Problem on the Cech Cohomology of a Sheaf over a Paracompact Space

Question

I am reading Rotman, Introduction to Homological Algebra, and $\textbf{Theorem 6.88.}$ reads:

Let $\mathcal{F}$ be a sheaf over a paracompact space $X$. If $X$ has an open cover $\mathcal{U}$ with $dim(\mathcal{N}(\mathcal{U}))<n$, then $\check{H}^q(\mathcal{F})=\{0\}$ for all $q\geq n+1$.

The proof of this theorem refers to Swan, The Theory of Sheaves, p.109, but I don't have this book.

However, $\mathcal{U}=\{X\}$ itself is an open cover where $dim(\mathcal{N}(\mathcal{U}))\leq 0$, and then by the above theorem, $\check{H}^q(\mathcal{F})=\{0\}$ for all $q\geq 1$, and this is definitely not true. This example even appears in Rotman too; it's $\textbf{Example 6.78.}$ (iii).

I think there is something wrong, but I don't know how to fix it. I think maybe there needs some kind of condition for an open cover $\mathcal{U}$: for example, every intersection of distinct sets in $\mathcal{U}$ is contractible. I am not sure if this is true if I add this condition. Or if every open cover $\mathcal{V}$ has a refinement $\mathcal{U}$ with $dim(\mathcal{N}(\mathcal{U}))<n$, then I am quite sure the theorem is true, but this is way too easier than the original theorem, so I don't think this is what Rotman wanted to state.

Thanks in advance.

Answer 1

There is indeed a mistake in the theorem. The correct version should be :

• if $\dim\mathcal{N(U)}<n$, then $\check H^q(\mathcal{U,F})=0$ for all $q\geq n+1$.
• in particular, if there exists $\mathcal{U}$ is a good cover for $\mathcal{F}$ (that is one such that any finite intersections $V=U_1\cap...\cap U_n$ satisfies $H^q(V,\mathcal{F})=0$ for all $q>0$) such that $\dim\mathcal{N(U)}<n$ then $\check H^q(\mathcal{F})=0$ for all $q\geq n+1$.