---
tag: 97MYM5X7N5
kind: concept
name: Čech cohomology
msc: [55N05, 14F06]
sources:
  - url: https://en.wikipedia.org/wiki/Čech_cohomology
same_as:
  wikidata: Q1095535
  wikipedia: Čech cohomology
---

## Summary

Čech cohomology computes cohomology of a sheaf from an open cover.
It uses the sections of the sheaf on the open sets of the cover and on their finite intersections.
It is defined by an explicit cochain complex, so it is often the practical way to compute [[sheaf cohomology|PQ6Q143ARJ]].
For a quasi-coherent sheaf on a separated scheme and an affine open cover, Čech cohomology equals sheaf cohomology.
The same idea applied to the nerve of a cover gives Čech cohomology of a topological space.

## Explanations

### L1 human

Cover a space by open sets on which everything is simple.
Record data on each piece, compare the data on each overlap of two pieces, and check compatibility on overlaps of three pieces.
Cohomology in degree $1$ counts ways to give compatible differences on overlaps that do not come from data on the pieces.
For 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 $1$.
This number detects the hole in the circle.