# Is the total ring of fractions mod the regular functions flasque?

I want to show that $$0\to \mathcal O_{\mathbb{P}_k^1}\to \mathcal K\to \mathcal K/\mathcal O_{\mathbb{P}_k^1}\to 0$$ is a flasque resolution of $$\mathcal O_{\mathbb{P}_k^1}$$ with $$k$$ infinite, but not necessarily algebraically closed. Where $$\mathcal K$$ is the total ring of fractions.

I was able to show that $$\mathcal K$$ is flasque, and by extension, the presheaf $$U\mapsto \mathcal K(U)/\mathcal O_{\mathbb{P}_k^1}(U)$$ is 'flasque' as a presheaf. However, being a quotient, we aren't guaranteed that this is a sheaf. I tried to show that it was, but I kept running into troubles. So, instead, I tried to show flasqueness of the sheafification, but again, I couldn't lift any of the maps. Additionally, $$\mathcal O_{\mathbb{P}_k^1}$$ isn't flasque; so, I'm stuck.

• What can you say about $H^1(U,\mathcal{O})$ if $U$ is any open subset of $\mathbb{P}^1$ ? – Roland Dec 3 '18 at 20:40
• Yes that is my point. There is two cases to be treated separately : $U=\mathbb{P}^1$ and $U\neq\mathbb{P}^1$. For the first one, it is a classical computation. For the second, use that $U$ is affine. – Roland Dec 3 '18 at 21:04