TorChase

One can use to show easily that if is exact with flat then it is pure exact. This requires knowing that is balanced (indeed even well defined) and so we give a simple diagram chase argument instead. We hope that this chase gives some indication as to how the chases proceed to prove the general theory of .

Let be an associative ring with , and a left -module. We say that is flat if given any exact sequence of right -modules , then is an exact sequence of Abelian groups. Similarly an exact sequence of right -modules is said to be pure exact if is an exact sequence of Abelian groups for every left -module .

Suppose is flat and is exact. We want to show it is pure exact, so we let be a right -module. The tensor product is already right exact so we at least have the following exact sequence: . By the general theory of , we know that , but we wish to prove this directly.

We consider a partial flat resolution of , that is an exact sequence with flat (for instance the free right -module ). We can form the somewhat large commutative diagram with exact rows and columns:

Let . Since , for some . Since we get , so for some . , so , so , so for some . , so , so . Thus .

Thus and is exact, and thus is pure.