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Completions of rings and modules August 25, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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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 {A} with a descending filtration (satisfying {A_0 =A}), which implies the {A_i} are ideals; as we saw, the completion will also be a ring. Most often, there will be an ideal {I} such that {A_i = I^i}, i.e. the filtration is {I}-adic. We have a completion functor from filtered rings to rings, sending {A \rightarrow \hat{A}}. Given a filtered {A}-module {M}, there is a completion {\hat{M}}, which is also a {\hat{A}}-module; this gives a functor from filtered {A}-modules to {\hat{A}}-modules. (more…)

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Topologies and the Artin-Rees lemma August 19, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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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…)