# Showing that compatible germs are the image of a section.

I'm currently self-studying Ravi Vakil's Rising Sea. I have been stuck on exercise 2.4.C, which ask one to prove that any compatible germs is the image of a section. The following definition etc are all taken from the notes. If I have understood it correctly we have the map $$I : \mathscr{F}(U) \rightarrow \prod_{p \in U} \mathscr{F}_p$$ given by $$s \mapsto (\bar{s})_p$$, i.e mapping a section to its germ in the respective points. Now germs $$(s_p)$$ are defined to be compatible if there exists a covering $$\{U_i\}$$ of $$U$$ and sections $$f_i \in U_i$$ such that the germ of $$f_i$$ for all $$p \in U_i$$ is $$s_p$$. I think that one is supposed to glue together the sections given in the definition of compatible germs, though I haven't been able to show that the given sections' restrictions are equal on the overlaps/intersections. Many thanks for any help or hint.

• Questions on this website should be as self-contained as possible. Please include the text of the exercise inside the question. – KReiser Mar 20 at 21:31
• I have now added an image of the relevent definition and exercise from the book. Hope that this is sufficient :) – Najonathan Mar 21 at 15:50
• Is this sufficient to open the question again or should I post a new question? – Najonathan Mar 23 at 10:30

Picking up where you left off, you want to show that, given two sections $$f_i \in \mathscr F(U_i)$$, $$f_j \in \mathscr F(U_j)$$, that they agree when restricted to $$U_i \cap U_j$$.
At any point $$p \in U_i \cap U_j$$, both $$(f_i)\mid_{U_i \cap U_j}$$ and $$(f_j)\mid_{U_i \cap U_j}$$ have the stalk $$s_p$$. But by 2 4.A, two sections being everywhere stalk-wise equal means that they are the same section.