I'm trying to prove that de Rham cohomology computes the cohomology of the constant sheaf $\mathbb{R}$ of real valued smooth functions on a manifold.
I would like to do this without using Čech cohomology, if possible.
Fix a smooth manifold $M$. Let $\mathbb{R}$ denote the contant sheaf, $\Omega^k$ the sheaf of $k$-forms on $M$. The Poincaré lemma tells us, that if $U$ (open) is contractible then the sequence $C^{\bullet}(U)$:
$\mathbb{R}(U) \to \Omega^0(U) \to \Omega^1(U) \to \dots$
is exact.
If $\mathcal{C}$ is the site of open sets of $M$, for each $x \in M$ define $\mathcal{C}_x$ to be the subcategory of contractible open neighbourhoods of $x$. The stalk $F_x$ of a sheaf $F$ on $\mathcal{C}$ is the colimit of the composition $\mathcal{C}_x^\mathrm{op} \xrightarrow{\subseteq} \mathcal{C}^\mathrm{op} \xrightarrow{F} \mathcal{Ab}$ as $M$ locally contractible.
Now consider the diagram $D_i: \mathcal{C}_x^\mathrm{op} \xrightarrow{\subseteq} \mathcal{C}^\mathrm{op} \xrightarrow{C^i} \mathcal{Ab}$. The sequence
$D_{-1} \to D_{0} \to D_{1} \to \dots$
of presheaves is exact as its sections are $C^\bullet(U)$.
As $\mathcal{C}_x^\mathrm{op}$ is filtered, the assignment $D \to \mathrm{colim} D$ is exact, so we obtain an exact sequence on stalks
$\mathbb{R}_x \to \Omega^0_x \to \Omega^1_x \to \dots$
for all $x \in M$. This is equivalent to the sequence of sheaves
$\mathbb{R} \to \Omega^0 \to \Omega^1 \to \dots$
being exact.
My questions are:
Is this argument correct?Do you know a reference that gives a similar proof?http://www3.nd.edu/~lnicolae/sheaves_coh.pdf- Is it possible to give a similarly elementary proof without using stalks?