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Let R be a commutative Noetherian ring, a an ideal of R, M a finitely generated R-module, and S a Serre subcategory of the category of R-modules. We introduce the concept of S-minimax R-modules and the notion of the S-finiteness dimension faS(M):=inf{faRp(Mp)|p∈SuppR(M/aM)andR/p∉S}and we will prove that: (i) If Ha0(M),⋯,Han-1(M) are S-minimax, then the set {p∈AssR(Han(M))|R/p∉S} is finite. This generalizes the main results of Brodmann-Lashgari (Proc Am Math Soc 128(10):2851-2853, 2000), Quy (Proc Am Math Soc 138:1965-1968, 2010), Bahmanpour-Naghipour (Proc Math Soc 136:2359-2363, 2008), Asadollahi-Naghipour (Commun Algebra 43:953-958, 2015), and Mehrvarz et al. (Commun Algebra 43:4860-4872, 2015). (ii) If S satisfies the condition Ca, then faS(M)=inf{i∈N0|Hai(M)is notS-minimax}.This is a formulation of Faltings’ Local-global principle for the S-minimax local cohomology modules. (iii) sup{i∈N0|Hai(M)is notS-minimax}=sup{i∈N0|Hai(M)is not inS}. [ABSTRACT FROM AUTHOR]