Given an very ample line bundle $L$ on a projective variety we embed it into a projective space such that pullback of $\mathcal{O}(1)$ is $L$. Then we can identify the two. Consider the full sub-category of vector bundles of the form $\bigoplus \mathcal{O}(n_i)$ where $n_i\geq 0$ (this is a finite direct sum, so these are finite rank vector bundles), obviously we can lift each object from the variety to the ambient projective space. For any variety $X$ we denote this category by $\mathcal{C}_X$.
Now consider a projective variety $X$ a subvariety $Z$ (assume $Z$ is a hyperplane section). Is it true that we can lift the morphisms between two objects $\mathcal{C}_Z$ to a neighborhood of $Z$ in $X$?(or formal completion of $Z$ along $X$). If so is it possible to give a functorial lift from the category of vector bundles $\mathcal{C}_Z$ on $Z$ to the formal completion $X_Z$? (or some neighborhood of $Z$ along $X$)
So the question is whether there is a functor from $\mathcal{C}_Z$ to $\mathcal{C}_{X_Z}$ denoted by $i$, such that $res\circ i=id$ where $res$ is the restriction functor from $\mathcal{C}_{X_Z}$ to $\mathcal{C}_Z$? If such a functor exists (maybe in certain cases) when is it an equivalence of categories?
Since the morphisms between two objects of $\mathcal{C}_Z$ splits into direct sum of global section of line bundles in the form $\mathcal{O}(n)$ we can deduce that problem is equivalent to asking when the ring $R_Z=\bigoplus_{n\geq 0}\Gamma(Z, \mathcal{O}_Z(n))$ split injects into the ring $R_{X_Z}=\bigoplus_{n\geq 0}\Gamma(X_Z, \mathcal{O}_{X_Z}(n))$? and when they are isomorphic. (This might also depend on the choice of $\mathcal{O}(1)$)
If these are not true in general I'd like to know cases that they can be true.
