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Tag 0098

Chapter 6: Sheaves on Spaces > Section 6.26: Morphisms of ringed spaces and modules

Lemma 6.26.4. Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_Y$-modules. Let $x \in X$. Then $$ (f^*\mathcal{G})_x = \mathcal{G}_{f(x)} \otimes_{\mathcal{O}_{Y, f(x)}} \mathcal{O}_{X, x} $$ as $\mathcal{O}_{X, x}$-modules where the tensor product on the right uses $f^\sharp_x : \mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$.

Proof. This follows from Lemma 6.20.3 and the identification of the stalks of pullback sheaves at $x$ with the corresponding stalks at $f(x)$. See the formulae in Section 6.23 for example. $\square$

    The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 3174–3188 (see updates for more information).

    \begin{lemma}
    \label{lemma-stalk-pullback-modules}
    Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$
    be a morphism of ringed spaces.
    Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_Y$-modules.
    Let $x \in X$. Then
    $$
    (f^*\mathcal{G})_x =
    \mathcal{G}_{f(x)}
    \otimes_{\mathcal{O}_{Y, f(x)}}
    \mathcal{O}_{X, x}
    $$
    as $\mathcal{O}_{X, x}$-modules where the tensor product on the right
    uses $f^\sharp_x : \mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$.
    \end{lemma}
    
    \begin{proof}
    This follows from Lemma \ref{lemma-stalk-tensor-sheaf-modules}
    and the identification of the stalks of pullback sheaves
    at $x$ with the corresponding stalks at $f(x)$. See the
    formulae in Section \ref{section-presheaves-structures-functorial}
    for example.
    \end{proof}

    Comments (2)

    Comment #746 by Anfang Zhou on June 27, 2014 a 12:38 am UTC

    Typo. It should be $\mathcal{F}_{f(x)} \otimes_{\mathcal{O}_{Y, f(x)}} \mathcal{O}_{X, x}$

    Comment #752 by Johan (site) on June 28, 2014 a 3:48 pm UTC

    Thanks for the comment. I have fixed this as well as the other typo you pointed out. See this commit.

    There is also 1 comment on Section 6.26: Sheaves on Spaces.

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