Let R be a commutative Noetherian ring. Let \fraka and
\frakb be ideals of R and let N be a finitely generated
R-module. We introduce a generalization of the
\frakb-finiteness dimension of f\frakb\fraka(N)
relative to \fraka in the context of generalized local
cohomology modules as
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f\frakb\fraka(M,N):inf{i ≥ 0| \frakb ⊆ |
√
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(0:RHi\fraka(M,N))
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}, |
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M is an R-module. We also show that
f\frakb\fraka(N) ≤ f\frakb\fraka(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.
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