Lemma 32.3.2. Let $S$ be a scheme. Let $I$ be a set and for each $i \in I$ let $f_ i : T_ i \to S$ be a surjective affine morphism. Then the product $T = \prod T_ i$ in the category of schemes over $S$ (Lemma 32.3.1) maps surjectively to $S$.

**Proof.**
Let $s \in S$. Choose $t_ i \in T_ i$ mapping to $s$. Choose a huge field extension $K/\kappa (s)$ such that $\kappa (s_ i)$ embeds into $K$ for each $i$. Then we get morphisms $\mathop{\mathrm{Spec}}(K) \to T_ i$ with image $s_ i$ agreeing as morphisms to $S$. Whence a morphism $\mathop{\mathrm{Spec}}(K) \to T$ which proves there is a point of $T$ mapping to $s$.
$\square$

## 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Your email address will not be published.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)