So in [another question of mine](https://mathoverflow.net/questions/94657/weil-kostant-integrality-result-as-stated-by-brylisnki) there is a sequence of complexes of sheaves which the author asserts is exact. 


Let $K^{\bullet} = \underline{\mathbb{C}}^\* \ \underrightarrow{d\ log} \ \underline{A}^1\_{M, \mathbb{C}}$ and so we have an exact sequence of complexes of sheaves:
$$0 \rightarrow {\mathbb{C}}^\* \rightarrow K^{\bullet}
 \rightarrow \underline{A^2}{M, cl}[-1] \rightarrow 0 $$


Where that nastily noted $\underline{A^2}{M, cl}[-1]$ means the two term complex with 0 in the first slot and closed 2 forms on $M$ in the second slot.


**The fact that this sequence is exact in itself seems to rely on the fact that the sheafification of the image of the contant $\mathbb{C}^\*$ sheaf is isomorphic to the sheaf of smooth functions** $\underline{\mathbb{C}^ \* }$ **right? Well that part is bothersome to me**



Is this Sequences of Complexes of Sheaves Exact?
This is not related to sheafification. The sheaf $\mathbb{C}^{\*}$ of locally constant functions on $M$ is already a sheaf, so sheafification will not change it. 


This sequence is not an exact sequence of complexes but it is an *exact triangle* of complexes. That is - it is an exact sequence of complexes, up to quasi-isomorphism. The obvious short exact sequence of complexes is 
$$
0 \to \mathbb{C}^{\*} \to \left[\begin{array}{c} \underline{\mathbb{C}}^{\*} \\ \downarrow \\ A^{1}\_{M} \end{array}
\right] \to \left[ \begin{array}{c} A^{1}\_{M,cl} \\ \downarrow \\ A^{1}\_{M} \end{array}\right] \to 0 .
$$
Now note that the last complex has an obvious surjective map 
$$
\left[ \begin{array}{c} A^{1}\_{M,cl} \\ \downarrow \\ A^{1}\_{M} \end{array}\right] \to \left[ \begin{array}{c} 0 \\ \downarrow \\ A^{2}\_{M,cl} \end{array} \right]
$$
and that this surjective map is a quasi-isomorphism. So up to a quasi-isomorphism, you can replace the last term in the short exact sequence with the complex you wanted.


