Finite generation from finitely many principal localizations

Suppose a $k$-algebra $A$ has elements $f_i,g_i$ with ${\sum }_i{g}_i{f}_i=1$, and each $A_{f_i}$ is a finitely generated algebra. Let $B\subseteq A$ contain the $f_i,g_i$ and numerators of finite generating sets of all $A_{f_i}$. Then $B$ is finitely generated and $B_{f_i}=A_{f_i}$. For $a\in A$, clear denominators so that all $f_i^{N}a$ belong to $B$. A unit-ideal identity for powers gives $1={\sum }_i{c}_i{f}_i^N$ in $B$, hence $a={\sum }_i{c}_i{f}_i^{N}a\in B$. Thus $A=B$.