---
date: 2022-04-05 23:42
modification date: Tuesday 5th April 2022 23:42:25
title: "O(D) for D a divisor"
aliases: ["O(D) for D a divisor", "O(D)", "O(1)", "O", twisting, twisting sheaf, "Serre's twisting sheaf"]
created: 2023-03-28T21:19
updated: 2023-03-28T21:19
---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [Serre's theorem](Serre's%20theorem)
	- [divisor](divisor.md)

---

# O(D) for D a divisor

![](2023-03-28-14.png)


Idea: for $X = \Proj S, S\in \gr_\ZZ \CRing$. define $\OO_{\Proj S}(d) = \Et{\Sigma^d S}$ where for $A\in \CRing$ and $M\in\mods{A}$, $\Et M$ is the sheaf of continuous sections of the etale space $$\Disjoint_{p\in \spec A} M\localize{p} \to \spec A.$$
![](attachments/Pasted%20image%2020220418124236.png)

![](attachments/Pasted%20image%2020220418100445.png)

- Idea: $\OO_{\PP^n\slice k}(d)$ are homogeneous polynomials of degree $d$.
- For $M = \oplus_k M_k$ a graded module and $H^0(X; \mcf) = M_0$, twisting yields $H^0(X; \mcf \tensor \OO(d)) = M_d$. homogeneous degree $d$ elements.
- For $\mcj$ the ideal sheaf of $Y \subseteq \PP^n\slice k$, $H^0(Y; \mcj \tensor \OO(d))$ are homogeneous polynomials of degree $d$ that vanish on $Y$.
- Def: $\OO(1) \da ?$
- Def: $\OO(n) \da \OO(1)^{\tensor n}$?