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What should affine and projective schemes over $\mathbf{F}_1$ be?

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Dear OP: Please define $\mathbb{F}_1$, otherwise I feel this question is meaningless. – Jason Starr May 9 '12 at 12:04
There is an overview paper on the arxiv: "Mapping F_1 land". – Lars May 9 '12 at 12:34
I vote to close. This question is not precise enough and not really specific. And every one can simply type the question into google and will see that a lot of work has been done in the last years on that. You don't need to ask here to find this. – Martin Brandenburg May 9 '12 at 16:28
And of course the paper mentioned by Lars is one of the many points of departure for the literature. – Martin Brandenburg May 9 '12 at 16:29

Your question sounds rather vague, but in any case, you might be interested in the paper "On the notion of geometry over $\mathbb{F}_1$" by Alain Connes and Caterina Consani, available on the arXiv (or on Connes' homepage

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