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The -adic completion of a module
Definition
Let be a commutative ring, let be an ideal, and let be an -module.
The -adic completion of is the inverse limit of the quotient system
When the ideal is fixed, this module is denoted simply by .
The associated completion map is
Depends on
Used by
- Completion need not be exact without a finiteness hypothesis Example
- Equivalent adic filtrations have canonically isomorphic completions Example
- Powers of an ideal give the same one-step adic completion Example
- Semilocal completion decomposes into completed local factors Example
- The p-adic completion map of the integers Example
- The p-adic integers as an inverse limit and as a completion Example
- Elements congruent to 1 modulo a defining ideal are units Proposition
- Adic completion is exact on finite modules over a Noetherian ring Theorem
- Finite modules over complete Noetherian rings are complete Theorem
- Kernel and universal property of adic completion Theorem
Dependency tree · two levels
3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., Proposition 22.8 (standard reference, not scraped)
- J. S. Milne, A Primer of Commutative Algebra, Lemma 24.1 (standard reference, not scraped)