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

TL;DR

The paper constructs examples of flat morphisms with regular fibers where $F$-rationality is not preserved, challenging assumptions about the stability of this property under certain algebraic operations.

Contribution

It provides explicit constructions of $F$-rational rings that lose $F$-rationality after base change or localization, demonstrating that $F$-rationality is not preserved under flat morphisms with regular fibers.

Findings

Constructed $F$-rational rings that are not $F$-rational after base change.

Provided examples where flat local homomorphisms do not preserve $F$-rationality.

Showed that tensor products of $F$-rational rings may fail to be $F$-rational.

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.