Skip to content
DraconiticResearch
Search Sources API
182,460 studies31 sourcesUpdated 15 min ago
PreprintCC BY 4.0arXiv

Localization and Affine Schemes over 𝔽1

  • Luqiao Xu
Open at sourcePDFDOICite

Abstract

We construct an affine geometry for commutative 𝔽1-algebras in the framework of Connes and Consani. Using their localization construction, we equip the Deitmar spectrum of the underlying multiplicative monoid A(1+) with a sheaf 𝒪A of 𝔽1-algebras. We prove that 𝒪A(D(f))≅ Af and, in particular, that global sections recover A. The resulting spectrum construction yields an anti-equivalence between commutative 𝔽1-algebras and absolute affine schemes. We compare our choice of covering families with the Connes--Consani topology and explain its role in affine reconstruction. Finally, we describe base change to ℤ, under which the absolute affine scheme associated to the Eilenberg--MacLane algebra HR recovers the classical affine scheme SpecR.

Related

  1. Bivariate Affine q-Krawtchouk Polynomials and Association Schemes over Galois Rings
    Yuta Watanabe
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  2. Non-vanishing of the structure-sheaf cohomology of affine henselian schemes and henselian affinoids
    Yoshinori Gongyo, Fumiharu Kato, Takayuki Uchiba
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. Tropical cohomology for band schemes over the tropical hyperfield
    Seungkyu Lee
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. Quasi-modularity of q-traces and integrals over Hilbert schemes
    Killian Hong-Minh, Sergey Mozgovoy
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. Log-concavity and unimodality of Hodge numbers of Hilbert schemes of points over a surface
    Anubhab Pahari
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{xu2026localization,
  title = {{Localization and Affine Schemes over $\mathbbF_1$}},
  author = {Luqiao Xu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.04843},
  eprint = {2607.04843},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.04843},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Localization and Affine Schemes over 𝔽_1
AU  - Luqiao Xu
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2607.04843
UR  - https://arxiv.org/abs/2607.04843
AB  - We construct an affine geometry for commutative 𝔽_1-algebras in the framework of Connes and Consani. Using their localization construction, we equip the Deitmar spectrum of the underlying multiplicative monoid A(1_+) with a sheaf 𝒪_A of 𝔽_1-algebras. We prove that 𝒪_A(D(f))≅ A_f and, in particular, that global sections recover A. The resulting spectrum construction yields an anti-equivalence between commutative 𝔽_1-algebras and absolute affine schemes. We compare our choice of covering families with the Connes--Consani topology and explain its role in affine reconstruction. Finally, we describe base change to ℤ, under which the absolute affine scheme associated to the Eilenberg--MacLane algebra HR recovers the classical affine scheme SpecR.
ER  -