Flat Modules and Homological Dimensions

Abstract

A right R-module P _R is called flat if P ⊗ _R — is an exact functor on ${}_R\mathfrak{M}$, the category of left R-modules. Specifically, this requires that, if A → B is injective in ${}_R\mathfrak{M}$, then P ⊗ _R A → P ⊗ _R B is injective also. Projective modules are flat, but flat modules enjoy an important property not shared by projective modules: they are closed w.r.t. direct limits. In particular, a module is flat if all f.g. submodules are flat. Over ℤ, the flat modules are just the torsion-free abelian groups.