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Tag 056A

Chapter 28: Morphisms of Schemes > Section 28.6: Scheme theoretic image

Example 28.6.4. If $A \to B$ is a ring map with kernel $I$, then the scheme theoretic image of $\mathop{\rm Spec}(B) \to \mathop{\rm Spec}(A)$ is the closed subscheme $\mathop{\rm Spec}(A/I)$ of $\mathop{\rm Spec}(A)$. This follows from Lemma 28.6.3.

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 855–861 (see updates for more information).

    \begin{example}
    \label{example-scheme-theoretic-image}
    If $A \to B$ is a ring map with kernel $I$, then the scheme theoretic image
    of $\Spec(B) \to \Spec(A)$ is the closed subscheme
    $\Spec(A/I)$ of $\Spec(A)$. This follows from
    Lemma \ref{lemma-quasi-compact-scheme-theoretic-image}.
    \end{example}

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