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Tag 00U8

Chapter 10: Commutative Algebra > Section 10.141: Étale ring maps

Lemma 10.141.9. Let $\varphi : R \to S$ be a ring map. If $R \to S$ is surjective, flat and finitely presented then there exist an idempotent $e \in R$ such that $S = R_e$.

First proof. Let $I$ be the kernel of $\varphi$. We have that $I$ is finitely generated by Lemma 10.6.3 since $\varphi$ is of finite presentation. Moreover, since $S$ is flat over $R$, tensoring the exact sequence $0 \to I \to R \to S \to 0$ over $R$ with $S$ gives $I/I^2 = 0$. Now we conclude by Lemma 10.20.5. $\square$

Second proof. Since $\mathop{\rm Spec}(S) \to \mathop{\rm Spec}(R)$ is a homeomorphism onto a closed subset (see Lemma 10.16.7) and is open (see Proposition 10.40.8) we see that the image is $D(e)$ for some idempotent $e \in R$ (see Lemma 10.20.3). Thus $R_e \to S$ induces a bijection on spectra. Now this map induces an isomorphism on all local rings for example by Lemmas 10.77.4 and 10.19.1. Then it follows that $R_e \to S$ is also injective, for example see Lemma 10.23.1. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 37468–37473 (see updates for more information).

    \begin{lemma}
    \label{lemma-surjective-flat-finitely-presented}
    Let $\varphi : R \to S$ be a ring map. If $R \to S$ is surjective, flat and
    finitely presented then there exist an idempotent $e \in R$ such that
    $S = R_e$.
    \end{lemma}
    
    \begin{proof}[First proof]
    Let $I$ be the kernel of $\varphi$.
    We have that $I$ is finitely generated by
    Lemma \ref{lemma-finite-presentation-independent}
    since $\varphi$ is of finite presentation.
    Moreover, since $S$ is flat over $R$, tensoring the exact sequence
    $0 \to I \to R \to S \to 0$ over $R$ with $S$
    gives $I/I^2 = 0$. Now we conclude by
    Lemma \ref{lemma-ideal-is-squared-union-connected}.
    \end{proof}
    
    \begin{proof}[Second proof]
    Since $\Spec(S) \to \Spec(R)$ is a homeomorphism
    onto a closed subset (see Lemma \ref{lemma-spec-closed}) and
    is open (see Proposition \ref{proposition-fppf-open}) we see that
    the image is $D(e)$ for some idempotent $e \in R$ (see
    Lemma \ref{lemma-disjoint-decomposition}). Thus $R_e \to S$
    induces a bijection on spectra. Now this map induces an isomorphism
    on all local rings for example by
    Lemmas \ref{lemma-finite-flat-local} and \ref{lemma-NAK}.
    Then it follows that $R_e \to S$ is also injective, for example
    see Lemma \ref{lemma-characterize-zero-local}.
    \end{proof}

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