Flat module

In homological algebra and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. In homological algebra and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Flatness was introduced by Serre (1956) in his paper Géometrie Algébrique et Géométrie Analytique. See also flat morphism. A module M over a ring R is called flat if the following condition is satisfied: for any injective map ϕ : K → L {displaystyle phi :K o L} of R-modules, the map induced by k ⊗ m ↦ ϕ ( k ) ⊗ m {displaystyle kotimes mmapsto phi (k)otimes m} is injective. This definition applies also if R is not necessarily commutative, and M is a left R-module and K and L right R-modules. The only difference is that in this case K ⊗ R M {displaystyle Kotimes _{R}M} and L ⊗ R M {displaystyle Lotimes _{R}M} are not in general R-modules, but only abelian groups. Since tensoring with M is, for any module M, a right exact functor (between the category of R-modules and abelian groups), M is flat if and only if the preceding functor is exact. It can also be shown in the condition defining flatness as above, it is enough to take L = R {displaystyle L=R} , the ring itself, and K {displaystyle K} a finitely generated ideal of R. Flatness is also equivalent to the following equational condition, which may be paraphrased by saying that R-linear relations that hold in M stem from linear relations which hold in R: for every linear dependency, r T x = ∑ i = 1 k r i x i = 0 {displaystyle r^{T}x=sum _{i=1}^{k}r_{i}x_{i}=0} with r i ∈ R {displaystyle r_{i}in R} and x i ∈ M {displaystyle x_{i}in M} , there exist a matrix A ∈ R k × j {displaystyle Ain R^{k imes j}} and an element y ∈ M j {displaystyle yin M^{j}} such that A y = x {displaystyle Ay=x} and r T A = 0. {displaystyle r^{T}A=0.} Furthermore, M is flat if and only if the following condition holds: for every map f : F → M , {displaystyle f:F o M,} where F {displaystyle F} is a finitely generated free R {displaystyle R} -module, and for every finitely generated R {displaystyle R} -submodule K {displaystyle K} of ker ⁡ f , {displaystyle ker f,} the map f {displaystyle f} factors through a map g to a free R {displaystyle R} -module G {displaystyle G} such that g ( K ) = 0 : {displaystyle g(K)=0:}