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Completions of fields September 1, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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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 {F} with an absolute value {\left \lvert . \right \rvert} like the rationals with the usual absolute value.

 Completions   

Definition 1 The completion {\hat{F}} of {F} is defined as the set of equivalence classes of Cauchy sequences:  (more…)

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Generalities on completions August 23, 2009

Posted by Akhil Mathew in algebra, commutative algebra, topology.
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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 {G} with a descending filtration of subgroups {\{G_i\}}. Because of this, we can consider “Cauchy sequences” and “convergence:” 

Definition 1

The sequence {\{x_i\} \subset G}, {i \in \mathbb{N}} is Cauchy if for each {A}, there exists {N} large enough that

 

\displaystyle i,j > N \quad \mathrm{implies} \quad x_i - x_j \in G_A.

The sequence {\{y_i\} \subset G} converges to {y} if for each {A}, there exists {N} large enough that

\displaystyle i>A \quad \mathrm{implies} \quad x_i -y \in G_A. (more…)