The Stacks project

Lemma 6.30.16. Let $f : X \to Y$ be a continuous map of topological spaces. Let $(\mathcal{C}, F)$ be a type of algebraic structures. Let $\mathcal{F}$ be a sheaf with values in $\mathcal{C}$ on $X$. Let $\mathcal{G}$ be a sheaf with values in $\mathcal{C}$ on $Y$. 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 morphism

\[ \varphi _ V^ U : \mathcal{G}(V) \longrightarrow \mathcal{F}(U) \]

of $\mathcal{C}$ compatible with restriction mappings. Then there is a unique $f$-map (see Definition 6.21.7 and the discussion of $f$-maps in Section 6.23) $\varphi : \mathcal{G} \to \mathcal{F}$ recovering $\varphi _ V^ U$ as the composition

\[ \mathcal{G}(V) \xrightarrow {\varphi _ V} \mathcal{F}(f^{-1}(V)) \xrightarrow {\text{restr.}} \mathcal{F}(U) \]

for every pair $(U, V)$ as above.

Proof. Let us first proves this for sheaves of sets. Fix $V \subset Y$ open. Pick $s \in \mathcal{G}(V)$. We are going to construct an element $\varphi _ V(s) \in \mathcal{F}(f^{-1}V)$. We can define a value $\varphi (s)_ x$ in the stalk $\mathcal{F}_ x$ for every $x \in f^{-1}V$ by picking a $U \in \mathcal{B}_ X$ with $x \in U \subset f^{-1}V$ and setting $\varphi (s)_ x$ equal to the equivalence class of $(U, \varphi _ V^ U(s))$ in the stalk. Clearly, the family $(\varphi (s)_ x)_{x \in f^{-1}V}$ satisfies condition $(*)$ because the maps $\varphi _ V^ U$ for varying $U$ are compatible with restrictions in the sheaf $\mathcal{F}$. Thus, by the proof of Lemma 6.30.6 we see that $(\varphi (s)_ x)_{x \in f^{-1}V}$ corresponds to a unique element $\varphi _ V(s)$ of $\mathcal{F}(f^{-1}V)$. Thus we have defined a set map $\varphi _ V : \mathcal{G}(V) \to \mathcal{F}(f^{-1}V)$. The compatibility between $\varphi _ V$ and $\varphi _ V^ U$ follows from Lemma 6.30.5.

We leave it to the reader to show that the construction of $\varphi _ V$ is compatible with restriction mappings as we vary $V \in \mathcal{B}_ Y$. Thus we may apply Lemma 6.30.14 above to “glue” them to the desired $f$-map.

Finally, we note that the map of sheaves of sets so constructed satisfies the property that the map on stalks

\[ \mathcal{G}_{f(x)} \longrightarrow \mathcal{F}_ x \]

is the colimit of the system of maps $\varphi _ V^ U$ as $V \in \mathcal{B}_ Y$ varies over those elements that contain $f(x)$ and $U \in \mathcal{B}_ X$ varies over those elements that contain $x$. In particular, if $\mathcal{G}$ and $\mathcal{F}$ are the underlying sheaves of sets of sheaves of algebraic structures, then we see that the maps on stalks is a morphism of algebraic structures. Hence we conclude that the associated map of sheaves of underlying sets $f^{-1}\mathcal{G} \to \mathcal{F}$ satisfies the assumptions of Lemma 6.23.1. We conclude that $f^{-1}\mathcal{G} \to \mathcal{F}$ is a morphism of sheaves with values in $\mathcal{C}$. And by adjointness this means that $\varphi $ is an $f$-map of sheaves of algebraic structures. $\square$


Comments (2)

Comment #6520 by Wet Lee on

In the second paragraph, should be ?

There are also:

  • 6 comment(s) on Section 6.30: Bases and sheaves

Post a comment

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 toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

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 input field. As a reminder, this is tag 009X. Beware of the difference between the letter 'O' and the digit '0'.