Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 113 / 4

Choose an ordering of the index set $I$. The degree-$q$ Čech cochain group isFor $s=(s_{i_0\cdots i_q})$, the Čech coboundary is the alternating sum of restrictionsThe identity ${\delta }^2=0$ makes this the Čech cochain complex, and the required Čech cohomology is

The cohomology of twisting sheaves on projective space isIn particular, $H^i(X,{\mathcal{O}}_X)=H^i(X,{\mathcal{O}}_X(1))=0$ for $i>0$, while $H^0(X,{\mathcal{O}}_X)=k$ and $H^0(X,{\mathcal{O}}_X(1))\cong k^{r+1}$.

Apply the long exact sequence in cohomology to the displayed Euler sequence. Its degree-zero part isand all later terms vanish. The first map sends $1$ to the tuple of homogeneous coordinates and is injective. Therefore

For the closed immersion $i:Y\hookrightarrow X$, the ideal sheaf of a closed subscheme givesBecause the projective hypersurface is cut out by one homogeneous polynomial $f$ of degree $d$, multiplication by $f$ identifiesThis is the ideal-sheaf sequence of a projective hypersurface.

Tensor the sequence from part (i) with the twisting sheaf on projective space ${\mathcal{O}}_X(n)$:Its long exact sequence in cohomology containsSince $\dim Y\ge 1$, we have $r\ge 2$, and the final group is intermediate cohomology of projective space. It vanishes by the cohomology of twisting sheaves on projective space. Hence

Set $n=0$ in part (ii). The constants give an injection $k\hookrightarrow H^0(Y,{\mathcal{O}}_Y)$ because the projective hypersurface $Y$ is nonempty, while part (ii) gives surjectivity. ThusA disconnected scheme has a nontrivial idempotent global regular function, equal to zero and one on its two clopen pieces. A field has no such idempotent, so the connectedness from global regular functions criterion gives

Use again the twisted ideal-sheaf sequence of a projective hypersurface. For $0<i<\dim Y=r-1$, the relevant part of its long exact sequence in cohomology isBoth outer terms are intermediate cohomology groups on ${\mathbb{P}}_k^r$, because $1\le i<i+1\le r-1$. They vanish by the cohomology of twisting sheaves on projective space. Therefore