Finite k-Projective Dimension and Generalized Auslander-Buchsbaum Inequality and Intersection Theorem

Abstract. 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-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.