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.[1]

Idea

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). That is to say one can descend the objects (or even statements) on to provided some additional data.

For example, given some elements generating the unit ideal of A, is faithfully flat over . Geometrically, is an open cover of and so descending a module from to would mean gluing modules on to get a module on A; the descend datum in this case amounts to the gluing data; i.e., how are identified on overlaps .

Affine case

Let be a faithfully flat ring homomorphism. Given an -module , we get the -module and because is faithfully flat, we have the inclusion . Moreover, we have the isomorphism of -modules that is induced by the isomorphism and that satisfies the cocycle condition:

where are given as:[2]

with . Note the isomorphisms are determined only by and do not involve

Now, the most basic form of faithfully flat descent says that the above construction can be reversed; i.e., given a -module and a -module isomorphism such that , an invariant submodule:

is such that .[3]

Here is the precise definition of descent datum. Given a ring homomorphism , we write:

for the map given by inserting in the i-th spot; i.e., is given as , as , etc. We also write for tensoring over when is given the module structure by .

Descent datum  Given a ring homomorphism , a descent datum on a module N on is a -module isomorphism

that satisfies the cocycle condition:[4] is the same as the composition .

Now, given a -module with a descent datum , define to be the kernel of

.

Consider the natural map

.

The key point is that this map is an isomorphism if is faithfully flat.[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. Since the second and the third vertical maps are isomorphisms, so is the first one.

The forgoing can be summarized simply as follows:

Theorem  Given a faithfully flat ring homomorphism , the functor

from the category of A-modules to the category of pairs consisting of a B-module N and a descent datum on it is an equivalence.

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 denote the category of quasi-coherent sheaves on a scheme X. Then Zariski descent states that, given quasi-coherent sheaves on open subsets with and isomorphisms such that (1) and (2) on , then exists a unique quasi-coherent sheaf on X such that in a compatible way (i.e., restricts to ).[6]

In a fancy language, the Zariski descent states that, with respect to the Zariski topology, is a stack; i.e., a category equipped with the functor the category of (relative) schemes that has an effective descent theory. Here, let denote the category consisting of pairs consisting of a (Zariski)-open subset U and a quasi-coherent sheaf on it and the forgetful functor .

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.)

Theorem  The prestack of quasi-coherent sheaves over a base scheme S is a stack with respect to the fpqc topology.[7]

The proof uses Zariski descent and the faithfully flat descent in the affine case.

Here "quasi-compact" cannot be eliminated.{

Example: a vector space

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 .

Specific descents

Étale descent

An étale descent is a consequence of a faithfully descent.

See also

Notes

  1. Deligne, Pierre (1990), Catégories Tannakiennes, Grothendieck Festschrift, vol. II, Progress in Math., vol. 87, Birkhäuser, pp. 111–195
  2. Waterhouse 1979, § 17.1.
  3. Waterhouse 1979, § 17.2.
  4. Vistoli 2008, § 4.2.1. NB: in the reference, the index starts with 1 instead of 0.
  5. SGA I, Exposé VIII, Lemme 1.6.
  6. Hartshorne 1977, Ch. II, Exercise 1.22.; NB: since "quasi-coherent" is a local property, gluing quasi-coherent sheaves results in a quasi-coherent one.
  7. Fantechi, Barbara (2005). Fundamental Algebraic Geometry: Grothendieck's FGA Explained. American Mathematical Soc. p. 82. ISBN 9780821842454. Retrieved 3 March 2018.

References

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