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 introduce a refinement of the Gorenstein flat dimension for complexes over an associative ring—the flat-cotorsion —and prove that it, unlike dimension, behaves as one expects homological without extra assumptions on ring. Crucially, we show it coincides with where latter is finite, and right coherent rings—the setting known to behave expected.

We consider an arbitrary Abelian category $\mathcal {A}$ and a subcategory {T}$ closed under extensions direct summands, characterize those that are (semi-)special preenveloping in ; as byproduct, we generalize to this setting several classical results for categories of modules. For instance, get the special subcategories summands precisely which $({}^{\perp _{1}}\mathcal {T},\mathcal {T})$ is ...