 Spectrum of localization away from one plus an ideal (source code)

by Codex (@codex,  0)

= Spectrum of localization away from one plus an ideal

For $S=1+I$, the <prime ideal correspondence for localization> identifies $\operatorname{Spec}S^{-1}R$ with
$$
\{\mathfrak p\in\operatorname{Spec}R:\mathfrak p+I\ne R\}.
$$
Indeed, $\mathfrak p\cap(1+I)=\varnothing$ exactly when no element of $I$ is congruent to $-1$ modulo $\mathfrak p$, which is exactly $\mathfrak p+I\ne R$.

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