= Affineness is invariant under nilpotent thickening
{title2=$X_0\text{ affine}\iff X\text{ affine}$}
Let $X_0\hookrightarrow X$ be a <nilpotent thickening> defined by $\mathcal I^m=0$. If $X$ is an <affine scheme>, its <closed subscheme> $X_0$ is an <affine scheme>. Conversely, suppose $X_0$ is an <affine scheme>. The underlying <topological spaces> are identical, so $X$ is a <quasi-compact scheme>. For any <quasi-coherent sheaf> $\mathcal F$, each subquotient $\mathcal I^j\mathcal F/\mathcal I^{j+1}\mathcal F$ is naturally a <quasi-coherent sheaf> on $X_0$. By <sheaf cohomology under a closed inclusion> and <vanishing of quasi-coherent cohomology on an affine scheme>, its first <sheaf cohomology> on $X$ vanishes. The <long exact sequence in sheaf cohomology> and descending <mathematical induction> through this finite filtration prove $H^1(X,\mathcal F)=0$. The <Serre criterion for affineness> now makes $X$ an <affine scheme>.
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