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.
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1$\begingroup$ Just out of curiosity, what's the counterexample? $\endgroup$– S. Carnahan ♦Oct 23, 2009 at 6:02
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$\begingroup$ Let the field be F_2(t). The two rings are A=F_2(t)[x,y]/(y^2+x^3+tx), B=F_2(t,s)/(s^2-t), which is actually a field. $\endgroup$– TJCMOct 24, 2009 at 5:34
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$\begingroup$ The notion of smooth and regular coincides when the residue field of the point is perfect. It seems that the condition in the problem is not enough $\endgroup$– user70159Apr 5, 2015 at 8:17
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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.
answered Oct 23, 2009 at 6:06