on the d-dimension of finitely generated modules

Let r be a commutative noetherian ring with identity and d a nonzero integer. this paper investigates the d-dimension, denoted dim(d, m), for finitely generated r-modules m, within the framework of d-local cohomology. we establish key inequalities and equalities relating the d-dimensions of modules under support conditions and in short exact sequences. in particular, it is shown that if supp(n) ⊆ supp(m), then dim(d, n) ≤ dim(d, m), and equality holds when the supports coincide. furthermore, for an exact sequence 0 → l → m → n → 0 of finitely generated modules, we prove that dim(d, m) = max{dim(d, l), dim(d, n)}. these results extend our understanding of the behavior of d-local cohomology and its associated dimension invariant.