Let $(R,\mathfrak{m},k)$ a noetherian local ring and $M$ a finitelly generated $R$-module. In this case, given $F_{\bullet}$ a free resolution of $M$, what is the relation between $F_{\bullet}$ and a minimal free resolution $G_{\bullet}$ of $M$?

In this papper

https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-44/issue-3/Betti-numbers-of-almost-complete-intersections/10.1215/ijm/1256060413.full

Dugger says that we can decompose $F_{\bullet}$ into $G_{\bullet} \oplus Q_{\bullet}$ where $Q_{\bullet}$ is a split exact resolution of zero. I'm new in this area. How I can make this decomposition?

Let $(R,\mathfrak{m},k)$ a Noetherian local ring and $M$ a finitely generated $R$-module. In this case, given $F_{\bullet}$ a free resolution of $M$, what is the relation between $F_{\bullet}$ and a minimal free resolution $G_{\bullet}$ of $M$?

In this paper

https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-44/issue-3/Betti-numbers-of-almost-complete-intersections/10.1215/ijm/1256060413.full

Dugger says that we can decompose $F_{\bullet}$ into $G_{\bullet} \oplus Q_{\bullet}$ where $Q_{\bullet}$ is a split exact resolution of zero. I'm new in this area. How I can make this decomposition?