# Geometric regularity for infinitely generated field extensions

Let $$k$$ be a field. Suppose that for a finite type $$k$$-algebra $$A$$, we define two following properties:

1. $$A\otimes_k k'$$ is a regular ring for all finitely generated field extensions $$k\subset k'$$.
2. $$A\otimes_k k'$$ is a regular ring for all field extensions $$k\subset k'$$.

Are these two properties equivalent? In other words, is the failure to be geometrically regular witnessed by a finitely generated field extension?

Note that the second definition makes sense because the base change of a finite type algebra is finite type (so Noetherian, in particular). Definitions are those used in Stacks project, in the case it matters.

(2) $$\Rightarrow$$ (1) is clear, and so it suffices to show the converse. Let $$k \subset K$$ be an arbitrary field extension; we want to show that $$A \otimes_k K$$ is regular assuming the condition in (1).
Proposiition [EGAIV$$_2$$, Prop. 5.13.7]. Let $$\{(A_\alpha,\varphi_{\beta\alpha})\}$$ be a filtered direct system of rings and set $$A = \varinjlim A_\alpha$$. Suppose that $$A_\alpha$$ is regular for every $$\alpha$$, and that $$\varphi_{\beta\alpha}$$ is flat for every $$\alpha \le \beta$$. If $$A$$ is noetherian, then $$A$$ is regular.
Now write $$K$$ as the direct limit $$\varinjlim k_\alpha$$ of the filtered direct system of finitely generated subfield extensions $$k \subset k_\alpha$$ of $$K$$ with transition homomorphisms $$\varphi_{\beta\alpha}$$. Then, for every $$\alpha$$, the ring $$A \otimes_k k_\alpha$$ is regular by assumption in (1). Note that the induced directed system $$\{(A \otimes_k k_\alpha,\mathrm{id}_A \otimes_k \varphi_{\beta\alpha})\}$$ has flat transition homomorphisms by base change. Since $$A \otimes_k K \simeq A \otimes_k \varinjlim k_\alpha \simeq \varinjlim \bigl(A \otimes_k k_\alpha\bigr),$$ the proposition therefore implies that $$A \otimes_k K$$ is regular. $$\blacksquare$$