Revista Matemática Iberoamericana


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Volume 19, Issue 2, 2003, pp. 355–366
DOI: 10.4171/RMI/351

Published online: 2003-08-31

Conservation of the noetherianity by perfect transcendental field extensions

Magdalena Fernández Lebrón[1] and Luis Narváez Macarro[2]

(1) Universidad de Sevilla, Spain
(2) Universidad de Sevilla, Spain

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 $k$.

Keywords: Perfect field, power series ring, noetherian ring, perfect closure, complete local ring

Fernández Lebrón Magdalena, Narváez Macarro Luis: Conservation of the noetherianity by perfect transcendental field extensions. Rev. Mat. Iberoamericana 19 (2003), 355-366. doi: 10.4171/RMI/351