The Stacks Project


Tag 009Y

Chapter 6: Sheaves on Spaces > Section 6.30: Bases and sheaves

Lemma 6.30.17. Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_Y$-modules. Let $\mathcal{B}_Y$ be a basis for the topology on $Y$. Let $\mathcal{B}_X$ be a basis for the topology on $X$. Suppose given for every $V \in \mathcal{B}_Y$, and $U \in \mathcal{B}_X$ such that $f(U) \subset V$ a $\mathcal{O}_Y(V)$-module map $$ \varphi_V^U : \mathcal{G}(V) \longrightarrow \mathcal{F}(U) $$ compatible with restriction mappings. Here the $\mathcal{O}_Y(V)$-module structure on $\mathcal{F}(U)$ comes from the $\mathcal{O}_X(U)$-module structure via the map $f^\sharp_V : \mathcal{O}_Y(V) \to \mathcal{O}_X(f^{-1}V) \to \mathcal{O}_X(U)$. Then there is a unique $f$-map of sheaves of modules (see Definition 6.21.7 and the discussion of $f$-maps in Section 6.26) $\varphi : \mathcal{G} \to \mathcal{F}$ recovering $\varphi_V^U$ as the composition $$ \mathcal{G}(V) \xrightarrow{\varphi_V} \mathcal{F}(f^{-1}(V)) \xrightarrow{\text{restrc.}} \mathcal{F}(U) $$ for every pair $(U, V)$ as above.

Proof. Similar to the above and omitted. $\square$

    The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 4328–4361 (see updates for more information).

    \begin{lemma}
    \label{lemma-f-map-basis-above-and-below-modules}
    Let $(f, f^\sharp) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$
    be a morphism of ringed spaces.
    Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules.
    Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_Y$-modules.
    Let $\mathcal{B}_Y$ be a basis for the topology on $Y$.
    Let $\mathcal{B}_X$ be a basis for the topology on $X$.
    Suppose given for every $V \in \mathcal{B}_Y$, and
    $U \in \mathcal{B}_X$ such that $f(U) \subset V$ a
    $\mathcal{O}_Y(V)$-module map
    $$
    \varphi_V^U :
    \mathcal{G}(V)
    \longrightarrow
    \mathcal{F}(U)
    $$
    compatible with restriction mappings. Here the
    $\mathcal{O}_Y(V)$-module structure on $\mathcal{F}(U)$
    comes from the $\mathcal{O}_X(U)$-module structure
    via the map $f^\sharp_V : \mathcal{O}_Y(V)
    \to \mathcal{O}_X(f^{-1}V) \to \mathcal{O}_X(U)$.
    Then there is a unique $f$-map of sheaves of modules (see
    Definition \ref{definition-f-map} and the discussion
    of $f$-maps in Section \ref{section-ringed-spaces-functoriality-modules})
    $\varphi : \mathcal{G} \to \mathcal{F}$
    recovering $\varphi_V^U$ as the composition
    $$
    \mathcal{G}(V) \xrightarrow{\varphi_V}
    \mathcal{F}(f^{-1}(V)) \xrightarrow{\text{restrc.}}
    \mathcal{F}(U)
    $$
    for every pair $(U, V)$ as above.
    \end{lemma}
    
    \begin{proof}
    Similar to the above and omitted.
    \end{proof}

    Comments (0)

    There are no comments yet for this tag.

    There are also 5 comments on Section 6.30: Sheaves on Spaces.

    Add a comment on tag 009Y

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?