I'm trying to prove, via the Cech cohomology, that $S=\mathbb A^2_k\setminus\{0\}$ with the induced Zariski topology is not an affine variety. Consider the structure sheaf $\mathcal O_{\mathbb A^2_k}\big|_S:=\mathcal O_S$ (which is quasi coherent), i must show that $\exists n$ such that $\check H^n(S,\mathcal O_S)\neq 0$. It is enough to prove that $\check H^n(\mathcal U,\mathcal O_S)\neq0$ for a certain affine cover of $S$ (and a certain $n$); so let's choose $\mathcal U=\{D(X), D(Y)\}$ where $D(X)=\{(x,y)\in S\,:\, x\neq 0\}$ and $D(Y)=\{(x,y)\in S\,:\, y\neq 0\}$. Clearly for $n\ge 2$ we have that $\check H^n(S,\mathcal O_S)=0$, so i must show that $\check H^1(\mathcal U,\mathcal O_S)\neq0$. The Cech complex is: $\mathcal O_S(D(X))\times\mathcal O_S(D(Y))=\Gamma(S)_X\times\Gamma(S)_Y\longrightarrow \mathcal O_S(D(X)\cap D(Y))=\Gamma(S)_{XY}\longrightarrow 0\cdots$
with the homomorphism: $d^0: (f,g)\mapsto g|_{{D(X)\cap D(Y)}}-f|_{{D(X)\cap D(Y)}}$. To complete the proof i should conclude that $d^0$ is not surjective, but why is this true?
thanks