Exercise 13, § 4

Let $p$ be a prime number, $q$ a power of $p$, and $A$ a reduced noetherian local ring of characteristic $p$. Denote by $A^q$ the subring of $A$ formed by the elements $a^q$ for $a \in A$. Prove that for $A$ to be regular, it is necessary and sufficient that the $A^q$-module $A$ be flat.

(One reduces to the case where $A$ is complete, hence a quotient of an algebra of formal power series $B = \kappa_A[[X_1, \ldots, X_n]]$ by an ideal $b$, which one may suppose contained in $m_B^2$. If $A$ is regular, one has $b = 0$. If $A$ is flat over $A^q$, let $x_i$ be the image of $X_i$ in $A$; prove that the family $(x_1^q, \ldots, x_n^q)$ is A-independent (exerc. 12), and deduce the equality $\lg_A(A/(x_1^q, \ldots, x_n^q)) = q^n$. Show that this entails $b \subset (X_1^{q^s}, \ldots, X_n^{q^s})$; prove analogously that one has $b \subset (X_1^{q^s}, \ldots, X_n^{q^s})$ for every s, whence finally $b = 0$.)