I am reading the lecture notes. I have a simple question about free resolution.
What is the meaning of the matrices over the arrows in a free resolution? For example, the matrices in the following free resolution.
\begin{align} 0 \to S^2(-3) \overset{\left( \begin{matrix} x_0 & x_1 \\ x_1 & x_2 \\ x_2 & x_3 \end{matrix} \right)}{\to} S^3(-2) \overset{( x_1 x_3 - x_2^2 \ -x_0x_3 + x_1 x_2 \ x_0x_2 - x_1^2 )}{\to} S. \end{align}
I have not take the course of algebraic geometry before. So it would be great if the answers are in detail. Any help would be greatly appreciated!