I-adic topology

Concept in commutative algebra

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 𝔞 of R determines a topology on M called the 𝔞-adic topology, characterized by the pseudometric

d ( x , y ) = 2 − sup { n ∣ x − y ∈ a n M } . {\displaystyle d(x,y)=2^{-\sup {\{n\mid x-y\in {\mathfrak {a}}^{n}M\}}}.}
The family
{ x + a n M : x ∈ M , n ∈ Z + } {\displaystyle \{x+{\mathfrak {a}}^{n}M:x\in M,n\in \mathbb {Z} ^{+}\}}
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 if

⋂ n > 0 a n M = 0 , {\displaystyle \bigcap _{n>0}{{\mathfrak {a}}^{n}M}=0{\text{,}}}
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 𝔞-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 ⋂ n > 0 a n = 0 {\displaystyle \bigcap _{n>0}{{\mathfrak {a}}^{n}}=0} 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.[2]

Completion

When M is Hausdorff, M can be completed as a metric space; the resulting space is denoted by M ^ {\displaystyle {\widehat {M}}} and has the module structure obtained by extending the module operations by continuity. It is also the same as (or canonically isomorphic to):

M ^ = lim ← ⁡ M / a n M {\displaystyle {\widehat {M}}=\varprojlim M/{\mathfrak {a}}^{n}M}
where the right-hand side is an inverse limit of quotient modules under natural projection.[3]

For example, let R = k [ x 1 , … , x n ] {\displaystyle R=k[x_{1},\ldots ,x_{n}]} be a polynomial ring over a field k and 𝔞 = (x1, ..., xn) the (unique) homogeneous maximal ideal. Then R ^ = k [ [ x 1 , … , x n ] ] {\displaystyle {\hat {R}}=k[[x_{1},\ldots ,x_{n}]]} , the formal power series ring over k in n variables.[4]

Closed submodules

As a consequence of the above, the 𝔞-adic closure of a submodule N ⊆ M {\displaystyle N\subseteq M} is N ¯ = ⋂ n > 0 ( N + a n M ) . {\textstyle {\overline {N}}=\bigcap _{n>0}{(N+{\mathfrak {a}}^{n}M)}{\text{.}}} [5] This closure coincides with N whenever R is 𝔞-adically complete and M is finitely generated.[6]

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

R is Zariski with respect to 𝔞 if and only if 𝔞 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. 148–151.
  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.