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  "body": "\n## Summary\n\nČech cohomology computes cohomology of a sheaf from an open cover.\nIt uses the sections of the sheaf on the open sets of the cover and on their finite intersections.\nIt is defined by an explicit cochain complex, so it is often the practical way to compute [[sheaf cohomology|PQ6Q143ARJ]].\nFor a quasi-coherent sheaf on a separated scheme and an affine open cover, Čech cohomology equals sheaf cohomology.\nThe same idea applied to the nerve of a cover gives Čech cohomology of a topological space.\n\n## Explanations\n\n### L1 human\n\nCover a space by open sets on which everything is simple.\nRecord data on each piece, compare the data on each overlap of two pieces, and check compatibility on overlaps of three pieces.\nCohomology in degree $1$ counts ways to give compatible differences on overlaps that do not come from data on the pieces.\nFor the circle covered by two arcs that overlap in two small arcs, with integer constants as data, this gives one free integer: the cohomology group $\\Z$ in degree $1$.\nThis number detects the hole in the circle.\n",
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