I-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 ๐ of R determines a topology on M called the ๐-adic topology, characterized by the pseudometric Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(x,y) = 2^{-\sup{\{n \mid x-y\in\mathfrak{a}^nM\}}}.} The family Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{x+\mathfrak{a}^nM:x\in M,n\in\mathbb{Z}^+\}} is a basis for this topology.[1]
An ๐-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 ifFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigcap_{n > 0}{\mathfrak{a}^nM} = 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 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigcap_{n > 0}{\mathfrak{a}^n} = 0} 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 ArtinโRees lemma.[2]
Completion
When M is Hausdorff, M can be completed as a metric space; the resulting space is denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R = k[x_1, \ldots, x_n]} be a polynomial ring over a field k and ๐ = (x1, ..., xn) the (unique) homogeneous maximal ideal. Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{R} = k[[x_1, \ldots, x_n]]} , the formal power series ring over k in n variables.[4]
Closed submodules
The ๐-adic closure of a submodule Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \subseteq M} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \overline{N} = \bigcap_{n > 0}{(N + \mathfrak{a}^n M)}\text{.}} [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]
References
- โ 1.0 1.1 Singh 2011, p. 147.
- โ Singh 2011, p. 148.
- โ Singh 2011, pp. 148โ151.
- โ Singh 2011, problem 8.16.
- โ Singh 2011, problem 8.4.
- โ Singh 2011, problem 8.8
- โ 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.