TITLE: Does regular field extension preserve regularity?
QUESTION [6 upvotes]: Let $k$ be an arbitrary field and suppose that $K/k$ is a regular field extension. Let $V$ be regular scheme of finite type over $\text{Spec }k$ (not necessarily smooth). Is it true that $\text{Spec }K\times_{\text{Spec }k}V$ is also regular?

REPLY [6 votes]: If $k \rightarrow K$ is formally smooth for the discrete topology (i.e. separable), by flat base change $A \rightarrow A\otimes_kK$ is formally smooth for any $k$-algebra $A$ essentially of finite type, and so, if $A$ is regular then $A\otimes_kK$ is regular.