In her thesis\, Karen Smith showed that if a variety in cha
racteristic zero had the property that its mod p reductions were F-rationa
l for some p >\;>\; 0\, then R had rational singularities in character
istic zero.

\n

\nSuppose now that R is a local ring in mixed charact
eristic. Suppose that R/p has F-rational singularities for some p. We
prove that R then has rational singularities in mixed characteristic and h
ence any localization to characteristic 0 also has rational singularities.
We also will discuss various generalizations to this statement.

\n**\nThis is joint work with Linquan Ma and uses recent advances in our und
erstanding of singularities in mixed characteristic.**