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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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What about something purely formal such as "group object in the category of locally ring space"? –  Tommaso Centeleghe Sep 28 '10 at 9:06
    
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). –  Sheikraisinrollbank Sep 28 '10 at 9:47
    
I will be more careful when making comments. Sorry. –  Tommaso Centeleghe Sep 28 '10 at 10:13
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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. –  Martin Brandenburg Sep 28 '10 at 12:19
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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. –  BCnrd Sep 28 '10 at 12:44
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