Lemma 27.8.8. Let $S$ be a graded ring. The scheme $\text{Proj}(S)$ is separated.

Proof. We have to show that the canonical morphism $\text{Proj}(S) \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is separated. We will use Schemes, Lemma 26.21.7. Thus it suffices to show given any pair of standard opens $D_{+}(f)$ and $D_{+}(g)$ that $D_{+}(f) \cap D_{+}(g) = D_{+}(fg)$ is affine (clear) and that the ring map

$S_{(f)} \otimes _{\mathbf{Z}} S_{(g)} \longrightarrow S_{(fg)}$

is surjective. Any element $s$ in $S_{(fg)}$ is of the form $s = h/(f^ ng^ m)$ with $h \in S$ homogeneous of degree $n\deg (f) + m\deg (g)$. We may multiply $h$ by a suitable monomial $f^ ig^ j$ and assume that $n = n' \deg (g)$, and $m = m' \deg (f)$. Then we can rewrite $s$ as $s = h/f^{(n' + m')\deg (g)} \cdot f^{m'\deg (g)}/g^{m'\deg (f)}$. So $s$ is indeed in the image of the displayed arrow. $\square$

## Comments (0)

There are also:

• 10 comment(s) on Section 27.8: Proj of a graded ring

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Your email address will not be published.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