Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 113 / 1

Take $X=\mathbb{R}$, $U=(0,1)$, and let $\mathcal{F}$ be the skyscraper sheaf with value a nonzero abelian group $A$ at $p=3/4$. Cover $V=(-1,1)$ by $V_1=(-1,2/3)$ and $V_2=(1/2,1)$. The presheaf gives $(j^p\mathcal{F})(V)=0=(j^p\mathcal{F})(V_1)$ and $(j^p\mathcal{F})(V_2)=A$. A nonzero section on $V_2$ and the zero section on $V_1$ agree on the overlap, whose skyscraper sections vanish, but cannot be glued on $V$. Thus $j^p\mathcal{F}$ need not be a sheaf.

Sheafification preserves stalks. If $P\in U$, neighborhoods contained in $U$ are cofinal, so $(j^p\mathcal{F}{)}_P={\mathcal{F}}_P$. If $P\notin U$, no neighborhood of $P$ lies in $U$, so every term defining the presheaf stalk is zero. Therefore

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On the defining presheaf, send a section over $V\subseteq U$ by the identityand use the unique zero map when $V⊈U$. These maps commute with restrictions and therefore sheafify to the natural counit

After restricting $j^p\mathcal{F}$ back to $U$, every open set lies in $U$, so one recovers the original sheaf $\mathcal{F}$. Equivalently, the natural map $\mathcal{F}\to j^{-1}{j}_{!}\mathcal{F}$ is an isomorphism on every stalk and hence an isomorphism of sheaves.

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Use the counit from part b for the first map and the restriction map $\mathcal{G}\to i_{\ast }{i}^{-1}\mathcal{G}$ for the second. Exactness can be checked on stalks. At $P\in U$ the sequence iswhereas at $P\in Z$ it isThusis a short exact sequence of sheaves.

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Let $X=\mathrm{Spec}k[x]$ and $U=D(x)$. A global section of $j_{!}{\mathcal{O}}_U$ is a regular function on the integral scheme $U$ whose support is closed in $X$ and contained in $U$. Every nonzero regular function on $U$ has support dense in $X$, soIf this sheaf were quasi-coherent, then on the affine scheme $X$ it would be the sheaf associated with this zero module and hence would vanish. Its stalks at points of $U$ are instead ${\mathcal{O}}_{U,P}\ne 0$ by part a. This contradiction proves that extension by zero need not preserve quasi-coherence.

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