F-regularity does not deform
We show that the property of F-regularity does not deform, and thereby settle this longstanding open question in the theory of tight closure. Specifically, we construct a three dimensional domain R which is not F-regular (or even F-pure), but has a quotient R/tR which is F-regular. Similar examples are also constructed over fields of characteristic zero. Our work has an immediate application to questions related to flat base change: Hochster and Huneke showed that if (A,m) -> (R,n) is a flat map with regular generic and closed fibers, R is excellent, and A is weakly F-regular, then R is weakly F-regular. They asked if the condition on the fibers may be relaxed by requiring instead that the two fibers are F-regular. Our result shows that this condition is not enough, even if the base ring A is a discrete valuation ring: taking A=K[[t]] as a subring of the ring R above, the closed fiber R/tR and the generic fiber R[1/t] are F-regular, whereas R is not.