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Suppose we have a module over a ring. We choose a projective resolution which allows us to define derived functors. In particular, sometimes we can choose our resolution to be free.

I always hear the term "free resolution" used, and that indicates to me that there are certain situations in which we'd like specifically a free resolution, not merely a projective one; otherwise we'd just say "projective resolution", and let free resolutions be a nice case.

What are the advantages of choosing a free resolution over a merely projective one?

Eric Auld
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1 Answers1

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As you said, free resolutions are projective resolutions. But it is important to remmember that in homological algebra you just need to take ANY projective resolution, in most cases.

The big difference is that it is easier (in a way) to construct a free resolution than a projective resolution. Just look at the prove for the existense of enough projective objects in the module category of a ring with unity, and convince yourself that it is harder to find a projective resolution than a free resolution, where you just need to find a generating set (not even minimal) of a module $M$.

Marco Armenta
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    Of course, it's not actually easier to construct a free resolution than a projective resolution, because a free resolution is a projective resolution. But you are correct that if you try to come up with a projective resolution, you will almost always get one that is actually also a free resolution. – Eric Wofsey Nov 07 '15 at 05:03