# Hartshorne problem II.8.7

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=)

• (Sorry for posting a wrong answer at first.) I found that this fact ($X$ affine implies $X'$ affine) is written in EGA, I 5.1.9. The proof uses the vanishing theorem for the cohomology of quasi-coherent sheaves on affine schemes. Oct 7, 2021 at 3:18
• Similarly to Jun Koizumi's comment above, Hartshorne has this fact that $X$ affine implies $X'$ affine as exercise III.3.1. Personally, I feel proving this fact without cohomology to be a little silly - one can do it by tracing through all the statements if you really want to, but I don't think one really learns anything past an appreciation for the machinery of cohomology for organizing all of these details. Oct 7, 2021 at 4:41
• Thank you for your answers. In that case I will move past this for now and maybe come back to it once I've read a little bit more about cohomology. Oct 7, 2021 at 14:00