Lemma 34.34.1. Let $S$ be a scheme. Let $\{ X_ i \to S\} _{i\in I}$ be an fpqc covering, see Topologies, Definition 33.9.1. Let $(V_ i/X_ i, \varphi _{ij})$ be a descent datum relative to $\{ X_ i \to S\} $. If each morphism $V_ i \to X_ i$ is affine, then the descent datum is effective.

**Proof.**
Being affine is a property of morphisms of schemes which is preserved under any base change, see Morphisms, Lemma 28.11.8. Hence Lemma 34.33.2 applies and it suffices to prove the statement of the lemma in case the fpqc-covering is given by a single $\{ X \to S\} $ flat surjective morphism of affines. Say $X = \mathop{\mathrm{Spec}}(A)$ and $S = \mathop{\mathrm{Spec}}(R)$ so that $R \to A$ is a faithfully flat ring map. Let $(V, \varphi )$ be a descent datum relative to $X$ over $S$ and assume that $V \to X$ is affine. Then $V \to X$ being affine implies that $V = \mathop{\mathrm{Spec}}(B)$ for some $A$-algebra $B$ (see Morphisms, Definition 28.11.1). The isomorphism $\varphi $ corresponds to an isomorphism of rings

as $A \otimes _ R A$-algebras. The cocycle condition on $\varphi $ says that

is commutative. Inverting these arrows we see that we have a descent datum for modules with respect to $R \to A$ as in Definition 34.3.1. Hence we may apply Proposition 34.3.9 to obtain an $R$-module $C = \mathop{\mathrm{Ker}}(B \to A \otimes _ R B)$ and an isomorphism $A \otimes _ R C \cong B$ respecting descent data. Given any pair $c, c' \in C$ the product $cc'$ in $B$ lies in $C$ since the map $\varphi $ is an algebra homomorphism. Hence $C$ is an $R$-algebra whose base change to $A$ is isomorphic to $B$ compatibly with descent data. Applying $\mathop{\mathrm{Spec}}$ we obtain a scheme $U$ over $S$ such that $(V, \varphi ) \cong (X \times _ S U, can)$ as desired. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)