= Idempotent–clopen correspondence
{title2=$e\longleftrightarrow D(e)=V(1-e)$}
A global <idempotent> on a <scheme> is zero or one in every <local ring>. Its one-locus is both open and closed. Conversely a clopen subset determines a global <idempotent> by gluing one on it and zero on its complement. Under this correspondence, decompositions into nonzero <orthogonal idempotents> correspond to disjoint clopen decompositions. When all <connected components> are open, the <primitive idempotents> correspond exactly to those components. Without orthogonality the assertion fails in characteristic two: $(1,0)=(1,1)+(0,1)$ in $\mathbb F_2\times\mathbb F_2$.
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