For an ordered <open cover> $\mathcal U=(U_i)$ and a <sheaf of sets> $\mathcal F$, the <Čech cochain group> is
$$
\check C^p(\mathcal U,\mathcal F)=\prod_{i_0<\cdots<i_p}\mathcal F(U_{i_0}\cap\cdots\cap U_{i_p}).
$$
Its <Čech coboundary> is
$$
(\delta c)_{i_0\ldots i_{p+1}}=\sum_{r=0}^{p+1}(-1)^r
c_{i_0\ldots\widehat{i_r}\ldots i_{p+1}}|.
$$
Terms in $\delta^2$ cancel in pairs. Hence
$$
\boxed{\check H^j(\mathcal U,\mathcal F)=\ker(\delta:\check C^j\to\check C^{j+1})/
\operatorname{im}(\delta:\check C^{j-1}\to\check C^j).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-118/2/a/solution