Consider a space $X$, $\{U_i\}_i$ an open covering, and $\mathcal{F}$ a sheaf on $X$. Consider $(c_{i_0,\ldots,i_n})$ a Cech $n$-cocycle. It is theorem that every Cech $n$-cocyle is cohomologous to a an alternating one, ie a cocycle, $(d_{i_0,\ldots,i_n})$ such that $d_{i_0,\ldots,i_k,i_{k+1},\ldots, i_n}=-d_{i_0,\ldots,i_{k+1},i_{k},\ldots, i_n}$ (switching two indices changes signs).

Define $c'_{i_0,\ldots,i_n}=-c_{i_0,\ldots,i_{k+1},i_k,\ldots,i_n}$ (ie $c$ with two indices switched). Is it true that $c'$ is a cocycle? And if so is it cohomologous to $c$? Of course this is true in the alternating case.

It seems there should be a way to do this using the fact that every cocycle is cohomologous to an alternating one, but I am getting stuck on the details.

Any thoughts or suggestions would be much appreciated.


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