Degenerate versions of Green's theorem for Hall modules

Young, Matthew B.

doi:10.1016/j.jpaa.2020.106557

Cited by 4 publications

(8 citation statements)

References 26 publications

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“…We remark that the K-theory of A-proj n and A-proj coincides. For earlier appearances of normal morphisms in F 1 -geometry, see [3,41,51]. The main results of this section determine the Grothendieck-Witt spaces GW ⊕ (A-proj n ) and GW Q (A-proj n ).…”

Section: Resultsmentioning

confidence: 98%

“…The reduction of H (A) ⊕h is then canonically isometric to H (coker(P(U 2 ) P(X /U 1 )). See also [51,Lemma 1.1].…”

Section: Proof An Isotropic Subobject Umentioning

confidence: 99%

“…The algebra H( A-proj n ), and its variations, have been studied by Szczesny [40][41][42]. In the setting of the representation theory of quivers over F 1 , which is combinatorial but not split, modules arising from the R • -construction have been studied in [51] where, in particular, a version of Green's theorem is proved.…”

Section: Remark 213mentioning

confidence: 99%

“…From the point of view of algebraic geometry over fields, monoid schemes can be seen as a direct generalization of toric geometry and Kato fans of logarithmic schemes; see [3,6,8,9,27] among others. The central position of monoid schemes within F 1 -geometry is confirmed by their numerous links to other areas of mathematics, such as Weyl groups as algebraic groups over F 1 [29,44], computational methods for toric geometry [7,8,14], a framework for tropical scheme theory [15], applications to representation theory [20,[40][41][42]51] and, last but not least, stable homotopy theory as K-theory over F 1 [3,10], a theme on which we dwell in this paper.…”

Section: Introductionmentioning

confidence: 99%

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Algebraic $$K\!$$-theory and Grothendieck–Witt theory of monoid schemes

2022

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We study the algebraic $$K\!$$ 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 $$K\!$$ K -theory space of an integral monoid scheme X in terms of its Picard group $${{\,\mathrm{Pic}\,}}(X)$$ Pic ( X ) and pointed monoid of regular functions $$\Gamma (X, {\mathcal {O}}_X)$$ Γ ( X , O X ) and a complete description of the Grothendieck–Witt space of X in terms of an additional involution on $${{\,\mathrm{Pic}\,}}(X)$$ Pic ( X ) . We also prove space-level projective bundle formulae in both settings.

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Section: Resultsmentioning

confidence: 98%

“…The reduction of H (A) ⊕h is then canonically isometric to H (coker(P(U 2 ) P(X /U 1 )). See also [51,Lemma 1.1].…”

Section: Proof An Isotropic Subobject Umentioning

confidence: 99%

Section: Remark 213mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Algebraic $$K\!$$-theory and Grothendieck–Witt theory of monoid schemes

2022

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“…We remark that the K-theory of A -proj n and A -proj coincides. For earlier appearances of normal morphisms in F 1 -geometry, see [2], [40], [49]. The main results of this section determine the Grothendieck-Witt spaces GW ⊕ (A -proj n ) and GW Q (A -proj n ).…”

Section: Introductionmentioning

confidence: 98%

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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