I have a question which probably is very straightforward but because of my lack of knowledge of formal schemes I'm asking it here. Let's assume we have a vector bundle $E$ on the formal completion $X_Z$ of a scheme $X$ along a closed subscheme $Z$. We know there is a coherent sheaf that extends $E$ on each affine. Does it follow that we can extend $E$ to a coherent sheaf?
Some of my thoughts: Let's assume for simplicity we are working with an affine chart consisting of two affines on $X$ this induces an affine chart on the formal completion too. Let $E_1$ and $E_2$ be the restriction of $E$ to each of them and $M_1$, $M_2$ be their coherent extensions. Restriction of $E$ to the intersection of two chart is denoted by $E_{12}$. The vector bundle $E$ fits into a short exact sequence on $X_Z$: $$0\rightarrow E \rightarrow E_1\oplus E_2 \rightarrow E_{12}\rightarrow 0 \hspace{1cm}(1)$$ By the definition of gluing vector bundles. I was thinking whether it is possible (or when it is possible) to define pushforward of $E_{12}$ from the formal completion in the intersection of two affines to the intersection of two affines? If we can define a pushforward that is a coherent sheaf denoted by $i_*E_{12}$. Then we have a morphism $M_1\oplus M_2 \rightarrow i_*(E_{12})$ that restricts on the formal completion to the surjection $E_1\oplus E_2 \rightarrow E_{12}$. Taking the kernel $0\rightarrow \text{kernel} \rightarrow M_1\oplus M_2 \rightarrow i_*(E_{12})$ and using the fact that restricting to formal completion is exact we get the exact sequence (1). So $\text{kernel}$ will be extending $E$.
