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For a module $M$ there is always a surjection $F \to M$ with $F$ free. As free modules are projective, there is always a surjection $P \to M$ with $P$ projective, and one may form the first syzygy $\Omega^1(M) := \text{ker}(P \to M)$. Iterating this process one obtains a projective resolution, and one can form the extension groups which can be shown to be independent of the choice of $P$ and the surjection.

The general impression I get from the stub Wikipedia article is that they here demand the modules to free, i.e. they form the syzygies (and thereby the extension groups) from a free resolution. While the word syzygy is never mentioned in Hatcher (I ctrl+F'ed it!), this seems to be the general approach taken in the formation of extension groups there as well.

As free implies projective, is this simply a case of "Yes, these things may be formed by projective modules, but for us it will be convenient to work with free modules" or is there a significant distinction to be made here? In particular, is it often the case that there is no surjection $P \to M$ with $P$ projective but not free?

Andrew Thompson
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  • For your last sentence: over a local ring, every projective module is free. – xyzzyz Dec 29 '15 at 11:49
  • That's a strong point in favor of those who define this in terms of free moduels. – Andrew Thompson Dec 29 '15 at 11:51
  • Moreover, over any PID (for example, $\mathbb{Z}$), every projective module is free. Correct me if I am wrong, but doesn't Hatcher use only $\mathbb{Z}$-modules? In such a case, there would be no distinction. – Crostul Dec 29 '15 at 11:57
  • @Crostul, in general yes. Some exercises (at least one) is based on doing things over other modules. – Andrew Thompson Dec 29 '15 at 12:01

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As an argument in favor of using projective instead of free resolutions, I would point to this paper of Auslander and Reiten, in which they define (on page 6) syzygies in term of minimal projective resolutions. (There may be better references, but it is the first one I could find.)

For finitely generated modules over Artin algebras, the kernels of all surjections $P\to M$ with $P$ projective are the same up to projective direct summands, so whichever definition you use will give the same result if you pass to the stable category (that is, the quotient of the module category by the ideal of morphism factoring through a projective module).

Pierre-Guy Plamondon
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