Hausdorff completion

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

The concept is named after Felix Hausdorff.

References

• Nicolas Bourbaki, Commutative algebra

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