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Under Spec \(\mathbb Z\). (Au-dessous de Spec \(\mathbb Z\).) (French. English summary) Zbl 1177.14022

The authors present a method to construct the category SCH of schemes from the category AB of abelian groups in a purely categorial language. This is not surprising, as a ring is just a monoidal object in the category AB and the tensor product \(\otimes \) in AB allows one to define schematic gluing. The input that this machine feeds on is just a monoidal category \((C,\otimes,1)\) which satisfies some harmless additional requirements. It yields interesting new results when applied to other monoidal categories. The “smallest” example is the category SET of sets with the cartesian product.
In this case the construction yields the category of schemes over the field of one element \({\mathbb F}_1\) as given by the reviewer [in: Number fields and function fields—two parallel worlds. Boston, MA: Birkhäuser. Progress in Mathematics 239, 87–100 (2005; Zbl 1098.14003)]. A category in between \({\mathbb F}_1\)-schemes and \({\mathbb Z}\)-schemes is obtained from the category \(({\mathbb N}-Mod,\otimes,{\mathbb N})\) of commutative monoids. Finally, the authors discuss the case of \({\mathbb S}\)-schemes, where \(\mathbb S\) is the category of ring spectra of integers. All these categories have base change to \(\mathbb Z\)-schemes. It is shown that the linear group \(\text{GL}_n\) and toric varieties can be defined in all cases. However, \(\text{GL}_n\) does not behave well under base change, which is a problem common to all \({\mathbb F}_1\)-theories so far.

MSC:

14A15 Schemes and morphisms
11G25 Varieties over finite and local fields
20G40 Linear algebraic groups over finite fields

Citations:

Zbl 1098.14003
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References:

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