We know that a coherent sheaf on a scheme is determined by its restriction on certain open coverings (satisfying compatibility condition). Now I wonder how about a closed covering. To do so I started with simple cases on a smooth complex projective varieity $X$ of dimension $n$.
Taking $X$ itself to be the covering is trivial, so I want to start with coverings of smalll dimensions. Firstly I think about the closed points (covering of dimension zero), but they are not a covering; also coherent sheaf with same stalks can be different.
Then it comes to dimension one coverings: let $X$ a smooth projective variety which can be covered by lines. Given two coherent sheaves $\mathcal{F}$ and $\mathcal{G}$ on $X$ such that
$$\iota^*_L\mathcal{F}\cong\iota^*_L\mathcal{G}$$
for any line $\iota_L:L\subset X$. What could one say about the relation between $\mathcal{F}$ and $\mathcal{G}$?
I think we should have $\mathcal{F}\cong\mathcal{G}$ for $X=\mathbb{P}^n$ by the structure of $\textbf{Coh}(\mathbb{P}^n)$ (edited: no we do not, here I should write $\mathbb{P}^1$ because the vector bundles on $\mathbb{P}^n$ are not necessarily decomposable).
What about the general case, for example a ruled surface?
I think such questions should be considered before. If so, any reference is welcome!
Best Answer
It is not true even for $\mathbb{P}^n$. For instance, the tangent bundle $T_{\mathbb{P}^n}$ restricts to each line as $$ T_{\mathbb{P}^n}\vert_L \cong \mathcal{O}_L(2) \oplus \mathcal{O}_L(1)^{\oplus (n-1)}, $$ and on the other hand $$ (\mathcal{O}_{\mathbb{P}^n}(2) \oplus \mathcal{O}_{\mathbb{P}^n}(1)^{\oplus (n-1)})\vert_L \cong \mathcal{O}_L(2) \oplus \mathcal{O}_L(1)^{\oplus (n-1)}, $$ however $T_{\mathbb{P}^n} \not\cong (\mathcal{O}_{\mathbb{P}^n}(2) \oplus \mathcal{O}_{\mathbb{P}^n}(1)^{\oplus (n-1)})$.