Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 113 / 3

A sheaf $\mathcal{F}$ of ${\mathcal{O}}_X$-modules is quasi-coherent when every affine open $U=\mathrm{Spec}A$ has $\mathcal{F}{|}_U\cong \tilde{M}$ for some $A$-module $M$.

Cover the target ${\mathbb{P}}_k^1$ by $D_{+}(y_0)$ and $D_{+}(y_1)$. On the first chart put $s=y_1/y_0$; its inverse image is $D_{+}(x_0)$ with coordinate $t=x_1/x_0$, and the morphism on rings isAs a $k[s]$-module,is free of rank two. The same calculation on the other standard chart uses $s^{-1}\mapsto t^{-2}$. Hence $f_{\ast }{\mathcal{O}}_X$ is locally free of rank two on the target.

The complement $Y$ is covered by the $m$ principal affine opens $D(f_1),\dots ,D(f_m)$. Every finite intersection is again a principal affine open, so this is an acyclic cover for ${\mathcal{O}}_Y$. Its Čech cochain complex has no degree-$m$ term because there is no intersection of $m+1$ distinct cover members. Therefore

Translate $p$ to the origin. The complement ${\mathbb{A}}_k^3\setminus \{0\}$ has the affine cover $D(x),D(y),D(z)$, and its second Čech cohomology isThe class of $x^{-1}{y}^{-1}{z}^{-1}$ is nonzero, so $H^2({\mathbb{A}}_k^3\setminus \{p\},\mathcal{O})\ne 0$. On the other hand, ${\mathbb{A}}_k^3\setminus \ell =D(x)\cup D(y)$, and the first part with $m=2$ gives $H^2({\mathbb{A}}_k^3\setminus \ell ,\mathcal{O})=0$. Since sheaf cohomology is invariant under scheme isomorphism, the two complements are not isomorphic.

Two nonisomorphic punctual schemes areTheir underlying spaces each have one point, but the second has a nonzero nilpotent element and the first is reduced.

After translating their common support to the origin, punctual closed subschemes of ${\mathbb{A}}_k^1$ correspond to ideals of $k[t]$ whose radical is $(t)$. Since $k[t]$ is a principal ideal domain, each such ideal is $(t^r)$ for a unique $r\ge 1$. Its coordinate ring has basis $1,t,\dots ,t^{r-1}$ and hence dimension $r$. Equal dimensions force equal exponents, so $Z$ and $Z^{\prime }$ are in fact the same closed subscheme after the common coordinate choice, and in particular are isomorphic.

In ${\mathbb{A}}_k^2$, the idealsdefine distinct punctual closed subschemes supported at the origin. Both quotient rings have dimension two over $k$, with bases $1,y$ and $1,x$ respectively. Thus they have equal-dimensional global-section spaces despite being distinct embedded closed subschemes.