Let $B$ be a curve (so separated, of finite type, and universally closed $1$-dimensional scheme over a field k) having a closed regular point $b$.
Futhermore let $\mathcal{E}$ be a loacally free sheaf of rank $2$. By consruction of relative Proj with $\mathcal{E}$ we get $S:= Proj(Sym^.\mathcal{E})$ with induced graduation. This provides us the structure morphism $f: S =Proj(Sym^.\mathcal{E}) \to B$.
Obviouly, $S$ has a canonically given taugological sheaf $\mathcal{O}_S(1)$.
Let $f^{-1}(b)$ be the closed fiber with property $f^{-1}(b) \cong \mathbb{P}^1$ as closed subscheme.
My question is why does in this case hold
$$\mathcal{O}_S(1) \vert _{\mathbb{P}^1} = \mathcal{O}_{\mathbb{P}^1}(1)$$
?
Obviously, for arbitrary closed subscheme $C \subset X$ of a scheme $X$ generally never holds $$\mathcal{O}_X \vert _C=\mathcal{O}_C$$
or like in our case
$$\mathcal{O}_S (1) \vert _C=\mathcal{O}_C(1)$$
is generally wrong.
But why does here
$$\mathcal{O}_S(1) \vert _{\mathbb{P}^1} = \mathcal{O}_{\mathbb{P}^1}(1)$$
hold? Have I to interpret this "restriction" as a pull back? And does this property (especially conservation of graduation) hold for pull backs?