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 $k$.
Publié le : 2003-09-14
Classification:
perfect field,
power series ring,
noetherian ring,
perfect closure,
complete local ring,
13E05,
13B35,
13A35
@article{1063050157,
author = {Fern\'andez-Lebr\'on, Magdalena and Narv\'aez, Luis},
title = {Conservation of the noetherianity by perfect transcendental field extensions},
journal = {Rev. Mat. Iberoamericana},
volume = {19},
number = {2},
year = {2003},
pages = { 355-366},
language = {en},
url = {http://dml.mathdoc.fr/item/1063050157}
}
Fernández-Lebrón, Magdalena; Narváez, Luis. Conservation of the noetherianity by perfect transcendental field extensions. Rev. Mat. Iberoamericana, Tome 19 (2003) no. 2, pp. 355-366. http://gdmltest.u-ga.fr/item/1063050157/