For the cover $\mathcal U=\{X_1,X_2\}$ of the <affine plane with doubled origin>, the overlap is the <punctured affine plane> $U$. Since $H^0(U,\mathcal O_U)=k[x,y]$, the Čech complex begins
$$
k[x,y]\oplus k[x,y]\longrightarrow k[x,y],
\qquad (a,b)\longmapsto b-a.
$$
This map is surjective, and the normalized complex has no terms in degrees at least two. Consequently
$$
\check H^p(\mathcal U,\mathcal O_Y)=0\qquad(p>0).
$$

The <Mayer-Vietoris sequence for sheaf cohomology> also gives $H^1(Y,\mathcal O_Y)=0$, but its next part gives
$$
H^2(Y,\mathcal O_Y)\cong H^1(U,\mathcal O_U).
$$
Part b with $f=1$ shows that the group on the right is infinite-dimensional. Thus
$$
\check H^2(\mathcal U,\mathcal O_Y)=0
\ne H^2(Y,\mathcal O_Y).
$$

This does not contradict the <acyclic cover theorem>. Although $X_1$ and $X_2$ are affine, their intersection $U$ is not acyclic: it has nonzero first structure-sheaf cohomology. Equivalently, this affine cover does not satisfy the theorem's hypotheses; the doubled-origin plane is not a <semi-separated scheme>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 113 / 3 / c / Solution

{wiki=Separated_scheme#Definitions}

A scheme is semi-separated when the intersection of any two affine open subsets is affine, equivalently when its diagonal is affine. This condition makes affine covers acyclic for quasi-coherent sheaves.


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 Incoming links: Semi-separated scheme