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David Zureick-Brown
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Equivalence of ordered and unordered cech cohomology.

Given a topological space X and a finite cover X = $\cup X_i$, one can define Cech cohomology of a sheaf of abelian groups F with respect to the cover $\{X_i\}$ in two different ways:

  1. (Ordered): The kth term of the Cech complex is $\bigoplus_{i_1 < \ldots < i_k} \Gamma(X_{i_1} \cap \ldots \cap X_{i_k}, F)$.
  2. (Unordered): The kth term of the Cech complex is $\bigoplus_{i_1, \ldots , i_k} \Gamma(X_{i_1} \cap \ldots \cap X_{i_k}, F)$.

In particular, the second description involves repetition and is non-zero in every degree. These two descriptions give isomorphic cohomology (the first maps you try to write down will likely be homotopy equivalences).


Question: Is there a canonical reference for this fact?

David Zureick-Brown
  • 10.5k
  • 3
  • 39
  • 96