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

A morphism $f:X\to S$ is a separated morphism when its diagonal morphismis a closed immersion.

For the requested example, take the affine plane with doubled origin: glue two copies $U,V\cong {\mathbb{A}}_k^2$ by the identity onThe opens $U$ and $V$ are affine, while their intersection is the punctured affine plane, which is not affine. Indeed, its regular functions are still $k[x,y]$; if it were affine, the canonical map to $\mathrm{Spec}k[x,y]={\mathbb{A}}_k^2$ would be an isomorphism, contrary to the missing origin. The resulting scheme is not separated: in a separated scheme, the intersection of two affine opens is the inverse image of the closed diagonal inside their affine product and is therefore affine.

On the standard charts $D_{+}(x)$ and $D_{+}(y)$ of ${\mathbb{P}}_k^1$, a section of the twisting sheaf on projective space $\mathcal{O}(d)$ is represented by a degree-zero element of the corresponding localization of $k[x,y](d)$. A global section is therefore a homogeneous polynomial of degree $d$ when $d\ge 0$. There are no nonzero global sections for $d<0$. Hence

Local isomorphism on every member of a fixed affine cover does not imply a global isomorphism: the local identifications may have different transition functions. For example, $\mathcal{O}$ and $\mathcal{O}(1)$ are both trivial on the two standard affine charts of ${\mathbb{P}}^1$, but they are not isomorphic because their spaces of global sections have dimensions one and two.

An injective map between line bundles need not be an isomorphism. Multiplication by a nonzero section giveson ${\mathbb{P}}^1$; its cokernel is a nonzero skyscraper sheaf supported at the zero of the section.

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 $\mathrm{Spec}k$.