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.

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]

An 𝔞-adic topology is a linear topology (a topology generated by some submodules).

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, ..., x_n) 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

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