The Stacks project

Lemma 27.10.4. Let $S$ be a graded ring. Set $X = \text{Proj}(S)$. Fix $d \geq 1$ an integer. The following open subsets of $X$ are equal:

  1. The largest open subset $W = W_ d \subset X$ such that each $\mathcal{O}_ X(dn)|_ W$ is invertible and all the multiplication maps $\mathcal{O}_ X(nd)|_ W \otimes _{\mathcal{O}_ W} \mathcal{O}_ X(md)|_ W \to \mathcal{O}_ X(nd + md)|_ W$ (see 27.10.1.1) are isomorphisms.

  2. The union of the open subsets $D_{+}(fg)$ with $f, g \in S$ homogeneous and $\deg (f) = \deg (g) + d$.

Moreover, all the maps $\widetilde M(nd)|_ W = \widetilde M|_ W \otimes _{\mathcal{O}_ W} \mathcal{O}_ X(nd)|_ W \to \widetilde{M(nd)}|_ W$ (see 27.10.1.5) are isomorphisms.

Proof. If $x \in D_{+}(fg)$ with $\deg (f) = \deg (g) + d$ then on $D_{+}(fg)$ the sheaves $\mathcal{O}_ X(dn)$ are generated by the element $(f/g)^ n = f^{2n}/(fg)^ n$. This implies $x$ is in the open subset $W$ defined in (1) by arguing as in the proof of Lemma 27.10.2.

Conversely, suppose that $\mathcal{O}_ X(d)$ is free of rank 1 in an open neighbourhood $V$ of $x \in X$ and all the multiplication maps $\mathcal{O}_ X(nd)|_ V \otimes _{\mathcal{O}_ V} \mathcal{O}_ X(md)|_ V \to \mathcal{O}_ X(nd + md)|_ V$ are isomorphisms. We may choose $h \in S_{+}$ homogeneous such that $x \in D_{+}(h) \subset V$. By the definition of the twists of the structure sheaf we conclude there exists an element $s$ of $(S_ h)_ d$ such that $s^ n$ is a basis of $(S_ h)_{nd}$ as a module over $S_{(h)}$ for all $n \in \mathbf{Z}$. We may write $s = f/h^ m$ for some $m \geq 1$ and $f \in S_{d + m \deg (h)}$. Set $g = h^ m$ so $s = f/g$. Note that $x \in D_{+}(g)$ by construction. Note that $g^ d \in (S_ h)_{-d\deg (g)}$. By assumption we can write this as a multiple of $s^{\deg (g)} = f^{\deg (g)}/g^{\deg (g)}$, say $g^ d = a/g^ e \cdot f^{\deg (g)}/g^{\deg (g)}$. Then we conclude that $g^{d + e + \deg (g)} = a f^{\deg (g)}$ and hence also $x \in D_{+}(f)$. So $x$ is an element of the set defined in (2).

The existence of the generating section $s = f/g$ over the affine open $D_{+}(fg)$ whose powers freely generate the sheaves of modules $\mathcal{O}_ X(nd)$ easily implies that the multiplication maps $\widetilde M(nd)|_ W = \widetilde M|_ W \otimes _{\mathcal{O}_ W} \mathcal{O}_ X(nd)|_ W \to \widetilde{M(nd)}|_ W$ (see 27.10.1.5) are isomorphisms. Compare with the proof of Lemma 27.10.2. $\square$


Comments (2)

Comment #8107 by Laurent Moret-Bailly on

Line 6 of proof: should contain . Line 9: should be .

There are also:

  • 2 comment(s) on Section 27.10: Invertible sheaves on Proj

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 01MU. Beware of the difference between the letter 'O' and the digit '0'.