Conservation of the noetherianity by perfect transcendental field extensions

Abstract

Let $k$ be a perfect field of characteristic $p>0$, $k(t)_{per}$ the perfect closure of $k(t)$ and $A$ a $k$-algebra. We characterize whether the ring $$ A\otimes_k k(t)_{per}=\bigcup_{m\geq 0}(A\otimes_k k(t^{\frac{1}{p^m}})) $$ is noetherian or not. As a consequence, we prove that the ring $A\otimes_k k(t)_{per}$ is noetherian when $A$ is the ring of formal power series in $n$ indeterminates over $k$.