Let $X$ be a closed subscheme of some $\mathbb{P}_A^n$ where $A$ is a ring. Then I know we can define twist sheaf on $\mathbb{P}_A^n$ by $\mathcal{O}(k):=\widetilde{R(k)}$ where $R=A[x_0,…,x_n]$.
First question: I'd like to ask if there exists a definition of twisted sheaf on $X$, maybe it can be defined as the pullback of $\mathcal{O}(k)$ under the closed immersion $i: X\rightarrow\mathbb{P}_A^n$. But $X$ maybe also embedding into other $\mathbb{P}_A^m$, we denote such a morphism $j: X\rightarrow\mathbb{P}_A^m$, then does $i^*\mathcal{O}(k)\cong j^*\mathcal{O}(k)$ hold?
Second question: How to define the twisted sheaf $\mathcal{O}(k)$ over $\mathbb{P}_S^n$ where $S$ is a general scheme? Is it defined by pulling back the sheaf $\mathcal{O}(k)$ over $\mathbb{P}_{\mathbb{Z}}^n$ under the projection $\mathbb{P}_S^n\rightarrow\mathbb{P}_{\mathbb{Z}}^n$?
Best Answer
Suppose $X = \operatorname{Proj} R_\bullet$, where $R_\bullet$ is a graded ring with $R_0 = A$, finitely generated in degree 1. Then we can define $\mathscr O(n)$ as the quasicoherent sheaf associated to the graded module $R(n)_\bullet$ (whose degree $k$ component is $R_{k+n}$). The sections of this sheaf over $D_+(f)$ are by definition the degree-n component of the localization $(R(n)_\bullet)_f$. For each $n$, this sheaf is a line bundle because of the assumptions on the graded ring. Now for any quasicoherent sheaf $\mathcal F$ on $X$, we can define the twist simply by $\mathcal F(n) = \mathcal F\otimes \mathscr O(n)$.
For the second question, you can globalize the construction above, by using the global $\operatorname{Proj}$ construction, which associates to every quasicoherent sheaf of graded $\mathscr O_S$-algebras over a scheme $S$ an $S$-scheme locally glued out of $\operatorname{Proj}$ of the graded $A$-algebras of sections over affine opens $\operatorname{Spec}A \subset S$. Assuming finite generation in degree 1, the global $\operatorname{Proj}$ construction comes with an $\mathscr O(1)$, locally glued out of the $\mathscr O(1)$'s from the previous paragraph. Then you can define $\mathscr O(n) = \mathscr O(1)^{\otimes n}$. Note that for the case of $\mathbb P^n_S$, the relevant sheaf of $\mathscr O_S$-algebras is $\mathscr O_S^{\oplus n+1}$.
These constructions are discussed in Vakil's FOAG 15.2 and 17.2, or in Hartshorne II.7 (starting page 160).