why_nex.tex

\documentclass{article}

\usepackage{amsmath}
\usepackage{tikz-cd}


\begin{document}
Let $U=\{U_0, U_1, U_2, U_3\}$ be an open covering indexed by $I$ and $V=\{V_0, V_1\}$ be another open covering indexed by $J$. Say $U$ refines $V$ by
\begin{equation*}
\begin{aligned}
  U_0 &\subseteq V_1, & U_1&\subseteq V_0\\
  U_2 &\subseteq V_1, & U_3&\subseteq V_0,
\end{aligned}
\end{equation*}
call this refinement $\tau$.

Consider the diagram
\begin{equation*}
\begin{tikzcd}
  C(U, 0) \arrow[r, "d"]                   & C(U, 1) \\
  C(V, 0) \arrow[r, "d"] \arrow[u, "\tau"] & C(V, 1) \arrow[u, "\tau"],
\end{tikzcd}
\end{equation*}
where $\tau$ stands for the refinement map
$$
\tau(f)({i_0,\dots,i_p})=f(\tau(i_0),\dots,\tau(i_p))\mid_{U_{i_0}\cap\dots\cap U_{i_p}}.
$$
Let $f\in C(V, 0)$, then $\tau d f\in C(U, 1)$, then
\begin{equation*}
\begin{aligned}
  \tau d f\{0,3\} &= d f\{0,1\}\mid_{U_0\cap U_3}\\
                  &= \left(f\{1\}\mid_{V_0\cap V_1}\right)\mid_{U_0\cap U_3} - 
                     \left(f\{0\}\mid_{V_0\cap V_1}\right)\mid_{U_0\cap U_3} \\
                  &= f\{1\}\mid_{U_0\cap U_3} - f\{0\}\mid_{U_0\cap U_3}, \\
  d\tau f\{0,3\}  &= \tau f\{3\}\mid_{V_0\cap V_3} -\tau f\{0\}\mid_{V_0\cap V_3}\\
                  &= \left(f\{0\}\mid_{U_0}\right)\mid_{V_0\cap V_3} -
                     \left(f\{1\}\mid_{U_0}\right)\mid_{V_0\cap V_3}.
\end{aligned}
\end{equation*}
They have different signs!
\end{document}