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Let be a proper ideal of a commutative Noetherian ring R of prime characteristic p and let Q() be the smallest positive integer m such that ${{(}^F)}^{\left[p^m\right]}{=}^{\left[p^m\right]}$, where ${}^F$ is the Frobenius closure of . This paper is concerned with the question whether the set $Q{(}^{\left[p^m\right]}):m\in \mathbb{N}₀$ is bounded. We give an affirmative answer in the case that the ideal is generated by an u.s.d-sequence c₁,..., cₙ for R such that (i) the map $R/{\sum }_{j=1}^{n}R{c}_j\to R/{\sum }_{j=1}^{n}Rc{²}_j$ induced by multiplication by c₁...cₙ is an R-monomorphism; (ii) for all $\in ass(c{₁}^j,...,c{ₙ}^j)$, c₁/1,..., cₙ/1 is a $R_{}$-filter regular sequence...

Let R be a commutative Noetherian ring. Let and be ideals of R and let N be a finitely generated R-module. We introduce a generalization of the -finiteness dimension of $f_{}^{}\left(N\right)$ relative to in the context of generalized local cohomology modules as $f_{}^{}(M,N):=infi\ge 0\left|\subseteq \surd \right(0{:}_R{H}_{}^i(M,N))$, where M is an R-module. We also show that $f_{}^{}\left(N\right)\le f_{}^{}(M,N)$ for any R-module M. This yields a new version of the Local-Global Principle for annihilation of local cohomology modules. Moreover, we obtain a generalization of the Faltings Lemma.

Let A be a Noetherian ring, let M be a finitely generated A-module and let Φ be a system of ideals of A. We prove that, for any ideal in Φ, if, for every prime ideal of A, there exists an integer k(), depending on , such that ${}^{k\left(\right)}$ kills the general local cohomology module $H_{{\Phi }_{}}^j\left(M_{}\right)$ for every integer j less than a fixed integer n, where ${\Phi }_{}:={}_{}:\in \Phi$, then there exists an integer k such that ${}^k{H}_{\Phi }^j\left(M\right)=0$ for every j < n.