# some examples of the Soft sheaves but not fine

As we know,a fine sheaf is also soft.So,I need some examples of the sheaves that are soft but not fine.Can the holomorphic sheaf $$\mathcal O(X)$$ be one?Any help and comments are accepted.Thanks a lot!

Example. Let $$\displaystyle X=\left\{z\in\mathbb{C}:|z|\leq\frac{1}{2}\right\}$$ be the "half unitary disk"; the holomorphic maps $$\begin{equation*} f(z)=\sum_{n=0}^{+\infty}z^n,\,g(z)=\sum_{n=1}^{+\infty}z^{n!} \end{equation*}$$ can not be extendend to the whole of $$\mathbb{C}$$, so $$\mathcal{O}_{\mathbb{C}}$$ is neither soft nor fine. $$\triangle$$
Considering the constant sheaf $$\mathcal{F}$$ to $$\mathbb{Z}$$ on $$\mathbb{A}^1_{\mathbb{C}}$$ with Zariski topology; $$\mathcal{F}$$ is flabby but not fine (of course), so it is soft not fine.
• I don't think the constant sheaf on $\mathbb{A}^1$ is soft : a section over the closed subset $\{0,1\}$ cannot necessarily be extended to the whole space. Aug 15, 2019 at 13:04
• @Armandoj18eos Sorry for my late reply. I checked several sources and it seems that there is two different notions of soft sheaves, or more accurately, there is two different notions of $\Gamma(Z,\mathcal{F})$ when $Z$ is a closed subset. One is $\varinjlim_{U\supset Z}\Gamma(U,\mathcal{F})$, the other is $\Gamma(Z,\mathcal{F}|_Z)$ (this is the one used for example in Kashiwara-Shapira, Godement...). These two notions agree if $Z$ has a fundamental system of paracompact (hence Hausdorff) neighborhood. The problem is : the first notion is ill-behaved for other spaces. Aug 21, 2019 at 19:28