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• CommentRowNumber1.
• CommentAuthorTim_Porter
• CommentTimeSep 18th 2017

I have added in some additional comments plus some references. In particular there is a paper by Barbara Osofsky (The subscript of $\aleph_n$ projective dimension, and the vanishing of $lim^{(n)}$_, Bull. Amer. Math. Soc. Volume 80, Number 1 (1974), 8-26.) which gives a neat discussion.