Amitsur complex

In algebra, the Amitsur complex is a natural complex associated to a ring homomorphism. It was introduced by . When the homomorphism is faithfully flat, the Amitsur complex is exact (thus determining a resolution), which is the basis of the theory of faithfully flat descent. 
The notion should be thought of as a mechanism to go beyond the conventional localization of rings and modules.
Definition.
Let formula_1 be a homomorphism of (not-necessary-commutative) rings. First define the cosimplicial set formula_2 (where formula_3 refers to formula_4, not formula_5) as follows. Define the face maps formula_6 by inserting 1 at the "i"-th spot:
Define the degeneracies formula_8 by multiplying out the "i"-th and ("i" + 1)-th spots:
They satisfy the "obvious" cosimplicial identities and thus formula_10 is a cosimplicial set. It then determines the complex with the augumentation formula_11, the Amitsur complex:
where formula_13
Exactness of the Amitsur complex.
Faithfully flat case.
In the above notations, if formula_11 is right faithfully flat, then a theorem of Alexander Grothendieck states that the (augmented) complex formula_15 is exact and thus is a resolution. More generally, if formula_11 is right faithfully flat, then, for each left "R"-module "M",
is exact.
"Proof":
Step 1: The statement is true if formula_1 splits as a ring homomorphism.
That "formula_11 splits" is to say formula_20 for some homomorphism formula_21 (formula_22 is a retraction and formula_11 a section). Given such a formula_22, define
by
An easy computation shows the following identity: with formula_27,
This is to say that "h" is a homotopy operator and so formula_29 determines the zero map on cohomology: i.e., the complex is exact.
Step 2: The statement is true in general.
We remark that formula_30 is a section of formula_31. Thus, Step 1 applied to the split ring homomorphism formula_32 implies:
where formula_34, is exact. Since formula_35, etc., by "faithfully flat", the original sequence is exact. formula_36
The case of the arc topology.
 show that the Amitsur complex is exact if "R" and "S" are (commutative) perfect rings, and the map is required to be a covering in the arc topology (which is a weaker condition than being a cover in the flat topology).