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Flores, Jaret; Lorscheid, Oliver; Szczesny, Matt

doi:10.1016/j.jalgebra.2017.04.029

Cited by 4 publications

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“…Since the tropical toric schemes considered in this paper have an F 1 model, it will be interesting to understand the relation of the results presented here to the theory of cohomology over F 1 2 in [16]. This will allow to show that other invariants of toric varieties also live over F 1 .…”

Section: Introductionmentioning

confidence: 91%

Picard groups for tropical toric schemes

2018

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From any monoid scheme X (also known as an F 1 -scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) X S by scalar extension to an idempotent semifield S. We prove that for a given irreducible monoid scheme X (satisfying some mild conditions) and an idempotent semifield S, the Picard group Pic(X) of X is stable under scalar extension to S (and in fact to any field K). In other words, we show that the groups Pic(X) and Pic(X S ) (and Pic(X K )) are isomorphic. In particular, if X C is a toric variety, then Pic(X) is the same as the Picard group of the associated tropical scheme. The Picard groups can be computed by considering the correct sheaf cohomology groups. We also define the group CaCl(X S ) of Cartier divisors modulo principal Cartier divisors for a cancellative semiring scheme X S and prove that CaCl(X S ) is isomorphic to Pic(X S ).

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Section: Introductionmentioning

confidence: 91%

Picard groups for tropical toric schemes

2018

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“…In the other direction, monoid schemes can be seen as a direct generalization of toric geometry and Kato fans of logarithmic schemes; see [26], [8], [5], [2], [7] among others. The central position of monoid schemes within F 1 -geometry is confirmed by the multiple links to other disciplines, such as Weyl groups as algebraic groups over F 1 [28], computational methods for toric geometry [6], [7], [13], a framework for tropical scheme theory [14], applications to representation theory [41], [19] and, last but not least, stable homotopy theory as K-theory over F 1 [9], [2], the theme on which we dwell in this paper.…”

Section: Introductionmentioning

confidence: 93%

Algebraic K-theory and Grothendieck-Witt theory of monoid schemes

Eberhardt¹,

Lorscheid²,

Young³

2020

Preprint

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We study the algebraic K-theory and Grothendieck-Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic K-theory space of an integral monoid scheme X in terms of its Picard group Pic(X) and pointed monoid of regular functions Γ(X, O X ) and a description of the Grothendieck-Witt space of X in terms of an additional involution on Pic(X). We also prove space-level projective bundle formulae in both settings.

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The topological shadow of $${{{\mathbb {F}}}_1}$$-geometry: congruence spaces

Lorscheid,

Ray

2024

Math. Z.

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Čech cohomology over F12

Cited by 4 publications

References 8 publications

Picard groups for tropical toric schemes

Picard groups for tropical toric schemes

Algebraic K-theory and Grothendieck-Witt theory of monoid schemes

The topological shadow of $${{{\mathbb {F}}}_1}$$-geometry: congruence spaces

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