So in  [another question of mine][1] 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**


  [1]: https://mathoverflow.net/questions/94657/weil-kostant-integrality-result-as-stated-by-brylisnki