# Can the property of essential finite type checked at a point?

Let $k$ be a field, and let $A$ be a commutative $k$-algebra which is noetherian.

Suppose that for each prime ideal $p$ of $A$, it holds that the field $k(p)$, the field of fractions of $A/p$ has finite transcendence degree over $k$.

Does this imply that $A$ is a localization of a finite type $k$-algebra?

• No. Take $k=\mathbb{Q}$, $A=\bar{\mathbb{Q}}$. – abx Apr 10 '14 at 10:14
• Another problem occurs when $A$ contains infinitely many nilpotents. – Will Sawin Apr 10 '14 at 21:33