8
$\begingroup$

Let $X$ be a stone space, i.e. a compact, totally disconnected hausdorff space. Then $H^1(X,\mathbb{Z}/2)=0$. Here's one way of proving this: $X$ with $\mathbb{Z}/2$ (the constant sheaf) is an affine scheme, now use the vanishing result for quasicoherent sheaves on affine schemes.

What happens if we also allow locally compact, totally disconnected hausdorff spaces? Then $X$ with $\mathbb{Z}/2$ is again a scheme, but it is not affine (unless $X$ is compact). A typical example would be an open subset of a stone space (actually this is generic), or the underlying topological space of a local field.

$\endgroup$
2
  • $\begingroup$ Consider Aleph_1 with its order topology. Do you know how to show that H^1(Aleph_1,Z/2)=0? I would actually bet it's non-trivial. $\endgroup$ Jun 4, 2010 at 13:47
  • $\begingroup$ I think $\aleph_1$ is a disjoint union of compact open subsets. Thus the cohomology vanishes. $\endgroup$ Jun 9, 2010 at 23:47

2 Answers 2

3
$\begingroup$

A search brought up Sheaf Cohomology of Locally Compact Totally Disconnected Spaces by R. Wiegand. Some topological invariants of Stone spaces by the same author might also be of interest.

$\endgroup$
2
  • $\begingroup$ Theorem 4.1 in the first paper is very useful. And in the second paper, the remark after Prop. 5.3 provides a counterexample for vanishing $H^1(X;\mathbb{Z}/2)$. $\endgroup$ Jun 4, 2010 at 14:31
  • $\begingroup$ @MartinBrandenburg, did you understand this example? The remark just says one can show but doesn't construct the cech cycle $\endgroup$ Jan 17, 2021 at 21:06
1
$\begingroup$

The vanishing theorem holds under reasonable topological hypotheses, say when we have a totally disconnected topological space $X$ that is paracompact, Hausdorff, locally compact and has a countable base (e.g. the $p$-adic rationals).

In that case the space is regular (since it is Hausdorff and locally compact) and hence satisfies the Lindelöf condition. Then for each closed $A$ and $B$ with empty intersection there is a clopen $U$ such that $A\subset U\subset X\setminus B$ (see e.g. theorem 6.2.7. in Engelking, General topology). So the constant sheaf with stalk $\mathbf{Z}$ or $\mathbf{Z}/2$ is soft and (using the paracompactness again) its higher cohomology vanishes.

$\endgroup$
4
  • $\begingroup$ interesting. what are the ring-theoretic properties of the corresponding boolean ring $C_0(X,\mathbb{Z}/2)$ (without unit), corresponding to your topological assumptions? $\endgroup$ Jun 5, 2010 at 6:46
  • $\begingroup$ Martin -- no idea, sorry! This came up as a weird example in a course I took a long time ago and luckily I still have the notes. $\endgroup$
    – algori
    Jun 5, 2010 at 14:30
  • $\begingroup$ Ok I will ask a new question. $\endgroup$ Jun 9, 2010 at 15:52
  • $\begingroup$ @MartinBrandenburg which one is the new question you created after this one? $\endgroup$ Nov 10, 2022 at 9:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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