Flat morphisms with regular fibers do not preserve $F$-rationality

Abstract

For each positive prime integer $p$ we construct a standard graded $F$-rational ring $R$, over a field $K$ of characteristic $p$, such that $R{\otimes }_K\overline{K}$ is not $F$-rational. By localizing we obtain a flat local homomorphism $(R,\mathfrak{m})\to (S,\mathfrak{n})$ such that $R$ is $F$-rational, $S/\mathfrak{m}S$ is regular (in fact, a field), but $S$ is not $F$-rational. In the process we also obtain standard graded $F$-rational rings $R$ for which $R{\otimes }_{K}R$ is not $F$-rational.