Let $B$ be a $k$-algebra such that there exist finitely many elements $b_1,\dots,b_n \in B$ satisfying:
- $(b_1,\dots,b_n) = B$ (equivalently, there are elements $c_1,\dots,c_n \in B$ such that $\sum_{i=1}^n b_i c_i = 1$)
- The localization $B_{b_i}$ at $b_i$ is a finitely generated $k$-algebra for all $b_i$'s.
Show that $B$ is finitely generated.
I am tempted to take the $b_i$'s and $c_i$'s as generators, but I am not really sure about what the finitely generated localizations tell me... any hints?
Write $B_i$ for the localization of $B$ at $b_i$. That $B_i$ be a finitely generated $k$-algebra means there are $a_{1,i},\ldots,a_{n_i,i}$ in $B$ such that for each $b\in B$, we can find $N>0$ such that $b_i^N b$ is a polynomial in the $a_{1,i},\ldots,a_{n_i,i}$. Now pick $b\in B$, and note that taking $N$ very large we can take $N$ that works for each $i$, so that $b_i^Nb$ is a polynomial in $a_{i,1},\cdots,a_{i,n_i}$. Now since the $b_i$ generate $1$, taking a large power shows that the $b_i^N$ do too. This means you can write $$b = 1\cdot b = \sum c_i b_i^N b $$
a polynomial in all the $a_{i,j}$.