Hausdorff completion

In algebra, the Hausdorff completion ${\displaystyle {\widehat {G}}}$ of a group G with filtration ${\displaystyle G_{n}}$ is the inverse limit ${\displaystyle \varprojlim G/G_{n}}$ of the discrete group ${\displaystyle G/G_{n}}$. A basic example is a profinite completion. The image of the canonical map ${\displaystyle G\to {\widehat {G}}}$ 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 ${\displaystyle \operatorname {gr} (G)\to \operatorname {gr} ({\widehat {G}})}$ is an isomorphism, where ${\displaystyle \operatorname {gr} (G)}$ is a graded module associated to the filtration.

The concept is named after Felix Hausdorff.

References

• Nicolas Bourbaki, Commutative algebra