Tate algebras and Frobenius non-splitting of excellent regular rings

An excellent ring of prime characteristic for which the Frobenius map is pure also split in many commonly occurring situations positive commutative algebra and algebraic geometry. However, using a fundamental construction from rigid geometry, we show that $F$-pure rings are not general, even Euclidean domains. Our uses existence complete non-Archimedean field $k$ $p$ with no nonzero continuous $k$-linear maps $k^{1/p} \to k$. explicit example such given based on ideas Gabber, may be independent interest. examples settle long-standing open question theory $F$-singularities whose origin can traced back to when Hochster Roberts introduced notion $F$-purity. The domains construct admit $R$-linear $R^{1/p} \rightarrow R$. These first illustrate $F$-purity splitting define different classes singularities domains, $p^{-1}$-linear maps. latter particularly interesting perspective test ideals.