ag.algebraic geometry - Who proved the exactness of Amitsur's complex ?

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
$Ato B$; then there is a cosimplicial complex associated to it whose $n-$cosimplices are
$B ^ {otimes {(n+1)} }$, and from there a complex obtains
$$ 0to A to B to B otimes B to ldots quad (AMITSUR) $$
For example the map $B to B otimes B$ is $bmapsto 1otimes b -b otimes 1$ 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 $Ato B$ has become $B to Botimes B: b mapsto 1otimes b$ , which HAS a retraction: just take the product $Botimes B to B: botimes b' mapsto bb'$ . 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...