I guess I should ask does the map even make sense? I mean, is essentially an infinite-tuple, and the map would require taking the last positions... but that might not even be possible. Perhaps the inverse limit of doesn't even exist...
Let be two sequences of positive integers such that for all , divides (but is not equal to) and divides but is not equal to . Let . Define maps for all such that (by the usual mappings). Also, define maps such that the following diagram commutes:
Where the down arrows are the maps I defined above for the and the standard maps for .
Let and . Does there exist a map such that the following diagram commutes:
I guess I should ask does the map even make sense? I mean, is essentially an infinite-tuple, and the map would require taking the last positions... but that might not even be possible. Perhaps the inverse limit of doesn't even exist...
Oh, and for my first example, I have
is a short exact sequence of inverse limit systems (provided the second and third systems actually produce inverse limits). I am mostly curious in the fourth inverse limit system (the right-most non-zero inverse limit system), but the documentation I am finding for inverse limits is becoming increasingly incomprehensible for me.