I-projective Modules And Their Properties

Let R be a commutative Noetherian local ring, M a finite R-module and I an ideal of R.In the chapter two, we will recall some basic concepts and properties in commutative algebra and homological algebra. In the section one of the chapter three,one important proposition is given firstly:The following conditions are equivalent: (l)Supp(Ext_R¹ (M, N))(?) V(I) for all f.g.R-modules N(2)for every short exact sequence P →|g L → 0 with P, L are f.g.R-modules, there exists an integer k ≥ 0,such that for every a ∈ I^k and every morphism f : M → L,there exists a morphism h : M → P with af = gh.The definition of I-projective module is given:M is said to be I-projective,if M satisfies either of the above equivalent conditions.In the section two and three,the definitions of I-projective dimension of M and global J-projective dimension of R are respectively given and their properties are discussed.The J-projective dimension of M is defined as:min{n| There exists an I-projective resolution of M: 0→ P_n →P_n-1 →……→ P₀→ M → 0}The global J-projective dimension of R is defined as:D_I(R) = sup{ Pd_I(M)| M is finite module }Finally the relation between the J-projective dimension and the relative filter depth on M w.r.t.I is presented.