Lemma 35.32.3. In the situation of Lemma 35.31.6 assume that $f : X' \to X$ is surjective and flat. Then the pullback functor is faithful.

**Proof.**
Let $(V_ i, \varphi _ i)$, $i = 1, 2$ be descent data for $X \to S$. Let $\alpha , \beta : V_1 \to V_2$ be morphisms of descent data. Suppose that $f^*\alpha = f^*\beta $. Our task is to show that $\alpha = \beta $. Note that $\alpha $, $\beta $ are morphisms of schemes over $X$, and that $f^*\alpha $, $f^*\beta $ are simply the base changes of $\alpha $, $\beta $ to morphisms over $X'$. Hence the lemma follows from Lemma 35.32.2.
$\square$

## 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Your email address will not be published.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)

There are also: