How do you center in the align environment?

I'd use array for that.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\[ \begin{array}{c *{4}{@{$\;$}c} }
0 \to \Omega^*(M) \to & \Omega^*(U) \oplus \Omega^*(V) & \to     & \Omega^*(U \cap V) & \to 0 \\
                      & (\omega,\tau)                  & \mapsto & \tau - \omega      &
\end{array} \]
\end{document}

An increasing number of mathematicians and physicists typeset these exact sequences with tikz-cd.

\documentclass{article}
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}[column sep=1em,row sep=0.5ex]
0\arrow[r] &\Omega^*(M) \arrow[r] &
\Omega^*(U)\oplus\Omega^*(V) \arrow[r]&\Omega^*(U\cap V) 
\arrow[r] & 0 \\
& & (\omega, \tau) \arrow[r,mapsto] & \tau-\omega & 
\end{tikzcd}
\end{document}

You can still use align, just you need to insert the items in the second line into a box the same width as their corresponding elements above. makebox with widthof from calc package can do this for you.

\documentclass{article}
\usepackage{amsmath}
\usepackage{calc}
\begin{document}

\newcommand{\sameas}[2]{\makebox[\widthof{$#1$}]{$#2$}}

\begin{align*}
0\to \Omega^*(M) \to \Omega^*(U)\oplus\Omega^*(V) &\mapsto \Omega^*(U\cap V) \to 0 \\
\sameas{\Omega^*(U)\oplus\Omega^*(V)}{(\omega,\tau)} &\to \sameas{\Omega^*(U\cap V)}{\tau-\omega}
\end{align*}

\end{document}