Čech cohomology

Definition

Let $X$ be a topological space, and let $\mathcal{F}$ be a sheaf of abelian groups on $X$. Let $\mathfrak{U} = (U_i)_{i \in I}$ be an open cover of $X$ with $I$ a well-ordered set. For $i_0, \dots, i_p \in I$ write $U_{i_0 \dots i_p} = U_{i_0} \cap \dots \cap U_{i_p}$. The Čech complex is the cochain complex with $$C^p(\mathfrak{U}, \mathcal{F}) = \prod_{i_0 < \dots < i_p} \mathcal{F}(U_{i_0 \dots i_p})$$ and differential $d \colon C^p \to C^{p+1}$ given by $$(d\alpha)_{i_0 \dots i_{p+1}} = \sum_{j=0}^{p+1} (-1)^j \, \alpha_{i_0 \dots \widehat{i_j} \dots i_{p+1}}|_{U_{i_0 \dots i_{p+1}}},$$ where the hat means that the index is omitted. The Čech cohomology of $\mathcal{F}$ with respect to $\mathfrak{U}$ is $\check{H}^p(\mathfrak{U}, \mathcal{F}) = H^p(C^\bullet(\mathfrak{U}, \mathcal{F}))$. The Čech cohomology of $X$ with coefficients in $\mathcal{F}$ is the direct limit $\check{H}^p(X, \mathcal{F}) = \varinjlim_{\mathfrak{U}} \check{H}^p(\mathfrak{U}, \mathcal{F})$ over refinements of covers.