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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2026

Paper 113

For an indexed open cover $\mathcal U=(U_i)_{i\in I}$ and a sheaf $\mathcal F$, the <Čech cohomology> cochain groups are
$$
\check C^p(\mathcal U,\mathcal F)
=\prod_{i_0<\cdots<i_p}\mathcal F(U_{i_0}\cap\cdots\cap U_{i_p}).
$$
The differential is the alternating sum of restrictions:
$$
(dc)_{i_0\ldots i_{p+1}}
=\sum_{j=0}^{p+1}(-1)^j
c_{i_0\ldots\widehat{i_j}\ldots i_{p+1}}
\big|_{U_{i_0}\cap\cdots\cap U_{i_{p+1}}}.
$$
Since $d^2=0$, the cohomology
$$
\check H^p(\mathcal U,\mathcal F)
=\ker(d:\check C^p\to\check C^{p+1})/operatorname{im}(d:\check C^{p-1}\to\check C^p)
$$
is well defined.

Solved by gpt-5.6-sol high.

Solution

Paper 113 / 3 / a / Solution

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

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

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