Faithfully flat descent

Faithfully flat descent is a technique from algebraic geometry, allowing one to draw conclusions about objects on the target of a faithfully flat morphism. Such morphisms, that are flat and surjective, are common, one example coming from an open cover.
In practice, from an affine point of view, this technique allows one to prove some statement about a ring or scheme after faithfully flat base change.
"Vanilla" faithfully flat descent is generally false; instead, faithfully flat descent is valid under some finiteness conditions (e.g., quasi-compact or locally of finite presentation).
A faithfully flat descent is a special case of Beck's monadicity theorem.
Basic form.
Let formula_1 be a faithfully flat ring homomorphism. Given an formula_2-module formula_3, we get the formula_4-module formula_5 and because formula_1 is faithfully flat, we have the inclusion formula_7. Moreover, we have the isomorphism formula_8 of formula_9-modules that is induced by the isomorphism formula_10 and that satisfies the cocycle condition:
where formula_12 are given as:
with formula_16. Note the isomorphisms formula_12 are determined only by formula_18 and do not involve formula_19
Now, the most basic form of faithfully flat descent says that the above construction can be reversed; i.e., given a formula_4-module formula_21 and a formula_9-module isomorphism formula_8 such that formula_11, an invariant submodule:
is such that formula_26.
Zariski descent.
The Zariski descent refers simply to the fact that a quasi-coherent sheaf can be obtained by gluing those on a (Zariski-)open cover. It is a special case of a faithfully flat descent but is frequently used to reduce the descent problem to the affine case.
In details, let formula_27 denote the category of quasi-coherent sheaves on a scheme "X". Then Zariski descent states that, given quasi-coherent sheaves formula_28 on open subsets formula_29 with formula_30 and isomorphisms formula_31 such that (1) formula_32 and (2) formula_33 on formula_34, then exists a unique quasi-coherent sheaf formula_35 on "X" such that formula_36 in a compatible way (i.e., formula_37 restricts to formula_38).
In a fancy language, the Zariski descent states that, with respect to the Zariski topology, formula_39 is a stack; i.e., a category formula_40 equipped with the functor formula_41 the category of (relative) schemes that has an effective descent theory. Here, let formula_39 denote the category consisting of pairs formula_43 consisting of a (Zariski)-open subset "U" and a quasi-coherent sheaf on it and formula_44 the forgetful functor formula_45.
Descent for quasi-coherent sheaves.
There is a succinct statement for the major result in this area: (the prestack of quasi-coherent sheaves over a scheme "S" means that, for any "S"-scheme "X", each "X"-point of the prestack is a quasi-coherent sheaf on "X".)
The proof uses Zariski descent and the faithfully flat descent in the affine case.
Here "quasi-compact" cannot be eliminated; see https://mathoverflow.net/questions/127362/counter-example-to-faithfully-flat-descent/