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 $X$ be an algebraic variety over an algebraically closed field $k$, and assume $H^1(X,\mathcal{I})=0$ for every coherent ideal sheaf $\mathcal{I}$. Given a closed point $P$, choose an affine open subscheme $U$ containing it and give $Y=X\setminus U$ its reduced induced subscheme structure. Since $P\notin Y$, evaluation at $P$ is a surjective morphism of sheaveswith kernel ${\mathcal{I}}_Y\cap {\mathcal{I}}_P$. Here $k_P$ is the skyscraper sheaf at the closed rational point, and the kernel is again a coherent ideal sheaf. The long exact sequence in sheaf cohomology therefore lifts $1\in k_P$ to a global section $f$ of ${\mathcal{I}}_Y$.

Viewed as a global regular function, $f$ vanishes on $Y$ and has $f(P)=1$. Thus $P\in X_f\subseteq U$, and $X_f=U_{f{|}_U}$ is a principal open subscheme of an affine scheme, hence affine. Such neighborhoods cover $X$: every nonempty closed subset of an algebraic variety contains a closed point. This is the local step in the cohomological criterion for affineness; the closed-point hypothesis makes the coherent skyscraper argument valid.

Let $X$ be a Noetherian scheme and suppose $H^1(X,\mathcal{I})=0$ for every coherent ideal sheaf $\mathcal{I}$. For a coherent sheaf $\mathcal{F}\subseteq {\mathcal{O}}_X^r$, projection onto the last coordinate has image an ideal sheaf $\mathcal{I}\subseteq {\mathcal{O}}_X$ and kernel ${\mathcal{F}}^{\prime }\subseteq {\mathcal{O}}_X^{r-1}$. Both are coherent sheaves, sois a short exact sequence of sheaves. Thus $\mathcal{F}$ is a sheaf extension of the quotient ideal $\mathcal{I}$ by the lower-rank submodule ${\mathcal{F}}^{\prime }$, with this order of the two terms.

For $r=1$, the assertion is exactly the assumed vanishing for coherent ideal sheaves. For larger $r$, mathematical induction gives $H^1(X,{\mathcal{F}}^{\prime })=0$, and the long exact sequence in sheaf cohomology places $H^1(X,\mathcal{F})$ between this zero group and $H^1(X,\mathcal{I})=0$. Hence $H^1(X,\mathcal{F})=0$.