Let $\mathcal F$ be a sheaf of abelian groups on a topological space $X$ and consider an open covering $\mathcal U=\{U_i\}_{i\in I}$ of $X$ . For the sake of simplicity denote $$ U_{i_0i_1\cdots i_q}=U_{i_0}\cap U_{i_1}\cap\dots\cap U_{i_q}. $$ The group $\check C^q(\mathcal U,\mathcal F)$ of Cech $q$ -cochains is the set of families $$ c=(c_{i_0i_1\cdots i_q})\in\prod_{(i_0,\dots,i_q)\in I^{q+1}}\mathcal F(U_{i_0i_1\cdots i_q}). $$ The group structure on $\check C^q(\mathcal U,\mathcal F)$ is the obvious one deduced from the addition law on sections of $\mathcal F$ .

The Cech differential $$ \delta^q\colon\check C^q(\mathcal U,\mathcal F)\to\check C^{q+1}(\mathcal U,\mathcal F) $$ is defined by the formula $$ (\delta^q c)_{i_0\cdots i_{q+1}}=\sum_{0\le j\le q+1}(-1)^j c_{i_0\cdots\widehat{i_j}\cdots i_{q+1}}|_{U_{i_0\cdots i_{q+1}}}, $$ and we set $\check C^{q}(\mathcal U,\mathcal F)=0$ , $\delta^q=0$ for $q<0$ . Easy computations show that $\delta^{q+1}\circ\delta^q=0$ . We get therefore a cochain complex $(\check C^\bullet(\mathcal U,\mathcal F),\delta)$ , called the complex of Cech cochains relative to the covering $\mathcal U$ .

The $q$ -th Cech cohomology group of $\mathcal F$ relative to $\mathcal U$ is $$ \check H^q(\mathcal U,\mathcal F)=H^q(\check C^\bullet(\mathcal U,\mathcal F)). $$