The Stacks project

Lemma 6.33.1. Let $X$ be a topological space. Let $X = \bigcup U_ i$ be an open covering. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of sets on $X$. Given a collection

\[ \varphi _ i : \mathcal{F}|_{U_ i} \longrightarrow \mathcal{G}|_{U_ i} \]

of maps of sheaves such that for all $i, j \in I$ the maps $\varphi _ i, \varphi _ j$ restrict to the same map $\mathcal{F}|_{U_ i \cap U_ j} \to \mathcal{G}|_{U_ i \cap U_ j}$ then there exists a unique map of sheaves

\[ \varphi : \mathcal{F} \longrightarrow \mathcal{G} \]

whose restriction to each $U_ i$ agrees with $\varphi _ i$.

Proof. For each open subset $U \subset X$ define

\[ \varphi _ U : \mathcal{F}(U) \to \mathcal{G}(U), \quad s \mapsto \varphi _ U(s) \]

where $\varphi _ U(s)$ is the unique section verifying

\[ (\varphi _ U(s))|_{U \cap U_ i} = (\varphi _ i)_{U \cap U_ i}(s|_{U \cap U_ i}). \]

Existence and uniqueness of such a section follows from the sheaf axioms due to the fact that

\begin{align*} ((\varphi _ i)_{U \cap U_ i}(s|_{U \cap U_ i}))|_{U \cap U_ i \cap U_ j} & = (\varphi _ i)_{U \cap U_ i \cap U_ j}(s|_{U \cap U_ i \cap U_ j})\\ & = (\varphi _ j)_{U \cap U_ i \cap U_ j}(s|_{U \cap U_ i \cap U_ j})\\ & = ((\varphi _ j)_{U \cap U_ j}(s|_{U \cap U_ j}))|_{U \cap U_ i \cap U_ j}. \end{align*}

This family of maps gives us indeed a map of sheaves: Let $V \subset U \subset X$ be open subsets then

\[ (\varphi _ U(s))|_ V = \varphi _ V(s|_ V) \]

since for each $i \in I$ the following holds

\begin{align*} (\varphi _ U(s))|_{V \cap U_ i} & = ((\varphi _ U(s))|_{U \cap U_ i})|_{V \cap U_ i}\\ & = ((\varphi _ i)_{U \cap U_ i}(s|_{U \cap U_ i}))|_{V \cap U_ i}\\ & = (\varphi _ i)_{V \cap U_ i}(s|_{V \cap U_ i})\\ & = \varphi _ V(s_{V})|_{V \cap U_ i}. \end{align*}

Furthermore, its restriction to each $U_ i$ agrees with $\varphi _ i$ since given $U \subset X$ open subset and $s \in \mathcal{F}(U \cap U_ i)$ then

\begin{align*} \varphi _{U \cap U_ i}(s) & = \varphi _{U \cap U_ i}(s)|_{U \cap U_ i}\\ & = (\varphi _ i)_{U \cap U_ i}(s|_{U \cap U_ i})\\ & = (\varphi _ i)_{U \cap U_ i}(s). \end{align*}

Comments (0)

There are also:

  • 3 comment(s) on Section 6.33: Glueing 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Your email address will not be published.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 04TN. Beware of the difference between the letter 'O' and the digit '0'.