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Help formatting a commutative diagram in TikZ

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I need some serious help getting this commutative diagram to look presentable. I've been messing around with it for literally weeks and this is the best I've got.

Help formatting a commutative diagram in TikZ

The only suppression that I could accept is swapping $H^{n-p}(F)$ for a suppression like $\mathcal{F}$. The code is:

\[\begin{tikzpicture}[scale=1.5] \node (sa) at (0,5) {$\ldots$}; \node (sb) at (4,5) {$\ldots$};  \node (a1) at (0,4.3) {$\mathop\bigoplus\limits_{p = 0}^n [H^p(U\cup V) \otimes H^{n-p}(F)]$}; \node (a2-1) at (-.12,3.3) {$\mathop\bigoplus\limits_{p = 0}^n [(H^p(U)\otimes H^{n-p}(F)$}; \node (a2-2) at (0,3) {$\oplus$}; \node (a2-3) at (.16,2.7) {$(H^p(V)\otimes H^{n-p}(F))]$}; \node (a3) at (0,1.7) {$\mathop\bigoplus\limits_{p = 0}^n [H^p(U\cap V)\otimes H^{n-p}(F)]$}; \node (a4) at (0,.7) {$\mathop\bigoplus\limits_{p = 0}^n [H^{p+1}(U\cup V) \otimes H^{n-p}(F)]$};  \node (b1) at (4,4.3) {$H^n(\pi^{-1}(U\cup V))$}; \node (b2-1) at (4,3.3) {$H^n(U\times F)$}; \node (b2-2) at (4,3) {$\oplus$}; \node (b2-3) at (4,2.7) {$H^n(V\times F)$}; \node (b3) at (4,1.7) {$H^n((U\cap V\times F))$}; \node (b4) at (4,.7) {$H^{n+1}(\pi^{-1}(U\cup V))$};  \node (ea) at (0,0) {$\ldots$}; \node (eb) at (4,0) {$\ldots$};  \path[->,font=\scriptsize,>=latex] ([xshift=5pt]sa.east) edge node[above]{$\Psi$} ([xshift=-5pt]sb.west) (a1) edge node[above]{$\Psi$} (b1) ([xshift= 36pt]a2-2.east) edge node[above]{$\Psi$} ([xshift= -20pt]b2-2.west) (a3) edge node[above]{$\Psi$} (b3) ([xshift=5pt]ea.east) edge node[above]{$\Psi$} ([xshift=-5pt]eb.west) (a4) edge node[above]{$\Psi$} (b4)  (sa) edge ([yshift= -3.6pt]a1.north) ([yshift=5pt]a1.south) edge node[left]{$\delta$} ([xshift= 3.43pt,yshift=-4pt]a2-1.north) ([xshift= -4.55pt]a2-3.south) edge node[left]{$\gamma$} ([yshift=-3.6pt]a3.north) ([yshift= 5pt]a3.south) edge node[left]{$d^*$} ([yshift= -3.6pt]a4.north) ([yshift= 5pt]a4.south) edge (ea)  (sb) edge (b1) (b1) edge node[right]{$\delta$} (b2-1) (b2-3) edge node[right]{$\gamma$} (b3) (b3) edge node[right]{$d^*$} (b4) (b4) edge (eb); \end{tikzpicture}\] 

Answers

Here's a possibility with tikz-cd; the Psi labels would better be aligned, though; there should be a way!

\documentclass{article}
\usepackage{amsmath,mathtools}
\usepackage{tikz-cd}

\begin{document}

\[
\newcommand{\sbop}{\mathop{\textstyle\bigoplus}\limits}
\begin{tikzcd}[column sep=6em,row sep=large]
\cdots \arrow[r,"\Psi"] \arrow[d] & \cdots \arrow[d]
\\
\smash{\sbop_{p = 0}^n}\, [H^p(U\cup V) \otimes H^{n-p}(F)]
  \arrow[r,"\Psi"] \arrow[d,swap,"\delta"]
  & H^n(\pi^{-1}(U\cup V)) \arrow[d,"\delta"]
\\
\begin{multlined}
  \smash[t]{\sbop_{p = 0}^n}\, [(H^p(U)\otimes H^{n-p}(F) \\
  \oplus (H^p(V)\otimes H^{n-p}(F))]
\end{multlined}
  \arrow[r,"\Psi"] \arrow[d,swap,"\gamma"]
  & H^n(U\times F) \oplus H^n(V\times F) \arrow[d,"\gamma"]
\\
\smash{\sbop_{p = 0}^n}\, [H^p(U\cap V)\otimes H^{n-p}(F)]
  \arrow[r,"\Psi"] \arrow[d,swap,"d^*"]
  & H^n((U\cap V\times F)) \arrow[d,"d^*"]
\\
\smash{\sbop_{p = 0}^n}\, [H^{p+1}(U\cup V) \otimes H^{n-p}(F)]
  \arrow[r,"\Psi"] \arrow[d]
  & H^{n+1}(\pi^{-1}(U\cup V)) \arrow[d]
\\
\cdots \arrow[r,"\Psi"] & \cdots
\end{tikzcd}
\]

\end{document}

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