The Stacks project

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. $\square$

Comments (1)

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.

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CNI. Beware of the difference between the letter 'O' and the digit '0'.