Twists of a quasi-coherent sheaf

Assume the Axiom of Choice (The Axiom of Choice). Let $S={⨁}_{d\ge 0}{S}_d$ be a commutative nonnegatively graded ring that is generated as an $S_0$-algebra by its degree one part $S_1$, and let $X=\mathrm{Proj}S$ with twisting sheaves ${\mathcal{O}}_X(d)=\widetilde{S(d)}$ (Twisting sheaf on Proj). By Invertible twists for degree-one generated rings each ${\mathcal{O}}_X(d)$ is invertible (Invertible sheaves), and the multiplication maps ${\mathcal{O}}_X(m){\otimes }_{{\mathcal{O}}_X}{\mathcal{O}}_X(n)\to {\mathcal{O}}_X(m+n)$ are isomorphisms.

Fixed embedding. The scheme $X$ is considered together with this fixed choice of degree-one generated presentation. Let $\mathcal{F}$ be an ${\mathcal{O}}_X$-module. For every integer $d\in \mathbb{Z}$ the $d$-th twist of $\mathcal{F}$ is the tensor product of sheaves of modules (Tensor product of sheaves of modules) $\mathcal{F}(d):=\mathcal{F}{\otimes }_{{\mathcal{O}}_X}{\mathcal{O}}_X(d)\mathrm{.}$ Thus $\mathcal{F}(d)$ is again an ${\mathcal{O}}_X$-module, and for $d=0$ one has $\mathcal{F}(0)=\mathcal{F}{\otimes }_{{\mathcal{O}}_X}{\mathcal{O}}_X$, with the canonical morphism $\mathcal{F}(0)\to \mathcal{F}$ induced by the multiplication maps of Twisting sheaf on Proj; this morphism is an isomorphism because ${\mathcal{O}}_X(0)={\mathcal{O}}_X$. For all integers $m,n$ the isomorphisms above induce canonical isomorphisms $\mathcal{F}(m){\otimes }_{{\mathcal{O}}_X}{\mathcal{O}}_X(n)\cong \mathcal{F}(m+n),{\mathcal{O}}_X(m){\otimes }_{{\mathcal{O}}_X}\mathcal{F}(n)\cong \mathcal{F}(m+n)\mathrm{.}$

Quasi-coherence. If $\mathcal{F}$ is quasi-coherent then so is every twist $\mathcal{F}(d)$: the invertible sheaf ${\mathcal{O}}_X(d)$ is quasi-coherent, and the tensor product of two quasi-coherent modules is quasi-coherent by Tensor product preserves quasi-coherence.

General ample line bundles. If instead $L$ is an invertible ${\mathcal{O}}_X$-module (Invertible sheaves) — for instance an ample one (Absolute ampleness by affine section opens) — then for $d\ge 0$ one writes $L^d:=L^{\otimes d}=L{\otimes }_{{\mathcal{O}}_X}\cdots {\otimes }_{{\mathcal{O}}_X}L$ ($d$ factors, $L^0={\mathcal{O}}_X$) and for $d<0$ one writes $L^d:=(L^{\vee }{)}^{-d}$, where $L^{\vee }=\mathcal{H}o{m}_{{\mathcal{O}}_X}(L,{\mathcal{O}}_X)$ is the inverse invertible sheaf of Invertible sheaves; the corresponding twist of $\mathcal{F}$ is $\mathcal{F}{\otimes }_{{\mathcal{O}}_X}{L}^d$. For these powers only the pair $(X,L)$ is used, with no presentation of $X$ implicit.

No silent change of embedding. The notation $\mathcal{F}(d)$ is reserved for the twisting determined by the fixed invertible sheaf ${\mathcal{O}}_X(1)$ of the chosen presentation $\mathrm{Proj}S$ (or, for a projectively embedded scheme, by the pullback of $\mathcal{O}(1)$ along the fixed embedding). When the embedding or the line bundle changes, the twist is written explicitly as $\mathcal{F}{\otimes }_{{\mathcal{O}}_X}{L}^d$, so that $\mathcal{F}(d)$ never silently switches the embedding.