Finite generation descends along a field extension
= Finite generation descends along a field extension
Let $L/k$ be a <field extension>. If $A\otimes_kL$ is a <finitely generated algebra> over $L$, then $A$ is finitely generated over $k$. Choose finitely many generators of $A\otimes_kL$ and collect the finitely many coefficients from $A$ occurring in them. The $k$-subalgebra $A_0$ generated by those coefficients satisfies $(A/A_0)\otimes_kL=0$, and <faithfully flat module>[faithful flatness] of $L/k$ gives $A=A_0$.