Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 113 / 4

For an open cover $\mathcal{U}=(U_i)$, the Čech cochain complex iswith differentialIts cohomology is ${\check{H}}^p(\mathcal{U},\mathcal{F})$.

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Suppose ${\mathbb{P}}_k^n$ had a cover by $n$ affine opens. Since projective space is separated, every finite intersection in this cover is affine. The acyclic cover theorem would therefore compute the cohomology of every quasi-coherent sheaf by its Čech complex. A cover with only $n$ members has no degree-$n$ cochains, so it would implyBut top cohomology of projective space givesa contradiction. Hence no such affine cover exists.

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Choose a frame of a line bundle on each $U_i$. On $U_i\cap U_j$, the frames differ by $g_{ij}\in {\mathcal{O}}_X^{\ast }(U_i\cap U_j)$, and compatibility on triple intersections is the Čech cocycle condition $g_{ij}{g}_{jk}=g_{ik}$. Replacing the local frames by units $h_i$ changes $g_{ij}$ by the Čech coboundary $h_i^{-1}{h}_j$. Conversely, a multiplicative one-cocycle glues the trivial line bundles ${\mathcal{O}}_{U_i}$ into a line bundle. Tensor product multiplies cocycles, so

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A Cartier divisor on an integral scheme is given by an open cover $(U_i)$ and rational functions $f_i\in K(X{)}^{\ast }$ such that $f_i/f_j\in {\mathcal{O}}_X^{\ast }(U_i\cap U_j)$. Two are linearly equivalent when their quotient is represented by one global rational function. The Cartier class group is the group of Cartier divisors modulo these principal divisors.

Let ${\mathcal{K}}_X^{\ast }$ be the sheaf of nonzero rational functions. Cartier divisors are the global sections of ${\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$, and the exact sequencegives a long exact cohomology sequence. On an integral scheme ${\mathcal{K}}_X^{\ast }$ is flasque: every nonempty restriction map is the identity on $K(X{)}^{\ast }$. Hence $H^1(X,{\mathcal{K}}_X^{\ast })=0$, and exactness gives

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