Local cohomology over Steenrod algebras

Abstract

Let 𝔽_q be a finite field with q = p^s elements. Let S = ⨁_{n ≥ 0} S_n be a non-negatively graded commutative, Noetherian 𝔽_q-algebra. Assume S has a structure of an unstable Steenrod algebra. Let M be a finitely generated graded unstable S-module. Let I be a homogeneous ideal in S. We prove that for i ≥ 0 the local cohomology modules Hⁱ_I(M) has a structure of a Cartan S-module. As application we give a direct and considerably simpler proof of D. Bourguiba and S. Zarati's generalization of Landweber-Stong conjecture. We also give a necessary and sufficient condition for S to have S_r-property of Serre in terms of Dickson polynomials.

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BibTeX RIS

BibTeX

@misc{puthenpurakal2026local,
  title = {{Local cohomology over Steenrod algebras}},
  author = {Tony J. Puthenpurakal},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08531},
  eprint = {2611.08531},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08531},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Local cohomology over Steenrod algebras
AU  - Tony J. Puthenpurakal
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.08531
UR  - https://arxiv.org/abs/2611.08531
AB  - Let 𝔽_q be a finite field with q = p^s elements. Let S = ⨁_n ≥ 0 S_n be a non-negatively graded commutative, Noetherian 𝔽_q-algebra. Assume S has a structure of an unstable Steenrod algebra. Let M be a finitely generated graded unstable S-module. Let I be a homogeneous ideal in S. We prove that for i ≥ 0 the local cohomology modules H^i_I(M) has a structure of a Cartan S-module. As application we give a direct and considerably simpler proof of D. Bourguiba and S. Zarati's generalization of Landweber-Stong conjecture. We also give a necessary and sufficient condition for S to have S_r-property of Serre in terms of Dickson polynomials.
ER  -

Abstract

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