There are 2 complexes computing Cech cohomology. The difference between them is that in the second one we require skew symmetry when you change the order of indices. How to show that they are quasi-isomorphic? Let $C(U,F)$ be the first and $C'(U,F)$ the second. Open sets are indexed by the set $I$ with some perfect order. There is inclusion $i: C'(U,F) \to C(U,F)$. There is projection $h: C(U,F) \to C'(U,F)$, defined by $$ (hc)_{i_0...i_n}=sgn(\varepsilon) c_{i_{\varepsilon_0}i_{\varepsilon_1}...} $$ I failed to construct a homotopy $k:C(U,F) \to C(U,F)[1]$ such that $$ 1-ih=dk+kd. $$ Could you help me?
2026-03-28 00:06:07.1774656367
Cech cohomology
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