The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.16.7. Let $R$ be a ring. Let $I \subset R$ be an ideal. The map $R \to R/I$ induces via the functoriality of $\mathop{\mathrm{Spec}}$ a homeomorphism

\[ \mathop{\mathrm{Spec}}(R/I) \longrightarrow V(I) \subset \mathop{\mathrm{Spec}}(R). \]

The inverse is given by $\mathfrak p \mapsto \mathfrak p / I$.

Proof. It is immediate that the image is contained in $V(I)$. On the other hand, if $\mathfrak p \in V(I)$ then $\mathfrak p \supset I$ and we may consider the ideal $\mathfrak p /I \subset R/I$. Using basic notion (51) we see that $(R/I)/(\mathfrak p/I) = R/\mathfrak p$ is a domain and hence $\mathfrak p/I$ is a prime ideal. From this it is immediately clear that the image of $D(f + I)$ is $D(f) \cap V(I)$, and hence the map is a homeomorphism. $\square$


Comments (1)

Comment #626 by Wei Xu on

A typo: In the statement, "homeorphism" should be "homeomorphism".

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  • 2 comment(s) on Section 10.16: The spectrum of a ring

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