arXiv:2611.08531v1 Announce Type: new 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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