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A locally ringed space is a common generalization of schemes and various manifolds. I am wondering about a locally ringed group which should be a common generalization of group schemes and various Lie groups.

Questions: did anyone ever work it out? are there any references?

Comment: when I try to reinvent it my wings take dream very soon: the multiplication is defined $G\times G \rightarrow G$ but $G\times G$ should not be a product of topological spaces, it is not for the schemes...

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  • $\begingroup$ What about something purely formal such as "group object in the category of locally ring space"? $\endgroup$
    – Tommaso Centeleghe
    Sep 28, 2010 at 9:06
  • $\begingroup$ Tommaso: presumably that wascally wabbit would object that with your definition groups schemes would not be ringed groups (see the last paragraph of his question). $\endgroup$
    – Sheikraisinrollbank
    Sep 28, 2010 at 9:47
  • $\begingroup$ I will be more careful when making comments. Sorry. $\endgroup$
    – Tommaso Centeleghe
    Sep 28, 2010 at 10:13
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    $\begingroup$ I don't think that there is a common generalization. LRS (locally ringed spaces) and Mfd (manifolds) have all products, but the inclusion Mfd -> LRS does not preserve products. $\endgroup$
    – Martin Brandenburg
    Sep 28, 2010 at 12:19
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    $\begingroup$ Dear Tom: Locally ringed topoi seems to be the concept you are getting at. The concept makes sense even without stalks (though the definition is not obvious at first glance) and it is sometimes convenient. $\endgroup$
    – BCnrd
    Sep 28, 2010 at 12:44

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