Steinberg symbol: Difference between revisions

Latest revision as of 04:50, 27 December 2013

In algebra, the Hausdorff completion $\hat{G}$ of a group G with filtration $G_{n}$ is the inverse limit $\lim_{\leftarrow} G / G_{n}$ of the discrete group $G / G_{n}$ . A basic example is a profinite completion. The image of the canonical map $G \to \hat{G}$ 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 $gr (G) \to gr (\hat{G})$ is an isomorphism.

The concept is named after Felix Hausdorff.

References

Nicolas Bourbaki, Commutative algebra

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+In algebra, the '''Hausdorff completion''' <math>\widehat{G}</math> of a [[group (mathematics)|group]] ''G'' with [[filtration (mathematics)|filtration]] <math>G_n</math> is the [[inverse limit]] <math>\varprojlim G/G_n</math> of the [[discrete group]] <math>G/G_n</math>.  A basic example is a [[profinite group|profinite completion]].  The image of the canonical map <math>G \to \widehat{G}</math> is a [[Hausdorff space|Hausdorff]] [[topological group]] and its [[kernel (algebra)|kernel]] is the intersection of all <math>G_n</math>: i.e., the [[closure (topology)|closure]] of the identity element.  The canonical [[homomorphism]] <math>\operatorname{gr}(G) \to \operatorname{gr}(\widehat{G})</math> is an [[isomorphism]].
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+The concept is named after [[Felix Hausdorff]].
+== References ==
+*[[Nicolas Bourbaki]], ''Commutative algebra''
+[[Category:Commutative algebra]]
+{{algebra-stub}}

Steinberg symbol: Difference between revisions

Latest revision as of 04:50, 27 December 2013

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