singh, Balwant (2011). Basic Commutative Algebra. Singapore/Hackensack, NJ: World Scientific. ISBNย 978-981-4313-61-2.
- Atiyah, M.ย F.; MacDonald, I.ย G. (1969). Introduction to Commutative Algebra. Reading, MA: Addison-Wesley.