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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?

If these are not true in general I'd like to know cases that they can be true.

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.

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 $n_i\geq 0$, 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 in the form $\bigoplus \mathcal{O}_Z(n_i)$ 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 of the aforementioned type 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?

If these are not true in general I'd like to know cases that they can be true.

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?

If these are not true in general I'd like to know cases that they can be true.

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. So in the category of vector bundles of the form $\bigoplus \mathcal{O}(n_i)$ where $n_i\geq 0$, obviously we can lift each object from the variety to the ambient projective space. 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 in the form $\bigoplus \mathcal{O}_Z(n_i)$ 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 of the aforementioned type on $Z$ to the formal completion $X_Z$? (or some neighborhood of $Z$ along $X$)

If these are not true in general I'd like to know cases that they can be true.

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 $n_i\geq 0$, 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 in the form $\bigoplus \mathcal{O}_Z(n_i)$ 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 of the aforementioned type 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?

If these are not true in general I'd like to know cases that they can be true.