Given a topological space $X$ and a presheaf of abelian groups on $X$, $A$, we can construct the set of germs at a point $x\in X$ by taking $\mathscr{A}_x=\lim\limits_{\rightarrow} A(U)$ where $x\in U\subseteq X$ and $U$ is open. In Sheaf Theory by Bredon, he claims there is a canonical group structure on $\mathscr{A}_x$.

What is the canonical group structure of the direct limit?

I would think if $[s]_x, [t]_x$ are germs then $[s]_x+[t]_x=[s+t]_x$ but, if $s,t$ are not in the same group, would we first need to restrict their sections until they are and then add them?

  • 1
    $\begingroup$ Right. It might be good to have notation that explicitly keeps track of $U$. Show that none of this depends on choices. $\endgroup$ – Hoot May 25 '15 at 17:25
  • $\begingroup$ @Hoot, that's what I was worried about haha. $\endgroup$ – Eoin May 25 '15 at 17:26

As Hoot suggested, the group structure is actually $[(s,U)]_x+[(t,V)]_x=[(s|_{U\cap V}+t|_{U\cap V}, U\cap V)]_x$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.