= Ordered Čech cochain complex
{title2=$\check C^q_+(\mathcal U,\mathcal F)=\prod_{i_0<\cdots<i_q}\mathcal F(U_{i_0}\cap\cdots\cap U_{i_q})$}
For a <well-order> on the indices of an <open cover>, the ordered <Čech cochain complex> uses only strictly increasing tuples, with the alternating restriction <Čech differential>. It computes the same <Čech cohomology> as the complex using all tuples. If every finite intersection is nonempty and the coefficient <constant sheaf> has the same <abelian group> $A$ on each intersection, inserting the least index $\alpha$ gives a <cochain homotopy>: set $(hs)_I=s_{\alpha,I}$ when $\alpha$ is not in $I$, and $(hs)_I=0$ otherwise. Expanding the alternating sums gives $\delta h+h\delta=\operatorname{id}$ in every positive degree, since the terms deleting an index of $I$ cancel and the term deleting $\alpha$ is $s_I$. Hence positive <Čech cohomology> vanishes in this situation; degree zero remains $A$.
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