So I've been trying to understand a proof of the fact that $H^p(M, \mathcal F)=0$ whenever $p\geq 1$ from page 42 of the book "Principles of Algebraic Geometry" by Griffiths and Harris.


The proof is carried out for the sheaf ${\mathcal a}^{r,s}$ of $C^{\infty}$ forms of type $(r,s)$ on $U$ and there is a remark after the proof of how one can prove the result for an arbitrary fine sheaf.


According to them, a fine sheaf is a sheaf for which we have maps $\eta\_\alpha:\mathcal F (U\_{\alpha})\rightarrow \mathcal F (U)$ for any $U=\cup U\_{\alpha}$, such that the support of $(\eta\_{\alpha}\sigma)$ is contained in $U\_\alpha$ and $\sum {\eta\_{\alpha}} (\sigma| \_{U\_\alpha})=\sigma$ for $\sigma\in \mathcal F (U)$.


So is $(\eta\_{\alpha}\sigma)$ a map from the manifold $M$ to $\mathbb R$? I don't understand how $(\eta\_{\alpha}\sigma)$ has $M$ as its domain.


Secondly how can they write $(\eta\_{\alpha}\sigma)$ when $\sigma\in \mathcal F (U)$? Isn't the domain of $\eta\_\alpha$ the group ${\mathcal F }(U\_\alpha)$?


Thirdly how does this formulation apply to the case when $\mathcal F$ is the sheaf ${\mathcal a}^{(r,s)}$? We would need maps $\eta\_{\alpha}: {\mathcal a}^{(r,s)}(U\_{\alpha})\rightarrow {\mathcal a}^{(r,s)}(U)$. But indstead the proof just uses the partion of unity $C^{\infty} $ maps $\rho\_{\alpha}$ subordinate to some open cover $\{U\_\alpha\}$. So it seems that they haven't used the fact that ${\mathcal a}^{r,s}$ is a fine sheaf.



Proof that higher cech cohomology groups vanish for fine sheaves.
About your third question. Using partition of unity $(\rho\_\alpha)\_\alpha$ subordinate to $(U\_\alpha)\_\alpha$ for $\mathcal F$ you can define maps $\eta\_\alpha:\mathcal F(U\_\alpha)\to\mathcal F(U)$ by taking $s\in\mathcal F(U\_\alpha)$ to $\rho\_\alpha s$. 




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About the first two points it seems to me that $(\eta\_\alpha\sigma):=\eta\_\alpha(\sigma\_{|U\_\alpha})$.




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EDIT: to summarize my answer a fine sheaf is, ["by definition"](http://en.wikipedia.org/wiki/Injective_sheaf#Fine_sheaves), a sheaf having partition of unity. 


