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Completions of fields
*September 1, 2009*

*Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.*

Tags: absolute values, Cauchy sequences, completions, p-adic numbers

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Tags: absolute values, Cauchy sequences, completions, p-adic numbers

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So again, we’re back to completions, though we’re going to go through it quickly. Except this time we have a field with an absolute value like the rationals with the usual absolute value.

**Completions **

Definition 1The(more…)completionof is defined as the set of equivalence classes of Cauchy sequences:

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Lifting idempotents à la Grothendieck
*August 29, 2009*

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

Tags: completions, connectedness, idempotents, lifting idempotents, schemes

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Tags: completions, connectedness, idempotents, lifting idempotents, schemes

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I am going to get back shortly to discussing algebraic number theory and discrete valuation rings. But this tidbit from EGA 1 that I just learned today was too much fun to resist. Besides, it puts the material on completions in more context, so I think the digression is justified.

**Lifting Idempotents **

The theorem says we can lift “approximate idempotents” in complete rings to actual ones. In detail:

Theorem 1Let be a ring complete with respect to the -adic filtration. Then if is idempotent (i.e. ) then there is an idempotent such that reduces to .(more…)

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The finite presentation trick and completions
*August 27, 2009*

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

Tags: abelian categories, completions, finite presentations, flatness, Noetherian rings

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Tags: abelian categories, completions, finite presentations, flatness, Noetherian rings

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The previous post got somewhat detailed and long, so today’s will be somewhat lighter. I’ll use completions to illustrate a well-known categorical trick using finite presentations.

**The finite presentation trick **

Our goal here is:

Theorem 1Let be a Noetherian ring, and an ideal. If we take all completions with respect to the -adic topology,

for any f.g. -module . (more…)

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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

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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…)

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Generalities on completions
*August 23, 2009*

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

Tags: Cauchy sequences, completions, inverse limits

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Tags: Cauchy sequences, completions, inverse limits

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Today I’ll discuss completions in their algebraic context. All this is really a version of Cauchy’s construction of the real numbers, but it’s also useful in algebra, since one can study a ring through its completions (e.g. in algebraic number theory, as I hope to get to soon).

**Generalities on Completions **

Suppose we have a filtered abelian group with a descending filtration of subgroups . Because of this, we can consider “Cauchy sequences” and “convergence:”

Definition 1The sequence , isCauchyif for each , there exists large enough that

The sequence

converges toif for each , there exists large enough that