Lifting strongly graded rings via overgroups

Abstract

Motivated by lifting problems for principal bundles, we study the following algebraic problem. Let smashĜ be a group with identity element e, let G ≤ smashĜ, and let S be a strongly G-graded unital ring with principal component R:=S_e. We ask whether the given grading extends to a strong smashĜ-grading without altering its G-homogeneous components. For every extension smashp̂ of the Picard homomorphism of S, we construct a characteristic class in the relative third cohomology group H³_smashp̂(smashĜ,G;U(Z(R))), where Z(R) denotes the center of R and U(Z(R)) its group of units. The~vanishing of this class is equivalent to the existence of a lift. When it vanishes, the relative second cohomology group H²_smashp̂(smashĜ,G;U(Z(R))) acts simply transitively on the equivalence classes of lifts with Picard homomorphism smashp̂. We also prove that an H-grading is strong if and only if both its restriction to a normal subgroup N trianglelefteq H and its induced H/N-grading are strong. We illustrate the theory through a range of examples.

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Cite

BibTeX RIS

BibTeX

@misc{husen2026lifting,
  title = {{Lifting strongly graded rings via overgroups}},
  author = {Emma Husen and Stefan Wagner},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09879},
  eprint = {2609.09879},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09879},
}

RIS

TY  - UNPB
TI  - Lifting strongly graded rings via overgroups
AU  - Emma Husen
AU  - Stefan Wagner
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.09879
UR  - https://arxiv.org/abs/2609.09879
AB  - Motivated by lifting problems for principal bundles, we study the following algebraic problem. Let smashĜ be a group with identity element e, let G ≤ smashĜ, and let S be a strongly G-graded unital ring with principal component R:=S_e. We ask whether the given grading extends to a strong smashĜ-grading without altering its G-homogeneous components. For every extension smashp̂ of the Picard homomorphism of S, we construct a characteristic class in the relative third cohomology group H^3_smashp̂(smashĜ,G;U(Z(R))), where Z(R) denotes the center of R and U(Z(R)) its group of units. The~vanishing of this class is equivalent to the existence of a lift. When it vanishes, the relative second cohomology group H^2_smashp̂(smashĜ,G;U(Z(R))) acts simply transitively on the equivalence classes of lifts with Picard homomorphism smashp̂. We also prove that an H-grading is strong if and only if both its restriction to a normal subgroup N trianglelefteq H and its induced H/N-grading are strong. We illustrate the theory through a range of examples.
ER  -

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