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# Hausdorff completion

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

The concept is named after Felix Hausdorff.

References

Nicolas Bourbaki, Commutative algebra

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