Logarithmic Kunz and Radu--André Theorems

Frobenius morphisms play a crucial role in algebraic geometry and commutative algebra in positive characteristic. In particular, $\mathit{\text{the}}\mathit{\text{Kunz}}\mathit{\text{theorem}}$, characterizing non-singularity via Frobenius morphisms, is a monumental work and has a relative version, called $\mathit{\text{the}}\mathit{\text{Radu–Andr}}\acute{\mathit{\text{e}}}\mathit{\text{theorem}}$. In this paper, we extend the Kunz theorem and the Radu–André theorem to logarithmic algebraic geometry in positive characteristic.