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I-adic topology

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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.

Definition

Let R be a commutative ring and M an R-module. Then each ideal ๐”ž of R determines a topology on M called the ๐”ž-adic topology, characterized by the pseudometric The family is a basis for this topology.[1]

Properties

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 ๐”ž-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 for any proper ideal ๐”ž of R. Thus under these conditions, for any proper ideal ๐”ž of R and any R-module M, the ๐”ž-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”ž-adic topology. The analogous result is not necessarily true for the submodule N itself: the subspace topology need not be the ๐”ž-adic topology. However, the two topologies coincide when R is Noetherian and M finitely generated. This follows from the Artin-Rees lemma.[2]

Completion

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.[3]

For example, let be a polynomial ring over a field k and ๐”ž = (x1, ..., xn) the (unique) homogeneous maximal ideal. Then , the formal power series ring over k in n variables.[4]

Closed submodules

As a consequence of the above, the ๐”ž-adic closure of a submodule is [5] This closure coincides with N whenever R is ๐”ž-adically complete and M is finitely generated.[6]

R is called Zariski with respect to ๐”ž if every ideal in R is ๐”ž-adically closed. There is a characterization:

R is Zariski with respect to ๐”ž if and only if ๐”ž is contained in the Jacobson radical of R.

In particular a Noetherian local ring is Zariski with respect to the maximal ideal.[7]

References

  1. ^ a b Singh 2011, p. 147.
  2. ^ Singh 2011, p. 148.
  3. ^ Singh 2011, pp. 148โ€“151.
  4. ^ Singh 2011, problem 8.16.
  5. ^ Singh 2011, problem 8.4.
  6. ^ Singh 2011, problem 8.8
  7. ^ Atiyah & MacDonald 1969, p. 114, exercise 6.

Sources

  • Singh, Balwant (2011). Basic Commutative Algebra. Singapore/Hackensack, NJ: World Scientific. ISBN 978-981-4313-61-2.
  • Atiyah, M. F.; MacDonald, I. G. (1969). Introduction to Commutative Algebra. Reading, MA: Addison-Wesley.