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What is the image of the restriction of the Spec functor (the functor from commutative rings to affine schemes) to commutative rings with the trivial monoid under multiplication?

Thanks very much

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    Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level.2012-10-18
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    Hi Julian, thanks for the comment. I want to find a subcategory of the category of affine schemes that is dual to the category of commutative rings with trivial monoid under multiplication which is equivalent to the category of abelian groups.2012-10-18
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    The dual of the category of abelian groups can be embedded in the category of locally compact abelian groups, by Pontryagin duality.2012-10-18

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The question doesn't make sense. The spec construction works for unital commutative rings (otherwise many properties break down, e.g. the equivalence between affine schemes). And these don't have a trivial multiplication (except for the zero ring).

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    The properties surely break down, but does the construction?2012-10-18
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    What's a prime ideal in a non-unital ring?2012-10-18
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    One such the quotient is a domain? There are domains without unit :-)2012-10-18
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    I really do not know if the construction makes sense (in particular, localizations of some form will be needed! (one should check the literature: it would not be surprising if people did come up with a sensible construction for localizations; universal localization might work, for example))2012-10-18
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    Localization at elements or prime ideals make no problems. In fact, they turn out to be unital. The structure sheaf can be written down as usual. However, the ring of global sections is unital and therefore cannot recover the given non-unital ring. Besides, the question was not about "how to develope algebraic geometry over non-unital rings" (which I had already asked on MO: http://mathoverflow.net/questions/59328/non-unital-algebraic-geometry).2012-10-18
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    Can you suggest where I could look in the literature to find an answer?2012-10-18
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    Have you read my answer?!2012-10-18
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    @MartinBrandenburg, the reason I added the comments above is that you claimed in your answer that «the construction works for *unital* rings» (your emphasis), and it seems (as you yourself wrote in a comment) that the construction actually works for non-unital case also. The question was not "how to develop AG for non-unital rings", you are right, but your answer would be more precise of you made a distinction between «the construction works only for unital rings» and «the construction works and has the desired properties only for unital rings»: *(cont.)*2012-10-22
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    *(cont.)* according o what you wrote, one of those statements is true and the other isn't!2012-10-22