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.
[1] Given a faithfully flat ring homomorphism
, the faithfully flat descent is, roughy, the statement that to give a module or an algebra over A is to give a module or an algebra over
together with the so-called descent datum (or data).
to get a module on A; the descend datum in this case amounts to the gluing data; i.e., how
be a faithfully flat ring homomorphism.
Now, the most basic form of faithfully flat descent says that the above construction can be reversed; i.e., given a
[3] Here is the precise definition of descent datum.
Descent datum — Given a ring homomorphism
-module isomorphism that satisfies the cocycle condition:[4]
[5] This is seen by considering the following: where the top row is exact by the flatness of B over A and the bottom row is the Amitsur complex, which is exact by a theorem of Grothendieck.
The cocycle condition ensures that the above diagram is commutative.
The forgoing can be summarized simply as follows: Theorem — Given a faithfully flat ring homomorphism
consisting of a B-module N and a descent datum
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.
denote the category of quasi-coherent sheaves on a scheme X.
Then Zariski descent states that, given quasi-coherent sheaves
, then exists a unique quasi-coherent sheaf
[6] In a fancy language, the Zariski descent states that, with respect to the Zariski topology,
the category of (relative) schemes that has an effective descent theory.
denote the category consisting of pairs
consisting of a (Zariski)-open subset U and a quasi-coherent sheaf on it and
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.)
Theorem — The prestack of quasi-coherent sheaves over a base scheme S is a stack with respect to the fpqc topology.
[8] Let F be a finite Galois field extension of a field k. Then, for each vector space V over F, where the product runs over the elements in the Galois group of