This question is regarding formal vector bundles on formal schemes. The formal schemes that I'm interested are formal completion of a scheme along a closed sub-scheme. The category of formal coherent sheaves form an abelian category. 1)Given a short sequence of formal vector bundles is it true that the sequence is exact iff its pullback to all $n$-th thickenings are exact? 2) Is it true that formal vector bundles are closed under extension? i.e. given a short exact sequence of formal coherent sheaves if the kernel and co-kernel are formal vector bundles then the same is true for the extension.
If these are not true then consider the following extra conditions. Now assume that the closed subvariety satisfies the effective lefschetz condition i.e. category of vector bundles on it is equivalent to the category of vector bundles on some neighborhood of the closed subscheme. Then 1) becomes obviously true, since you can lift the exact sequence to a nbhd. How about 2?
If true I wonder whether there is a reference for these.