Choose an ordering of the index set $I$. The degree-$q$ <Čech cochain group> is
$$
\check C^q(\mathcal U,\mathcal F)
=\prod_{i_0<\cdots<i_q}\mathcal F(U_{i_0}\cap\cdots\cap U_{i_q}).
$$
For $s=(s_{i_0\cdots i_q})$, the <Čech coboundary> is the alternating sum of restrictions
$$
(\delta s)_{i_0\cdots i_{q+1}}
=\sum_{j=0}^{q+1}(-1)^j
s_{i_0\cdots\widehat{i_j}\cdots i_{q+1}}
\big|_{U_{i_0}\cap\cdots\cap U_{i_{q+1}}}.
$$
The identity $\delta^2=0$ makes this the <Čech cochain complex>, and the required <Čech cohomology> is
$$
\boxed{\check H^q(\mathcal U,\mathcal F)=\ker(\delta:\check C^q\to\check C^{q+1})/\operatorname{im}(\delta:\check C^{q-1}\to\check C^q).}
$$

Solution

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

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
$$
\boxed{\check H^j(\mathcal U,\mathcal F)=\ker(\delta:\check C^j\to\check C^{j+1})/
\operatorname{im}(\delta:\check C^{j-1}\to\check C^j).}
$$

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

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

The degree-$p$ Čech cochain group for an ordered cover is $\prod_{i_0<\cdots<i_p}\mathcal F(U_{i_0}\cap\cdots\cap U_{i_p})$.

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