Help formatting a commutative diagram in TikZ

Question

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.

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}\]

Answer 1

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}