Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 113 / 1 / c / Solution

For $n\ge 2$, the omitted point has codimension at least two in the normal integral scheme ${\mathbb{P}}_k^n$. Regular functions extend across such a subset, soThuson $\mathrm{Spec}k$, and it is a coherent sheaf because it corresponds to the one-dimensional $k$-vector space $k$.

The complement of a rational point in ${\mathbb{P}}_k^1$ is ${\mathbb{A}}_k^1$. Its regular functions are $k[t]$, which is infinite-dimensional over $k$. Its pushforward to $\mathrm{Spec}k$ is therefore quasi-coherent but not coherent.

For an example on $X$, choose a projective line $L\subset {\mathbb{P}}_k^n$ through $p$. Thenis closed in $X$. For the closed immersion $i:C\hookrightarrow X$, the sheaf $\mathcal{F}=i_{\ast }{\mathcal{O}}_C$ is coherent, butConsequently ${\pi }_{\ast }\mathcal{F}$ is not coherent on $\mathrm{Spec}k$.