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PreprintarXiv licencearXiv

Exact sequences of fundamental group schemes for G-torsors

  • Lingguang Li
  • Hao Wang
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Abstract

Let k be a field, G an affine k-group scheme, X a connected scheme proper over k, f:(Y,y)β†’(X,x) a pointed G-torsor, π’žX and $\mathcal{C}_Y $ the Tannakian categories of vector bundles on X and Y, such that f*π’žXβŠ† π’žY, we give the necessary and sufficient conditions for the exactness of the natural sequence of k-group schemes 1β†’ Ο€(π’žY,y) β†’ Ο€(π’žX,x) β†’ G β†’ 1. This provides a common framework of exact sequences for the S-, Nori, F-, Γ©tale fundamental group schemes and their extended variants. Finally, an isogeny of an elliptic curve shows that the relevant preservation property fails for the local, extended local, and unipotent categories.

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Cite

BibTeXRIS
BibTeX
@misc{li2026exact,
  title = {{Exact sequences of fundamental group schemes for $G$-torsors}},
  author = {Lingguang Li and Hao Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.09757},
  eprint = {2608.09757},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.09757},
}
RIS
TY  - UNPB
TI  - Exact sequences of fundamental group schemes for G-torsors
AU  - Lingguang Li
AU  - Hao Wang
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2608.09757
UR  - https://arxiv.org/abs/2608.09757
AB  - Let k be a field, G an affine k-group scheme, X a connected scheme proper over k, f:(Y,y)β†’(X,x) a pointed G-torsor, π’ž_X and $\mathcal{C}_Y $ the Tannakian categories of vector bundles on X and Y, such that f^*π’ž_XβŠ† π’ž_Y, we give the necessary and sufficient conditions for the exactness of the natural sequence of k-group schemes 1β†’ Ο€(π’ž_Y,y) β†’ Ο€(π’ž_X,x) β†’ G β†’ 1. This provides a common framework of exact sequences for the S-, Nori, F-, Γ©tale fundamental group schemes and their extended variants. Finally, an isogeny of an elliptic curve shows that the relevant preservation property fails for the local, extended local, and unipotent categories.
ER  -