I am interested in the proof of Leray's theorem that relates Čech cohomology and sheaf cohomology.
The theorem states that if we have a space $X$, a sheaf $\mathcal{F}$ and a covering of $X$ such that our sheaf is acyclic on the covering, i.e. Čech cohomology is $0$ on every finite intersection of elements in the covering, then these two cohomologies agree.
Now, my question. All the proves that I've found are by induction and assume that cohomology vanishes in every degree, so $\check{\mathrm{H}}^p(\mathcal{U}, \mathcal{F})=0$ for every $p>0$.
This looks like a rather strong assumption to check, especially if one is interested in computing cohomology just in low degrees (in particular I am interested in degree 0 and 1).
Looking at the proves that I found, indeed, it seems to me that we just need that Čech cohomology vanishes up to degree $p$ to prove the isomorphism in this degree (for example Climbing Mount Bourbaki).
I may be very wrong on this point, so the question is
- Can someone provide a counter example in which we really need that cohomology vanishes in all the degrees and not just up to what we are interested in?
or
- Can someone point out a reference which clearly states that we can just check up to degree $p$ (or $p+1$, depending on your indexing) to prove the isomorphism?
Thank you in advance,
Davide
EDIT: Despite the 5 "ups" I received no answer, so I decided to ask the same question also on Math Overflow.