Completions of fieldsSeptember 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…)

Generalities on completionsAugust 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…)