On the spectrum of monoids and semilattices

Pirashvili, Ilia

doi:10.48550/arxiv.1112.0023

Cited by 1 publication

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“…There are also some remarks on incidence geometries that come from F 1schemes, see [19], [99] and [116]. It turns out that apartments of buildings (and other incidence geometries) are naturally attached to representations of Chevalley groups over F 1 on projective space.…”

Section: Results and New Directionsmentioning

confidence: 99%

A blueprinted view on $\mathbb F_1$-geometry

Lorscheid¹

2016

Absolute Arithmetic and $\Mathbb F_1$-Geometry

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This overview paper has two parts. In the first part, we review the development of F1-geometry from the first mentioning by Jacques Tits in 1956 until the present day. We explain the main ideas around F1, embedded into the historical context, and give an impression of the multiple connections of F1-geometry to other areas of mathematics.In the second part, we review (and preview) the geometry of blueprints. Beyond the basic definitions of blueprints, blue schemes and projective geometry, this includes a theory of Chevalley groups over F1 together with their action on buildings over F1; computations of the Euler characteristic in terms of F1-rational points, which involve quiver Grassmannians; K-theory of blue schemes that reproduces the formula Ki(F1) = π st i (S 0 ); models of the compactifications of Spec Z and other arithmetic curves; and explanations about the connections to other approaches towards F1 like monoidal schemes after Deitmar, B1-algebras after Lescot, Λ-schemes after Borger, relative schemes after Toën and Vaquié, log schemes after Kato and congruence schemes after Berkovich and Deitmar.

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Section: Results and New Directionsmentioning

confidence: 99%

A blueprinted view on $\mathbb F_1$-geometry

Lorscheid¹

2016

Absolute Arithmetic and $\Mathbb F_1$-Geometry

View full text Add to dashboard Cite

show abstract

On the spectrum of monoids and semilattices

Cited by 1 publication

References 6 publications

A blueprinted view on $\mathbb F_1$-geometry

A blueprinted view on $\mathbb F_1$-geometry

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