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Tag 01R9

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

Lemma 28.6.6. Let $$ \xymatrix{ X_1 \ar[d] \ar[r]_{f_1} & Y_1 \ar[d] \\ X_2 \ar[r]^{f_2} & Y_2 } $$ be a commutative diagram of schemes. Let $Z_i \subset Y_i$, $i = 1, 2$ be the scheme theoretic image of $f_i$. Then the morphism $Y_1 \to Y_2$ induces a morphism $Z_1 \to Z_2$ and a commutative diagram $$ \xymatrix{ X_1 \ar[r] \ar[d] & Z_1 \ar[d] \ar[r] & Y_1 \ar[d] \\ X_2 \ar[r] & Z_2 \ar[r] & Y_2 } $$

Proof. The scheme theoretic inverse image of $Z_2$ in $Y_1$ is a closed subscheme of $Y_1$ through which $f_1$ factors. Hence $Z_1$ is contained in this. This proves the lemma. $\square$

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 914–933 (see updates for more information).

    \begin{lemma}
    \label{lemma-factor-factor}
    Let
    $$
    \xymatrix{
    X_1 \ar[d] \ar[r]_{f_1} & Y_1 \ar[d] \\
    X_2 \ar[r]^{f_2} & Y_2
    }
    $$
    be a commutative diagram of schemes. Let $Z_i \subset Y_i$, $i = 1, 2$ be
    the scheme theoretic image of $f_i$. Then the morphism
    $Y_1 \to Y_2$ induces a morphism $Z_1 \to Z_2$ and a
    commutative diagram
    $$
    \xymatrix{
    X_1 \ar[r] \ar[d] & Z_1 \ar[d] \ar[r] & Y_1 \ar[d] \\
    X_2 \ar[r] & Z_2 \ar[r] & Y_2
    }
    $$
    \end{lemma}
    
    \begin{proof}
    The scheme theoretic inverse image of $Z_2$ in $Y_1$
    is a closed subscheme of $Y_1$ through
    which $f_1$ factors. Hence $Z_1$ is contained in this.
    This proves the lemma.
    \end{proof}

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