finite k-projective dimension and generalized auslander-buchsbaum inequality and intersection theorem

Let r be a commutative noetherian ring, m be a finitely generated r-module and a be an ideal of r. for an arbitrary integer k ≥ −1, we introduce the concept of k-projective dimension of m denoted by k-pdrm . we show that the finite k-projective dimension of m is at least k-depth(a, r) − k-depth(a, m ). as a generalization of the intersection theorem, we show that for any finitely generated r-module n, in certain conditions, k-pdrm is nearer upper bound for dimn than pdrm . finally, if m is k-perfect, dimn ≤ k-gradem that generalizes the strong intersection theorem.