Cohomological criterion for affineness ($X\text{ affine}⟺H^1(X,\mathcal{I})=0\text{ for all coherent ideals }\mathcal{I}$)

A Noetherian scheme is affine exactly when every coherent ideal sheaf has zero first sheaf cohomology. One direction is vanishing of quasi-coherent cohomology on an affine scheme. For the converse, ideal-sheaf vanishing produces affine principal neighbourhoods from ideal-sheaf vanishing. A finite such cover yields a unit-ideal certificate from a principal affine cover, and those affine charts glue to the spectrum of a commutative ring of global sections. The criterion, including the finite-type ideal version for quasi-compact quasi-separated schemes, is recorded in Stacks Project, Section 30.3.

If finitely many $X_{f_i}$ cover $X$, the morphism ${\mathcal{O}}_X^r\to {\mathcal{O}}_X$ with coefficients $f_i$ is onto. If its kernel has vanishing first sheaf cohomology, the long exact sequence in sheaf cohomology lifts the global section $1$ to coefficients $g_i$. This turns a geometric cover into a unit ideal in the ring of global sections.

Let $P$ have an affine open neighbourhood $U$, and let $Y=X\setminus U$. Evaluation gives ${\mathcal{I}}_Y\twoheadrightarrow k_P$ with kernel ${\mathcal{I}}_Y\cap {\mathcal{I}}_P$. If first sheaf cohomology of every coherent ideal sheaf vanishes, a global section $f$ of ${\mathcal{I}}_Y$ can be chosen with $f(P)=1$. Then $X_f\subseteq U$ is a principal open subset of an affine variety and is affine.

If $H^1(X,\mathcal{I})=0$ for every coherent ideal sheaf, projection to the last coordinate makes any coherent $\mathcal{F}\subseteq {\mathcal{O}}_X^r$ an extension of a coherent submodule of ${\mathcal{O}}_X^{r-1}$ by an ideal sheaf. The long exact sequence in sheaf cohomology proves the assertion by mathematical induction on $r$.