ag.algebraic geometry - Can isomorphisms of schemes be constructed on formal neighborhoods?

No. Let A be k[[t]]. Let X be A^1 setminus {-1,0,1} and Y be A^1 setminus {1,t,-1}. In explicit equations, X = Spec k[[t]][x, y]/y(x-1)x(x+1)-1 and Y = Spec k[[t]][x, y]/y(x-1)(x-t)(x+1)-1.

Over k[[t]]/t^{n+1}, the reductions of X and Y are isomorphic because all infinitesimal deformations of a smooth affine scheme are trivial. (See corollary 4.7 in Hartshorne's notes on deformation theory.)

However, X and Y are not isomorphic because the two fibers over the general point are not: For any field K, if we have P_K^1 setminus {a,b,c,d} for {a,b,c,d} in P^1(K), then the cross ratio of a,b,c and d is a well defined element of K. In particular, this applies when K=k((t)).