Consider an affine curve C over a field k and let R be its coordinate ring. When $k=\bar k$ it is known that R is factorial iff C is rational and smooth. This paper considers the case $k\ne \bar k$. Theorem 2.1 gives some equivalent conditions for C to be rational and factorial; one of them is the following: there exist f,g$\in R$ such that the k-rational points of C are precisely the proper non-zero ideals of the form $(\lambda f+\mu g)R$, $\lambda$,$\mu\in k$. {\S} 3 contains some interesting results in the case $k={\bbfR}$. [H.Matsumura]
Rational and factorial curves over an arbitrary field
ROGGERO, Margherita
1984-01-01
Abstract
Consider an affine curve C over a field k and let R be its coordinate ring. When $k=\bar k$ it is known that R is factorial iff C is rational and smooth. This paper considers the case $k\ne \bar k$. Theorem 2.1 gives some equivalent conditions for C to be rational and factorial; one of them is the following: there exist f,g$\in R$ such that the k-rational points of C are precisely the proper non-zero ideals of the form $(\lambda f+\mu g)R$, $\lambda$,$\mu\in k$. {\S} 3 contains some interesting results in the case $k={\bbfR}$. [H.Matsumura]
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