= Exactness of the unit-ideal localization Čech complex
{title2=$0\to M\to\prod_iM_{f_i}\to\prod_{i<j}M_{f_if_j}\to\cdots$}
For a finite family generating the <unit ideal> in a <commutative ring>, this augmented <Čech cochain complex> of any <module> is exact. One proof clears the finitely many denominators and cocycle relations by powers $f_i^N$, writes $1=\sum_i a_if_i^N$, and contracts an alternating cocycle by the weighted insertions $\sum_i a_if_i^Nc_{iI}$. The argument uses <vanishing criterion in a module localization> and does not require the <module> to be finitely generated.
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