Lemma 32.3.3 (0CNK)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 32: Limits of Schemes
Section 32.3: Infinite products
Lemma 32.3.3 (cite)

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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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