Topos points of quasi-coherent sheaves over monoid schemes

Autor(en): Pirashvili, Ilia
Stichwörter: Mathematics; SPECTRUM
Erscheinungsdatum: 2020
Herausgeber: CAMBRIDGE UNIV PRESS
Journal: MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY
Volumen: 169
Ausgabe: 1
Startseite: 31
Seitenende: 74
Zusammenfassung: 
Let X be a monoid scheme. We will show that the stalk at any point of X defines a point of the topos Qc(X) of quasi-coherent sheaves over X. As it turns out, every topos point of Qc(X) is of this form if X satisfies some finiteness conditions. In particular, it suffices for M/Mx to be finitely generated when X is affine, where M x is the group of invertible elements. This allows us to prove that two quasi-projective monoid schemes X and Y are isomorphic if and only if Qc(X) and Qc(Y) are equivalent. The finiteness conditions are essential, as one can already conclude by the work of A. Connes and C. Consani [3]. We will study the topos points of free commutative monoids and show that already for N8, there are `hidden' points. That is to say, there are topos points which are not coming from prime ideals. This observation reveals that there might be a more interesting `geometry of monoids'.
ISSN: 03050041
DOI: 10.1017/S0305004119000069

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