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What are the obstacles that prevent from defining Spec$\mathbb{Z}$ in absolute geometry? By absolute geometry I mean the geometry over the field with one element F1.

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What do you mean by absolute geometry? –  Martin Brandenburg Jan 4 '12 at 13:38
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I am no expert, but the question seems rather broad. Is there a more focused version that you could ask, e.g. "why does this idea not work?" –  Yemon Choi Jan 4 '12 at 14:58
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The problem, as I understand it, is not in defining spec Z as an object over $F_1$. There are actually quite a few definitions of $F_1$ and the first property that any putative definition must have is that it must map to Z. The difficulty is in finding a setup where spec Z has various desired properties, such as admitting a nice compactification that behaves like a curve over $F_1$ (I think this is called the Deninger program). –  Jeffrey Giansiracusa Jan 4 '12 at 15:06
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The problem is probably not to define Spec Z in absolute geometry but to define the correct variant of absolute geometry where this question and some others will have an obvious and expected answer. –  Zoran Skoda Jan 4 '12 at 15:14
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