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Completions of rings and modules
*August 25, 2009*

*Posted by Akhil Mathew in algebra, commutative algebra.*

Tags: Artin-Rees lemma, completions, exact functors, Hilbert basis theorem, Noetherian rings

2 comments

Tags: Artin-Rees lemma, completions, exact functors, Hilbert basis theorem, Noetherian rings

2 comments

So, we saw in the previous post that completion can be defined generally for abelian groups. Now, to specialize to rings and modules.

**Rings **

The case in which we are primarily interested comes from a ring with a descending filtration (satisfying ), which implies the are ideals; as we saw, the completion will also be a ring. Most often, there will be an ideal such that , i.e. the filtration is -adic. We have a completion functor from filtered rings to rings, sending . Given a filtered -module , there is a completion , which is also a -module; this gives a functor from filtered -modules to -modules. (more…)

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Topologies and the Artin-Rees lemma
*August 19, 2009*

*Posted by Akhil Mathew in algebra, commutative algebra.*

Tags: Artin-Rees lemma, filtered modules, filtered rings, filtrations, I-adic filtration

2 comments

Tags: Artin-Rees lemma, filtered modules, filtered rings, filtrations, I-adic filtration

2 comments

Today I’ll continue the series on graded rings and filtrations by discussing the resulting topologies and the Artin-Rees lemma.

All filtrations henceforth are **descending**.

**Topologies **

Recall that a **topological group** is a topological space with a group structure in which the group operations of composition and inversion are continuous—in other words, a group object in the category of topological spaces. (more…)