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.
Any advice is greatly appreciated! Thank you all in advance :).