---
tag: JR83AKA184
kind: definition
name: Serre's twisting sheaf
msc: [14F06, 14C20]
defines: V93Y0XYPZD
uses: [CAPQBQDRA7, A39XQHCVNW, KPKND0K7W6, 7JCF2TE3V5]
sources:
  - url: https://en.wikipedia.org/wiki/Proj_construction
---

## Definition

Let $S = \bigoplus_{d \ge 0} S_d$ be a [[graded ring|CAPQBQDRA7]], and let $X = \operatorname{Proj} S$ be given by the [[Proj construction|A39XQHCVNW]].
For an integer $n$, let $S(n)$ be the graded $S$-module with $S(n)_d = S_{n + d}$, where $S_e = 0$ for $e < 0$.
Let $M$ be a graded $S$-module, and let $f \in S$ be homogeneous of positive degree.
Let $\widetilde{M}$ be the sheaf on $X$ with sections $M_{(f)}$ over $D_+(f)$, where $M_{(f)}$ is the degree zero part of $M_f$.
The *twisting sheaf* of degree $n$ is the [[quasi-coherent sheaf|KPKND0K7W6]] $\mathcal{O}_X(n) = \widetilde{S(n)}$.
*Serre's twisting sheaf* is $\mathcal{O}_X(1)$.

## Remarks

Let $S = k[x_0, \dots, x_r]$ over a field $k$, with $r \ge 1$, so that $X = \mathbb{P}^r_k$.
Then $\mathcal{O}_X(n)$ is an [[invertible sheaf|7JCF2TE3V5]], and its global sections are the homogeneous polynomials of degree $n$ for $n \ge 0$, and $0$ for $n < 0$.
On $D_+(x_i)$, the section $x_i^n$ generates $\mathcal{O}_X(n)$, which gives the local isomorphism with $\mathcal{O}_X$.
If $S$ is generated by $S_1$ as an $S_0$-algebra, then each $\mathcal{O}_X(n)$ is invertible and $\mathcal{O}_X(n) \otimes \mathcal{O}_X(m) \cong \mathcal{O}_X(n + m)$.
Without that hypothesis $\mathcal{O}_X(1)$ need not be invertible.
Take $S = k[x_0, x_1, x_2]$ with $x_0, x_1$ of degree $1$ and $x_2$ of degree $2$, and the point $p$ where $x_0 = x_1 = 0$.
On $D_+(x_2)$, the sections of $\mathcal{O}_X(1)$ need the two generators $x_0$ and $x_1$ near $p$, so $\mathcal{O}_X(1)$ is not invertible at $p$.
For a quasi-coherent sheaf $\mathcal{F}$, one writes $\mathcal{F}(n) = \mathcal{F} \otimes \mathcal{O}_X(n)$, the twist of $\mathcal{F}$ by $n$.