Twisting sheaf on Proj

Assume the Axiom of Choice (The Axiom of Choice), inherited from the associated-sheaf construction of Associated sheaf of a graded module on Proj. Let $S={⨁}_{d\ge 0}{S}_d$ be a commutative nonnegatively graded ring, let $X=\mathrm{Proj}S$ be the scheme of Proj carries a scheme structure, and for an integer $n$ let $S(n)$ be the graded $S$-module with $S(n{)}_d=S_{n+d}$ (Graded shift convention for Proj). The twisting sheaf ${\mathcal{O}}_X(n)$ is the associated sheaf of the graded module $S(n)$: ${\mathcal{O}}_X(n)=\widetilde{S(n)}$ in the sense of Associated sheaf of a graded module on Proj. Thus on a standard open $D_{+}(f)$ with $f$ homogeneous of positive degree, $\mathrm{\Gamma }(D_{+}(f),{\mathcal{O}}_X(n))=S(n{)}_{(f)}=\{a\mathrm{/}{f}^k\in S(n)[f^{-1}]:a\in S(n)\text{ homogeneous of degree }kd\},$ the degree-zero part of the homogeneous localisation of $S(n)$, with restriction maps induced by homogeneous localisation.

Because $S(0)=S$ and $\tilde{S}={\mathcal{O}}_X$ by construction of the structure sheaf, one has ${\mathcal{O}}_X(0)={\mathcal{O}}_X$. For every pair of integers $m,n$ the multiplication of the graded ring defines sheaf morphisms ${\mathcal{O}}_X(m){\otimes }_{{\mathcal{O}}_X}{\mathcal{O}}_X(n)⟶{\mathcal{O}}_X(m+n),$ induced on $D_{+}(f)$ by the $S_{(f)}$-bilinear maps $S(m{)}_{(f)}\times S(n{)}_{(f)}\to S(m+n{)}_{(f)}$, $(a\mathrm{/}{f}^k,b\mathrm{/}{f}^l)\mapsto ab\mathrm{/}{f}^{k+l}$, which are compatible with the restriction maps; the induced maps ${\mathcal{O}}_X(0)\otimes {\mathcal{O}}_X(n)\to {\mathcal{O}}_X(n)$ are the canonical identifications.

No invertibility is asserted here. For an arbitrary nonnegatively graded ring the sheaf ${\mathcal{O}}_X(n)$ need not be invertible, and the multiplication maps above need not be isomorphisms. The precise positive statement is Invertible twists for degree-one generated rings: if $S$ is generated as an $S_0$-algebra by $S_1$, then every ${\mathcal{O}}_X(n)$ is invertible and every such multiplication map is an isomorphism. The sign convention $S(n{)}_d=S_{n+d}$ of Graded shift convention for Proj is used throughout this page.

With this convention, the frames in the later example Twist transitions on the projective line ↗ transform by $e_1=t^n{e}_0$.