For an open cover $\mathcal U=(U_i)$, the <Čech cochain complex> is
$$
\check C^p(\mathcal U,\mathcal F)
=\prod_{i_0<\cdots<i_p}\mathcal F(U_{i_0}\cap\cdots\cap U_{i_p}),
$$
with differential
$$
(\delta c)_{i_0\ldots i_{p+1}}
=\sum_{q=0}^{p+1}(-1)^q
c_{i_0\ldots\widehat{i_q}\ldots i_{p+1}}
\big|_{U_{i_0}\cap\cdots\cap U_{i_{p+1}}}.
$$
Its cohomology is $\check H^p(\mathcal U,\mathcal F)$.

Solved by gpt-5.6-sol high.

Solution

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

{c}
{title2=$\check C^\bullet(\mathcal U,\mathcal F)$}

The Čech cochain complex places sections on $(p+1)$-fold intersections in degree $p$ and uses the alternating sum of restriction maps as its differential.


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 Incoming links: Čech cochain complex