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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.

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.

For a closed point $P$ with residue field $k(P)$, evaluation gives a morphism from the structure sheaf onto the skyscraper sheaf $k_P$. Its kernel is the ideal sheaf of $P$. On a Noetherian scheme it is a coherent ideal sheaf.

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 $r$.

For $r=1$, the assertion is the assumed vanishing for coherent ideal sheaves. For $r>1$, project $\mathcal{F}\subseteq {\mathcal{O}}_X^r$ onto the last component. Its image $\mathcal{I}\subseteq {\mathcal{O}}_X$ is a coherent ideal sheaf, and its kernel ${\mathcal{F}}^{\prime }$ is a coherent sheaf contained in ${\mathcal{O}}_X^{r-1}$. Here images and kernels are coherent because a variety is Noetherian. The short exact sequencegives an exact segment $H^1(X,{\mathcal{F}}^{\prime })\to H^1(X,\mathcal{F})\to H^1(X,\mathcal{I})$ in the long exact sequence in sheaf cohomology. The outer terms vanish by mathematical induction and the hypothesis, so the middle term vanishes. This is ideal-sheaf vanishing for a coherent submodule of a trivial bundle.

Let ${\mathcal{I}}_Y$ be the coherent ideal sheaf of $Y$, and let ${\mathfrak{m}}_P$ be the ideal sheaf of a closed point. Because $P\notin Y$, the stalk $({\mathcal{I}}_Y{)}_P$ is ${\mathcal{O}}_{X,P}$. Evaluation at $P$ therefore gives a surjective morphism of sheaves ${\mathcal{I}}_Y\to k_P$ to the skyscraper sheaf at $P$. Its kernel $\mathcal{J}={\mathcal{I}}_Y\cap {\mathfrak{m}}_P$ is again a coherent ideal sheaf. Fromand $H^1(X,\mathcal{J})=0$, the long exact sequence in sheaf cohomology shows that $\Gamma (X,{\mathcal{I}}_Y)\to k$ is onto. Choose $f$ mapping to $1$. It vanishes on $Y$ and satisfies $f(P)=1$. Hence $X_f\subseteq U$, and inside the affine variety $U$ it is the principal open subset defined by $f{|}_U$. A principal open of an affine variety is affine. This gives affine principal neighbourhoods from ideal-sheaf vanishing.

For an affine variety, every coherent ideal sheaf is quasi-coherent; vanishing of quasi-coherent cohomology on an affine scheme therefore gives $H^1(X,\mathcal{I})=0$.

For the projective-space complement, assume $n\ge 1$ and choose two distinct closed points $P,Q\in X$. Take the ideal sheaf of two closed points $\mathcal{I}={\mathfrak{m}}_P\cap {\mathfrak{m}}_Q$ on $X$. The codimension-two extension of regular functions on a normal variety givesOne can see this directly: on every standard affine chart of ${\mathbb{P}}^n$, a rational function written in lowest terms cannot have a nonconstant denominator, because an irreducible polynomial factor of the denominator would define a pole along a codimension-one hypersurface, and such a hypersurface is not removed by $Z$. The extended function is constant because every global regular function on projective space is constant. Now the short exact sequencesends $k$ diagonally into $k^2$ on global sections. Its cokernel is $k$, and the long exact sequence in sheaf cohomology injects that cokernel into $H^1(X,\mathcal{I})$. ThusThe assumption $n\ge 1$ is necessary: ${\mathbb{P}}^0$ is already affine and has no such example.