Past exam of the introduction to schemes course of the University of Oxford / 2017 / 3

Write ${\mathbb{P}}^r={\mathbb{P}}_k^r$ over its ground field; the argument also works over any Noetherian affine base. Since $\iota$ is a closed immersion, $X$ is a noetherian scheme and a quasi-compact scheme. LetHere the pullback of a sheaf of modules makes $L$ an invertible sheaf, the restriction of the twisting sheaf on projective space. It is an ample line bundle.

By global generation by sufficiently positive twists, choose $n\ge 0$ such that $F(n)$ is a globally generated sheaf. Finitely many global sections suffice: at each point, finite generation of the stalk of the coherent sheaf $F(n)$ lets us select finitely many generating germs of global sections. Those sections generate on a neighbourhood, because their cokernel sheaf is a coherent sheaf with zero stalk at that point. A finite subcover, using that $X$ is a quasi-compact scheme, supplies finitely many global sections generating everywhere. This is finite global generation on a quasi-compact scheme, and gives a surjective sheaf morphism

Tensor with $(L^{\vee }{)}^{\otimes n}$, where $L^{\vee }$ is the dual of a sheaf. Here tensoring with an invertible sheaf is exact: locally the tensor product of sheaves is tensoring with a free module of rank one. Using the permitted identity $L\otimes L^{\vee }\cong {\mathcal{O}}_X$, we obtainA finite direct sum of sheaves of invertible sheaves is a locally free sheaf of finite rank, and it is a coherent sheaf on the noetherian scheme $X$. This constructs the required $G$. No regularity or smoothness assumption on $X$ is needed.

Use homogeneous coordinates $x_0,\dots ,x_r$ on projective space and letThese $r+1$ open subschemes cover $X$. Each is a closed subscheme of the affine scheme $D_{+}(x_i)$, hence is an affine scheme.

Every nonempty finite intersection is affine as well. Indeed, inside $D_{+}(x_{i_0})$ the intersection with $D_{+}(x_{i_1}),\dots ,D_{+}(x_{i_p})$ is the principal open subscheme obtained by invertingIntersecting it with the closed subscheme $X$ gives $U_{i_0}\cap \cdots \cap U_{i_p}$, again an affine scheme.

For any quasi-coherent sheaf $F$, vanishing of quasi-coherent cohomology on an affine scheme makes every such intersection acyclic for $F$. The acyclic cover theorem therefore identifies the sheaf cohomology of $F$ with the Čech cohomology of this cover. Its alternating Čech cochain complex has termswhich vanish for $p>r$: one cannot choose $p+1$ distinct indices from $r+1$ indices. ConsequentlyThe choice $N=\max \{1,r\}$ meets the strict requirement $N>0$ even when $r=0$. It is independent of $F$, and applies to all quasi-coherent sheaves, including those which are not coherent sheaves. This is the cohomological dimension bound from an affine cover.

Fix a coherent sheaf $F$ and an integer $k>M$. Set $K_0=F$. Repeatedly apply a to construct short exact sequences of sheaveswhere each $G_j$ is a coherent sheaf and a locally free sheaf. Each kernel sheaf $K_{j+1}$ is a coherent sheaf because $X$ is a noetherian scheme. Thus the construction can be repeated as many times as needed; it does not require the kernel sheaves themselves to be locally free sheaves.

For every integer $q>M$, the long exact sequence in sheaf cohomology containsBoth outer groups vanish by the hypothesis, since $q$ and $q+1$ exceed $M$. Hence ${\delta }_j$ is an isomorphism. Iterating this dimension shifting in sheaf cohomology gives

Choose $s\ge 0$ with $k+s>N$, where $N$ is the uniform bound from b. The coherent sheaf $K_s$ is a quasi-coherent sheaf, so b makes the last group zero. ThereforeThe argument uses a finite initial segment of a locally free resolution. There is no need to assume that every coherent sheaf has a finite locally free resolution: the uniform bound from b is what terminates the argument.