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 $M$?
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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?