The Stacks project

Lemma 7.8.6. Let $\mathcal{C}$ be a category. Let $\mathcal{V} = \{ V_ j \to U\} _{j \in J} \to \mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a morphism of families of maps with fixed target of $\mathcal{C}$ given by $\text{id} : U \to U$, $\alpha : J \to I$ and $f_ j : V_ j \to U_{\alpha (j)}$. Let $\mathcal{F}$ be a presheaf on $\mathcal{C}$. If

  1. the fibre products $U_ i \times _ U U_{i'}$, $U_ i \times _ U V_ j$, $V_ j \times _ U V_{j'}$ exist,

  2. $\mathcal{F}$ satisfies the sheaf condition with respect to $\mathcal{V}$, and

  3. for every $i \in I$ the map $\mathcal{F}(U_ i) \to \prod _{j \in J} \mathcal{F}(V_ j \times _ U U_ i)$ is injective.

Then $\mathcal{F}$ satisfies the sheaf condition with respect to $\mathcal{U}$.

Proof. By Lemma 7.8.5 the map $\mathcal{F}(U) \to \prod \mathcal{F}(U_ i)$ is injective. Suppose given $s_ i \in \mathcal{F}(U_ i)$ such that $s_ i|_{U_ i \times _ U U_{i'}} = s_{i'}|_{U_ i \times _ U U_{i'}}$ for all $i, i' \in I$. Set $s_ j = f_ j^*(s_{\alpha (j)}) \in \mathcal{F}(V_ j)$. Since the morphisms $f_ j$ are morphisms over $U$ we obtain induced morphisms $f_{jj'} : V_ j \times _ U V_{j'} \to U_{\alpha (i)} \times _ U U_{\alpha (i')}$ compatible with the $f_ j, f_{j'}$ via the projection maps. It follows that

\[ s_ j|_{V_ j \times _ U V_{j'}} = f_{jj'}^*(s_{\alpha (j)}|_{U_{\alpha (j)} \times _ U U_{\alpha (j')}}) = f_{jj'}^*(s_{\alpha (j')}|_{U_{\alpha (j)} \times _ U U_{\alpha (j')}}) = s_{j'}|_{V_ j \times _ U V_{j'}} \]

for all $j, j' \in J$. Hence, by the sheaf condition for $\mathcal{F}$ with respect to $\mathcal{V}$, we get a section $s \in \mathcal{F}(U)$ which restricts to $s_ j$ on each $V_ j$. We are done if we show $s$ restricts to $s_ i$ on $U_ i$ for any $i \in I$. Since $\mathcal{F}$ satisfies (3) it suffices to show that $s$ and $s_ i$ restrict to the same element over $U_ i \times _ U V_ j$ for all $j \in J$. To see this we use

\[ s|_{U_ i \times _ U V_ j} = s_ j|_{U_ i \times _ U V_ j} = (\text{id} \times f_ j)^*s_{\alpha (j)}|_{U_ i \times _ U U_{\alpha (j)}} = (\text{id} \times f_ j)^*s_ i|_{U_ i \times _ U U_{\alpha (j)}} = s_ i|_{U_ i \times _ U V_ j} \]

as desired. $\square$

Comments (0)

There are also:

  • 6 comment(s) on Section 7.8: Families of morphisms with fixed target

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Your email address will not be published.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 0G1L. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 7.8.6 as pdf