Faithfully flat case
In the above notations, if is right faithfully flat, then a theorem of Alexander Grothendieck states that the (augmented) complex is exact and thus is a resolution. More generally, if is right faithfully flat, then, for each left -module ,
is exact.
Proof:
Step 1: The statement is true if splits as a ring homomorphism.
That " splits" is to say for some homomorphism ( is a retraction and a section). Given such a , define
by
An easy computation shows the following identity: with ,
- .
This is to say that is a homotopy operator and so determines the zero map on cohomology: i.e., the complex is exact.
Step 2: The statement is true in general.
We remark that is a section of . Thus, Step 1 applied to the split ring homomorphism implies:
where , is exact. Since , etc., by "faithfully flat", the original sequence is exact.