Lemma 10.23.2. Let $R$ be a ring, and let $f_1, f_2, \ldots f_ n\in R$ generate the unit ideal in $R$. Then the following sequence is exact:

\[ 0 \longrightarrow R \longrightarrow \bigoplus \nolimits _ i R_{f_ i} \longrightarrow \bigoplus \nolimits _{i, j}R_{f_ if_ j} \]

where the maps $\alpha : R \longrightarrow \bigoplus _ i R_{f_ i}$ and $\beta : \bigoplus _ i R_{f_ i} \longrightarrow \bigoplus _{i, j} R_{f_ if_ j}$ are defined as

\[ \alpha (x) = \left(\frac{x}{1}, \ldots , \frac{x}{1}\right) \text{ and } \beta \left(\frac{x_1}{f_1^{r_1}}, \ldots , \frac{x_ n}{f_ n^{r_ n}}\right) = \left(\frac{x_ i}{f_ i^{r_ i}}-\frac{x_ j}{f_ j^{r_ j}}~ \text{in}~ R_{f_ if_ j}\right). \]

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