Cech cohomology of $\mathbb A^2_k\setminus\{0\}$ 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
 A: First  notice  that the restriction morphism $\Gamma(\mathbb A^2_k,\mathcal O_{ \mathbb A^2_k})\to \Gamma(S, \mathcal O_S)$ is bijective because the affine plane $\mathbb{A}^2_k$ is normal ("Hartogs phenomenon").
Hence we may identify $\Gamma(S, \mathcal O_S)$ with the polynomial ring $k[X,Y]$
a) The open set $D(X)$ is isomorphic to  $\mathbb G_m\times \mathbb A^1_k$ where $\mathbb G_m=\operatorname  {Spec} k[T,T^{-1}]$, the affine line with origin deleted.
Hence  $\Gamma(D(X),\mathcal O_{ A^2_k})=k[X,X^{-1},Y]$.
b) Similarly  $D(Y)$ is isomorphic to  $\mathbb A^1_k \times \mathbb G_m$.
Hence  $\Gamma(D(Y),\mathcal O_{ A^2_k})=k[X,Y, Y^{-1}]$.
c) Finally the open set $D(X)\cap D(Y)$ is isomorphic to the product $\mathbb G_m\times_k \mathbb G_m$ .
Hence $\Gamma(D(X)\cap D(Y),\mathcal O_{ A^2_k})=k[X,X^{-1}]\otimes _k k[Y,Y^{-1}]= k[X,X^{-1},Y,Y^{-1}]$.
d) With these identifications established, the first cohomology group $\check H^1(\mathcal U,\mathcal O_S)$ of the structural sheaf is the cohomology of the complex
$$ k[X,X^{-1},Y]\times k[X,Y,Y^{-1}]    \to           k[X,X^{-1},Y,Y^{-1}] \to 0      $$  where the non trivial map is $$(f(X,X^{-1},Y),g(X,Y,Y^{-1}))\mapsto g(X,Y,Y^{-1})-f(X,X^{-1},Y)$$
e) Hence we  see that the required cohomology  is the following infinite dimensional $k$-vector space , spectacularly violating vanishing  of cohomology for affine schemes, which $S$ is thus not.
Final result
$$    \check H^1(\mathcal U,\mathcal O_S)=\check H^1(S,\mathcal O_S)=\oplus _{i,j\gt 0} \; k\cdot X^{-i} Y^{-j}                             $$
