The topological shadow of F1 -geometry: congruence spaces

Oliver Lorscheid; Samarpita Ray

doi:10.1007/s00209-023-03425-0

The topological shadow of F₁ -geometry: congruence spaces

Oliver Lorscheid, Samarpita Ray^*

^*Corresponding author for this work

Dynamical Systems, Geometry and Mathematical Physics

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Abstract

In this paper we introduce congruence spaces, which are topological spaces that are canonically attached to monoid schemes and that reflect closed topological properties. This leads to satisfactory topological characterizations of closed morphisms and closed immersions as well as separated and proper morphisms. We study congruence spaces thoroughly and extend standard results from usual scheme theory to monoid schemes: a closed immersion is the same as an affine morphism for which the pullback of sections is surjective; a morphism is separated if and only if the image of the diagonal is a closed subset of the congruence space; a valuative criterion for separated and proper morphisms.

Original language	English
Article number	30
Number of pages	46
Journal	Mathematische zeitschrift
Volume	306
Issue number	2
DOIs	https://doi.org/10.1007/s00209-023-03425-0
Publication status	Published - 18-Jan-2024

Keywords

Congruences
Monoid schemes
The field with one element
Topological properties

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10.1007/s00209-023-03425-0

The topological shadow of F1-geometry: congruence spacesFinal publisher's version, 721 KBLicence: Taverne

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abstract = "In this paper we introduce congruence spaces, which are topological spaces that are canonically attached to monoid schemes and that reflect closed topological properties. This leads to satisfactory topological characterizations of closed morphisms and closed immersions as well as separated and proper morphisms. We study congruence spaces thoroughly and extend standard results from usual scheme theory to monoid schemes: a closed immersion is the same as an affine morphism for which the pullback of sections is surjective; a morphism is separated if and only if the image of the diagonal is a closed subset of the congruence space; a valuative criterion for separated and proper morphisms.",

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AU - Lorscheid, Oliver

AU - Ray, Samarpita

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N2 - In this paper we introduce congruence spaces, which are topological spaces that are canonically attached to monoid schemes and that reflect closed topological properties. This leads to satisfactory topological characterizations of closed morphisms and closed immersions as well as separated and proper morphisms. We study congruence spaces thoroughly and extend standard results from usual scheme theory to monoid schemes: a closed immersion is the same as an affine morphism for which the pullback of sections is surjective; a morphism is separated if and only if the image of the diagonal is a closed subset of the congruence space; a valuative criterion for separated and proper morphisms.

AB - In this paper we introduce congruence spaces, which are topological spaces that are canonically attached to monoid schemes and that reflect closed topological properties. This leads to satisfactory topological characterizations of closed morphisms and closed immersions as well as separated and proper morphisms. We study congruence spaces thoroughly and extend standard results from usual scheme theory to monoid schemes: a closed immersion is the same as an affine morphism for which the pullback of sections is surjective; a morphism is separated if and only if the image of the diagonal is a closed subset of the congruence space; a valuative criterion for separated and proper morphisms.

KW - Congruences

KW - Monoid schemes

KW - The field with one element

KW - Topological properties

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JO - Mathematische zeitschrift

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ER -

The topological shadow of F₁ -geometry: congruence spaces

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The topological shadow of F1 -geometry: congruence spaces

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The topological shadow of F₁ -geometry: congruence spaces