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When referring to syzygies, some books refer to free resolution and some books refer to projective resolution. Are they equivalent in some sense? Is it true, for instance, that the $n$-th syzygy in a finite projective resolution is the $n$-th syzygy in a finite free resolution?

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No.

A module can have different projective and free dimensions. Very nice examples can be found here. The free dimension of a projective module can be as big as one.

If the ring satisfies some hypotheses, then the answer becomes yes. For example, over a local commutative ring, or a non-negatively graded connected ring (and you are talking about graded modules and homogeneous maps)