Let $\mathcal{A}$ be an abelian category. Let $M\in\operatorname{Ob}(\mathcal{A})$ If there is an exact sequence $$\xymatrix{0\ar[r]&S\ar[r]&P_n\ar[r]&{\cdots}\ar[r]&P_1\ar[r]&P_0\ar[r]&M\ar[r]&0},$$ where each $P_i$ is a projective object in $\mathcal{A}$ then we call $S$ an $n$ syzygy of $M$

If $S$ is itself projective, then the projective dimension of $M$ $\operatorname{pd}(M)$ is less than or equal to $n$

Remark.

1. The word ``syzygy'' is used in astromony to describe three celestial objects (usually the Sun, Moon and Earth, or the Sun, Earth and Moon) being collinear.
2. Any two $n$ syzygies of a given object are projectively equivalent.