Completions of fields September 1, 2009Posted by Akhil Mathew in algebraic number theory, algebra, number theory, commutative algebra.
Tags: completions, absolute values, Cauchy sequences, 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.
Definition 1 The completion of is defined as the set of equivalence classes of Cauchy sequences: (more…)
Lifting idempotents à la Grothendieck August 29, 2009Posted by Akhil Mathew in algebra, algebraic geometry, commutative algebra.
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.
The theorem says we can lift “approximate idempotents” in complete rings to actual ones. In detail:
Theorem 1 Let 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…)
The finite presentation trick and completions August 27, 2009Posted by Akhil Mathew in algebra, category theory, commutative algebra.
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 1 Let be a Noetherian ring, and an ideal. If we take all completions with respect to the -adic topology,
for any f.g. -module . (more…)
Completions of rings and modules August 25, 2009Posted by Akhil Mathew in algebra, commutative algebra.
Tags: Artin-Rees lemma, completions, exact functors, Hilbert basis theorem, Noetherian rings
So, we saw in the previous post that completion can be defined generally for abelian groups. Now, to specialize to rings and modules.
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…)
Generalities on completions August 23, 2009Posted by Akhil Mathew in algebra, commutative algebra, topology.
Tags: Cauchy sequences, completions, inverse limits
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 , is Cauchy if for each , there exists large enough that
The sequence converges to if for each , there exists large enough that