Clarification on Smooth de Rham Theorem – Differential Topology

Question

I am misunderstanding something in Theorem 2.1.9 in Dimca’s Sheaves in Topology:

Let $X$ be a real smooth manifold. Then the natural morphism from the constant sheaf to the de Rham complex

$$\mathbb{R}_X \rightarrow \Omega_X^\bullet$$

is an acyclic resolution of $\mathbb{R}_X$.

I am getting confused about in which sense $\Omega_X^\bullet$ is acyclic: if it is acyclic as a complex, then it has no cohomology above $H^0$ by definition. But, of course a smooth real manifold can still have cohomology above $H^0$!

So I would like to confirm that acyclic here means a resolution using $\Gamma$-acyclic sheaves, i.e., something like injective or flasque sheaves.

Edit: Maybe we want hypercohomology of both sides, since Poincar\’e’s lemma says that the de Rham complex $\Omega_X^\bullet$ really is acyclic as a complex. But then the statement

$$ H^k(X,\mathbb{R}) \cong \frac{\text{ker } d: \Omega^k(X) \rightarrow \Omega^{k+1}(X)}{\text{Im }d: \Omega^{k-1}(X) \rightarrow \Omega^k(X)} $$

makes no sense to me, because the right hand side here should be zero for $k>0$ if $\Omega^\bullet(X)$ is acyclic as a complex

Answer 1

The following statements are true:

• The complex $0 \to \mathbb{R}_X \to \Omega^0_X \to \Omega^1_X \to \dots$ is an acyclic complex (of sheaves), i.e. it is exact at each step. This is the content of the Poincare lemma, and it is what it means for $\Omega^\bullet$ to be a resolution of $\mathbb{R}_X$.
• Each individual $\Omega^i$ is flasque, and therefore an acyclic sheaf (i.e. its sheaf cohomology vanishes in positive degree). This is a different meaning of the word "acyclic" from the above; but specifying "acyclic resolution" means this sense of the word acyclic (a resolution by acyclic sheaves), since the other kind of acyclicity is already implicit in the use of the word "resolution". (I don't blame you for finding this confusing!)

However, it is not true that either of these statements has anything to do with the exactness of the sequence of global sections $\Omega^0(X) \to \Omega^1(X) \to \dots$, which very frequently has interesting cohomology in all possible degrees (up to $\dim X$). (But why would you expect it to be this? There is much more information in a sheaf than just its global sections.)