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\(\mathbb{F}_{1}\) for Everyone

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Abstract

This text serves as an introduction to \({\mathbb {F}_{1}}\)-geometry for the general mathematician. We explain the initial motivations for \({\mathbb {F}_{1}}\)-geometry in detail, provide an overview of the different approaches to \({\mathbb {F}_{1}}\) and describe the main achievements of the field.

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Notes

  1. To be precise, Tits considers in [61] only semi-simple algebraic groups and he considers \(\operatorname {PGL}(n)\) in place of \(\operatorname {GL}(n)\). However, we can illustrate Tits’ idea in the case of either group and we will allow ourselves this inaccuracy for the sake of a simplified account.

  2. This is, again, slightly inaccurate. In general, one can consider the Weyl group for any torus of a matrix group. However, if the torus is not specified, it is assumed that the torus is of maximal rank. For \(G=\operatorname {GL}(n)\), the diagonal torus is of maximal rank, but this is not true for all matrix groups.

  3. In order to avoid a digression into technicalities, we do not introduce sheaves. The reader can safely omit all details concerning sheaves.

  4. Please note that we face a clash of notation at this point: while we denote by \(\mathbb {G}_{m,\mathbb {Z}}\) the spectrum of the polynomial ring \(\mathbb {Z}[T]^{+}\), the very same notation is also used for the spectrum of the free blueprint \((\{aT^{i}\},\mathbb {Z}[T]^{+})\) in the definition of \(\mathbb {G}_{m,B}\) in the case \(B=\mathbb {Z}\). However, for the sake of a more intuitive notation, we do not dissolve this contradiction, but refer the reader to [42] and [45] for a more sophisticated treatment.

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    Oliver Lorscheid

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Lorscheid, O. \(\mathbb{F}_{1}\) for Everyone. Jahresber. Dtsch. Math. Ver. 120, 83–116 (2018). https://doi.org/10.1365/s13291-018-0177-x

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