O(D) for D a divisor

Idea: for $X = \mathop{\mathrm{Proj}}S, S\in {\mathsf{gr}\,}_{\mathbf{Z}}\mathsf{CRing}$. define ${\mathcal{O}}_{\mathop{\mathrm{Proj}}S}(d) = \text{Ét}{\Sigma^d S}$ where for $A\in \mathsf{CRing}$ and $M\in {}_{A}{\mathsf{Mod}} $, $\text{Ét}M$ is the sheaf of continuous sections of the etale space \begin{align*}\coprod_{p\in \operatorname{Spec}A} M \left[ { \scriptstyle { {p}^{-1}} } \right] \to \operatorname{Spec}A.\end{align*} attachments/Pasted%20image%2020220418124236.png

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Idea: ${\mathcal{O}}_{{\mathbf{P}}^n_{/ {k}} }(d)$ are homogeneous polynomials of degree $d$.
For $M = \oplus_k M_k$ a graded module and $H^0(X; {\mathcal{F}}) = M_0$, twisting yields $H^0(X; {\mathcal{F}}\otimes{\mathcal{O}}(d)) = M_d$. homogeneous degree $d$ elements.
For ${\mathcal{J}}$ the ideal sheaf of $Y \subseteq {\mathbf{P}}^n_{/ {k}} $, $H^0(Y; {\mathcal{J}}\otimes{\mathcal{O}}(d))$ are homogeneous polynomials of degree $d$ that vanish on $Y$.
Def: ${\mathcal{O}}(1) \coloneqq?$
Def: ${\mathcal{O}}(n) \coloneqq{\mathcal{O}}(1)^{\otimes n}$?