= Milnor patching of projective modules
{c}
A compatible triple $(P_1,P_2,\phi)$ patches to $\{(x_1,x_2):\phi(\bar x_1)=\bar x_2\}$. It is finitely generated projective and recovers $P_i$ after <extension of scalars>. For <free modules> glued by $g$, the block identity $\operatorname{diag}(g,g^{-1})\in E_\infty(A_0)$ lifts through the surjective side and makes the <direct sum> with inverse gluing free. Complements reduce general projective triples to that free case. Conversely an <idempotent> over $A$ is exactly a compatible pair of <idempotents> over its two factors.
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