tikz Commutative Diagrams - Compiling and Best Practice

Here there is my version with tikz-cd. It is very easy to create simple and complex commutative diagrams.

Advice: do not worry about the background because It does not appear. Your initial code it is into this link.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{tikz-cd}
\begin{document}

\begin{tikzcd}
V \arrow[r] \arrow[rdd] & FAI(V) \arrow[dd, dotted] \\
 &  \\
 & CL(V,q)
\end{tikzcd}

\end{document}

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
C \arrow[d, "\langle f_i \rangle_{i \in I}"', dotted] \arrow[rd, "f_i"] &  \\
\prod_{i \in I} A_i \arrow[r, "\pi_i"'] & A_i
\end{tikzcd}
\end{document}

Your code was almost correct, you just forgot the text label of the node.

Yours was like this:

\draw[->] (V) to node (FAI);

but it must be like this:

\draw[->] (V) to node {} (FAI);

or like this: (thanks @PaulGaborit)

\draw[->] (V) to (FAI);

I made three of these corrections.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{tikz}
\begin{document}

$$\begin{tikzpicture}[node distance=2cm, auto]
  \node (V) {$V$};
  \node (FAI) [right of= V] {$FAI(V)$};
  \node (CL) [below of= FAI] {$CL(V,q)$};
  \draw[->] (V) to (FAI);% corrected here
  \draw[->, dashed] (FAI) to (CL);% here
  \draw[->] (V) to (CL);% and here
\end{tikzpicture}$$

\end{document}