Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 38 min ago
Peer-reviewedCC BYEuropean Journal of Mathematics

𝔸1-homotopy theory of log schemes

  • Doosung Park1
  1. 1Fakultät Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstrasse 20, 42119, Wuppertal, Germany
Open at sourcePDFDOICite

Abstract

We construct the 𝔸1 A 1 -local stable motivic homotopy categories of fs log schemes. For schemes with the trivial log structure, our construction is equivalent to the original construction of Morel–Voevodsky. We prove that our construction satisfies the localization property. As a consequence, we obtain the Grothendieck six-functor formalism for strict morphisms of fs log schemes. We extend 𝔸1 A 1 -invariant cohomology theories of schemes to fs log schemes. In particular, we define motivic cohomology, homotopy K-theory, and algebraic cobordism of fs log schemes. For any fs log scheme log smooth over a scheme, we express cohomology of its boundary in terms of cohomology of schemes.

Related

  1. â„™1 Homotopy Theory of Schemes
    Dipankar Maity
    HALPreprintCC BY 4.0Full textAlgebra & number theoryGeometry & topologyHALarXiv
  2. On the internal homotopy theory of motivic categories
    Dipankar Maity
    HALPreprintLicence not statedAlgebra & number theoryGeometry & topologyHAL
  3. Compact quotients of homogeneous spaces and homotopy theory of sphere bundles
    Fanny Kassel, Yosuke Morita, Nicolas Tholozan
    HALPreprintLicence not statedGeneral mathematicsHAL
  4. Metastable homotopy theory via Tate coalgebras
    Marco Nervo
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  5. Rationality of varieties and 𝔸1-homotopy theory
    Utsav Choudhury, Biman Roy
    arXivPreprintCC BY-NC-SA 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@article{park2026𝔸1homotopy,
  title = {{$$\mathbb A^1$$-homotopy theory of log schemes}},
  author = {Doosung Park},
  journal = {European Journal of Mathematics},
  year = {2026},
  month = aug,
  doi = {10.1007/s40879-026-00916-2},
  url = {https://doi.org/10.1007/s40879-026-00916-2},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - 𝔸^1-homotopy theory of log schemes
AU  - Doosung Park
PY  - 2026
DA  - 2026/08/06
JO  - European Journal of Mathematics
DO  - 10.1007/s40879-026-00916-2
UR  - https://doi.org/10.1007/s40879-026-00916-2
AB  - We construct the 𝔸^1 A 1 -local stable motivic homotopy categories of fs log schemes. For schemes with the trivial log structure, our construction is equivalent to the original construction of Morel–Voevodsky. We prove that our construction satisfies the localization property. As a consequence, we obtain the Grothendieck six-functor formalism for strict morphisms of fs log schemes. We extend 𝔸^1 A 1 -invariant cohomology theories of schemes to fs log schemes. In particular, we define motivic cohomology, homotopy K-theory, and algebraic cobordism of fs log schemes. For any fs log scheme log smooth over a scheme, we express cohomology of its boundary in terms of cohomology of schemes.
SN  - 2199-675X
SN  - 2199-6768
ER  -