Degree-one cohomology of metacyclic $2$-groups: a low-bit dichotomy

Abstract

We study the subalgebra generated by degree-one mod-$2$ cohomology for metacyclic $2$-groups G=⟨ a,b| a^{2^α}=1, b^{2^m}=a^c, b⁻¹ab=a^r⟩ with unique normal forms, where α,m≥2 and r≡−1pmod4. Presentation compatibility forces c=0 or 2^α−1 and G^ab≅ C₂× C_2^m. If c≡0pmod4, the degree-one-generated algebra is 𝔽₂ [x,y]/(y²,x²+xy), a four-dimensional algebra shared by the four nonmodular types of Aaboun--Zekhnini. The unique exceptional case α=2, c=2 is modular and has infinite-dimensional degree-one algebra 𝔽₂[x,y]/(y²,x²y). The regular algebra is PBW, Koszul, universally Koszul, and strongly Koszul, whereas the modular algebra is not Koszul. We realize the regular algebra by inflation from the common quotient C₄rtimes₋₁C₄ and identify its quadratic dual with the characteristic-two Jordan plane. For the inversion tower C₄rtimes₋₁C_2^m, we give uniform presentations of the full mod-$2$ cohomology rings and determine the kernels and images of inflation in every degree. Their direct limit yields the continuous cohomology ring of C₄rtimes₋₁ℤ₂. We also record an arithmetic consequence for Galois groups arising over cyclotomic ℤ₂-extensions of real quadratic fields.

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BibTeX RIS

BibTeX

@misc{palaisti2026degreeone,
  title = {{Degree-one cohomology of metacyclic $2$-groups: a low-bit dichotomy}},
  author = {Marina Palaisti},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36401},
  eprint = {2610.36401},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36401},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Degree-one cohomology of metacyclic $2$-groups: a low-bit dichotomy
AU  - Marina Palaisti
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.36401
UR  - https://arxiv.org/abs/2610.36401
AB  - We study the subalgebra generated by degree-one mod-$2$ cohomology for metacyclic $2$-groups G=⟨ a,b| a^2^α=1, b^2^m=a^c, b^−1ab=a^r⟩ with unique normal forms, where α,m≥2 and r≡−1pmod4. Presentation compatibility forces c=0 or 2^α−1 and G^ab≅ C_2× C_2^m. If c≡0pmod4, the degree-one-generated algebra is 𝔽_2 [x,y]/(y^2,x^2+xy), a four-dimensional algebra shared by the four nonmodular types of Aaboun--Zekhnini. The unique exceptional case α=2, c=2 is modular and has infinite-dimensional degree-one algebra 𝔽_2[x,y]/(y^2,x^2y). The regular algebra is PBW, Koszul, universally Koszul, and strongly Koszul, whereas the modular algebra is not Koszul. We realize the regular algebra by inflation from the common quotient C_4rtimes_−1C_4 and identify its quadratic dual with the characteristic-two Jordan plane. For the inversion tower C_4rtimes_−1C_2^m, we give uniform presentations of the full mod-$2$ cohomology rings and determine the kernels and images of inflation in every degree. Their direct limit yields the continuous cohomology ring of C_4rtimes_−1ℤ_2. We also record an arithmetic consequence for Galois groups arising over cyclotomic ℤ_2-extensions of real quadratic fields.
ER  -

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