Central algebraic cohomology of Galois gerbes
Abstract
We study central algebraic cohomology of Galois gerbes over local fields of characteristic zero and number fields, with coefficients in a connected reductive group and an algebraic central subgroup, not necessarily finite or connected. For gerbes admitting a comparison morphism from Kaletha's rigid gerbe, we obtain fiber product formulas for abelianized cohomology in terms of algebraic fundamental groups of finite central quotients and homomorphisms from the band to the central subgroup. Such comparisons exist whenever the band is of finite type. Over number fields, independently of the comparison hypothesis, the full cohomology set is recovered from its abelianization and its real localizations through a cartesian square. Over non-Archimedean local fields and totally imaginary number fields, abelianization is bijective. We also specialize the formulas to Kottwitz's local and global gerbes.
Related
- Hochschild cohomology and AS-Gorenstein property of weak Hopf Galois extensions
- On definable Galois theory and definable Galois cohomology in the totally transcendental setting
- R-trivial Galois cohomology classes and cohomological p-dimension
- A Galois-equivariant Dade-Glauberman-Nagao correspondence with central defect
- The Lange-Stuhler Theorem via Gerbes
BibTeX
@misc{kang2026central,
title = {{Central algebraic cohomology of Galois gerbes}},
author = {Taeyeoup Kang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01159},
eprint = {2610.01159},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01159},
}
RIS
TY - UNPB
TI - Central algebraic cohomology of Galois gerbes
AU - Taeyeoup Kang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01159
UR - https://arxiv.org/abs/2610.01159
AB - We study central algebraic cohomology of Galois gerbes over local fields of characteristic zero and number fields, with coefficients in a connected reductive group and an algebraic central subgroup, not necessarily finite or connected. For gerbes admitting a comparison morphism from Kaletha's rigid gerbe, we obtain fiber product formulas for abelianized cohomology in terms of algebraic fundamental groups of finite central quotients and homomorphisms from the band to the central subgroup. Such comparisons exist whenever the band is of finite type. Over number fields, independently of the comparison hypothesis, the full cohomology set is recovered from its abelianization and its real localizations through a cartesian square. Over non-Archimedean local fields and totally imaginary number fields, abelianization is bijective. We also specialize the formulas to Kottwitz's local and global gerbes.
ER -