Lemma 32.3.3. Let $S$ be a scheme. Let $I$ be a set and for each $i \in I$ let $f_ i : T_ i \to S$ be an integral morphism. Then the product $T = \prod T_ i$ in the category of schemes over $S$ (Lemma 32.3.1) is integral over $S$.

**Proof.**
Omitted. Hint: On affine pieces this reduces to the following algebra fact: if $A \to B_ i$ is integral for all $i$, then $A \to \otimes _ A B_ i$ is integral.
$\square$

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