The Stacks project

Lemma 61.10.2. Let $X \to Y$ be a morphism of affine schemes. The following are equivalent

  1. $\{ X \to Y\} $ is a standard V covering (Topologies, Definition 34.10.1),

  2. $X = \mathop{\mathrm{lim}}\nolimits X_ i$ is a directed limit of affine schemes over $Y$ such that $\{ X_ i \to Y\} $ is a ph covering for each $i$, and

  3. $X = \mathop{\mathrm{lim}}\nolimits X_ i$ is a directed limit of affine schemes over $Y$ such that $\{ X_ i \to Y\} $ is an h covering for each $i$.

Proof. Proof of (2) $\Rightarrow $ (1). Recall that a V covering given by a single arrow between affines is a standard V covering, see Topologies, Definition 34.10.7 and Lemma 34.10.6. Recall that any ph covering is a V covering, see Topologies, Lemma 34.10.10. Hence if $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as in (2), then $\{ X_ i \to Y\} $ is a standard V covering for each $i$. Thus by Lemma 61.10.1 we see that (1) is true.

Proof of (3) $\Rightarrow $ (2). This is clear because an h covering is always a ph covering, see More on Flatness, Definition 38.34.2.

Proof of (1) $\Rightarrow $ (3). This is the interesting direction, but the interesting content in this proof is hidden in More on Flatness, Lemma 38.34.1. Write $X = \mathop{\mathrm{Spec}}(A)$ and $Y = \mathop{\mathrm{Spec}}(R)$. We can write $A = \mathop{\mathrm{colim}}\nolimits A_ i$ with $A_ i$ of finite presentation over $R$, see Algebra, Lemma 10.127.2. Set $X_ i = \mathop{\mathrm{Spec}}(A_ i)$. Then $\{ X_ i \to Y\} $ is a standard V covering for all $i$ by (1) and Topologies, Lemma 34.10.6. Hence $\{ X_ i \to Y\} $ is an h covering by More on Flatness, Definition 38.34.2. This finishes the proof. $\square$


Comments (0)


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Your email address will not be published.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