Dual Bass Numbers, Co-Cohen Macaulay Modules And Co-Gorenstein Modules

Let R be a commutative ring with unit element, X(?)Spec R be saturated set ofprimes, I be an ideal of R and M be a strongly representable linearly compact R-module.We show that Cograde_X(I,M)=inf {i|Cos_R(Tor_i^R(R/I,M))(?)X} equal to the lengthof any maximal M-filter-co-regular sequence with respect to X in I, and any M-filter-co-regular sequence with respect to X in I of finite length can be extend to a maximal Mfilter co-regular sequence with respect to X in I. Let p∈Spec R, J be an ideal of R_p, thenCograde_Rp(J, Hom_R(R_p, M))=inf{i|Tor_i^R_p(R_p/J, Hom_R(R_p, M))≠0} equal to the lengthof any maximal Hom_R(R_p, M)-poor-co-regular sequence in J, and any Hom_R(R_p, M)-poor-co-regular sequence in J of finite length can be extend to a maximal Hom^R(R_p, M)-poor-co-regular sequence in J.We study the vanishing properties of dual Bass numbers, and get the theorem of van-ishing properties of dual Bass numbers. Let R be a U ring, M be an Artin R-module,p∈Cos_RM, ifπ_i(p,M)＞0, then Cograde_RpHom_R(R_p,M)≤i≤fd(R_p)Hom_R(R_p,M).whereπ_i(p, M)=dim_k(p)Tor_i^R_p(k(p),Hom_R(R_p,M)) be the i-th dual Bass numbers ofM with respect to p, fd_RpHom_R(R_p, M) might be infinite. If Cograde_RpHom_R(R_p, M)=s, fd_RpHom_R(R_p,M)=t＜∞, thenπ_s(p,M)＞0,π_t(p,M)＞0.We also study the properties of The Co-localization of Co-Cohen Macaulay modules. Weprove that The Co-localization of Co-Cohen Macaulay modules preserve Co-Cohen Macaulay-ness in a certain condition. In addition, we give a elaborate characterization of Co-CohenMacaulay modules in terms of vanishing property of the dual Bass numbers.Finally, Co-Gorenstein modules is introduced and characterized.