Infinite products of affine schemes exist and are affine.

Lemma 32.3.1. 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 affine morphism. Then the product $T = \prod T_ i$ exists in the category of schemes over $S$. In fact, we have

$T = \mathop{\mathrm{lim}}\nolimits _{\{ i_1, \ldots , i_ n\} \subset I} T_{i_1} \times _ S \ldots \times _ S T_{i_ n}$

and the projection morphisms $T \to T_{i_1} \times _ S \ldots \times _ S T_{i_ n}$ are affine.

Proof. Omitted. Hint: Argue as in the discussion preceding the lemma and use Lemma 32.2.2 for existence of the limit. $\square$

Comment #3022 by Brian Lawrence on

Suggested slogan: We can take infinite products of schemes affine over a given base, and the infinite product is affine over any finite sub-product.

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