MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
show/hide this revision's text 2 added 109 characters in body

When I teach Alg Top I

(This is a corrected version of my original, I do the Kunneth and UC Theorems for rings off-the--cuff answer).

If R of hom has proj dim 1 and C is a flat chain complex, since that's all then you use to get the short exact UCT sequence. I seem To get the splitting, you need to recall also assume that C is projective, so that the unnatural splitting requires R map C -->> B from the complex to be the subcomplex of boundaries has a PIDsplitting. (Sorry I don't have time to check this right now.) But this suggests how unnatural

A spectrum level construction of the splitting really is (assuming this is correct)seems unlikely since you don't have kernels and cokernels, but only fibers and cofibers there.
show/hide this revision's text 1

When I teach Alg Top I, I do the Kunneth and UC Theorems for rings R of hom dim 1, since that's all you use to get the short exact sequence. I seem to recall that the unnatural splitting requires R to be a PID. (Sorry I don't have time to check this right now.) But this suggests how unnatural the splitting really is (assuming this is correct).