The Stacks project

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 $s$. $\square$

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