R-trivial Galois cohomology classes and cohomological p-dimension
Abstract
Given a prime p, a field F of characteristic different from p and a finite p-primary Galois module V, a Galois cohomology class in Hn(F,V) is called R-trivial if it can be connected to $0$ by a class defined over an open subscheme of the projective line. All classes of Hn(F,V) are R-trivial if F has cohomological p-dimension at most n, by the Bloch-Kato conjecture. We show that this bound is sharp: working over fields of iterated Laurent series, we produce, for every n≥ 1, a field F of cohomological p-dimension n+1 and a Galois module V such that Hn(F,V) contains classes which are not R-trivial. The module V is a quotient of the permutation module on the affine hyperplanes of an elementary abelian p-group, arising as a quotient of the absolute Galois group of F. We also describe R-trivial classes as sums of corestrictions of cup products involving a Kummer class.
Related
- Cohomology of psl(n|n) in trivial and in adjoint representations
- Hochschild cohomology and AS-Gorenstein property of weak Hopf Galois extensions
- On the order of a finite group with trivial centre and bounded conjugacy classes
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- On the arithmetic rank and cohomological dimension of some residual intersections
BibTeX
@misc{clercq2026rtrivial,
title = {{$R$-trivial Galois cohomology classes and cohomological $p$-dimension}},
author = {Charles De Clercq},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22758},
eprint = {2610.22758},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22758},
}
RIS
TY - UNPB
TI - R-trivial Galois cohomology classes and cohomological p-dimension
AU - Charles De Clercq
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22758
UR - https://arxiv.org/abs/2610.22758
AB - Given a prime p, a field F of characteristic different from p and a finite p-primary Galois module V, a Galois cohomology class in H^n(F,V) is called R-trivial if it can be connected to $0$ by a class defined over an open subscheme of the projective line. All classes of H^n(F,V) are R-trivial if F has cohomological p-dimension at most n, by the Bloch-Kato conjecture. We show that this bound is sharp: working over fields of iterated Laurent series, we produce, for every n≥ 1, a field F of cohomological p-dimension n+1 and a Galois module V such that H^n(F,V) contains classes which are not R-trivial. The module V is a quotient of the permutation module on the affine hyperplanes of an elementary abelian p-group, arising as a quotient of the absolute Galois group of F. We also describe R-trivial classes as sums of corestrictions of cup products involving a Kummer class.
ER -