Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 113 / 3

An ${\mathcal{O}}_X$-module $\mathcal{F}$ is a quasi-coherent sheaf when every affine open $V=\mathrm{Spec}A\subseteq X$ has $\mathcal{F}{|}_V\cong \tilde{M}$ for some $A$-module $M$. For a morphism $f:X\to Y$, its direct image sheaf iswhile the pullback of a sheaf of modules is

If $i:Z\hookrightarrow X$ is a closed immersion of Noetherian schemes, then on an affine open $V=\mathrm{Spec}A\subseteq X$ one has $Z\cap V=\mathrm{Spec}(A/I)$. The restriction of $i_{\ast }{\mathcal{O}}_Z$ corresponds to the cyclic $A$-module $A/I$, so it is finitely generated. Hence $i_{\ast }{\mathcal{O}}_Z$ is a coherent sheaf; more generally this is the direct image of a coherent sheaf under a closed immersion.

Coherence need not survive an arbitrary pushforward. Letbe the open immersion. The sheaf ${\mathcal{O}}_{D(x)}$ is coherent, butis not a finitely generated $k[x]$-module. Therefore $j_{\ast }{\mathcal{O}}_{D(x)}$ is not coherent.

Choose a finite affine cover $\mathcal{U}=(U_j)$ of the Noetherian scheme $X$. Because $X$ is separated, every finite intersection $U_J$ is affine. Its inverse image $V_J=Z\cap U_J$ under the closed immersion is also affine. By the definition of the direct image sheaf,The Čech complexes for $i_{\ast }\mathcal{F}$ on $\mathcal{U}$ and for $\mathcal{F}$ on the induced cover $(Z\cap U_j)$ are therefore identical, including their restriction maps. Both affine covers are acyclic for the relevant quasi-coherent sheaves, so the acyclic cover theorem givesfor every $q$. This is cohomology under a closed immersion.

For $X={\mathbb{P}}_k^n$, use its $n+1$ standard affine opens. The induced cover of $Z$ is still acyclic, and its Čech complex has no cochains in degrees greater than $n$. The cohomological dimension bound from an affine cover therefore yields

Let the degree-$d$ projective hypersurface $X_d$ be defined by the homogeneous polynomial $F$. Multiplication by $F$ identifies its ideal sheaf with ${\mathcal{O}}_{{\mathbb{P}}^n}(-d)$, giving the ideal-sheaf sequence of a projective hypersurfaceIn particular, the kernel in the question is ${\mathcal{O}}_{{\mathbb{P}}^n}(-d)$.

For $n=2$, additivity of the Euler characteristic in this short exact sequence and the line-bundle formulagiveHere $H^2(X_d,{\mathcal{O}}_{X_d})=0$ by the preceding cohomological-dimension argument. Moreover $H^0({\mathbb{P}}^2,\mathcal{O}(-d))=H^1({\mathbb{P}}^2,\mathcal{O}(-d))=0$ for $d>0$, so the long exact sequence gives $H^0(X_d,{\mathcal{O}}_{X_d})\cong k$. Consequentlywhich is the genus-degree formula for a projective plane curve.