The Stacks project

Lemma 34.5.4. Let $T$ be an affine scheme. Let $\{ T_ i \to T\} _{i \in I}$ be a smooth covering of $T$. Then there exists a smooth covering $\{ U_ j \to T\} _{j = 1, \ldots , m}$ which is a refinement of $\{ T_ i \to T\} _{i \in I}$ such that each $U_ j$ is an affine scheme, and such that each morphism $U_ j \to T$ is standard smooth, see Morphisms, Definition 29.34.1. Moreover, we may choose each $U_ j$ to be open affine in one of the $T_ i$.

Proof. Omitted, but see Algebra, Lemma 10.137.10. $\square$

Comments (0)

There are also:

  • 3 comment(s) on Section 34.5: The smooth topology

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Your email address will not be published.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 0222. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 34.5.4 as pdf