The theory of local homology modules was initiated by Matlis in 1974. It is a dual version of the theory of local cohomology modules. Mohammadi and Divaani-Aazar (2012) studied the connection between local homology and Gorenstein flat modules by using Gorenstein flat resolutions. In this paper, we introduce generalized local homology modules for complexes and we give several ways for computing ...

let r be an associative ring with identity, c(r) be the category of com-plexes of r-modules and flat(c(r)) be the class of all at complexes of r-modules. we show that the at cotorsion theory (flat(c(r)); flat(c(r))−)have enough injectives in c(r). as an application, we prove that for each atcomplex f and each complex y of r-modules, exti (f,x)= 0, whenever ris n-perfect and i > n.

Let R be an associative ring with identity, C(R) be the category of com-plexes of R-modules and Flat(C(R)) be the class of all at complexes of R-modules. We show that the at cotorsion theory (Flat(C(R)); Flat(C(R))−)have enough injectives in C(R). As an application, we prove that for each atcomplex F and each complex Y of R-modules, Exti (F,X)= 0, whenever Ris n-perfect and i > n.

we observe some new characterizations of $n$-presented modules. using the concepts of $(n,0)$-injectivity and $(n,0)$-flatness of modules, we also present some characterizations of right $n$-coherent rings, right $n$-hereditary rings, and right $n$-regular rings.

If R is a ring of coefficients and G a finite group, then a flat RG-module which is projective as an R-module is necessarily projective as an RG-module. More generally, if H is a subgroup of finite index in an arbitrary group Γ, then a flat RΓmodule which is projective as an RH-module is necessarily projective as an RΓ-module. This follows from a generalization of the first theorem to modules o...