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arXiv · 2611.08531

Local cohomology over Steenrod algebras

Tony J. Puthenpurakal
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Abstract

Let $\mathbb{F}_q$ be a finite field with $q = p^s$ elements. Let $S = \bigoplus_{n \geq 0} S_n$ be a non-negatively graded commutative, Noetherian $\mathbb{F}_q$-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 $i \geq 0$ 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.

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Tony J. Puthenpurakal. 2026-11-04. Local cohomology over Steenrod algebras. https://arxiv.org/abs/2611.08531

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