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58.16 Faithfully flat descent

Definition 58.16.1. 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 family $(\mathcal{F}_ i, \varphi _{ij})_{i, j\in I}$ where

  1. for all $i$, $\mathcal{F}_ i$ is a quasi-coherent sheaf on $T_ i$, and

  2. for all $i, j \in I$ the map $\varphi _{ij} : \text{pr}_0^* \mathcal{F}_ i \cong \text{pr}_1^* \mathcal{F}_ j$ is an isomorphism on $T_ i \times _ T T_ j$ such that 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 $\varphi _ i : t_ i^* \mathcal{F} \cong \mathcal{F}_ i$ satisfying the cocycle condition, 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 58.16.2. 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 58.16.3. 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 58.16.4. 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 58.16.5. 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 58.16.6. 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 58.16.2 is a consequence the following result.

Theorem 58.16.7. 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. $\square$

Remarks 58.16.8. The results on descent of modules have several applications:

  1. The exactness of the Čech complex in positive degrees for the covering $\{ \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)\} $ where $A \to B$ is faithfully flat. This will give some vanishing of cohomology.

  2. If $(N, \varphi )$ is a descent datum with respect to a faithfully flat map $A \to B$, then the corresponding $A$-module is given by

    \[ M = \mathop{\mathrm{Ker}}\left( \begin{matrix} N & \longrightarrow & B \otimes _ A N \\ n & \longmapsto & 1 \otimes n - \varphi (n \otimes 1) \end{matrix} \right). \]

    See Descent, Proposition 35.3.9.

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