Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 16 / 2

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.

The quasi-compactness of $X$ gives a finite cover by the affine principal neighbourhoods constructed above. Because at every point some $f_i$ is a unit in the local ring, the map of coherent sheavesis surjective. Its kernel $\mathcal{F}$ is a coherent submodule of the trivial bundle, so $H^1(X,\mathcal{F})=0$ by part (a). The long exact sequence in sheaf cohomology makes the map on global sections surjective. Lifting $1$ suppliesThis is the unit-ideal certificate from a principal affine cover.

The localization of global sections on a principal open gives $A_{f_i}=\Gamma (X_{f_i},{\mathcal{O}}_X)$. Each of these rings is a finitely generated algebra because $X_{f_i}$ is an affine variety. Choose finite generators of every $A_{f_i}$ and write them as $a_{ij}/f_i^{e_{ij}}$ with $a_{ij}\in A$. Let $B$ be the $k$-subalgebra of $A$ generated by all $f_i,g_i,a_{ij}$. It is a finitely generated algebra, and $B_{f_i}=A_{f_i}$ since these localizations contain the chosen generators and $f_i^{-1}$.

For any integer $N\ge 1$, raise ${\sum }_i{g}_i{f}_i=1$ to the power $r(N-1)+1$. Every resulting monomial contains some $f_i^N$, so it gives a unit-ideal identity for powersFor an arbitrary $a\in A$, equality $A_{f_i}=B_{f_i}$ implies $f_i^{N_i}a\in B$ for some $N_i$: multiply by an additional power if equality of localized fractions requires it. Choose a common $N$ and the identity above. Then $a={\sum }_i{c}_i(f_i^{N}a)\in B$. Consequently $A=B$, so $A$ is a finitely generated algebra. It is a reduced ring, since a nilpotent element global regular function has zero germ everywhere on the reduced variety $X$. This is finite generation from finitely many principal localizations.

Let $Y$ be the affine variety with coordinate ring $A$. Its principal opens $D(f_i)$ cover $Y$ because ${\sum }_i{g}_i{f}_i=1$. On $X_{f_i}$, the localization of global sections on a principal open identifies its coordinate ring with $A_{f_i}$, giving an isomorphismOn an overlap the two maps are induced by the same elements of $A$, or equivalently by the same identification with $A_{f_i{f}_j}$, so they agree. Glue them to $\varphi :X\to Y$. The inverses agree on the overlaps as well and glue to its inverse. At a closed point $P$, this is the map corresponding to the evaluation ring homomorphism $A\to k$, $a\mapsto a(P)$. This proves the cohomological criterion for affineness by an explicit global isomorphism.

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.