Hausdorff completion

In algebra, the Hausdorff completion [math]\displaystyle{ \widehat{G} }[/math] of a group G with filtration [math]\displaystyle{ G_n }[/math] is the inverse limit [math]\displaystyle{ \varprojlim G/G_n }[/math] of the discrete group [math]\displaystyle{ G/G_n }[/math]. A basic example is a profinite completion. The image of the canonical map [math]\displaystyle{ G \to \widehat{G} }[/math] is a Hausdorff topological group and its kernel is the intersection of all [math]\displaystyle{ G_n }[/math]: i.e., the closure of the identity element. The canonical homomorphism [math]\displaystyle{ \operatorname{gr}(G) \to \operatorname{gr}(\widehat{G}) }[/math] is an isomorphism, where [math]\displaystyle{ \operatorname{gr}(G) }[/math] is a graded module associated to the filtration.

The concept is named after Felix Hausdorff.

References

• Nicolas Bourbaki, Commutative algebra

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