Why might church leaders and Europe's traditional nobility have cooperated for so long

Besides the obvious "God wants me to support the church so that's what I'll do," nobles and clergy both believed in a natural hierarchal system known as "The Great Chain of Being." The Great Chain of Being essentially stated that everything in the universe had its place. God was at the top of the chain, and inanimate objects such as metals and dirt were at the bottom with everything else falling somewhere in between.

The nobility and the clergy where both very invested in maintaining the Great Chain, as it allowed them to retain the power that they enjoyed over the people of medieval Europe, thus they cooperated to retain that power structure even though they may have often been at odds with each other politically.