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 de- noted by k-pdR M . 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-pdR M is nearer upper bound for dimN than pdR M . Finally, if M is k-perfect, dimN ≤ k -gradeM that generalizes the Strong Intersection Theorem.