Faithfully flat descent

Short description: Technique from algebraic geometry

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]

Basic form

Let $\displaystyle{ A \to B }$ be a faithfully flat ring homomorphism. Given an $\displaystyle{ A }$-module $\displaystyle{ M }$, we get the $\displaystyle{ B }$-module $\displaystyle{ N = M \otimes_A B }$ and because $\displaystyle{ A \to B }$ is faithfully flat, we have the inclusion $\displaystyle{ M \hookrightarrow M \otimes_A B }$. Moreover, we have the isomorphism $\displaystyle{ \varphi : N \otimes B \overset{\sim}\to N \otimes B }$ of $\displaystyle{ B^{\otimes 2} }$-modules that is induced by the isomorphism $\displaystyle{ B^{\otimes 2} \simeq B^{\otimes 2}, x \otimes y \mapsto y \otimes x }$ and that satisfies the cocycle condition:

$\displaystyle{ \varphi^1 = \varphi^0 \circ \varphi^2 }$

where $\displaystyle{ \varphi^i : N \otimes B^{\otimes 2} \overset{\sim}\to N \otimes B^{\otimes 2} }$ are given as:[2]

$\displaystyle{ \varphi^0(n \otimes b \otimes c) = \rho^1(b) \varphi(n \otimes c) }$
$\displaystyle{ \varphi^1(n \otimes b \otimes c) = \rho^2(b) \varphi(n \otimes c) }$
$\displaystyle{ \varphi^2(n \otimes b \otimes c) = \varphi(n \otimes b) \otimes c }$

with $\displaystyle{ \rho^i(x)(y_0 \otimes \cdots \otimes y_r) = y_0 \cdots y_{i-1} \otimes x \otimes y_i \cdots y_r }$. Note the isomorphisms $\displaystyle{ \varphi^i : N \otimes B^{\otimes 2} \overset{\sim}\to N \otimes B^{\otimes 2} }$ are determined only by $\displaystyle{ \varphi }$ and do not involve $\displaystyle{ M. }$

Now, the most basic form of faithfully flat descent says that the above construction can be reversed; i.e., given a $\displaystyle{ B }$-module $\displaystyle{ N }$ and a $\displaystyle{ B^{\otimes 2} }$-module isomorphism $\displaystyle{ \varphi : N \otimes B \overset{\sim}\to N \otimes B }$ such that $\displaystyle{ \varphi^1 = \varphi^0 \circ \varphi^2 }$, an invariant submodule:

$\displaystyle{ M = \{ n \in N | \varphi(n \otimes 1) = n \otimes 1 \} \subset N }$

is such that $\displaystyle{ M \otimes B = N }$.[3]

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 $\displaystyle{ \mathcal{Q}coh(X) }$ denote the category of quasi-coherent sheaves on a scheme X. Then Zariski descent states that, given quasi-coherent sheaves $\displaystyle{ F_i }$ on open subsets $\displaystyle{ U_i \subset X }$ with $\displaystyle{ X = \bigcup U_i }$ and isomorphisms $\displaystyle{ \varphi_{ij} : F_i |_{U_i \cap U_j} \overset{\sim}\to F_j |_{U_i \cap U_j} }$ such that (1) $\displaystyle{ \varphi_{ii} = \operatorname{id} }$ and (2) $\displaystyle{ \varphi_{ik} = \varphi_{jk} \circ \varphi_{ij} }$ on $\displaystyle{ U_i \cap U_j \cap U_k }$, then exists a unique quasi-coherent sheaf $\displaystyle{ F }$ on X such that $\displaystyle{ F|_{U_i} \simeq F_i }$ in a compatible way (i.e., $\displaystyle{ F|_{U_j} \simeq F_j }$ restricts to $\displaystyle{ F|_{U_i \cap U_j} \simeq F_i|_{U_i \cap U_j} \overset{\varphi_{ij}}\underset{\sim}\to F_j|_{U_i \cap U_j} }$).[4]

In a fancy language, the Zariski descent states that, with respect to the Zariski topology, $\displaystyle{ \mathcal{Q}coh }$ is a stack; i.e., a category $\displaystyle{ \mathcal{C} }$ equipped with the functor $\displaystyle{ p : \mathcal{C} \to }$ the category of (relative) schemes that has an effective descent theory. Here, let $\displaystyle{ \mathcal{Q}coh }$ denote the category consisting of pairs $\displaystyle{ (U, F) }$ consisting of a (Zariski)-open subset U and a quasi-coherent sheaf on it and $\displaystyle{ p }$ the forgetful functor $\displaystyle{ (U, F) \mapsto U }$.

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.[5]

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/

Notes

1. Deligne, Pierre (1990), Catégories Tannakiennes, Grothendieck Festschrift, vol. II, Progress in Math., 87, Birkhäuser, pp. 111–195
2. Waterhouse 1979, § 17.1.
3. Waterhouse 1979, § 17.2.
4. 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.
5. Fantechi, Barbara (2005). Fundamental Algebraic Geometry: Grothendieck's FGA Explained. American Mathematical Soc.. p. 82. ISBN 9780821842454. Retrieved 3 March 2018.

References

• SGA 1, Exposé VIII – this is the main reference (but it depends on a result from Giraud (1964), which replaced (in much more general form) the unpublished Exposé VII of SGA1).
• Giraud, Jean (1964), Méthode de la descent, Mém. Soc. Math. France, 2, doi:10.24033/msmf.2
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9
• Street, Ross (20 Mar 2003). "Categorical and combinatorial aspects of descent theory". arXiv:math/0303175. (a detailed discussion of a 2-category)
• Angelo Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory (Updated September 2, 2008)
• Waterhouse, William (1979), Introduction to affine group schemes, Graduate Texts in Mathematics, 66, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-6217-6, ISBN 978-0-387-90421-4