Let $R$ be a commutative ring such that $X:=\mathrm{Spec}(R)=\{p\}$, where $p$ is a prime ideal. Then, obviously, the affine variety $X$ consists of one point only. However, can we decide if this variety is smooth in the Zariski topology? Moreover, if the above ring $R$ is in addition graded, can we consider the affine variety corresponding to the underlying non-graded ring $R$ as a smooth projective variety?
2026-03-25 19:10:28.1774465828
Smoothness of affine variety consisting of finitely many points
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