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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thisidea not work?" – Yemon Choi Jan 4 '12 at 14:58`$F_1$`

. 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