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
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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…)
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
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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…)