In his article Noncommutative differential geometry (Inst. Hautes Études Sci. Publ. Math. No. 62 (1985), 257–360) A. Connes gives in Lemma 44 (p. 343f) a projective topological resolution of the module $C^{\infty}(V)$ over $C^{\infty}(V \times V)$ for a smooth compact manifold $V$.
The basic construction is as follows:
One considers the complex bundles $E_k$ on $V \times V$ given by the pull back of the projection $\mathrm{pr}_2\colon V \times V \to V$ onto the second coordinate of the exterior power $\bigwedge^k T_{\mathbb{C}}^* V$ of the complexified cotangent bundle of $V$. Now, if the Euler characteristic of $V$ vanishes, one finds a section $X$ of $E_1^*$ which does not vanish outside the diagonal. Connes then states that a resolution is given by
$$C^{\infty}(V) \xleftarrow{\Delta^*} C^{\infty}(V \times V) \xleftarrow{\iota_X} C^{\infty}(V^2, E_1) \xleftarrow{\iota_X} \cdots \leftarrow C^{\infty}(V^2, E_n) \leftarrow 0,$$
where $n$ is the dimension of $V$.
For the proof Connes chooses smooth cut-offs $\chi$ and $\chi'$ in $C^{\infty}(V \times V)$ such that $X(a,b) = \exp_b^{-1}(a)$ for $(a,b)$ in the support of $\chi'$ and $\chi' = 1$ on the support of $\chi$ which itsself satisfies $\chi = 1$ in a neighbourhood of the diagonal $\Delta$.
We do not understand a conclusion at the very end of the proof. Connes defines the section $$s(\omega) = \chi' \int_0^1 \varphi_t^*(d_b(\chi \omega)) \frac{dt}{t} + (1-\chi) \omega' \wedge \omega.$$ Then he proves that for every form $\omega_1 \in E_k$ vanishing off the support of $\chi$ and statisfying $\omega_1(a,a) = 0$ (i.e. $\omega_1$ vanishes on the diagonal) one has $$\int_0^1 \phi_t^* d(\iota_X \omega_1) + \iota_X \int_0^1 \varphi_t^* d\omega_1 \frac{dt}{t} = \omega_1.$$ Finally, he applies the above identiy to the form $\omega_1 = \chi \omega$, a smooth cut-off of an arbitray given form $\omega \in E_k$. But why does $\omega_1$ vanish on the diagonal and one can therefore apply the above identity?
Can anyone clarify this step?
Searching through the web, everyone seems to give more or less this proof due to Connes and therefore further literature was no help.