If the dominating number equals aleph_n, then the n-th derived limit lim^n A[Z^(aleph_n)] is nonzero; and it is consistent that lim^k A is nonzero for every finite k >= 2, with continuum aleph_{omega+2}.
Infinitary combinatorics in condensed mathemat- ics and strong homology
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Simultaneously nonvanishing higher derived limits
If the dominating number equals aleph_n, then the n-th derived limit lim^n A[Z^(aleph_n)] is nonzero; and it is consistent that lim^k A is nonzero for every finite k >= 2, with continuum aleph_{omega+2}.