Suppose we stack coaxially, vertically a large number of rings, made of some high temperature superconducting material, and start a current through each of the rings in alternate directions (e.g. the 1st ring CW, 2nd ring CCW, 3rd ring CW, etc). Each ring will repel the rings adjacent to it, but since the repulsion is inversely proportional to the distance between rings to the 4th power, the effect on non-adjacent rings should be very minor. Using some simple mechanical restraining to keep the rings from sliding away horizontally from co-axiality, we should be able to see the condensed stack starting to rise up and each ring hovering above the one below it. Then, once such a tower is erected high enough, there could be various ways to attach "climbers" so as to use it as a space elevator.
The lower rings may require some cooling but above a certain altitude the low ambient temperature should make various materials superconducting without any additional cooling. Alternatively, extremely strong and lightweight conducting materials such as carbon nanotubes could be used so that the reduction in weight requires less current to hold up the tower.
What are the flaws in the above reasoning? (I presume there are many...)