k is a perfect field. X and Y are two regular varieties over k. Does their fiber product over k remain to be regular?


Note: When k is algebraically closed it's true by Jacobian criterion. When k is not perfect there's counter-example.



Does the fiber product of two regular varieties over perfect field remain regular?
The answer is yes. Indeed, over a perfect field the notions of smooth and regular coincide so it follows from the fact that base change and composition preserve smoothness.


