Generalized local cohomology and Serre cohomological dimension

‎Let R be a commutative Noetherian ring, I,~J be two ideals of R, and  M,~N be two R-modules. Let S be a Serre subcategory of the category of R-modules.We introduce Serre cohomological dimension of N, M with respect to (I,J), as{\rm cd}_S(I, J, N, M)=\sup\{i\in \Bbb N_۰: H_{I,J}^{i}(N, M)\not\in S\}.We study some properties of {\rm cd}_S(I, J, N, M), and we getsome formulas and upper bounds for it.

Keywords:

‎Cohomological dimension‎ , ‎Local cohomology‎ , ‎Serre Subcategory