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The structure of Deitmar schemes. I. (English) Zbl 1329.14009

Summary: We explain how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, \(\mathbb{F}_{1}\)) to a so-called “loose graph” (which is a generalization of a graph). Several properties of the Deitmar scheme can be proven easily from the combinatorics of the (loose) graph, and it also appears that known realizations of objects over \(\mathbb{F}_{1}\) (such as combinatorial \(\mathbb{F}_{1}\)-projective and \(\mathbb{F}_{1}\)-affine spaces) exactly depict the loose graph which corresponds to the associated Deitmar scheme. This idea is then conjecturally generalized so as to describe all Deitmar schemes in a similar synthetic manner.

MSC:

14A15 Schemes and morphisms
14G15 Finite ground fields in algebraic geometry
11G25 Varieties over finite and local fields
14G20 Local ground fields in algebraic geometry
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References:

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