59.16 Faithfully flat descent
In this section we discuss faithfully flat descent for quasi-coherent modules. More precisely, we will prove quasi-coherent modules satisfy effective descent with respect to fpqc coverings.
Definition 59.16.1. Let \mathcal{U} = \{ t_ i : T_ i \to T\} _{i \in I} be a family of morphisms of schemes with fixed target. A descent datum for quasi-coherent sheaves with respect to \mathcal{U} is a collection ((\mathcal{F}_ i)_{i \in I}, (\varphi _{ij})_{i, j \in I}) where
\mathcal{F}_ i is a quasi-coherent sheaf on T_ i, and
\varphi _{ij} : \text{pr}_0^* \mathcal{F}_ i \to \text{pr}_1^* \mathcal{F}_ j is an isomorphism of modules on T_ i \times _ T T_ j,
such that the cocycle condition holds: the diagrams
\xymatrix{ \text{pr}_0^*\mathcal{F}_ i \ar[dr]_{\text{pr}_{02}^*\varphi _{ik}} \ar[rr]^{\text{pr}_{01}^*\varphi _{ij}} & & \text{pr}_1^*\mathcal{F}_ j \ar[dl]^{\text{pr}_{12}^*\varphi _{jk}} \\ & \text{pr}_2^*\mathcal{F}_ k }
commute on T_ i \times _ T T_ j \times _ T T_ k. This descent datum is called effective if there exist a quasi-coherent sheaf \mathcal{F} over T and \mathcal{O}_{T_ i}-module isomorphisms \varphi _ i : t_ i^* \mathcal{F} \cong \mathcal{F}_ i compatible with the maps \varphi _{ij}, namely
\varphi _{ij} = \text{pr}_1^* (\varphi _ j) \circ \text{pr}_0^* (\varphi _ i)^{-1}.
In this and the next section we discuss some ingredients of the proof of the following theorem, as well as some related material.
Theorem 59.16.2. If \mathcal{V} = \{ T_ i \to T\} _{i\in I} is an fpqc covering, then all descent data for quasi-coherent sheaves with respect to \mathcal{V} are effective.
Proof.
See Descent, Proposition 35.5.2.
\square
In other words, the fibered category of quasi-coherent sheaves is a stack on the fpqc site. The proof of the theorem is in two steps. The first one is to realize that for Zariski coverings this is easy (or well-known) using standard glueing of sheaves (see Sheaves, Section 6.33) and the locality of quasi-coherence. The second step is the case of an fpqc covering of the form \{ \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)\} where A \to B is a faithfully flat ring map. This is a lemma in algebra, which we now present.
Descent of modules. If A \to B is a ring map, we consider the complex
(B/A)_\bullet : B \to B \otimes _ A B \to B \otimes _ A B \otimes _ A B \to \ldots
where B is in degree 0, B \otimes _ A B in degree 1, etc, and the maps are given by
\begin{eqnarray*} b & \mapsto & 1 \otimes b - b \otimes 1, \\ b_0 \otimes b_1 & \mapsto & 1 \otimes b_0 \otimes b_1 - b_0 \otimes 1 \otimes b_1 + b_0 \otimes b_1 \otimes 1, \\ & \text{etc.} \end{eqnarray*}
Lemma 59.16.3. If A \to B is faithfully flat, then the complex (B/A)_\bullet is exact in positive degrees, and H^0((B/A)_\bullet ) = A.
Proof.
See Descent, Lemma 35.3.6.
\square
Grothendieck proves this in three steps. Firstly, he assumes that the map A \to B has a section, and constructs an explicit homotopy to the complex where A is the only nonzero term, in degree 0. Secondly, he observes that to prove the result, it suffices to do so after a faithfully flat base change A \to A', replacing B with B' = B \otimes _ A A'. Thirdly, he applies the faithfully flat base change A \to A' = B and remark that the map A' = B \to B' = B \otimes _ A B has a natural section.
The same strategy proves the following lemma.
Lemma 59.16.4. If A \to B is faithfully flat and M is an A-module, then the complex (B/A)_\bullet \otimes _ A M is exact in positive degrees, and H^0((B/A)_\bullet \otimes _ A M) = M.
Proof.
See Descent, Lemma 35.3.6.
\square
Definition 59.16.5. Let A \to B be a ring map and N a B-module. A descent datum for N with respect to A \to B is an isomorphism \varphi : N \otimes _ A B \cong B \otimes _ A N of B \otimes _ A B-modules such that the diagram of B \otimes _ A B \otimes _ A B-modules
\xymatrix{ {N \otimes _ A B \otimes _ A B} \ar[dr]_{\varphi _{02}} \ar[rr]^{\varphi _{01}} & & {B \otimes _ A N \otimes _ A B} \ar[dl]^{\varphi _{12}} \\ & {B \otimes _ A B \otimes _ A N} }
commutes where \varphi _{01} = \varphi \otimes \text{id}_ B and similarly for \varphi _{12} and \varphi _{02}.
If N' = B \otimes _ A M for some A-module M, then it has a canonical descent datum given by the map
\begin{matrix} \varphi _\text {can}:
& N' \otimes _ A B
& \to
& B \otimes _ A N'
\\ & b_0 \otimes m \otimes b_1
& \mapsto
& b_0 \otimes b_1 \otimes m.
\end{matrix}
Definition 59.16.6. A descent datum (N, \varphi ) is called effective if there exists an A-module M such that (N, \varphi ) \cong (B \otimes _ A M, \varphi _\text {can}), with the obvious notion of isomorphism of descent data.
Theorem 59.16.2 is a consequence the following result.
Theorem 59.16.7. If A \to B is faithfully flat then descent data with respect to A\to B are effective.
Proof.
See Descent, Proposition 35.3.9. See also Descent, Remark 35.3.11 for an alternative view of the proof.
\square
Comments (2)
Comment #6514 by Janos Kollar on
Comment #6570 by Johan on