= Finite generation descends along a field extension
Let $L/k$ be a <field extension> and $A$ a $k$-<algebra>. If $A\otimes_kL$ is a <finitely generated algebra> over $L$, then $A$ is finitely generated over $k$.
Choose finitely many generators of $A\otimes_kL$. Each is a finite <sum> of pure <tensors>, so only finitely many coefficients from $A$ occur. Let $A_0\subseteq A$ be the $k$-<subalgebra> generated by those coefficients. The image of $A_0\otimes_kL$ contains every chosen generator and all scalars, hence equals $A\otimes_kL$. Therefore
$$
(A/A_0)\otimes_kL=0.
$$
Here $A/A_0$ is a <quotient vector space>, not a quotient algebra: $A_0$ need not be an <ideal>. The extension $L$ is a <faithfully flat module> over $k$, so this vanishing forces $A/A_0=0$. Thus $A=A_0$, proving finite generation.
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