MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The question concerns a sheaf $S$ of abelian groups over a compact space $X$. Suppose each stalk $S_x$ is finite rank free. Is the group of global sections free?

share|cite|improve this question
Well, each stalk $S_x$ is generated by a finite number of sections over an open containing $x$. We can cover $X$ by such opens, and then find a finite subcover. This should, I hope, give some information about the group of global sections. – David Roberts Sep 13 '11 at 4:53
Is your space at least Hausdorff or you are working in an algebraic category? – Anton Fonarev Sep 13 '11 at 7:15
For the constant sheaf $\mathbb Z$, it is true: the global sections may be identified with a subgroup of the group of bounded functions $X\to\mathbb Z$, and the latter group is free (G. Nöbeling, Verallgemeinerung eines Satzes von Herrn E. Specker, Invent. Math. 6 (1968) 41-55). – user2035 Sep 13 '11 at 13:43
The space is not necessarily Hausdorff, but an affirmative answer in the Hausdorff case would be interesting. The case of the constant sheaf mentioned is the motivation for this question. Specker's proof of freeness in that case doesn't seem to go over very well to this more general case. – GMark Sep 13 '11 at 15:24
Interesting question. Concerning a-fortiori's comment, here is a online-copy of Nöbeling's article… – Martin Brandenburg Sep 15 '11 at 16:16

Your Answer


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

Browse other questions tagged or ask your own question.