Čech cochain complex (${\check{C}}^{\bullet }(\mathcal{U},\mathcal{F})$)

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.

A Čech cochain is a cocycle when its alternating coboundary vanishes. For a multiplicative one-cochain $(g_{ij})$, this says $g_{ij}{g}_{jk}=g_{ik}$ on triple intersections.

A Čech coboundary is the image of a cochain in the preceding degree. Multiplicatively, a zero-cochain $(h_i)$ changes a one-cocycle by $g_{ij}\mapsto h_i^{-1}{g}_{ij}{h}_j$.