Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $\mathcal F_i, i\in I$, be a directed family of sheaves of abelian groups on a paracompact Hausdorff space $X$. Let $\mathcal F=\varinjlim F_i$ denote the direct limit sheaf. Is it true that $\mathcal F(X)=\varinjlim \mathcal F_i(X)$? If not, is it true under extra assumptions on $\mathcal F_i$'s?

share|cite|improve this question

1 Answer 1

In general not: take $X$ to be $\mathbb{Z}_{\geq 0}$ and $\mathcal{F}_i$ the constant sheaf supported at $X\cap [0,i]$. The direct limit will be the constant sheaf on $X$ and will have many more sections than there are in the direct limit of the sections of the $\mathcal{F}_i$'s.

However, if you consider compactly supported cohomology (or if $X$ itself is compact), then everything is fine. See e.g. Godement, theorem 4.12.1.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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