The Stacks project

Lemma 32.5.1. Let $W$ be a quasi-affine scheme of finite type over $\mathbf{Z}$. Suppose $W \to \mathop{\mathrm{Spec}}(R)$ is an open immersion into an affine scheme. There exists a finite type $\mathbf{Z}$-algebra $A \subset R$ which induces an open immersion $W \to \mathop{\mathrm{Spec}}(A)$. Moreover, $R$ is the directed colimit of such subalgebras.

Proof. Choose an affine open covering $W = \bigcup _{i = 1, \ldots , n} W_ i$ such that each $W_ i$ is a standard affine open in $\mathop{\mathrm{Spec}}(R)$. In other words, if we write $W_ i = \mathop{\mathrm{Spec}}(R_ i)$ then $R_ i = R_{f_ i}$ for some $f_ i \in R$. Choose finitely many $x_{ij} \in R_ i$ which generate $R_ i$ over $\mathbf{Z}$. Pick an $N \gg 0$ such that each $f_ i^ Nx_{ij}$ comes from an element of $R$, say $y_{ij} \in R$. Set $A$ equal to the $\mathbf{Z}$-algebra generated by the $f_ i$ and the $y_{ij}$ and (optionally) finitely many additional elements of $R$. Then $A$ works. Details omitted. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 32.5: Absolute Noetherian Approximation

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Your email address will not be published.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 01Z7. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 32.5.1 as pdf