TITLE: intuition about the "section after base-change" for flat descent and exactness of the Amitsur complex
QUESTION [10 upvotes]: Suppose $A \rightarrow B$ is a faithfully flat map of rings. Then the Amitsur complex is exact:
$0 \rightarrow A \rightarrow B \rightarrow B \otimes_A B \rightarrow \dots$
(the second map is $id \otimes 1 - 1 \otimes id$, and the subsequent maps are alternating sums of the different ways of putting in a 1.)
That this is exact makes sense! Its saying that element of $b\in B$ such that $1\otimes b=b\otimes 1$ comes from $A$. The way to prove it is exact is roughly the following (the details can be found in Milne's online notes on étale cohomology.)
First, one can check that the sequence is exact if the first map has a section. Second, it is exact iff it is exact after a faithfully flat base change. Finally, if we base change by $B$, we have a section to the first map: the map we want to find a section to is $B \rightarrow B\otimes_A B, b \mapsto b\otimes 1$, and this has a section given by multiplication.
Why do we care? This fact is important in showing schemes are sheaves on the étale site. (Again, this can be found in Milne's notes.)
Okay, so my question is: what is really going on behind the scenes? I know a section is supposed to somehow "solve an equation", but I have no idea what is really being solved here. The proof feels like magic.
(One note: Its probably instructive to look at the special case of $A \rightarrow \prod A_f$ , a cover by basic open affines. I believe this proof is showing that the sequence is exact simultaneously on each member of the cover, and because exactness is a local property we have that the sequence is exact.)

REPLY [4 votes]: The more general name this sort of argument goes by is descent theory, and chapter six of Bosch, Lutkebohmert, and Raynaud's "Neron Models" is a good introduction.  The general idea is that you have some object (say, a quasi-coherent sheaf) defined on the "open sets" in a cover (it could be a Zariski cover, or an etale cover, or an fpqc cover, or...) of $X$, and you want to know whether you can glue it in some unique way to get something on $X$ itself.
That was vague, so let's consider the case when we have a quasi-coherent sheaf of modules $M_f$ on $\text{Spec}A_f$, where $A\rightarrow \prod A_f$ is a cover by open affines.  Let's set $B=\oplus A_f$ and $M'=\oplus M_f$.  We want a quasi-coherent sheaf $M$ on $\text{Spec} A$.  What's the most obvious requirement on the $M_f$?  Well, they have to agree on overlaps.  That is, if $U=\text{Spec}B=\coprod\text{Spec}A_f$, and $p_1,p_2$ are the projections $U\times_A U\rightarrow U$, we need an isomorphism 
$p_1^* M'\rightarrow p_2^* M'$, because $U\times_A U$ is the disjoint union of all the overlaps.  Another way of saying this is that if $M'=M\otimes_A B$, then $M\rightarrow M\otimes_A B\rightrightarrows M\otimes_A B\otimes_A B$ is exact.  Note that this is not a sufficient condition for $M$ to exist!  Line bundles are locally trivial, so of course there are isomorphisms on the overlaps, but not every choice of such isomorphisms is compatible with triple overlaps ($U\times_A U\times_A U$).
As for what the assumption of a section is doing there, it's saying that locally, your map of schemes $\text{Spec}B\rightarrow \text{Spec}A$ has a section.  Not Zariski-locally, fpqc locally, and that's important.  One reason the Zariski topology isn't so great to work with is that a lot of morphisms that "look like" they should be fiber bundles (that is, locally trivial in some sense) really aren't.  But they tend to be in finer topologies, like the etale topology and the fpcq topology.  The "base change to get a section" trick you mention above is saying that fpqc-locally, $\text{Spec}B\rightarrow \text{Spec}A$ behaves like a fiber bundle, in that it has lots of sections to work with.  If the map happens to be etale, that's just saying that locally on the base, the total space is a bunch of disjoint copies of the base, which is definitely a topological property we want.
In the specific application you mention, showing that schemes are sheaves in the etale topology, what you're actually going to do is show that they're sheaves in the fpqc topology, and deduce that they're sheaves in any coarser topology, like the etale topology.