Again I'm stuck on a problem in Hartshorne. The situation is as follows: Let $X=\text{Spec}(A)$ be an affine, non-singular scheme which is finite of some field $k$. Let also $\mathcal{F}$ be a coherent sheaf on $X$. Let $X'$ be an infinitesimal extension of $X$ by $\mathcal{F}$. This means $X'$ has a sheaf of ideals $\mathcal{J}$ such that $\mathcal{J^2}=0$ and there is an isomorphism $(X',\mathcal{O}_{X'}/\mathcal{J})\cong (X,\mathcal{O}_X)$ under which $\mathcal{J}$ (which is a $\mathcal{O}_{X'}/\mathcal{J}$ module) corresponds to $\mathcal{F}$. We have to show that the extension is trivial meaning $\mathcal{O}_{X'}=\mathcal{O}_X\oplus \mathcal{F}$.
My idea was to try and show that $X'$ is affine. If it is we could apply global sections functor to the exact sequence $$0\to \mathcal{J}\to \mathcal{O}_{X'}\to i_*\mathcal{O}_X\to 0$$ of quasi-cogerent $\mathcal{O}_{X'}$-modules (where $i:X\to X'$ is a homeomorphism) to get an exact sequence $$0\to J\to B\to A\to 0$$ of $B$-modules where $J=\Gamma(\mathcal{J})$ and $B=\Gamma(\mathcal{O}_{X'})$. We could then use the infinitesimal lifting property from the previous problem to see that $B$ is the trivial extension of $A$ and $J$.
If anyone has an idea on how to show $X'$ is affine or want to share another approach to this problem I would be very grateful as always=)
