A foundational result in Grothendieck's descent theory and in his étale cohomology is the exactness of Amitsur's complex. More precisely, suppose we have an $A$-algebra
$A\to B$; then there is a cosimplicial complex associated to it whose $n-$cosimplices are
$B\ ^ {\otimes {(n+1)} }$, and from there a complex obtains
$$ 0\to A \to B \to B \otimes B \to \ldots \quad (AMITSUR) $$
For example the map $B \to B \otimes B$ is $b\mapsto 1\otimes b -b \otimes 1$ and the following maps are obtained similarly by inserting $1$'s in tensor products of copies of $B$ and taking alternating sums. The key result is that this Amitsur complex is exact if the initial algebra $A \to B$ is faithfully flat.
The proof is splendid: "one" (ah, that's the point!) remarks that if the structural map has an $A$- linear retraction, then it is easy to conclude by constructing a homotopy. And then one reduces to this case by a bold gambit: since one doesn't know how to prove exactness of $(AMITSUR)$ one tensors with $B$ and gets the even more complicated complex $(AMITSUR)\otimes B$ . But now the initial map $A\to B$ has become $B \to B\otimes B: b \mapsto 1\otimes b$ , which HAS a retraction: just take the product $B\otimes B \to B: b\otimes b' \mapsto bb'$ . So the tensored complex is exact and the initial complex was necessarily exact by faithful flatness.
Question: who proved this? I suspect the argument I sketched is due to Grothendieck since I couldn't find a reference to Amitsur in EGA nor in SGA. So, what exactly did Amitsur prove in this context and how did he do it? I have a vague intuition that he didn't express himself in terms of faithful flatness, but my Internet search failed miserably. So, dear mathoverflow participants, you are my last hope...
