The Stacks project

Lemma 34.8.2. Let $\{ f_ j : U_ j \to T\} _{j = 1, \ldots , m}$ be a standard ph covering. Let $T' \to T$ be a morphism of affine schemes. Then $\{ U_ j \times _ T T' \to T'\} _{j = 1, \ldots , m}$ is a standard ph covering.

Proof. Let $f : U \to T$ be proper surjective and let an affine open covering $U = \bigcup _{j = 1, \ldots , m} U_ j$ be given as in Definition 34.8.1. Then $U \times _ T T' \to T'$ is proper surjective (Morphisms, Lemmas 29.9.4 and 29.41.5). Also, $U \times _ T T' = \bigcup _{j = 1, \ldots , m} U_ j \times _ T T'$ is an affine open covering. This concludes the proof. $\square$

Comments (0)

There are also:

  • 3 comment(s) on Section 34.8: The ph 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Your email address will not be published.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 0DBE. Beware of the difference between the letter 'O' and the digit '0'.