Exactness of the unit-ideal localization Čech complex ($0\to M\to {\prod }_i{M}_{f_i}\to {\prod }_{i<j}{M}_{f_i{f}_j}\to \cdots$)

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 $1={\sum }_i{a}_i{f}_i^N$, and contracts an alternating cocycle by the weighted insertions ${\sum }_i{a}_i{f}_i^N{c}_{iI}$. The argument uses vanishing criterion in a module localization and does not require the module to be finitely generated.