Milnor square

Summary

A Milnor square is a commutative square of rings in which one ring is the pullback of two others over a third. In addition, one of the two maps to the third ring is surjective. Milnor showed that projective modules over the pullback can be built by patching modules over the two corners. He also showed that $K_0$ and $K_1$ of the four rings fit into a Mayer–Vietoris sequence. It is the algebraic analogue of covering a space by two closed pieces.